Proof. We construct a positive supersolution associated with eq. 2

\[{g}({x}) = \sin \frac{\pi x_{1}}{2\eta}.\] Note that this function vanishes on \({x}_{1} = 0\), is positive in \({x}_{1} > 0\), and satisfies \[\begin{cases} \Delta {g} + {c}( {x} ){g} < 0 & \text{in } \Omega, \\ \nu \cdot \nabla {g} \geq 0 & \text{on } \Gamma_{1}. \end{cases}\] Applying the maximum principle in [27], we obtain \({v} \geq 0\) in \(\Omega\). ◻

Proof. The proof is given in [3]. ◻

Proof. This result follows from Serrin’s lemma [2], [3]. For further details, see the proof of [21]. ◻

Proof. By the definition of the moving domain \({D}_{\lambda, \vartheta, \vartheta_{1}}\), the closure of the union of \({D}_{\lambda, \vartheta, \vartheta_{1}}\) and its reflection \(({D}_{\lambda, \vartheta, \vartheta_{1}})'\) (with respect to \({T}_{\lambda, \vartheta}\)) is convex. Hence, the second condition in 1 is satisfied. Let \(\eta > 0\) be the constant provided by 1, so that the maximum principle holds for mixed boundary value problems in domains of sufficiently small width.

Step 1: Initiation of the Moving Plane Method. By the definition of \({D}_{\lambda, \vartheta, \vartheta_{1}}\), there exists a constant \(\varepsilon_{1} > 0\) such that for every \({\lambda} \in ({\lambda}_{M}(\vartheta) - \varepsilon_{1}, {\lambda}_{M}(\vartheta) )\) we have \[{D}_{\lambda, \vartheta, \vartheta_{1}}\subset \big\{{x} \in \mathbb{R}^{2}: \; \operatorname{dist}({x}, {T}_{\lambda, \vartheta}) < {\eta}\big\}.\] Then by applying the maximum principle with mixed boundary conditions (see 1), one deduces that \({w}^{\lambda, \vartheta}\) is strictly positive in \({D}_{\lambda, \vartheta, \vartheta_{1}}\). Moreover, applying the Hopf boundary lemma to \({w}^{\lambda, \vartheta}\) yields 3 . Therefore, 6 and 3 hold for all \({\lambda}\) in the range \({\lambda}_{M}(\vartheta) - \varepsilon_{1} < {\lambda} < {\lambda}_{M}(\vartheta)\).

Step 2: 6 and 3 hold for every \({\lambda} \geq \Lambda\). We argue by contradiction. Define \[\bar{\lambda} = \inf\{\lambda' > 0: \; \text{\eqref{LY305b} and \eqref{LY303a} hold for every }\lambda \geq \lambda'\}.\] Suppose that \(\bar{\lambda} > \Lambda\). By continuity, we have \[{w}^{\bar{\lambda}, \vartheta} \geq 0 \text{ in } {D}_{\bar{\lambda}, \vartheta, \vartheta_{1}}.\] Then, by the strong maximum principle and the Hopf boundary lemma, we deduce that \({w}^{\bar{\lambda}, \vartheta}\) is strictly positive in \({D}_{\bar{\lambda}, \vartheta, \vartheta_{1}}\) and on the smooth portion of \(\Gamma_{\bar{\lambda}, \vartheta, \vartheta_{1}}^{2}\). Even though \(\Gamma_{\lambda, \vartheta, \vartheta_{1}}^{2}\) may contain non-smooth boundary points (namely, corners formed by two lines), one may verify (see [29]) that \({w}^{\lambda, \vartheta}\) is also positive at these points. Hence, for every \({\lambda} \geq \Lambda\) one obtains \[\label{LY317a} \begin{align} {w}^{\lambda, \vartheta} > 0 & \text{ in } \overline{{D}_{\lambda, \vartheta, \vartheta_{1}}} \setminus {T}_{\lambda, \vartheta} \text{ if }\vartheta \in (\pi/2 - \alpha, \pi), \\ {w}^{\lambda, \vartheta} > 0 & \text{ in } \overline{{D}_{\lambda, \vartheta, \vartheta_{1}}} \setminus ({T}_{\lambda, \vartheta} \cup \Gamma_{\lambda, \vartheta, \vartheta_{1}}^{1}) \text{ if }\vartheta = \pi/2 - \alpha, \end{align}\tag{7}\] and \[\nabla {u} \cdot \mathbf{e}_{\vartheta + \alpha} > 0 \text{ on } {T}_{\bar{\lambda}, \vartheta} \cap \Omega.\]

Assume that the intersection \(\Gamma_{D} \cap {T}_{\bar{\lambda}, \vartheta}\) is nonempty (the case \(\Gamma_{D} \cap {T}_{\bar{\lambda}, \vartheta} = \emptyset\) can be treated similarly). Define \({\bar{x}} = (\bar{x}_{1}, \bar{x}_{2})\) as the unique point in \(\Gamma_{D} \cap {T}_{\bar{\lambda}, \vartheta}\subset \partial {D}_{\bar{\lambda}, \vartheta, \vartheta_{1}}\), and set \({P}_{\bar{\lambda}} = (\bar{\lambda}\sin\alpha, - \bar{\lambda}\cos\alpha)\). Fix any \(\delta_{1}, \delta_{2} \in (0, {\eta})\) with \(\delta_{2} < \delta_{1}\), and choose \(\delta_{3} > 0\) such that the intersection \({T}_{\bar{\lambda}, \vartheta} \cap {L}_{0}\) is contained in \(\Omega \cap \{{x}: \; - \delta_{1} < ({x} - {\bar{x}}) \cdot \mathbf{e}_{0} < - \delta_{2}\}\), where the line \[{L}_{0} = \{{x}: \; ({x} - {\bar{x}}) \cdot \mathbf{e}_{\vartheta + \alpha - \pi/2} = - \delta_{3} \}\] is defined to be perpendicular to \({T}_{\bar{\lambda}, \vartheta}\). Since 3 holds for \({\lambda} \geq \bar{\lambda}\), the continuity of \(\nabla {u}\) implies the existence of a small constant \(\delta_{4} \in (0, {\eta}/3)\) such that \[\nabla {u} \cdot \mathbf{e}_{\vartheta + \alpha} > 0 \text{ on } \big\{{x} \in {L}_{0}: \; |({x} - \bar{x}) \cdot \mathbf{e}_{\vartheta + \alpha}| \leq 3\delta_{4}\big\},\] and hence, \[\label{LY318a} {w}^{\lambda, \vartheta} > 0 \text{ in } {D}_{\lambda, \vartheta, \vartheta_{1}} \cap \big\{{x} \in {L}_{0}: \; |({x} - \bar{x}) \cdot \mathbf{e}_{\vartheta + \alpha}| \leq \delta_{4}\big\} \; \text{ for } |\lambda - \bar{\lambda}| < \delta_{4}.\tag{8}\] Set \[\begin{align} \mathcal{D}_{1} & = \big\{{x} \in \Omega: \; \big|({x} - \bar{x}) \cdot \mathbf{e}_{\vartheta + \alpha}\big| \leq \delta_{4}, \quad ({x} - \bar{x}) \cdot \mathbf{e}_{\vartheta + \alpha - \pi/2} \leq - \delta_{3}\big\}, \\ \mathcal{D}_{2} & = \big\{{x} \in \Omega: \; ({x} - \bar{x}) \cdot \mathbf{e}_{\vartheta + \alpha} \leq \delta_{4}, \quad ({x} - \bar{x}) \cdot \mathbf{e}_{0} \geq - \delta_{1}\big\}, \\ \mathcal{D}_{3} & = \big\{{x} \in \Omega: \; ({x} - \bar{x}) \cdot \mathbf{e}_{\vartheta + \alpha} \leq - \delta_{4}, \quad ({x} - \bar{x}) \cdot \mathbf{e}_{0} \leq - \delta_{2}\big\}. \end{align}\] It is clear that \(\mathcal{D}_{1}\cup\mathcal{D}_{2}\cup\mathcal{D}_{3} = \{{x} \in \Omega: \; ({x} - {P}_{\bar{\lambda}}) \cdot \mathbf{e}_{\vartheta + \alpha} \leq \delta_{4} \}\); see 2. In particular, from 7 the function \({w}^{\bar{\lambda}, \vartheta}\) is positive in the compact set \(\overline{\mathcal{D}_{3}} \cap \overline{{D}_{\bar{\lambda}, \vartheta, \vartheta_{1}}}\). By continuity, there exists a small constant \(\varepsilon_{2} \in (0, \delta_{4})\) such that \[\label{LY318b} {w}^{\lambda, \vartheta} > 0 \text{ in } \overline{{D}_{\lambda, \vartheta, \vartheta_{1}}} \cap \overline{\mathcal{D}_{3}} \quad \text{if } {\lambda} \in (\bar{\lambda} - \varepsilon_{2}, \bar{\lambda}).\tag{9}\] Now, let \({\lambda} \in (\bar{\lambda} - \varepsilon_{2}, \bar{\lambda})\). From 8 and 9 , the function \({w}^{\lambda, \vartheta}\) satisfies \[\label{LY319} \begin{cases} \Delta {w}^{\lambda, \vartheta} + {c}^{\lambda, \vartheta} {w}^{\lambda, \vartheta} = 0 & \text{in } {D}_{\lambda, \vartheta, \vartheta_{1}} \cap \mathcal{D}_{j}, \\ {w}^{\lambda, \vartheta} \geq 0, \quad {w}^{\lambda, \vartheta}\not\equiv 0 & \text{on } \partial ({D}_{\lambda, \vartheta, \vartheta_{1}} \cap \mathcal{D}_{j} ) \setminus \Gamma_{\lambda}^{2}, \\ \nabla {w}^{\lambda, \vartheta} \cdot {\nu} \geq 0 & \text{on } \partial ({D}_{\lambda, \vartheta, \vartheta_{1}} \cap \mathcal{D}_{j} ) \cap \Gamma_{\lambda}^{2}, \end{cases}\tag{10}\] for \(j = 1\). By applying the maximum principle (see 1), we deduce that \[\label{LY318c} {w}^{\lambda, \vartheta} > 0 \text{ in } {D}_{\lambda, \vartheta, \vartheta_{1}} \cap \mathcal{D}_{j},\tag{11}\] for \(j = 1\). Using this result together with 8 and 9 , one shows that \({w}^{\lambda, \vartheta}\) satisfies 10 for \(j = 2\). Hence, by the maximum principle from 1 we conclude that 11 holds for \(j = 2\). Furthermore, the Hopf lemma implies that 6 and 3 remain valid for \({\lambda} \in (\bar{\lambda} - \varepsilon_{2}, \bar{\lambda})\).

This contradicts the definition of \(\bar{\lambda}\). Hence, we conclude that \(\bar{\lambda} \leq \Lambda\) and consequently, 6 and 3 hold for every \({\lambda} > \Lambda\). Moreover, by referring back to the argument at the beginning of Step 2 one deduces that 6 and 3 also hold for \({\lambda} = \Lambda\). This completes the proof. ◻

Proof. We only prove the case \(\alpha \in (0, \pi/2)\); the case \(\beta \in (0, \pi/2)\) follows analogously.

Step 1: Monotonicity of \({u}\) in the interior of \(\Omega\). Let \(\vartheta = \pi/2\) and set \(\vartheta_{1} = 0\) (see the left picture in 1). By the definition of \({\Phi}_{1}\) in ?? , for every \({\lambda} \geq {\Phi}_{1}\) the set \(\Gamma_{\lambda, \vartheta, \vartheta_{1}}^{2}\) and its reflection \((\Gamma_{\lambda, \vartheta, \vartheta_{1}}^{2} )'\) (with respect to \({T}_{\lambda, \vartheta}\)) are contained in \(\Gamma_{N}^{ - }\). Moreover, \({w}^{\lambda, \vartheta} > 0\) on \(\Gamma_{\lambda, \vartheta, \vartheta_{1}}^{1}\). This implies that the conditions in ?? are fulfilled for \({\lambda} \geq {\Phi}_{1}\). Hence, by applying 4 we conclude that 5 and 3 hold for \(\vartheta = \pi/2\) and \({\lambda} \geq {\Phi}_{1}\).

Step 2: Monotonicity of \({u}\) along the lower Neumann side. Since \({w}^{\lambda, \pi/2} > 0\), Serrin’s boundary point lemma yields the non-vanishing of the tangential derivative of \({u}\) along \(\Gamma_{N}^{ - }\); see also 3. Hence, ?? holds for every \(\lambda \geq \Phi_{1}\). ◻

Proof. We focus on the case \(\alpha < \pi/2\); the case \(\beta < \pi/2\) follows analogously.

Step 1. We first claim that 3 holds for \(\lambda \geq \Phi_{2}\) and \(\vartheta = \alpha_{*}\). Indeed, by 5 and the definition of \(\alpha_{*}, \Phi_{2}\), the claim is already valid for \(\alpha \in [\pi/4, \pi/2)\). It thus remains to consider \(\alpha < \pi/4\).

Step 1.1: In the case \(\alpha < \pi/4\), 3 holds for \(\lambda \geq \Phi_{2}\) and \(\vartheta \in [\pi/2, 3\pi/4]\). Define \[{J}_{k}: = \Big\{ \frac{\pi}{2} + \frac{j\pi}{2^{k + 1}}: \; j = 0, 1, \dotsc, 2^{k - 1} \Big\}, \quad {J}_{ \infty}: \; = \bigcup_{k = 1}^{ \infty} {J}_{k}.\] Let \(\vartheta = 3\pi/4\), with \(\vartheta_1 = \pi/2\), \(\vartheta_3 = \pi\). By 5 and 12 , the boundary conditions ?? are satisfied for \(\lambda \geq \Phi_2\), and 4 applies. Therefore, 3 holds for all \(\vartheta \in {J}_1\). Assume 3 holds for all \(\vartheta \in {J}_{k}\). Take any \(\vartheta \in {J}_{k + 1} \setminus {J}_{k}\), and write \[\vartheta = \frac{\pi}{2} + (2j + 1)\frac{\pi}{2^{k + 2}}, \quad \text{with } \vartheta_1 = \frac{\pi}{2} + \frac{j\pi}{2^{k + 1}}, \quad \vartheta_3 = \frac{\pi}{2} + \frac{(j + 1)\pi}{2^{k + 1}}.\] Then \(\vartheta_1, \vartheta_3 \in {J}_{k}\) and \(\vartheta_3 - \vartheta = \vartheta - \vartheta_1 > 0\). Hence, ?? is satisfied and by 4, 3 holds for \(\vartheta \in {J}_{k + 1}\). By induction, 3 holds for all \(\vartheta \in {J}_{ \infty}\).

Since \({J}_{ \infty}\) is dense in the interval \([\pi/2, 3\pi/4]\) and \(\nabla {u}\) is continuous, it follows that \[\label{LY326} \nabla {u} \cdot \mathbf{e}_{\vartheta + \alpha} \geq 0 \text{ on } \Omega \cap {T}_{\lambda, \vartheta}\tag{13}\] holds for all \(\vartheta \in [\pi/2, 3\pi/4]\) and \({\lambda} \geq {\Phi}_{2}\). To obtain strict positivity, fix any \(\vartheta \in (\pi/2, 3\pi/4)\), and choose \(\vartheta_{1} \in [\pi/2, 3\pi/4]\), \(\vartheta_{3} \in {J}_{ \infty}\) with \(\vartheta_{3} - \vartheta = \vartheta - \vartheta_{1} > 0\). Then ?? holds and 4 yields strict monotonicity. Hence, 3 holds for all \(\vartheta \in [\pi/2, 3\pi/4]\), concluding Step 1.1. In particular, this completes Step 1 for the case \(\alpha \in [\pi/8, \pi/4)\).

Step 1.2: In the case \(\alpha \in (0, \pi/8)\), 3 holds for \(\lambda \geq \Phi_{2}\) and \(\pi/2 \leq \vartheta \leq \pi - 2\alpha\). For this purpose, define for \(k \geq 1\) the sets \[\tilde{J}_{k} = \big[\frac{\pi}{2}, \pi - \frac{\pi}{2^{k + 1}}\big] \cap \big[\frac{\pi}{2}, \pi - 2\alpha\big].\] From Step 1.1, we have already established 3 for \(\vartheta \in \tilde{J}_{1}\). Assume that 3 holds for all \(\vartheta \in \tilde{J}_{k}\) for some \(k\ge1\). Let \(\vartheta \in \tilde{J}_{k + 1} \setminus \tilde{J}_{k}\) and set \(\vartheta_{3} = \pi\) and \(\vartheta_{1} = 2\vartheta - \vartheta_{3}\). Then \(\vartheta_{1} \in \tilde{J}_{k}\), and ?? is valid for \(\lambda \geq \Phi_{2}\). By 4 it follows that 5 and 3 \(\lambda \geq \Phi_{2}\). By induction, we deduce that 3 holds for all \(\vartheta \in \cup _{k = 1}^{ \infty}\tilde{J}_{k} = [\pi/2, \pi - 2\alpha]\). This concludes Step 1.

Step 2. We claim that 3 holds for \(\lambda \geq \Phi_{2}\) and \(\vartheta \in [\pi/2 - \alpha, \alpha_{*}]\).

Step 2.1: This step holds for \(\vartheta = \alpha_{*}/2\). Since \(\alpha_{*} \geq \pi - 2\alpha\), we have \(\alpha_{*}/2 \geq \pi/2 - \alpha\). For \(\vartheta = \alpha_{*}/2\), set \(\vartheta_{1} = 0\) and \(\vartheta_{3} = \alpha_{*}\). Then, for \({\lambda} \geq {\Phi}_{2}\), by 12 the function \({w}^{\lambda, \vartheta}\) satisfies ?? . Applying 4, we conclude that 5 and 3 hold for \(\vartheta = \alpha_{*}/2\) and \(\lambda \geq \Phi_{2}\).

Step 2.2: This step holds for \(\vartheta \in [\alpha_{*}/2, \alpha_{*}]\). The argument proceeds analogously to Step 1.1, so we omit further details.

Step 2.3: This step holds for \(\vartheta \in [\pi/2 - \alpha, \alpha_{*}]\). This case is handled similarly to Step 1.2; details are omitted for brevity. This completes the proof. ◻

Proof. We focus on the first assertion; the second one can be proved in the same way. Note that \({T}_{\lambda, \pi/2} \cap \Gamma_{D} \neq \emptyset\) if and only if \(\lambda \in [\Psi_{0} \cos\gamma, \Phi_{0}]\).

Step 1. We first claim that 5 and 3 hold for all \((\vartheta, {\lambda})\) satisfying \[\label{LY329a} \vartheta \in [\bar{\vartheta}, \pi - \alpha ) \quad \text{and} \quad \lambda \geq {\Psi}_{0}\sin\bigl(\vartheta - \gamma\bigr)\csc\vartheta,\tag{15}\] where \(\bar{\vartheta} < \pi - \alpha\) is any fixed number satisfying \[\label{LY329b} 2\bar{\vartheta} - \gamma \geq \pi - \alpha, \quad 2\bar{\vartheta} - \pi \geq \pi/2 - \alpha, \quad \text{and} \quad {\Psi}_{0}\sin\bigl(\bar{\vartheta} - \gamma\bigr) \csc\bar{\vartheta} \geq {\Phi}_{2}.\tag{16}\] Under the condition 15 (with the choice \(\vartheta_{3} = \pi\)) the definition 12 implies that \[\Gamma_{\lambda, \vartheta, \vartheta_{1}}^{2B} = \emptyset \quad \text{and} \quad \Gamma_{\lambda, \vartheta, \vartheta_{1}}^{2A}\subset {T}_{\lambda, 2\vartheta - \pi},\] with \[2\vartheta - \pi \in [\pi/2 - \alpha, \alpha_{*}] \quad \text{and} \quad \lambda \geq {\Phi}_{2}.\] See 3. Hence, by applying 6 we deduce that 5 and 3 hold under 15 .

Step 2. For any fixed \({\bar{\phi}} \in \bigl[{\Phi}, {\Phi}_{0}\bigr)\), let \({\bar{x}} = ({\bar{x}}_{1}, {\bar{x}}_{2})\) be the unique intersection point of the line \({T}_{{\bar{\phi}}, \pi/2}\) and \(\Gamma_{D}\). We aim to establish that \[\label{LY330} \nabla {u} \cdot \mathbf{e}_{\vartheta + \alpha} > 0 \text{ on } \Omega \cap \{ {x}: \; ({x} - {\bar{x}}) \cdot \mathbf{e}_{\vartheta + \alpha} = 0 \}\tag{17}\] for every \(\vartheta \in [\pi/2, \bar{\vartheta}]\). To prove this, consider the angular derivative of \({u}\) about \({\bar{x}}\), defined by \[\label{LY331a} ({R}_{\bar{x}}{u})({x}) = ({x}_{1} - \bar{x}_{1}) \partial_{x_{2}}{u}({x}) - ({x}_{2} - \bar{x}_{2}) \partial_{x_{1}}{u}({x}),\tag{18}\] in the domain \[\mathcal{D} = \bigl\{{x} \in \Omega: \; ({x} - {\bar{x}}) \cdot \mathbf{e}_{\pi/2 + \alpha} < 0, \quad ({x} - {\bar{x}}) \cdot \mathbf{e}_{\bar{\vartheta} + \alpha} > 0\bigr\}.\] Then \({R}_{\bar{x}}{u} \in {C}^{1}(\overline{\mathcal{D}})\) satisfies the corresponding linearized equation in the weak sense: \[\label{LY331b} [\Delta + {f}'({u})] {R}_{\bar{x}}{u} = 0 \text{ in } \mathcal{D} \quad \text{and} \quad {R}_{\bar{x}}{u} \leq, \not \equiv 0 \text{ on } \partial\mathcal{D},\tag{19}\] where \({f}'({u})\) is defined a.e. in \(\Omega\) and belongs to \(L^{\infty}(\Omega)\) (since \({f} \in \mathrm{Lip}_{loc}(\mathbb{R})\) and \({u}\) is bounded), and the boundary inequality in 19 follows from Step 1 and 5. Furthermore, since \[[\Delta + {f}'({u})] (\nabla {u} \cdot \mathbf{e}_{\pi/2 + \alpha}) = 0 \text{ in } \mathcal{D} \quad \text{and} \quad \nabla {u} \cdot \mathbf{e}_{\pi/2 + \alpha} > 0 \text{ in } \overline{\mathcal{D}} \setminus \partial\Omega,\] we apply the maximum principle (see, e.g., [46]) to 19 and deduce that \({R}_{\bar{x}}{u} < 0\) in \(\mathcal{D}\), hence establishing 17 .

Combining the results of Step 1 and Step 2, we conclude that 17 holds for any \(\vartheta \in [\pi/2, \pi - \alpha)\) and for any \({\bar{x}} \in \bigl( \Gamma_{D} \cap \{ {x}: \; {x} \cdot \mathbf{e}_{\alpha - \pi/2} \geq {\Phi}\}\bigr)\). In particular, this implies that 3 holds for \(\vartheta \in [\pi/2, \pi - \alpha)\) and \(\lambda \geq \Upsilon_{\Phi, \vartheta}\). This completes the proof. ◻

Proof. We focus on the case \(\alpha < \pi/2\); the case \(\beta < \pi/2\) follows similarly. Suppose first that \(\gamma \leq \alpha\). Then from ?? , we have \(\Phi_{1} = \Psi_{0}\cos\gamma\). Thus, the conclusion follows from 5 and 7. We mainly consider \(\gamma > \alpha\). Then from ?? , we have \(\Phi_{1} = \Phi_{0}/2\). Define \(\Lambda_{0}: = \Phi_{0}\) and \[\Lambda_{j + 1}: \; = \frac{1}{2} \Upsilon_{\Lambda_{j}, \pi - \gamma} = \frac{1}{2} \left( (\Phi_{0} - \Lambda_{j}) \tan\alpha \cot\gamma + \Lambda_{j} \right) \; \text{ for } j \in \mathbb{N}.\] It can be shown that \(\Lambda_{1} = \Phi_{1}\), and that the sequence \(\{ \Lambda_{j} \}\) is strictly decreasing and converges to \[\Lambda_{ \infty} = \Psi_{0} \cos\gamma.\] By 5 and 7, 3 holds for \(\vartheta \in [\pi/2, \pi - \alpha )\) and \(\lambda \geq \Upsilon_{\Lambda_{1}, \vartheta}\). Now, assume that the claim holds for some \(k\ge1\); that is, 3 holds for all \(\vartheta \in [\pi/2, \pi - \alpha )\) and \(\lambda \geq \Upsilon_{\Lambda_{k}, \vartheta}\). Let \[\label{LY332b} \vartheta = \pi/2 \; \text{ and } \; \lambda \geq \Lambda_{k + 1},\tag{20}\] Then \(\Gamma_{\lambda, \vartheta, \vartheta_{1}}^{2B}\subset{T}_{2\lambda, \pi - \gamma}\) with \(\pi - \gamma \in [\pi/2, \pi)\) and \(2\lambda \geq 2\Lambda_{k + 1} = \Upsilon_{\Lambda_{k}, \pi - \gamma}\). Hence, ?? is satisfied. According to 4, we conclude that 5 and 3 hold under 20 . Combining this with 7, 3 holds for \(\vartheta \in [\pi/2, \pi - \alpha )\) and \(\lambda \geq \Upsilon_{\Lambda_{k + 1}, \vartheta}\). By mathematical induction, the claim is valid for every \(j\), implying that 3 holds for \(\vartheta \in [\pi/2, \pi - \alpha)\) and \(\lambda > \Upsilon_{\Lambda_{ \infty}, \vartheta}\).

Thus, we obtain that 3 holds for \(\vartheta \in [\pi/2, \pi - \alpha)\) and \(\lambda > \Psi_{0}\csc\vartheta\sin(\vartheta - \gamma)\). It remains to show the case \(\lambda = \Psi_{0}\csc\vartheta\sin(\vartheta - \gamma)\). By continuity, 13 holds \(\vartheta \in [\pi/2, \pi - \alpha)\) and \(\lambda = \Psi_{0}\csc\vartheta\sin(\vartheta - \gamma)\). Let \({R}_{\bar{x}}{u}\) be the angular derivative of \({u}\) about \(\bar{x} = (1, \cot\beta)\) (see 18 ), and define \[\mathcal{D} = \{ {x} \in \Omega: \; ({x} - \bar{x}) \cdot \mathbf{e}_{\pi/2 + \alpha} < 0\}.\] Since \({R}_{\bar{x}}{u} \leq, \not \equiv 0\) on \(\mathcal{D}\) and satisfies the linear equation 19 , the strong maximum principle yields \({R}_{\bar{x}}{u} < 0\) in \(\mathcal{D}\). Hence, 3 holds for \[\vartheta \in (\pi/2, \pi - \alpha) \text{and} \lambda = \Psi_{0}\csc\vartheta\sin(\vartheta - \gamma).\] In particular, 5 and 3 hold for \(\vartheta = \pi/2\) and \(\lambda \geq \Psi_{0}\cos\gamma\); moreover, by 7, 4 holds for \(\vartheta \in (\pi - \beta, \pi/2 + \gamma]\) and \(\lambda \geq \Psi_{0}\). This completes the proof. ◻

Proof. Note that the function \({w}^{\lambda, \vartheta}\) may fail to satisfy the Neumann boundary condition on \(\Gamma_{\lambda, \vartheta}^{2B}\) for certain values of \(\vartheta\) (for example, when \(\vartheta\) is slightly less than \(\pi/2 + \gamma\)). To establish \[{w}^{\lambda, \vartheta}( {x} ) = {u}({x}^{\lambda, \vartheta}) - {u}( {x} ) > 0 \text{ for } {x} \in \overline{ {D}_{\lambda, \vartheta} } \setminus {T}_{\lambda, \vartheta},\] we follow the approach in [29]. From the assumptions of this lemma, we have \[\begin{align} \tag{21} {w}^{\lambda, \pi/2 + \gamma} > 0 & \; \text{ in } \overline{ {D}_{\lambda, \pi/2 + \gamma} } \setminus {T}_{\lambda, \pi/2 + \gamma}, \\ \tag{22} {w}^{\lambda, \gamma} > 0 & \; \text{ in } \overline{ {D}_{\lambda, \gamma} } \setminus {T}_{\lambda, \gamma}, \end{align}\] and \[\label{LY334c} \normalsize \text{ \({u}( {x} )\) is increasing as the angle of \(\overrightarrow{{P}_{\lambda}{x}}\) and \(\overrightarrow{{P}_{\lambda}{z}_{0}}\) decreases along each arc \({S}({P}_{\lambda}, {r})\)},\tag{23}\] where \({P}_{\lambda} = (\lambda\sin\alpha, - \lambda\cos\alpha)\) and \[{S}({P}_{\lambda}, {r}) = \{ {x} \in \overline{\Omega}: \; |{x} - {P}_{\lambda}| = {r}, \; ({x} - {P}_{\lambda}) \cdot \mathbf{e}_{\pi/2 - \beta} \geq 0 \geq ({x} - {P}_{\lambda}) \cdot \mathbf{e}_{ - \beta} \}.\] The condition \(\beta \leq \pi/2\) guarantees that \(\gamma \geq \pi/2 - \alpha\), so that \({S}({P}_{\lambda}, {r})\) is connected and lies on a circular arc. Fix any \(\vartheta \in (0, \pi/2 + \gamma)\) and let \({x}\) be an arbitrary point in \(\overline{ {D}_{\lambda, \vartheta} } \setminus {T}_{\lambda, \vartheta}\). Then there exist two angles \(\psi_{1}, \psi_{2} \in [0, \pi]\) such that \[{x} \in {T}_{\lambda, \psi_{1}} \quad \text{and} \quad {y}: = {x}^{\lambda, \vartheta} \in {T}_{\lambda, \psi_{2}},\] where we note that \(\psi_{1} = 2\vartheta - \psi_{2} < \vartheta\). We consider the following four cases.

Case 1: \(\psi_{2}, \psi_{1} \in [\gamma, \pi/2 + \gamma]\). By the monotonicity in 23 , we directly obtain: \[{u}( {x} ) < {u}({y}).\]

Case 2: \(\psi_{2} \in [\gamma, \pi/2 + \gamma]\) and \(\psi_{1}\not \in [\gamma, \pi/2 + \gamma]\). Reflect \({x}\) with respect to the line \({T}_{\lambda, \gamma}\) to obtain a point \({x}^{\lambda, \gamma}\). Then \({x}^{\lambda, \gamma} \in {T}_{\lambda, \psi_{1}'}\) with \(\psi_{1}' = 2\gamma - \psi_{1}\). Since \(\gamma > \pi/2 - \alpha\), we deduce that \({x}^{\lambda, \gamma} \subset\{{x}_{1} < 1\}\). Since \(\psi_{1}' - \psi_{2} = 2(\gamma - \vartheta) < 0\) and \(\psi_{2} \leq \pi/2 + \gamma\), we deduce that \({x}^{\lambda, \gamma}\) lies below the line containing \(\Gamma_{N}^{ - }\). Thus, \({x}^{\lambda, \gamma} \in \overline{\Omega}\). Then from 22 and 23 , we obtain: \[{u}( {x} ) < {u}({x}^{\lambda, \gamma}) \quad \text{and} \quad {u}({x}^{\lambda, \gamma}) < {u}({y}).\]

Case 3: \(\psi_{2}\not \in [\gamma, \pi/2 + \gamma]\) and \(\psi_{1} \in [\gamma, \pi/2 + \gamma]\). Reflect \({y}\) with respect to the line \({T}_{\lambda, \pi/2 + \gamma}\) to obtain a point \({y}^{\lambda, \pi/2 + \gamma}\). Then \({y}^{\lambda, \pi/2 + \gamma} \in {T}_{\lambda, \psi_{2}'}\) with \(\psi_{2}' = \pi + 2\gamma - \psi_{2}\). It follows that \({y}^{\lambda, \pi/2 + \gamma}\) lies on or below the line containing \(\Gamma_{N}^{ + }\). Moreover, since \(\psi_{2}' - \psi_{1} = 2(\pi/2 + \gamma - \vartheta) > 0\) and \(\psi_{1} \geq \pi/2 - \alpha\), we deduce that \({y}^{\lambda, \pi/2 + \gamma} \in \overline{\Omega}\). Thus, from 23 and 21 , we obtain \[{u}( {x} ) < {u}({y}^{\lambda, \pi/2 + \gamma}) \quad \text{and} \quad {u}({y}^{\lambda, \pi/2 + \gamma}) < {u}({y}).\]

Case 4: \(\psi_{2}, \psi_{1}\not \in [\gamma, \pi/2 + \gamma]\). In this case, we have \({y}^{\lambda, \pi/2 + \gamma} \in {T}_{\lambda, \psi_{2}'}\) and \({x}^{\lambda, \gamma} \in {T}_{\lambda, \psi_{1}'}\). Since \(0 \leq \psi_{1} < \psi_{2} \leq \pi\), we deduce that \(\psi_{1} \geq \max\{2\vartheta - \pi, 0\}\) and hence \(\psi_{2}' - \psi_{1}' = 2(\pi/2 + \psi_{1} - \vartheta) \geq 0\). Thus, one obtains \[0 \leq \psi_{1} < \gamma < \psi_{1}' \leq \psi_{2}' < \pi/2 + \gamma < \psi_{2} \leq \pi.\] It follows that both \({x}^{\lambda, \gamma}\) and \(({x}^{\lambda, \vartheta})^{\lambda, \pi/2 + \gamma}\) belong to \(\overline{\Omega}\). Consequently, from 22 , 23 and 21 we have \[{u}( {x} ) < {u}({x}^{\lambda, \gamma}), \quad {u}({x}^{\lambda, \gamma}) \leq {u}({y}^{\lambda, \pi/2 + \gamma}), \quad \text{and} \quad {u}({y}^{\lambda, \pi/2 + \gamma}) < {u}({y}).\] In all cases, we have shown that \({u}( {x} ) < {u}({y})\). This concludes the proof. ◻

Proof. We focus on the first assertion; the second one can be proved in the same way.

Step 1. Assume that both \(\alpha\) and \(\beta\) are acute. Then 3 holds for \[\lambda \geq \Phi_{0} - \varepsilon_{1} \quad \text{and} \quad \pi/2 - \alpha \leq \vartheta \leq \pi - \alpha,\] where \(\varepsilon_{1}\in(0, \Phi_{0} - \Phi_{2})\) is chosen so that, setting \(\Phi_{3}: = \Phi_{0} - \varepsilon_{1}\) and \(\Psi_{3}: = \Phi_{3}\sin\alpha\, \csc\beta\), the following hold: \[\tag{24} \begin{align} \tag{25} &\text{\eqref{LY303a} holds for \(\vartheta \in [\tfrac{\pi}{2} - \alpha, \alpha_{**}]\) and \(\lambda \geq \Phi_{3}\), } \\ \tag{26} &\text{\eqref{LY303b} holds for \(\vartheta \in [\tfrac{\pi}{2} - \beta, \beta_{**}]\) and \(\lambda \geq \Psi_{3}\).} \end{align}\] The existence of such an \(\varepsilon_{1}\) follows from 6 and 7, together with the fixed constants \(\alpha_{**}\) and \(\beta_{**}\) satisfying \[\max\{\beta/2 + \gamma, \;\pi - 2\alpha\} < \alpha_{**} < \pi - \alpha, \quad \pi - 2\beta < \beta_{**} = \pi - 2\alpha_{**} + 2\gamma < \pi - \beta.\]

For \(\vartheta = \pi - \alpha\) and \(\lambda \geq \Phi_{3}\), we have \(\Gamma_{\lambda, \vartheta}^{2A} \subset {T}_{\lambda, \pi - 2\alpha}\) and \(\Gamma_{\lambda, \vartheta}^{2B} \subset {T}_{\hat{\lambda}, \pi - 2\beta}\), where \(\hat{\lambda} = \lambda\sin\alpha\csc\beta \geq \Psi_{3}\). By 24 , \(w^{\lambda, \vartheta}\) satisfies the strict boundary condition in ?? on \(\Gamma_{\lambda, \vartheta}^{2}\). Hence, by 4, 5 and 3 hold for \(\vartheta = \pi - \alpha\) and \(\lambda \geq \Phi_{3}\).

It remains to consider \(\vartheta\in(\alpha_{**}, \pi - \alpha)\) with \(\lambda \geq \Phi_{3}\). In this case, \(\Gamma_{\lambda, \vartheta}^{2B}\subset {T}_{\hat{\lambda}, \hat{\vartheta}}\), where \[\hat{\lambda} = \hat{\lambda}(\lambda, \vartheta) > \Psi_{3} \quad \text{and} \quad \hat{\vartheta} = \pi - 2\vartheta + 2\gamma \in [\pi - 2\beta, \beta_{**}].\] It then follows from 26 that \(w^{\lambda, \vartheta}\) satisfies the strict boundary condition in ?? on \(\Gamma_{\lambda, \vartheta}^{2B}\). Therefore, arguing as in Step 1.1 of 6, we deduce that 5 and 3 hold for \(\vartheta\in(\alpha_{**}, \pi - \alpha)\) and \(\lambda \geq \Phi_{3}\). This completes Step 1.

Step 2. We claim that 3 holds when \(\vartheta = \tilde{\vartheta}_{*}\) and \(\Phi_{0} - \lambda\) is positive and sufficiently small. Here, we define \(\tilde{\vartheta}_{*} = \tilde{\vartheta}_{\kappa}\), where the integer \(\kappa\) is chosen such that \(\tilde{\vartheta}_{\kappa} \geq 2\gamma > \tilde{\vartheta}_{\kappa - 1}\), i.e., \(\tilde{\vartheta}_{\kappa + 1} \geq \gamma + \pi/2 > \tilde{\vartheta}_{\kappa }\), and \[\tilde{\vartheta}_{0} = \pi - \alpha, \quad \tilde{\vartheta}_{j} = (\pi + \tilde{\vartheta}_{j - 1})/2, \; j \in \mathbb{N}^{+}.\]

By Step 1, the claim is valid when \(\tilde{\vartheta}_{*} \leq \pi - \alpha\), that is, when \(2\gamma \leq \pi - \alpha\) or equivalently \(\gamma \leq \beta\). Now we assume that \(\gamma > \beta\) (hence \(\gamma > \pi/4\)). Applying 9, we obtain \[\label{LY337a} {w}^{\lambda, \vartheta} > 0 \text{ in } \overline{{D}_{\lambda, \vartheta, 2\vartheta - \pi}} \setminus {T}_{\lambda, \vartheta}\tag{27}\] for \(\vartheta \in [\pi - \alpha/2, \pi/2 + \gamma]\) and \(\lambda \geq \Phi_{0}\). Based on this and the monotonicity ?? on the upper Neumann boundary \(\Gamma_{N}^{ + }\) (see details in 8 and Step 2 of 5), it follows by continuity that there exists \(\varepsilon_{2} \in (0, \varepsilon_{1})\) such that \[{w}^{\lambda, \vartheta} > 0 \text{ on } \Gamma_{\lambda, \vartheta, 2\vartheta - \pi}^{2B} \; \text{ for } \lambda > \Phi_{0} - \varepsilon_{2} \; \text{ and } \; \vartheta \in \{\tilde{\vartheta}_{1}, \tilde{\vartheta}_{2}, \ldots, \tilde{\vartheta}_{\kappa}\}.\]

Now, set \(\vartheta = \tilde{\vartheta}_{1}\) and \(\vartheta_{1} = 2\vartheta - \pi\). Then \(\vartheta_{1} = \tilde{\vartheta}_{0}\) and \[\begin{cases} \Delta {w}^{\lambda, \vartheta} + {c}^{\lambda, \vartheta} {w}^{\lambda, \vartheta} = 0 & \text{in } {D}_{\lambda, \vartheta}, \\ {w}^{\lambda, \vartheta} \geq 0 & \text{on } \partial {D}_{\lambda, \vartheta} \setminus \Gamma_{\lambda, \vartheta}^{2A}, \\ \partial_{\nu}{w}^{\lambda, \vartheta} \geq, \not \equiv 0 & \text{on } \Gamma_{\lambda, \vartheta}^{2A}, \end{cases}\] which is similar to ?? . Following an argument similar to that in 4, we conclude that 27 and 3 hold for \(\lambda > \Phi_{0} - \varepsilon_{2}\) with \(\vartheta = \tilde{\vartheta}_{1}\). By mathematical induction, we conclude that 27 and 3 hold for all \(\lambda > \Phi_{0} - \varepsilon_{2}\) and \(\vartheta \in \{\tilde{\vartheta}_{1}, \tilde{\vartheta}_{2}, \ldots, \tilde{\vartheta}_{\kappa}\}\).

Step 3. We claim that 3 holds for \(\vartheta \in [\pi - \alpha, \pi/2 + \gamma]\) and \(\lambda > \Phi_{0} - \varepsilon_{2}\). Indeed, this follows by an argument analogous to Step 2 in the proof of 7. For any fixed \(\lambda \in [\Phi_{0} - \varepsilon_{2}, \Phi_{0})\) and \(\bar{x} = (\bar{x}_{1}, \bar{x}_{2})\) with \(\bar{x}_{1} = \lambda\sin\alpha\), \(\bar{x}_{2} = - \lambda\cos\alpha\), define \[\mathcal{D} = \{ {x} \in \Omega: \; ({x} - \bar{x}) \cdot \mathbf{e}_{\tilde{\vartheta}_{*} + \alpha} < 0, \; ({x} - \bar{x}) \cdot \mathbf{e}_{\tilde{\vartheta}_{0} + \alpha} > 0\}.\] Then the angular derivative \({R}_{\bar{x}}{u}\), introduced in 18 , belongs to \({C}^{1}(\overline{\mathcal{D}})\) and satisfies \[\label{LY338} [\Delta + {f}'({u})] {R}_{\bar{x}}{u} = 0 \text{ in } \mathcal{D} \quad \text{and} \quad {R}_{\bar{x}}{u} \geq, \not \equiv 0 \text{ on } \partial\mathcal{D}\tag{28}\] as established in Steps 1 and 2. Recalling that \[[\Delta + {f}'({u})] (\nabla {u} \cdot \mathbf{e}_{\tilde{\vartheta}_{*} + \alpha}) = 0 \text{ in } \mathcal{D} \quad \text{and} \quad (\nabla {u} \cdot \mathbf{e}_{\tilde{\vartheta}_{*} + \alpha}) < 0 \text{ in } \overline{\mathcal{D}} \setminus \partial\Omega,\] we apply the maximum principle (see, e.g., [46]) to conclude that \({R}_{\bar{x}}{u} > 0\). In particular, this establishes 3 for \(\vartheta \in [\pi/2 - \alpha, \tilde{\vartheta}_{*}]\) and \(\lambda \geq \Phi_{0} - \varepsilon_{2}\).

Now, let \(\vartheta = \pi/2 + \gamma\), \(\vartheta_{1} = 2\vartheta - \pi = 2\gamma\) with \(\lambda \geq \Phi_{0} - \varepsilon_{2}\). Observe that \(\Gamma_{\lambda, \vartheta}^{2B}\) and its reflection \((\Gamma_{\lambda, \vartheta}^{2B})'\) belong to \(\Gamma_{N}^{ + }\). By the definition of \(\tilde{\vartheta}_{*}\), we derive \(2\gamma \in [\pi/2 - \alpha, \tilde{\vartheta}_{*}]\), which ensures that \({w}^{\lambda, \vartheta}\) satisfies ?? . Applying 4, we conclude that both 5 and 3 hold for \(\vartheta = \pi/2 + \gamma\) and \(\lambda \geq \Phi_{0} - \varepsilon_{2}\).

Finally, as argued at the beginning of this step, it follows that 3 remains valid for \(\vartheta \in [\pi - \alpha, \pi/2 + \gamma]\) with \(\lambda \geq \Phi_{0} - \varepsilon_{2}\). This completes the proof. ◻

Proof. The case \(\gamma = \pi/2\) is already established in 2; we thus assume \(\gamma < \pi/2\). Let \((\bar{t}, \infty)\) denote the largest open interval of positive values \({t}\) for which 3 holds for \(\vartheta \in [\gamma, \pi/2 + \gamma]\) and \(\lambda \geq {t} \Phi_{0}\), and for which 4 holds for \(\vartheta \in [\gamma, \pi/2 + \gamma]\) and \(\lambda \geq {t} \Psi_{0}\). By 10, \(\bar{t}\) is well defined and \(\bar{t} < 1\). Assume by contradiction that \(\bar{t} > 0\), and define \(\bar{\Phi}: = \bar{t}\Phi_{0}\).

Step 1. We first establish that 3 holds for \(\vartheta \in [\gamma, \pi/2 + \gamma]\) and \(\lambda \geq \bar{t} \Phi_{0}\), while 4 remains valid for \(\vartheta \in [\gamma, \pi/2 + \gamma]\) and \(\lambda \geq \bar{t} \Psi_{0}\). In fact, we set \[\bar{x}: = (\bar{x}_{1}, \bar{x}_{2}), \quad \text{where} \quad \bar{x}_{1}: = \bar{\Phi}\sin\alpha, \quad \bar{x}_{2}: = - \bar{\Phi}\cos\alpha,\] and \[\mathcal{D}: = \{ {x} \in \Omega: \; ({x} - \bar{x}) \cdot \mathbf{e}_{\pi/2 - \beta} > 0, \quad ({x} - \bar{x}) \cdot \mathbf{e}_{ - \beta} < 0 \}.\] By the monotonicity near the Dirichlet boundary (see 2), the angular derivative \({R}_{\bar{x}}{u}\), defined in 18 , does not vanish identically in \(\mathcal{D}\). From the definition of \(\bar{t}\) and continuity, we know that the nonstrict inequality 13 holds for \(\vartheta \in [\gamma, \pi/2 + \gamma]\) and \(\lambda \geq \bar{\Phi} = \bar{t} \Phi_{0}\), and hence \({R}_{\bar{x}}{u} \geq 0\) in \(\mathcal{D}\). Applying the strong maximum principle to the linear elliptic equation satisfied by the nonnegative function \({R}_{\bar{x}}{u}\) (see 28 ), we conclude that \({R}_{\bar{x}}{u} > 0\) in \(\mathcal{D}\), implying that 3 holds strictly for \(\vartheta \in (\gamma, \pi/2 + \gamma)\) and \(\lambda = \bar{\Phi}\).

For \(\vartheta = \gamma\) and \(\lambda \geq \bar{\Phi}\), we observe that \(\Gamma_{\lambda, \gamma}^{2B}\subset {T}_{2\lambda, \gamma}\), and \[\Gamma_{\lambda, \gamma}^{2A} \subset {T}_{\lambda, 0}, \quad (\Gamma_{\lambda, \gamma}^{2A})'\subset {T}_{\lambda, 2\gamma} \quad \text{with} \quad 2\gamma \in [\gamma, \pi/2 + \gamma].\] Thus, 5 and 3 hold for \(\vartheta = \gamma\) and \(\lambda \geq \bar{\Phi}\).

Similarly, for \(\vartheta = \pi/2 + \gamma\) and \(\lambda \geq \bar{\Phi}\), we have \(\Gamma_{\lambda, \gamma}^{2B} \subset \Gamma_{N}^{ + }\), and \[(\Gamma_{\lambda, \pi/2 + \gamma}^{2A})'\subset {T}_{\lambda, \pi}, \quad \Gamma_{\lambda, \pi/2 + \gamma}^{2A}\subset {T}_{\lambda, 2\gamma} \quad \text{with} \quad 2\gamma \in [\gamma, \pi/2 + \gamma].\] Thus, 5 and 3 hold for \(\vartheta = \pi/2 + \gamma\) and \(\lambda \geq \bar{\Phi}\). Furthermore, by Step 2 of 5, \({u}\) is strictly monotone along the upper Neumann boundary \(\Gamma_{N}^{ + }\); in particular, ?? holds for \(\lambda \geq \bar{\Phi}\cos\gamma\). In summary, we have:

• 3 holds for \(\vartheta \in [\gamma, \pi/2 + \gamma]\) and \(\lambda \geq \bar{\Phi}\), and ?? is true for \(\lambda \geq \bar{\Phi}\cos\gamma\).

• Similarly, 4 holds for \(\vartheta \in [\gamma, \pi/2 + \gamma]\) and \(\lambda \geq \bar{t} \Psi_{0}\), and ?? is true for \(\lambda \geq \bar{t} \Psi_{0}\cos\gamma\).

Step 2. We next show that 3 and 4 remain valid for \(\vartheta \in [\gamma, \pi/2 + \gamma]\) and \(0 < \bar{t}\Phi_{0} - \lambda\ll1\), similarly, 4 holds for \(\vartheta \in [\gamma, \pi/2 + \gamma]\) and \(0 < \bar{t}\Psi_{0} - \lambda\ll1\). Indeed, let \(\bar{\vartheta}\) be the unique angle such that \({T}_{\bar{\Phi}, \bar{\vartheta}}\) passes through the upper mixed boundary point. Since \(\bar{\Phi} < \Phi_{0}\), it follows that \(\bar{\vartheta} \in (\gamma, \pi - \alpha)\). Choose \(\delta > 0\) sufficiently small so that \[\label{LY339a} 0 < \delta < \min\big\{ \tfrac{\bar{\vartheta} - \gamma}{3}, \; \tfrac{\pi/2 + \gamma - \bar{\vartheta}}{5}, \; \tfrac{\pi/2 + 2\gamma + \beta - 2\bar{\vartheta}}{2} \big\} \quad \text{and} \quad \hat{\lambda}(\bar{\Phi}, \bar{\vartheta} + \delta) > \Psi_{2}.\tag{29}\] Then there exists \(\varepsilon_{1} > 0\) small enough so that \[\label{LY339b} \hat{\lambda}(\lambda, \vartheta) > \Psi_{2} \quad \text{and} \quad \check{\lambda}(\lambda, \vartheta) > \bar{\Phi} \quad \text{for all } |\vartheta - \bar{\vartheta}| \leq \delta \; \text{ and } \; \lambda \geq \bar{\Phi} - \varepsilon_{1}.\tag{30}\] Note that \({u}\) is strictly monotone along the Neumann boundary (see ?? for \(\lambda \geq \bar{\Phi}\) and ?? for \(\lambda \geq \bar{\Phi}\cos\gamma\)) as well as near the Dirichlet boundary (see 2). By continuity, there exists a small constant \(\varepsilon_{2} \in (0, \varepsilon_{1})\) such that 3 holds for \[\vartheta \in [\gamma, \bar{\vartheta} - \delta] \cup [\bar{\vartheta} + \delta, \pi/2 + \gamma] \quad \text{and} \quad \lambda \geq \bar{\Phi} - \varepsilon_{2}.\] Now consider the case where \(|\vartheta - \bar{\vartheta}| < \delta\) and \(\lambda \geq \bar{\Phi} - \varepsilon_{2}\). Define \(\vartheta_{1} = \bar{\vartheta} - 3\delta\) and \(\vartheta_{3} = 2\vartheta - \vartheta_{1}\). By 29 , both \(\vartheta_{1}\) and \(\vartheta_{3}\) belong to \([\gamma, \bar{\vartheta} - \delta] \cup [\bar{\vartheta} + \delta, \pi/2 + \gamma]\), ensuring that \({w}^{\lambda, \vartheta}\) satisfies the strict boundary condition on \(\Gamma_{\lambda, \vartheta, \vartheta_{1}}^{2A}\). Furthermore, \(\Gamma_{\lambda, \vartheta}^{2B}\subset \hat{T}_{\hat{\lambda}, \hat{\vartheta}} = {T}_{\check{\lambda}, \check{\vartheta}}\) with \[\begin{align} & \hat{\lambda} > \Psi_{2} \quad \text{and} \quad \hat{\vartheta} = \pi - 2\vartheta + 2\gamma \in [\pi/2 - \beta, \pi/2] \quad \text{if } \vartheta \geq \pi/4 + \gamma, \\ & \check{\lambda} > \bar{\Phi} \quad \text{and} \quad \check{\vartheta} = 2\vartheta - \gamma \in [\gamma, \pi/2 + \gamma] \quad \text{if } \vartheta < \pi/4 + \gamma, \end{align}\] where 30 is used. Therefore, \({w}^{\lambda, \vartheta}\) satisfies the strict boundary condition on \(\Gamma_{\lambda, \vartheta, \vartheta_{1}}^{2B}\). Consequently, \({w}^{\lambda, \vartheta}\) satisfies ?? . Applying 4, we deduce that 6 and 3 remain valid for all \(\vartheta \in [\bar{\vartheta} - \delta, \bar{\vartheta} + \delta]\), with \(\vartheta_{1} = \bar{\vartheta} - 3\delta\) and for \(\lambda \geq \bar{\Phi} - \varepsilon_{2}\). Thus, 3 holds for every \(\vartheta \in [\gamma, \pi/2 + \gamma]\) and \(\lambda \geq \bar{\Phi} - \varepsilon_{2}\). A similar argument applies to 4 , ensuring its validity for all \(\vartheta \in [\gamma, \pi/2 + \gamma]\) when \(0 < \bar{t} \Psi_{0} - \lambda \ll 1\).

This contradicts the definition of \(\bar{t}\), implying that \(\bar{t} = 0\). The proof is complete. ◻

Proof. Without loss of generality, we assume that \(\alpha \geq \pi/2\). Set \[\bar{\Phi}: = \inf\{\Phi > 0: \; \eqref{LY303a} \text{ holds for every } \vartheta \in (0, \pi/2 + \gamma] \; \text{ and } \; \lambda \geq \Phi\}.\] We argue by contradiction and suppose that \(\bar{\Phi} > 0\). By 8, this infimum is well-defined and satisfies \(\bar{\Phi} \leq \Phi_{0}\). Using the same reasoning as in Step 1 of 5, it follows that 3 holds for all \(\vartheta \in (0, \pi/2 + \gamma]\) and \(\lambda \geq \bar{\Phi}\). More precisely, for each \(\lambda \geq \bar{\Phi}\), we have: \[{w}^{\lambda, \pi/2 + \gamma} > 0 \text{ in } \overline{ {D}_{\lambda, \pi/2 + \gamma} } \setminus {T}_{\lambda, \pi/2 + \gamma},\] and \[\text{ \({u}\) is increasing as the angle between \(\overrightarrow{ {P}_{\lambda} {x} }\) and \(\overrightarrow{ {P}_{\lambda} {z}_{0}}\) decreases on each arc \({S}({P}_{\lambda}, {r})\)},\] where \({S}({P}_{\lambda}, {r})\) is a connected arc of a circle defined by \[{S}({P}_{\lambda}, {r}) = \{ {x} \in \overline{\Omega}: \; |{x} - {P}_{\lambda}| = {r}, \; ({x} - {P}_{\lambda}) \cdot \mathbf{e}_{\pi/2 - \beta} \geq 0 \}.\]

Step 1. For \(\vartheta \in (0, \pi/2 + \gamma]\) and \(\lambda \geq \bar{\Phi}\), the inequality \[\label{LY343} {w}^{\lambda, \vartheta} > 0 \text{ in } \overline{ {D}_{\lambda, \vartheta} } \setminus {T}_{\lambda, \vartheta}\tag{31}\] holds. Details are omitted as the argument parallels that in 9.

Step 2. Both 5 and 3 hold at \(\vartheta = \pi/2\) for all \(\lambda \in (\bar{\Phi} - \varepsilon_{1}, \bar{\Phi}]\) for some small constant \(\varepsilon_{1} > 0\). This follows from the same argument as in Step 2 of the proof of 10.

Step 3. We now show that 3 holds for \[\label{LY345} 0 < \vartheta \leq \tilde{\vartheta}_{0} \; \text{ and } \; \lambda \geq \bar{\Phi} - \varepsilon_{2},\tag{32}\] where \(\tilde{\vartheta}_{0} = \max\{\pi/4 + \gamma, \bar{\vartheta} + \delta\}\) and \(\bar{\vartheta}\) is the unique angle such that \({T}_{\bar{\Phi}, \bar{\vartheta}}\) contains the upper mixed boundary point. The constants \(\delta > 0\) and \(\varepsilon_{2} \in (0, \varepsilon_{1})\) are chosen to satisfy: \[\label{LY346} \bar{\vartheta} + \delta < \pi/4 + \gamma + \beta/2, \quad \hat{\lambda}(\bar{\Phi} - \varepsilon_{2}, \bar{\vartheta} + \delta) > \Psi_{2}, \quad (\bar{\Phi} - \varepsilon_{2})(1 + \sin\gamma) > \bar{\Phi}.\tag{33}\] Note that the existence of such \(\delta\) and \(\varepsilon_{2}\) is ensured by the fact that \(\bar{\vartheta} \leq \pi - \alpha < \pi/4 + \gamma + \beta/2\).

Step 3.1. We verify that the strict boundary condition of \({w}^{\lambda, \vartheta}\) holds on \(\Gamma_{\lambda, \vartheta}^{2B}\) under the assumptions in 32 . In fact, we have:

1. If \(\vartheta \in (0, \gamma/2]\), then \(\Gamma_{\lambda, \vartheta}^{2B} = \emptyset\) is always valid.

2. If \(\vartheta \in (\gamma/2, \pi/4 + \gamma]\), then \(\Gamma_{\lambda, \vartheta}^{2B} = {T}_{\check{\lambda}, \check{\vartheta}}\) with \[\check{\vartheta} = 2\vartheta - \gamma \in (0, \pi/2 + \gamma] \; \text{ and } \; \check{\lambda} = \lambda + \frac{\lambda\sin\gamma}{\sin(2\vartheta - \gamma)} > \bar{\Phi}.\]

3. If \(\vartheta \in (\pi/4 + \gamma, \pi/4 + \gamma + \beta/2]\), then \(\Gamma_{\lambda, \vartheta}^{2B}\subset\hat{T}_{\hat{\lambda}, \hat{\vartheta}}\) with \[\hat{\vartheta} = \pi - 2\vartheta + 2\gamma \in [\pi/2 - \beta, \pi/2) \; \text{ and } \; \hat{\lambda} = \frac{\lambda\sin\vartheta}{\sin(\vartheta - \gamma)} > \Psi_{2}.\]

By the definition of \(\bar{\Phi}\) and applying 6, we conclude that the boundary condition of \({w}^{\lambda, \vartheta}\) is satisfied on \(\Gamma_{\lambda, \vartheta}^{2B}\).

Step 3.2. 31 and 3 hold for \(\vartheta \in (0, \pi/4]\) and \(\lambda \geq \bar{\Phi} - \varepsilon_{2}\). This follows directly from the argument used in Step 1.1 of 6.

Step 3.3. 31 and 3 to \(\vartheta \in (0, \min\{\pi/2, \tilde{\vartheta}_{0}\}]\) and \(\lambda \geq \bar{\Phi} - \varepsilon_{2}\). This is achieved by the same reasoning as in Step 1.2 of 6.

Step 3.4. Finally, 31 and 3 hold for \(\vartheta \in (0, \tilde{\vartheta}_{0}]\) and \(\lambda \geq \bar{\Phi} - \varepsilon_{2}\). Again, this follows from arguments similar to those in Step 1.2 of 6.

Step 4. We show that 3 holds for \(\vartheta = \tilde{\vartheta}_{*}\) and \(\lambda \geq \bar{\Phi} - \varepsilon_{3}\) for some \(\varepsilon_{3} \in (0, \varepsilon_{2})\). Here, we define \(\tilde{\vartheta}_{*} = \tilde{\vartheta}_{\kappa}\) where the integer \(\kappa\) is chosen such that \(\tilde{\vartheta}_{\kappa} \geq 2\gamma > \tilde{\vartheta}_{\kappa - 1}\), i.e., \(\tilde{\vartheta}_{\kappa + 1} \geq \gamma + \pi/2 > \tilde{\vartheta}_{\kappa }\), and \[\tilde{\vartheta}_{0} = \max\{\pi/4 + \gamma, \bar{\vartheta} + \delta\}, \quad \tilde{\vartheta}_{j} = (\pi + \tilde{\vartheta}_{j - 1})/2, \; j \in \mathbb{N}^{+}.\] Since the proof follows the same argument as in Step 2 of 10, the details are omitted.

Step 5. We conclude that 3 holds for \(\vartheta \in (0, \pi/2 + \gamma]\) and \(\lambda \geq \bar{\Phi} - \varepsilon_{3}\). Since this follows the same reasoning as Step 3 in 10, we omit the details.

By combining these steps, we obtain a contradiction to the definition of \(\bar{\Phi}\). Therefore, we must have \(\bar{\Phi} = 0\). This completes the proof. ◻

Proof. Define \[\bar{\Lambda}: = \inf\{\Lambda > 0: \; \text{\eqref{LY303a} and \eqref{LY303b} hold for } \vartheta \in [\gamma/2, \pi/2 + \gamma/2], \; \lambda > \Lambda\}.\] By Step 1 of 10, this infimum is well defined and satisfies \(\bar{\Lambda} < \Phi_{0}\). To prove the theorem, it suffices to show that \(\bar{\Lambda} = 0\). We argue by contradiction and assume that \(\bar{\Lambda} > 0\). By continuity, for all \(\vartheta \in [\gamma/2, \pi/2 + \gamma/2]\) and \(\lambda \geq \bar{\Lambda}\), we have \[\nabla {u} \cdot \mathbf{e}_{\vartheta + \alpha} \geq 0 \text{ on } \Omega \cap {T}_{\lambda, \vartheta}, \quad \text{and} \quad \nabla {u} \cdot \mathbf{e}_{ - \vartheta - \beta} \geq 0 \text{ on } \Omega \cap \hat{T}_{\lambda, \vartheta}.\] Moreover, \[{w}^{\lambda, \gamma/2} > 0 \text{ in } {D}_{\lambda, \gamma/2} \quad \text{and} \quad {w}^{\lambda, \pi/2 + \gamma/2} > 0 \text{ in } {D}_{\lambda, \pi/2 + \gamma/2},\] Thus, 3 and 4 hold for \(\vartheta \in [\gamma/2, \pi/2 + \gamma/2]\) and \(\lambda \geq \bar{\Lambda}\). By 9, we obtain \[{w}^{\lambda, \pi/2} > 0 \text{ in } \overline{{D}_{\lambda, \pi/2}} \setminus {T}_{\lambda, \pi/2} \quad \text{for all } \lambda \geq \bar{\Lambda}.\]

Let \(\bar{\vartheta}\) be the angle such that \({T}_{\bar{\Lambda}, \bar{\vartheta}}\) passes through the upper mixed boundary point. Since \(\bar{\Lambda} < \Phi_{0}\), we have \(\bar{\vartheta} \in (\gamma, \pi - \alpha)\). Choose a small constant \(\delta > 0\) such that \[\label{LY402a} 0 < \delta < \min\big\{ \tfrac{\bar{\vartheta} - \gamma}{3}, \; \tfrac{\pi + \gamma - 2\bar{\vartheta}}{10}, \; \tfrac{2\pi + 3\gamma - 4\bar{\vartheta}}{4} \big\}.\tag{34}\] We then choose \(\varepsilon_{1} > 0\) small enough so that \[\label{LY402b} \hat{\lambda}(\lambda, \vartheta) > \bar{\Lambda} \; \text{ and } \; \check{\lambda}(\lambda, \vartheta) > \bar{\Lambda} \; \text{ for } |\vartheta - \bar{\vartheta}| \leq \delta, \; \lambda \geq \bar{\Lambda} - \varepsilon_{1}.\tag{35}\] By continuity, there exists a small constant \(\varepsilon_{2} \in (0, \varepsilon_{1})\) such that 3 holds for \[\vartheta \in [\gamma, \bar{\vartheta} - \delta] \cup [\bar{\vartheta} + \delta, \pi/2 + \gamma/2] \; \text{ and } \; \lambda \geq \bar{\Lambda} - \varepsilon_{2}.\] Furthermore, applying the same reasoning as in Step 2 of 6, we conclude that 3 also holds for \(\vartheta \in [\gamma/2, \gamma]\) and \(\lambda \geq \bar{\Lambda} - \varepsilon_{2}\). Now for \(|\vartheta - \bar{\vartheta}| < \delta\) and \(\lambda \geq \bar{\Lambda} - \varepsilon_{2}\), take \(\vartheta_{1} = \bar{\vartheta} - 3\delta\) and \(\vartheta_{3} = 2\vartheta - \vartheta_{1}\). The condition 34 ensures that \[\vartheta_{1} \in [\gamma, \bar{\vartheta} - \delta] \quad \text{and} \quad \vartheta_{3} \in [\bar{\vartheta} + \delta, \pi/2 + \gamma/2].\] From 35 , we find that \(\Gamma_{\lambda, \vartheta}^{2B}\subset \hat{T}_{\hat{\lambda}, \hat{\vartheta}} = {T}_{\check{\lambda}, \check{\vartheta}}\) with \[\begin{align} \hat{\lambda} > \bar{\Lambda} & \; \text{ and } \; \hat{\vartheta} = \pi - 2\vartheta + 2\gamma \in [\gamma/2, \pi/2 + \gamma/2] \text{ if } \vartheta \geq \pi/4 + 3\gamma/4, \\ \check{\lambda} > \bar{\Lambda} & \; \text{ and } \; \check{\vartheta} = 2\vartheta - \gamma \in [\gamma/2, \pi/2 + \gamma/2] \text{ if } \vartheta < \pi/4 + 3\gamma/4. \end{align}\] Hence, \({w}^{\lambda, \vartheta}\) satisfies the strict boundary conditions on both \(\Gamma_{\lambda, \vartheta, \vartheta_{1}}^{2A}\) and \(\Gamma_{\lambda, \vartheta, \vartheta_{1}}^{2B}\), and satisfies ?? . Applying 4, we conclude that 6 and 3 hold for all \(\lambda \geq \bar{\Lambda} - \varepsilon_{2}\), \(\vartheta \in (\bar{\vartheta} - \delta, \bar{\vartheta} + \delta)\) and \(\vartheta_{1} = \bar{\vartheta} - 3\delta\).

In summary, 3 holds for \(\vartheta \in [\gamma/2, \pi/2 + \gamma/2]\) and \(\lambda \geq \bar{\Lambda} - \varepsilon_{2}\). Similarly, 4 remains valid for \(\vartheta \in [\gamma/2, \pi/2 + \gamma/2]\) and \(\bar{\Lambda} - \lambda\ll1\). This contradicts the minimality of \(\bar{\Lambda}\), hence \(\bar{\Lambda} = 0\). Consequently, both 3 and 4 hold for \(\vartheta \in [\gamma/2, \pi/2 + \gamma/2]\) and \(\lambda > 0\).

To conclude the symmetry, observe that \[{w}^{\lambda, \gamma/2}( {x} ) = {u}({x}_{1}, - 2\lambda\sin(\gamma/2) - x_{2}) - {u}({x}_{1}, x_{2}) > 0 \text{ for } {x} \in \Omega \text{ with } x_{2} < - \lambda\sin(\gamma/2).\] Similarly, one can deduce that \[{u}({x}_{1}, 2\lambda\sin(\gamma/2) - x_{2}) - {u}({x}_{1}, x_{2}) > 0 \text{ for } {x} \in \Omega \text{ with } x_{2} > \lambda\sin(\gamma/2).\] Taking the limit as \(\lambda \to 0^{ + }\), we obtain \[{u}({x}_{1}, - x_{2}) - {u}({x}_{1}, x_{2}) = 0 \; \text{ for } {x} \in \Omega,\] which proves the symmetry of \({u}\) with respect to the \(x_{1}\)-axis. ◻

Proof. Without loss of generality, we assume that \(\alpha \geq \beta\), which implies \[\label{LY501b} \alpha \geq \pi/4.\tag{36}\]

Part 1. We claim that 3 holds for \(\vartheta \in [\pi/2 - \alpha, \pi/2]\) and \(\lambda \geq 0\). To prove this, we argue by contradiction and assume that \(\bar{\Phi} > 0\), where we set \[\bar{\Phi}: = \inf \{\Phi > 0: \; \eqref{LY303a} \text{ holds for } \vartheta \in [\pi/2 - \alpha, \pi/2] \text{ and } \lambda > \Phi\}.\] By Step 1 of 10, this infimum is well-defined and satisfies \(\bar{\Phi} < \Phi_{0}\). For \(\vartheta = \pi/2\) and \(\lambda \geq \bar{\Phi}/2\), we have \(\Gamma_{\lambda, \pi/2}^{2B}\subset {T}_{2\lambda, \pi - \gamma}\) and \(\pi - \gamma \in [\pi/2 - \alpha, \pi/2]\). Therefore, \({w}^{\lambda, \pi/2}\) satisfies the boundary condition ?? . By 4, both 5 and 3 hold for \(\vartheta = \pi/2\) and \(\lambda \geq \bar{\Phi}/2\). Moreover, the condition 36 implies that \(\pi/4 \geq \pi/2 - \alpha\). We can apply the arguments from Step 1.1 and Step 2 of 6 to deduce that 3 with \(\lambda \geq \bar{\Phi}/2\) holds for \(\vartheta \in [\pi/4, \pi/2]\) and for \(\vartheta \in [\pi/2 - \alpha, \pi/4]\). This contradicts the definition of \(\bar{\Phi}\). Hence, \(\bar{\Phi} = 0\), and 3 holds for all \(\vartheta \in [\pi/2 - \alpha, \pi/2]\) and \(\lambda \geq 0\). This completes Part 1.

Part 2. We now claim that 3 holds for \(\vartheta = \gamma\) and \(\lambda > 0\). To prove this, define \(\tilde{u}\) by reflecting \({u}\) across the line \({T}_{0, 0}\), which contains the lower Neumann boundary \(\Gamma_{N}^{ - }\), as follows: \[\label{LY504} \tilde{u}( {x} ) = \begin{cases} {u}( {x} ), & \text{ if } {x} \in \overline{\Omega}, \\ {u}({x}^{0, 0}), & \text{ if }{x}^{0, 0} \in \overline{\Omega}, \end{cases}\tag{37}\] where \({x}^{0, 0}\) denotes the reflection of \({x}\) with respect to \(\Gamma_{N}^{ - }\). Then \(\tilde{u}\) is a positive solution of 1 in the doubled domain \(\tilde{\Omega}: = \Omega \cup \Omega' \cup \Gamma_{N}^{ - }\), where \(\Omega'\) is the reflection of \(\Omega\) with respect to \(\Gamma_{N}^{ - }\); see 4. From the assumptions \(\gamma > \pi/2\) and \(\alpha \geq \pi/4\), it follows that \(\pi - \gamma \in [\pi/2 - \alpha, \pi/2]\). Thus, by Part 1, we obtain that \(\nabla {u} \cdot \mathbf{e}_{\alpha - \gamma} < 0\) on \(\Omega \cap {T}_{0, \pi - \gamma}\), that is, \[\nabla \tilde{u} \cdot \mathbf{e}_{ - \beta} < 0 \text{ on } \tilde{\Omega} \cap \hat{T}_{0, 0},\] where \(\hat{T}_{0, 0}\) is the line containing \(\Gamma_{N}^{ + }\). Consequently, \(\tilde{u}\) satisfies \[\begin{cases} \Delta \tilde{u} + {f}(\tilde{u}) = 0 & \text{ in } \tilde{\Omega} \cap \{{x} \cdot \mathbf{e}_{ - \beta} > 0\}, \\ \tilde{u} = 0 & \text{ on } \partial\tilde{\Omega} \cap \{{x} \cdot \mathbf{e}_{ - \beta} > 0\}, \\ \nabla\tilde{u} \cdot \mathbf{e}_{ - \beta} \leq 0 & \text{ on } \overline{\tilde{\Omega}} \cap \{{x} \cdot \mathbf{e}_{ - \beta} = 0\}. \end{cases}\] Applying the moving plane method in the direction \(\mathbf{e}_{ - \beta}\), it is easy to derive that \[\begin{gather} \tilde{u}({x}^{\lambda, \gamma}) - \tilde{u}( {x} ) > 0 \text{ for } {x} \in \tilde{\Omega} \text{ with } \lambda\sin\gamma < {x} \cdot \mathbf{e}_{ - \beta} \leq 2\lambda\sin\gamma, \\ \nabla \tilde{u} \cdot \mathbf{e}_{ - \beta} < 0 \text{ on } \tilde{\Omega} \cap {T}_{\lambda, \gamma} \end{gather}\] for every \(\lambda > 0\). In particular, this implies that \(\nabla {u} \cdot \mathbf{e}_{ - \beta} < 0\) in \(\Omega\). Therefore, 3 holds for all \(\vartheta \in [\pi/2 - \alpha, \gamma]\) and \(\lambda > 0\), and 4 holds for all \(\vartheta \in [\pi - \beta, \pi]\) and \(\lambda \geq \Psi_{0}\). This completes the proof. ◻

Proof. The isosceles case (i.e., \(\alpha = \beta\)) has been established in 7. Without loss of generality, we assume that \(\alpha > \beta\), which implies \[\label{LY506b} \gamma \geq \pi/2 + \beta.\tag{38}\] To prove the result, we define \[\bar{\Phi}: = \inf \{\Phi > 0: \; \text{ \eqref{LY303a} holds for \(\vartheta \in [\pi/2 - \alpha, \gamma]\) and \(\lambda > \Phi\)} \}.\] By Step 1 of 10, this infimum is well-defined and satisfies \(\bar{\Phi} < \Phi_{0}\). Suppose, for contradiction, that \(\bar{\Phi} > 0\).

Part 1. We first show that 3 holds for all \(\vartheta \in [\pi/2 - \alpha, \gamma]\) and \(\lambda \geq \bar{\Phi}\). In fact, by continuity and the strong maximum principle applied to the angular derivative of \({u}\), we obtain 3 for all \(\vartheta \in (\pi/2 - \alpha, \gamma)\) and \(\lambda \geq \bar{\Phi}\); see Step 1 of 5. For the endpoint \(\vartheta = \pi/2 - \alpha\), we take \(\vartheta_{1} = 0\), \(\vartheta_{3} = \pi - 2\alpha\). Note that 38 implies that \(\vartheta_{3} \in [\pi/2 - \alpha, \gamma]\) and \(\Gamma_{\lambda, \vartheta}^{2B} = \emptyset\). Since \(\lambda \geq \bar{\Phi}\), we have \({w}^{\lambda, \vartheta}\) satisfies ?? . Hence, both 5 and 3 hold for \(\vartheta = \pi/2 - \alpha\) and \(\lambda \geq \bar{\Phi}\). For the other endpoint \(\vartheta = \gamma\), we take \(\vartheta_{3} = \pi\), \(\vartheta_{1} = 2\gamma - \pi\), so that \(\Gamma_{\lambda, \gamma}^{2B}\subset {T}_{2\lambda, \gamma}\) and \(\Gamma_{\lambda, \gamma}^{2A}\subset {T}_{\lambda, 2\gamma - \pi}\). By 38 , \(2\gamma - \pi \in [\pi/2 - \alpha, \gamma]\), and so \({w}^{\lambda, \vartheta}\) satisfies ?? . Thus, 5 and 3 hold for \(\vartheta = \gamma\) and \(\lambda \geq \bar{\Phi}\).

Part 2. We prove that for \(\lambda \geq \bar{\Phi}\), \[\label{LY508} \nabla {u} \cdot \mathbf{e}_{\alpha - \pi/2} < 0 \text{ on } \Gamma_{N}^{ - } \cap {T}_{\lambda, \pi/2}.\tag{39}\] Note that it is difficult to derive the positivity of the function \({w}^{\lambda, \pi/2}\), the argument employed in Step 2 of 5 is no longer applicable. We therefore adopt a different strategy based on the local asymptotic analysis of the angular derivative developed in [31]. The required details are presented below. Let \(\lambda \in [\bar{\Phi}, \Phi_{0})\), and define \[\bar{x}: = (\bar{x}_{1}, \bar{x}_{2}), \quad \text{with } \; \bar{x}_{1} = \lambda\sin\alpha \text{ and } \bar{x}_{2} = - \lambda\cos\alpha.\] The angular derivative \({R}_{\bar{x}}{u}\) about \(\bar{x}\), defined in 18 , satisfies \[\Delta {R}_{\bar{x}}{u} + {f}'({u}) {R}_{\bar{x}}{u} = 0 \text{ in } \Omega, \quad \text{and} \quad {R}_{\bar{x}}{u} = 0 \text{ on } \Gamma_{N}^{ - }.\] Let \(({r}, \vartheta)\) denote polar coordinates centered at \(\bar{x}\), where \(\vartheta\) is the polar angle measured from the ray \(\bar{x}{z}_{1}\) to the ray \(\bar{x}{x}\). From [44], [47], there exist an integer \({l} \geq 1\) and a constant \({C}_{0} \in \mathbb{R}\setminus \{0\}\), such that \[\label{LY509a} ({R}_{\bar{x}}{u})({r}, \vartheta) = {C}_{0}{r}^{l}\sin(l\vartheta) + O({r}^{l + 1}),\tag{40}\] with \({R}_{\bar{x}}{u}\) has exactly \({l} - 1\) nodal domains near \(\bar{x}\) in \(\Omega\). Here, the term \(O({r}^{l + 1})/{r}^{l + 1}\) is bounded as \({r}\to0\). By Part 1, we have that \[({R}_{\bar{x}}{u})({r}, \vartheta) > 0 \; \text{ for } \vartheta \in [\pi/2 - \alpha, \gamma],\] so combining with 40 yields \[\label{LY509c} {C}_{0}\sin({l}\vartheta) > 0 \; \text{ for } \vartheta \in (\pi/2 - \alpha, \gamma).\tag{41}\] Therefore, the interval \((\pi/2 - \alpha, \gamma)\) must be contained in a nodal interval of the function \(\sin({l}\vartheta)\), whose length is at most \(\pi/{l}\). Combining this with the fact that \[\label{LY506c} \gamma > \pi/2 \quad \text{and} \quad \alpha > \beta,\tag{42}\] we deduce that \(\beta < (\alpha + \beta)/2 = (\pi - \gamma)/2 < \pi/4\), and \[{l} \leq \frac{\pi}{\gamma - (\pi/2 - \alpha)} = \frac{\pi}{\pi/2 - \beta} < 4.\] Since \(\pi/2 \in (\pi/2 - \alpha, \gamma)\), it follows from 41 that \({C}_{0}\sin({l}\pi/2) > 0\), so \({l}\) is odd.

We claim that \({l} \neq 3\). Indeed, if \({l} = 3\), then 41 becomes \({C}_{0}\sin (3\vartheta) > 0\) for \(\vartheta \in (\pi/2 - \alpha, \gamma)\). Combining this with the fact that the interval \((3(\pi/2 - \alpha), 3\gamma)\) contains \(3\pi/2\), we conclude that \({C}_{0} < 0\) and \[3(\pi/2 - \alpha) \geq \pi, \quad 3\gamma \leq 2\pi,\] that is, \(\alpha \leq \pi/6\) and \(\gamma \leq 2\pi/3\). From this and the condition \(\alpha > \beta\) in 42 , we obtain \(\alpha > (\alpha + \beta)/2 = (\pi - \gamma)/2 > \pi/6\), which yields a contradiction.

Since \({l} < 4\), \({l}\) is odd, and \({l}\neq3\), we conclude \({l} = 1\) and \({C}_{0} > 0\). Therefore, \({R}_{\bar{x}}{u} > 0\) near \(\bar{x}\) in \(\Omega\). Applying Hopf lemma to \({R}_{\bar{x}}{u}\), it follows that the outward normal derivative of \({R}_{\bar{x}}{u}\) at \(\bar{x}\) is negative, i.e., \(\partial_{\nu}{R}_{\bar{x}}{u}(\bar{x}) < 0\). Therefore, \(\eqref{LY508}\) holds.

Part 3. We show that 3 holds for \(\vartheta \in [\pi/2 - \alpha, \gamma]\) and \(\bar{\Phi} - \lambda \ll1\). Since \({u}\) is strictly monotone near the lower Neumann boundary (by \(\eqref{LY508}\)), near the Dirichlet boundary (see 2), and in the interior of \(\Omega\) (by Part 1), it follows by continuity that \(\eqref{LY303a}\) holds for \[\vartheta \in [\pi - 2\alpha, \gamma] \quad \text{and} \quad \lambda \geq \bar{\Phi} - \varepsilon\] for some small constant \(\varepsilon > 0\). Using the same argument as in Step 2 of 6, we deduce that 3 holds for \(\vartheta \in [\pi/2 - \alpha, \pi - 2\alpha]\) and \(\lambda \geq \bar{\Phi} - \varepsilon\). This contradicts the definition of \(\bar{\Phi}\), and we conclude \(\bar{\Phi} = 0\). This completes the proof. ◻

Proof. Since the nonpositive function \(\tilde{v}\) is negative near \(z_{1}\) (e.g., 6) and satisfies \([\Delta + {f}'({u})] \tilde{v} = 0\) in \(\Omega\), the strong maximum principle implies that \(\tilde{v} < 0\) in \(\Omega\). The proof is divided into two parts.

Part 1. We first show that assumption ?? implies that 3 holds for every \(\lambda \geq 0\) and for all \(\vartheta \in [\pi/2 - \alpha, \gamma]\). Moreover, it follows that \[\label{LY512} \nabla {u} \cdot \mathbf{e}_{\alpha - \pi/2} < 0 \text{ on } \Gamma_{N}^{ - }.\tag{43}\] Indeed, since \(\alpha > \beta\), it follows that \(\gamma/2 > \pi/2 - \alpha\). For \(\vartheta = \gamma/2\), take \(\vartheta_{1} = 0\), \(\vartheta_{3} = \gamma\). Then, by ?? and for any \(\lambda \geq 0\), the function \({w}^{\lambda, \vartheta}\) satisfies the boundary condition ?? . Applying 4, we deduce that both 6 and 3 hold. Following Step 1.1 of 6, we conclude that 3 holds for all \(\lambda \geq 0\) and \(\vartheta \in [\gamma/2, \gamma]\). A similar argument from Step 1.2 of 6 shows that 3 remains valid for \[\vartheta \in \cup_{k = 1}^{ \infty}[\min\{2^{ - k}\gamma, \; \pi/2 - \alpha\}, \; \gamma].\] In particular, 3 holds for \(\lambda \geq 0\) and for \(\vartheta = \pi/2\), this is, \[\nabla {u}({x}) \cdot \mathbf{e}_{\alpha - \pi/2} < 0 \; \text{ for } {x} \in \Omega \text{ with } {x} \cdot \mathbf{e}_{\alpha - \pi/2} \geq 0.\] Consequently, the even extension \(\tilde{u}\) of \({u}\) with respect to the Neumann boundary \(\Gamma_{N}^{ - }\), defined in 37 , satisfies \[\nabla \tilde{u}({x}) \cdot \mathbf{e}_{\alpha - \pi/2} \leq, \not\equiv 0 \; \text{ for } {x} \in \tilde{\Omega} \text{ with } {x} \cdot \mathbf{e}_{\alpha - \pi/2} \geq 0,\] where \(\tilde{\Omega}\) denotes the reflected (or doubled) domain obtained by reflecting \(\Omega\) across the lower Neumann boundary \(\Gamma_{N}^{ - }\); see 4. Since the function \(\nabla \tilde{u} \cdot \mathbf{e}_{\alpha - \pi/2}\) satisfies the linear equation \[[\Delta + {f}'(\tilde{u})] (\nabla \tilde{u} \cdot \mathbf{e}_{\alpha - \pi/2}) = 0 \text{ in } \tilde{\Omega},\] the strong maximum principle guarantees that \(\nabla \tilde{u} \cdot \mathbf{e}_{\alpha - \pi/2}\) must be negative in \(\tilde{\Omega} \cap \{{x} \cdot \mathbf{e}_{\alpha - \pi/2} \geq 0\}\). This proves 43 .

Part 2. We now show that the origin \({z}_{0}\) is not an extremum point of \({u}\).

Let \(({r}, \theta)\) denote the standard polar coordinates centered at the origin. By classical regularity theory for solutions in domains with conical points (e.g., Theorem  6.4.2.5 in [43]), the expansion of \({u}\) near the origin is \[\label{LY515u} {u}({r}, \theta) = {c}_{0} - {c}_{1} {r}^{\omega}\cos\big(\omega(\theta - \alpha + \pi/2)\big) - \tfrac{1}{4}{c}_{2} {r}^{2} + o({r}^{2})\tag{44}\] where \(\omega = \pi/\gamma \in (1, 2)\), \({c}_{0} = {u}({z}_{0}) > 0\), and \({c}_{2} = {f}({c}_{0})\). Define the angular derivative \[{v}( {x} ): = x_{1}\partial_{x_{2}}{u}( {x} ) - x_{2}\partial_{x_{1}}{u}( {x} ).\] From Part 1, we know that 3 holds for \(\lambda = 0\) and \(\vartheta \in [\pi/2 - \alpha, \gamma]\). That is \[\label{LY514} {v}( {x} ) > 0 \; \text{ for } {x} \in \Omega \text{ with } 0 \leq x_{2} < x_{1}\cot\beta.\tag{45}\] Note that \({v}\) satisfies linear equation \[[\Delta + {f}'({u})] {v} = 0 \text{ in } \Omega \quad \text{and} \quad {v} = 0 \text{ on } \Gamma_{N},\] and \({v}\not\equiv 0\) in \(\Omega\) (in any neighborhood of \({z}_{0}\)). By [44], [45], there exist an integer \({l} \geq 1\) and a constant \(\tilde{c}_{l} \neq 0\) such that \[{v}({r}, \theta) = {l}\omega \tilde{c}_{l} {r}^{l\omega}\sin\big({l}\omega(\theta - \alpha + \pi/2)\big) + o({r}^{{l}\omega}).\] Moreover, the nodal set \(\mathcal{Z}({v}) = \overline{\{{x} \in \Omega: \; {v}({x}) = 0\}}\) has exactly \({l} - 1\) branches near the origin, each tangent to the line \[x_{1} \sin(\frac{j\gamma}{l} + \alpha) - x_{2}\cos(\frac{j\gamma}{l} + \alpha) = 0\] at the origin \({z}_{0}\) for \(j = 1, \ldots, {l} - 1\). Combining this with 45 , we obtain \(\gamma/{l} \geq \pi/2 - \beta\), hence \({l} \leq \gamma/(\pi/2 - \beta)\). Since \(\alpha > \beta\) implies \(\gamma = \pi - \alpha - \beta < \pi - 2\beta\), we conclude \({l} < 2\), thus \({l} = 1\). Hence, \[{v}({r}, \theta) = \tilde{c}_{1}\omega {r}^{\omega}\sin\big(\omega(\theta - \alpha + \pi/2)\big) + o({r}^{2}).\] From 45 , we deduce that \(\tilde{c}_{1} > 0\) and \({v} > 0\) in a neighborhood of the origin \({z}_{0}\). Therefore, \({c}_{1} > 0\), which implies that the origin is not an extremum point of \({u}\). ◻

Proof. Define the set of non-vertex critical points by \[\operatorname{crit}_{\mathrm{nv}}({u}): = \{{x} \in \overline{\Omega}: \; |\nabla {u}({x})| = 0 \text{ and } {x} \text{ is not a vertex}\}.\]

Step 1. We show that \(\operatorname{crit}_{\mathrm{nv}}({u}) \cap \Gamma_{N}\) contains at most one point, and at that point the second tangential derivative of \({u}\) is negative.

Let \({v}\) be the directional derivative of \({u}\) in the direction parallel to \(\Gamma_{N}^{ + }\), namely, \[{v}: = \nabla {u} \cdot \mathbf{e}_{\pi/2 - \beta}.\] Applying the Hopf lemma to \({u}\) on \(\Gamma_{D}\) and using 5, we obtain \[\label{LY604} \begin{gather} {v} < 0 \text{ on } \operatorname{Int}(\Gamma_{D}), \\ {v} < 0 \text{ in } \mathcal{O}_{{z}_{2}} \cap (\Omega \cup \Gamma_{N}^{ + }) \quad \text{and} \quad {v} > 0 \text{ on } \mathcal{O}_{{z}_{1}} \cap \Gamma_{N}^{ - }, \end{gather}\tag{46}\] where \(\mathcal{O}_{{z}_{i}}\) denotes a small neighborhood of \({z}_{i}\) for \(i = 1, 2\). Note that \[\Delta {v} + \mu {v} = 0 \text{ in } \Omega \quad \text{and} \quad \partial_{\nu} {v} = 0 \text{ on } \Gamma_{N}^{ + }.\] By classical nodal set theory (see, e.g., [47]), for any \({p} \in \operatorname{crit}_{\mathrm{nv}}({u}) \cap \Gamma_{N}^{ + }\), there exists an integer \({n} \in \mathbb{N}^{ + }\) such that exactly \({n}\) smooth arcs of \(\mathcal{Z}({v})\) emanate from \({p}\), intersecting \(\Gamma_{N}^{ + }\) at angles \((2j - 1)/(2{n})\), \(j = 1, 2, \ldots, {n}\). Thus, \(\operatorname{crit}_{\mathrm{nv}}({u}) \cap \Gamma_{N}^{ + }\) consists of finitely many isolated points. An analogous argument for \(\Gamma_{N}^{ - }\) shows that \({u}\) has only finitely many critical points on \(\Gamma_{N}^{ - }\), and thus on \(\partial\Omega\). Let \(\overline{\Gamma}_{N}^{\pm}\) denote the closures of the two Neumann sides \(\Gamma_{N}^{\pm}\).

Step 1.1. \(\overline{\Gamma}_{N}^{ + } \cup \mathcal{Z}({v})\) cannot contain a loop, and \(\mathfrak{D}_{ - }: = \{{x} \in \Omega: {v}({x}) < 0\}\) is connected.

If there were a loop in \(\overline{\Gamma}_{N}^{ + } \cup \mathcal{Z}({v})\), it would enclose a nodal domain \(\mathfrak{D}_{1}\) with \(\partial\mathfrak{D}_{1} \subset \overline{\Gamma}_{N}^{ + } \cup \mathcal{Z}({v})\). Hence, \(\lambda_{1}^{\mathrm{mix}}\bigl(\mathfrak{D}_{1}, \partial\mathfrak{D}_{1} \cap \Gamma_{N}^{ + }\bigr) = \mu\). By the variational characterization of the first eigenvalue, \[\lambda_{1}^{\mathrm{mix}}\bigl(\mathfrak{D}_{1}, \partial\mathfrak{D}_{1} \cap \Gamma_{N}^{ + }\bigr) > \lambda_{1}^{\mathrm{mix}}(\Omega, \Gamma_{N}) = \mu,\] which is a contradiction. Thus, no such loop can occur.

Suppose, for contradiction, that \(\mathfrak{D}_{ - }\) is disconnected. By 46 , one component of \(\mathfrak{D}_{ - }\) meets \(\operatorname{Int}(\Gamma_{D})\), so there is another component \(\mathfrak{D}_{*}\) whose boundary does not meet \(\operatorname{Int}(\Gamma_{D})\). Since \(\overline{\Gamma}_{N}^{ + }\cup\mathcal{Z}({v})\) has no loop, \(\partial\mathfrak{D}_{*} \cap \Gamma_{D} = \emptyset\) and \(\Gamma_{*}: = \partial\mathfrak{D}_{*} \cap \overline{\Gamma}_{N}^{ - } \neq \emptyset\). Define the auxiliary function \[\phi({x}) = \begin{cases} {v}({x}) & \text{ if } {x} \in \mathfrak{D}_{*}, \\ 0 & \text{ if } {x} \in \Omega \setminus \mathfrak{D}_{*}. \end{cases}\] Then \(\phi\) satisfies \[\label{LY605} \phi \in W^{1, 2}(\Omega) \quad \text{with} \quad \phi = 0 \text{ on } \Gamma_{D},\tag{47}\] and \(\partial_{\nu}\phi = 0\) on \(\Gamma_{N}^{ + }\). Integration by parts gives \[\int_{\Omega} |\nabla \phi|^{2} dx = \mu \int_{\Omega} |\phi|^{2} dx + \int_{\partial\Omega} \phi \partial_{\nu}\phi ds_x = \mu \int_{\Omega} |\phi|^{2} dx + \int_{\Gamma_{*}} {v}\partial_{\nu}{v} ds_x.\] To handle \(\int_{\Gamma_{*}} {v} \partial_{\nu}{v} ds_{x}\), we use a technique inspired by Terence Tao [36]. Denote the tangential and normal vectors of \(\Gamma_{N}^{ - }\) by \[\mathit{e} = (\sin\alpha, - \cos\alpha), \quad \mathit{e}^{\perp} = (\cos\alpha, \sin\alpha),\] and write \({v} = {c}_{1}\partial_{\mathit{e}}{u} + {c}_{2}\partial_{\mathit{e}^{\perp}}{u}\), with \({c}_{1} = \cos \gamma < 0\) and \({c}_{2} = \sin \gamma > 0\). A direct calculation yields \[{v}\partial_{\nu}{v} = ({c}_{1}\partial_{\mathit{e}} {u}) \cdot ( - {c}_{2} (\partial_{\mathit{e}^{\perp}})^{2} {u}) = {c}_{1}{c}_{2} \partial_{\mathit{e}} {u} \cdot ((\partial_{\mathit{e}})^{2} {u} + \mu {u}).\] Since \({v} < 0\) in \(\mathfrak{D}_{*}\), one infers \(\partial_{\mathit{e}} {u} \geq 0\) on \(\Gamma_{*}\), hence \[\int_{\Gamma_{*}} {u} \partial_{\mathit{e}} {u} ds_{x} \geq 0.\] Moreover, \(\mathcal{Z}({v}) \cap \overline{\Gamma}_{N}^{ - }\) is a finite set of critical points of \({u}\), so \(\Gamma_{*}\) is a union of finitely many isolated points and finitely many line segments \({q}_{2i}{q}_{2i + 1}\), \(i = 0, \ldots, {m}\), where \({q}_{i}\) are distinct critical points of \({u}\) and \({q}_{i}\) lies to the left of \({q}_{i + 1}\). Therefore \[\begin{align} \int_{\Gamma_{*}} \partial_{\mathit{e}} {u} (\partial_{\mathit{e}})^{2}{u} ds_x = \frac{1}{2} \int_{\Gamma_{*}} \partial_{\mathit{e}}\big(\partial_{\mathit{e}}{u}\big)^{2} ds_x = \frac{1}{2}\sum_{i = 0}^{m}(|\partial_{\mathit{e}} {u}({q}_{2i + 1})|^{2} - |\partial_{\mathit{e}} {u}({q}_{2i})|^{2} ) = 0. \end{align}\] Hence, \[\int_{\Gamma_{*}} {v}\partial_{\nu}{v} ds_{x} = {c}_{1}{c}_{2} \int_{\Gamma_{*}} \partial_{\mathit{e}}{u} (\partial_{\mathit{e}})^{2}{u} ds_{x} + {c}_{1}{c}_{2} \mu \int_{\Gamma_{*}} {u} \partial_{\mathit{e}}{u} ds_{x} \leq 0,\] and so \(\phi\) satisfies 47 and \[\int_{\Omega} |\nabla \phi|^{2} d{x} \leq \mu \int_{\Omega} |\phi|^{2} d{x}.\] By the variational characterization of the first mixed eigenvalue \(\mu = \lambda_{1}^{\mathrm{mix}}(\Omega, \Gamma_{N})\), \(\phi\) must be a multiple of \({u}\), contradicting the definition of \(\phi\). Therefore, \(\mathfrak{D}_{ - }\) is connected.

Step 1.2. \(\operatorname{crit}_{\mathrm{nv}}({u}) \cap \Gamma_{N}^{ + }\) contains at most one point, and at that point the second-order tangential derivative of \({u}\) is strictly negative. It is well-known (cf. Section 2 of [48]) that the nodal line \(\mathcal{Z}({v})\) decomposes into immersed \({C}^{1}\) loops and properly immersed \({C}^{1}\) arcs. By Step 1.1, no loop can occur, so \(\mathcal{Z}({v})\) is a finite union of properly immersed \({C}^{1}\) arcs. Each such arc has exactly two distinct endpoints, one on \(\overline{\Gamma}_{N}^{ - }\) and one on \(\Gamma_{N}\). Since \(\mathfrak{D}_{ - }\) is connected, there is at most one arc ending on \(\Gamma_{N}^{ + }\). Standard nodal line theory (see [47]) then shows that \({v}\) has at most one zero on \(\Gamma_{N}^{ + }\), and at that zero the second tangential derivative of \({u}\) is negative.

Step 1.3. \({u}\) has at most one critical point on \(\Gamma_{N}^{ + } \cup \Gamma_{N}^{ - }\). Suppose for contradiction that there are two such points. A similar argument as in Steps 1.1 and 1.2 implies that the restriction of \({u}\) to \(\Gamma_{N}^{ - }\) has at most one critical point, which is non-degenerate. Hence, \({u}\) has two critical points, \({p} \in \Gamma_{N}^{ + }\) and \({q} \in \Gamma_{N}^{ - }\). From 46 , \[{v} < 0 \text{ on } \operatorname{Int}(\Gamma_{D}) \cup \operatorname{Int}({z}_{2}{p}) \cup \operatorname{Int}({z}_{0}{q}) \quad \text{and} \quad {v} > 0 \text{ on } \operatorname{Int}({p}{z}_{0}) \cup \operatorname{Int}({q}{z}_{1}).\] Let \(\mathfrak{D}_{2}\) be the nodal domain of \({v}\) whose closure contains the line segment \({p}{z}_{0}\). Since \(\mathfrak{D}_{ - } = \{{x} \in \Omega: {v}({x}) < 0\}\) is connected (by Step 1.1), its boundary must contain both the line segments \({z}_{2}{p}\) and \({z}_{0}{q}\). Hence, \(\mathfrak{D}_{2}\) is bounded by the line segment \({p}{z}_{0}\) together with a portion of \(\mathcal{Z}({v})\), contradicting Step 1.1. This contradiction completes the proof of Step 1.

Step 2. \(\operatorname{crit}_{\mathrm{nv}}({u})\) has at most one point, which lies on the longer Neumann side.

Step 2.1. \({u}\) is strictly monotone in some direction. By Step 1, \({u}\) has at most one non-vertex critical point on \(\partial\Omega\). Without loss of generality, assume there is no critical point on \(\Gamma_{N}^{ - }\). Consider the directional derivative \(\tilde{v} = \nabla {u} \cdot \mathbf{e}_{ - \beta}\) in the direction perpendicular to \(\Gamma_{N}^{ + }\). Then \(\tilde{v} \leq, \not\equiv 0\) on \(\partial\Omega\). If \(\tilde{v}\) were positive somewhere in \(\Omega\), its positive nodal domain \(\mathfrak{D}_{3}\) would satisfy \[\Delta \tilde{v} + \mu \tilde{v} = 0 \text{ in } \mathfrak{D}_{3}, \quad \tilde{v} = 0 \text{ on } \partial\mathfrak{D}_{3},\] forcing \(\mu = \lambda_{1}(\mathfrak{D}_{3}) > \lambda_{1}(\Omega) > \lambda_{1}^{\mathrm{mix}}(\Omega, \Gamma_{N}) = \mu\), a contradiction. Hence, \(\tilde{v} \leq 0\) in \(\Omega\), and by the strong maximum principle, \(\tilde{v} < 0\) in \(\Omega\). Furthermore, applying the Hopf lemma to \(\tilde{v}\) on \(\Gamma_{N}^{ + }\), we conclude that the inner normal derivative of \({u}\) on \(\Gamma_{N}^{ + }\) is negative. Consequently, the unique non-vertex critical point of \({u}\), if it exists, is a non-degenerate maximum.

Step 2.2. The global maximum of \({u}\) cannot be attained in the interior of the strictly shorter Neumann side. Without loss of generality, suppose \(\alpha > \beta\). Then the function \({w}^{0, \gamma/2}\), defined in 5 , satisfies \[\begin{cases} \Delta {w}^{0, \gamma/2} + \mu {w}^{0, \gamma/2} = 0 & \text{in } {D}_{0, \gamma/2}, \\ \partial_{\nu}{w}^{0, \gamma/2} = 0 & \text{on } \Gamma_{0, \gamma/2}^{2} \subset \Gamma_{N}^{ - }, \\ {w}^{0, \gamma/2} \geq, \not\equiv 0 & \text{on } \partial{D}_{0, \gamma/2} \setminus \Gamma_{0, \gamma/2}^{2}. \end{cases}\] If \({w}^{0, \gamma/2}\) were negative somewhere in \(\mathfrak{D}_{4}\), its negative nodal component \(\mathfrak{D}_{4}\) would satisfy \[\Delta {w}^{0, \gamma/2} + \mu {w}^{0, \gamma/2} = 0 \text{ in } \mathfrak{D}_{4}, \quad {w}^{0, \gamma/2} = 0 \text{ on } \partial\mathfrak{D}_{4} \setminus \Gamma_{N}^{ - }, \quad \partial_{\nu}{w}^{0, \gamma/2} = 0 \text{ on } \partial\mathfrak{D}_{4} \cap \Gamma_{N}^{ - }.\] This gives \(\mu = \lambda_{1}^{\mathrm{mix}}(\mathfrak{D}_{4}, \partial\mathfrak{D}_{4} \cap \Gamma_{N}^{ - })\). On the other hand, \[\lambda_{1}^{\mathrm{mix}}(\mathfrak{D}_{4}, \partial\mathfrak{D}_{4} \cap \Gamma_{N}^{ - }) > \lambda_{1}^{\mathrm{mix}}(\Omega, \Gamma_{N}^{ - }) > \lambda_{1}^{\mathrm{mix}}(\Omega, \Gamma_{N}) = \mu,\] a contradiction. Thus, \({w}^{0, \gamma/2} \geq 0\) in \({D}_{0, \gamma/2}\). By the strong maximum principle and the Hopf lemma, we obtain \[\label{LY608} {w}^{0, \gamma/2} > 0 \text{ in } \overline{{D}_{0, \gamma/2}} \setminus {T}_{0, \gamma/2} \quad \text{ and } \quad\nabla {u} \cdot \mathbf{e}_{{\vartheta} + \alpha} > 0 \text{ on } \Omega \cap {T}_{0, {\vartheta}} \; \text{for } {\vartheta} = \gamma/2.\tag{48}\] Hence, the global maximum cannot occur in the interior of the strictly shorter Neumann side.

Combining all the steps above, we conclude that \({u}\) has at most one non-vertex critical point, which is a non-degenerate global maximum located on the longer Neumann side. Moreover, \({u}\) is monotone in the direction perpendicular to the longer Neumann side. For \(\alpha \neq \beta\), 13 implies that the global maximum is not located at the Neumann vertex, while for \(\alpha = \beta\), 7 implies that the global maximum is uniquely located at the Neumann vertex. This completes the proof. ◻