January 01, 1970
We study a quadratic nonlocal variational problem on a hybrid domain formed by a compact interval and finitely many discrete points. The associated energy splits into continuous, discrete, and interface contributions. Our main estimate shows that the interface term yields a coercive coupling between the two phases and provides an equivalent hybrid norm. As a consequence, we prove existence and uniqueness of a minimizer for the corresponding variational problem and characterize it as the unique weak solution of the associated hybrid Euler–Lagrange system. The latter combines a nonlocal integral equation on the continuous component with a finite nonlocal algebraic system on the discrete nodes.
MSC 2020: Primary 26A33, 35R11, 46E35; Secondary 34B37, 34N05, 49J45.
Keywords: nonlocal variational problem, hybrid continuous-discrete domain, interface coercivity, weak solution, Euler–Lagrange system.
Fractional energies arise naturally in the analysis of nonlocal equations, variational problems, and regularity theory. In the standard continuous setting they are generated by the Gagliardo seminorm and lead to the classical fractional Sobolev spaces; see, for instance, [1]–[3]. On the other hand, the calculus on time scales was introduced to provide a common language for continuous and discrete models; see [4]–[6].
From the measure-theoretic point of view, the time-scale framework already provides a natural notion of Lebesgue \(\Delta\)-measure and multidimensional \(\Delta\)-integration. In particular, Bohner and Guseinov developed a multiple Lebesgue integration theory on products of time scales, while Cabada and Vivero showed how the Lebesgue \(\Delta\)-integral can be expressed in terms of an ordinary Lebesgue integral on the real line; see [7], [8]. These results justify the use of measure-based function spaces on \(\mathbb{T}\) and on \(\mathbb{T}\times\mathbb{T}\).
On this basis, first-order Lebesgue and Sobolev spaces on time scales have already been studied in several directions. Basic properties of Sobolev spaces endowed with the Lebesgue \(\Delta\)-measure were established in [9], and generalized Lebesgue and Sobolev spaces with variable exponent on bounded time scales were investigated in [10]. Thus, at order one, the functional-analytic setting is already available in the literature.
The fractional situation is different. Existing fractional theories on time scales are developed predominantly through specific notions of fractional differentiation, such as conformable derivatives or Riemann–Liouville-type derivatives; see, for example, [11]–[13]. By contrast, a genuinely nonlocal Gagliardo-type approach based on a double-integral energy over \(\mathbb{T}\times\mathbb{T}\) appears to be much less explored on time scales, especially when one has in mind variable-order constructions. This is the point of view from which the present work originates.
The purpose of the present paper is to study a nonlocal variational problem on a simple hybrid continuous-discrete domain and to identify the coercive role of the interface contribution. More precisely, for an integer \(N\ge 2\), we consider \[\mathbb{T}=[0,1]\cup\{2,3,\dots,N+1\},\] endowed with the measure \[\mu=\mathbf{1}_{[0,1]}(x)\,dx+\sum_{k=2}^{N+1}\delta_k.\] This model consists of a continuous phase and a finite discrete phase separated by a positive gap. For functions \(u:\mathbb{T}\to\mathbb{R}\), the nonlocal quadratic energy generated by the kernel \(\lvert x-y\rvert^{-1-2\alpha}\) splits into three terms:
a continuous–continuous interaction on \([0,1]\times[0,1]\),
a discrete–discrete interaction on \(\{2,\dots,N+1\}^2\),
a continuous–discrete interface contribution.
The first two terms correspond to the pure continuous and pure discrete components. The third one is the key feature of the model: it couples the continuous profile to the discrete variables and yields a coercive transmission mechanism between the two phases.
Our main estimate shows that the interface contribution is not a lower-order perturbation. It controls simultaneously the \(L^2\) oscillation of the continuous component around its mean and the distance of each discrete value to that same mean. Consequently, the full energy is equivalent to a natural hybrid norm. This provides a robust variational framework for the boundary-value-type problem associated with the quadratic functional \[\mathcal{J}(u):=\frac{1}{2}\mathcal{E}_\alpha(u)+\frac{\lambda}{2}\lVert u\rVert_{L^2(\mathbb{T},\mu)}^2-\int_{\mathbb{T}}fu\,d\mu, \qquad \lambda>0.\] Within this framework we prove existence and uniqueness of a minimizer, define the corresponding notion of weak solution, and derive the associated hybrid Euler–Lagrange system. We interpret 12 as the weak formulation of a hybrid nonlocal transmission problem on the continuous-discrete domain \(\mathbb{T}\). The latter consists of a nonlocal integral equation on the continuous phase coupled with a finite nonlocal algebraic system on the discrete nodes.
The paper is organized as follows. Section 2 introduces the hybrid measure space, the energy, its decomposition, and the main coercivity estimate. Section 3 formulates the variational problem and establishes the equivalence between minimizers and weak solutions, together with existence and uniqueness. Section 4 derives the hybrid transmission system in weak form and discusses its formal pointwise counterpart. The final section presents a concrete variational example on a hybrid time scale and illustrates the continuous-discrete transmission structure of the model.
While the measure-theoretic and first-order Sobolev frameworks on time scales are already available, our purpose here is different: we introduce a nonlocal Gagliardo-type energy on the hybrid continuous-discrete domain under consideration and use it to formulate the transmission problem studied in the sequel.
Fix an integer \(N\ge 2\) and define \[\mathbb{T}=[0,1]\cup D, \qquad D:=\{2,3,\dots,N+1\}.\] We equip \(\mathbb{T}\) with the positive finite Borel measure \[\label{eq:def-mu} \mu(A):=\int_{A\cap[0,1]}dx+\sum_{k=2}^{N+1}\delta_k(A),\tag{1}\] for every Borel set \(A\subset \mathbb{T}\). Equivalently, \[\int_{\mathbb{T}} f\,d\mu=\int_0^1 f(x)\,dx+\sum_{k=2}^{N+1} f(k)\] for every integrable function \(f\) on \(\mathbb{T}\).
A function \(u:\mathbb{T}\to\mathbb{R}\) will be written as \[u=(v,a_2,\dots,a_{N+1}),\] where \[v=u|_{[0,1]},\qquad a_k=u(k)\quad (k=2,\dots,N+1).\] Thus the continuous and discrete phases are encoded separately.
Let \(\alpha\in(0,1)\). For \(u\in L^2(\mathbb{T},\mu)\), define the quadratic energy \[\label{eq:def-energy} \mathcal{E}_\alpha(u):= \iint_{\mathbb{T}\times\mathbb{T}} \frac{\lvert u(x)-u(y)\rvert^2}{\lvert x-y\rvert^{1+2\alpha}} \,d\mu(x)\,d\mu(y).\tag{2}\] Since the discrete phase is finite and separated from \([0,1]\) by a positive distance, the mixed and discrete contributions are automatically finite whenever the restriction \(v\) belongs to \(H^\alpha(0,1)\).
Definition 1. We define the hybrid energy space by \[\mathcal{H}_\alpha(\mathbb{T}):= \Big\{u=(v,a_2,\dots,a_{N+1}): v\in H^\alpha(0,1),\;a_k\in\mathbb{R}\Big\}.\] It is endowed with the norm \[\label{eq:def-Hnorm} \lVert u\rVert_{\mathcal{H}_\alpha(\mathbb{T})}^2:=\lVert u\rVert_{L^2(\mathbb{T},\mu)}^2+\mathcal{E}_\alpha(u).\tag{3}\]
The continuous–continuous part of 2 is the standard Gagliardo seminorm squared on \((0,1)\), namely \[\label{eq:gagliardo-seminorm} [v]_{H^\alpha(0,1)}^2:=\int_0^1\int_0^1\frac{\lvert v(x)-v(y)\rvert^2}{\lvert x-y\rvert^{1+2\alpha}}\,dx\,dy.\tag{4}\]
Lemma 1 (Equivalence with the product norm). There exist constants \(c_0,C_0>0\), depending only on \(\alpha\) and \(N\), such that for every \[u=(v,a_2,\dots,a_{N+1})\in \mathcal{H}_\alpha(\mathbb{T}),\] one has \[c_0\Bigl(\lVert v\rVert_{H^\alpha(0,1)}^2+\sum_{k=2}^{N+1}\lvert a_k\rvert^2\Bigr) \le \lVert u\rVert_{\mathcal{H}_\alpha(\mathbb{T})}^2 \le C_0\Bigl(\lVert v\rVert_{H^\alpha(0,1)}^2+\sum_{k=2}^{N+1}\lvert a_k\rvert^2\Bigr).\] Consequently, \(\mathcal{H}_\alpha(\mathbb{T})\) is a Hilbert space.
Proof. The upper bound follows from the decomposition in Proposition 1. The continuous–continuous term is exactly \([v]_{H^\alpha(0,1)}^2\). Since the discrete phase is finite and satisfies \(\operatorname{dist}([0,1],D)=1\), one has \[1\le \lvert k-x\rvert\le N+1\qquad (x\in[0,1],\;k\in D),\] which gives \[\mathcal{E}_{\mathrm{cd}}(v,a) \le \sum_{k=2}^{N+1}\int_0^1 \lvert a_k-v(x)\rvert^2\,dx \le 2\sum_{k=2}^{N+1}\lvert a_k\rvert^2+2N\lVert v\rVert_{L^2(0,1)}^2.\] Similarly, since \(D\) is finite, \[\mathcal{E}_{\mathrm{dd}}(a) \le 2\sum_{\substack{i,j=2\\ i\neq j}}^{N+1}(\lvert a_i\rvert^2+\lvert a_j\rvert^2) \le 4N\sum_{k=2}^{N+1}\lvert a_k\rvert^2.\] Hence \[\mathcal{E}_\alpha(u)\le C\Bigl([v]_{H^\alpha(0,1)}^2+\lVert v\rVert_{L^2(0,1)}^2+\sum_{k=2}^{N+1}\lvert a_k\rvert^2\Bigr)\] for some \(C>0\), which yields the upper bound.
For the lower bound, we simply note that \[\lVert u\rVert_{\mathcal{H}_\alpha(\mathbb{T})}^2=\lVert u\rVert_{L^2(\mathbb{T},\mu)}^2+\mathcal{E}_\alpha(u) \ge \lVert u\rVert_{L^2(\mathbb{T},\mu)}^2 =\lVert v\rVert_{L^2(0,1)}^2+\sum_{k=2}^{N+1}\lvert a_k\rvert^2\] and also \(\mathcal{E}_\alpha(u)\ge [v]_{H^\alpha(0,1)}^2\). Combining these estimates gives \[\lVert u\rVert_{\mathcal{H}_\alpha(\mathbb{T})}^2 \ge \lVert v\rVert_{L^2(0,1)}^2+[v]_{H^\alpha(0,1)}^2+\sum_{k=2}^{N+1}\lvert a_k\rvert^2 =\lVert v\rVert_{H^\alpha(0,1)}^2+\sum_{k=2}^{N+1}\lvert a_k\rvert^2,\] which proves the claim. ◻
The energy 2 splits naturally into three contributions.
Proposition 1 (Decomposition of the energy). For every \(u=(v,a_2,\dots,a_{N+1})\in \mathcal{H}_\alpha(\mathbb{T})\), \[\label{eq:energy-decomposition} \mathcal{E}_\alpha(u)=\mathcal{E}_{\mathrm{cc}}(v)+2\mathcal{E}_{\mathrm{cd}}(v,a)+\mathcal{E}_{\mathrm{dd}}(a),\qquad{(1)}\] where \[\begin{align} \mathcal{E}_{\mathrm{cc}}(v)&:=\int_0^1\int_0^1\frac{\lvert v(x)-v(y)\rvert^2}{\lvert x-y\rvert^{1+2\alpha}}\,dx\,dy, \label{eq:Ecc}\\ \mathcal{E}_{\mathrm{cd}}(v,a)&:=\sum_{k=2}^{N+1}\int_0^1\frac{\lvert a_k-v(x)\rvert^2}{\lvert k-x\rvert^{1+2\alpha}}\,dx, \label{eq:Ecd}\\ \mathcal{E}_{\mathrm{dd}}(a)&:=\sum_{\substack{i,j=2\\ i\neq j}}^{N+1}\frac{\lvert a_i-a_j\rvert^2}{\lvert i-j\rvert^{1+2\alpha}}. \label{eq:Edd} \end{align}\] {#eq: sublabel=eq:eq:Ecc,eq:eq:Ecd,eq:eq:Edd}
Proof. Using 1 , the product measure \(\mu\otimes\mu\) decomposes into the sum of the continuous–continuous, continuous–discrete, discrete–continuous, and discrete–discrete parts. Substituting this decomposition into 2 , we obtain \[\begin{align} \mathcal{E}_\alpha(u) &=\int_0^1\int_0^1\frac{\lvert v(x)-v(y)\rvert^2}{\lvert x-y\rvert^{1+2\alpha}}\,dx\,dy \\ &\quad +\sum_{k=2}^{N+1}\int_0^1\frac{\lvert a_k-v(x)\rvert^2}{\lvert k-x\rvert^{1+2\alpha}}\,dx +\sum_{k=2}^{N+1}\int_0^1\frac{\lvert v(x)-a_k\rvert^2}{\lvert x-k\rvert^{1+2\alpha}}\,dx \\ &\quad +\sum_{\substack{i,j=2\\ i\neq j}}^{N+1}\frac{\lvert a_i-a_j\rvert^2}{\lvert i-j\rvert^{1+2\alpha}}. \end{align}\] Since \(\lvert k-x\rvert=\lvert x-k\rvert\) and \(\lvert a_k-v(x)\rvert=\lvert v(x)-a_k\rvert\), the two mixed terms coincide, which yields ?? . ◻
Lemma 2 (Interface estimate). There exist constants \(c_1,c_2>0\), depending only on \(\alpha\) and \(N\), such that for every \(u=(v,a_2,\dots,a_{N+1})\in\mathcal{H}_\alpha(\mathbb{T})\), \[\label{eq:interface-equivalence} c_1\sum_{k=2}^{N+1}\int_0^1\lvert a_k-v(x)\rvert^2\,dx \le \mathcal{E}_{\mathrm{cd}}(v,a) \le c_2\sum_{k=2}^{N+1}\int_0^1\lvert a_k-v(x)\rvert^2\,dx.\tag{5}\]
Proof. Fix \(k\in\{2,\dots,N+1\}\) and \(x\in[0,1]\). Then \[k-1\le k-x\le k.\] Therefore, \[\frac{1}{k^{1+2\alpha}}\le \frac{1}{\lvert k-x\rvert^{1+2\alpha}}\le \frac{1}{(k-1)^{1+2\alpha}}.\] Multiplying by \(\lvert a_k-v(x)\rvert^2\) and integrating over \((0,1)\), we obtain \[\frac{1}{k^{1+2\alpha}}\int_0^1 \lvert a_k-v(x)\rvert^2\,dx \le \int_0^1 \frac{\lvert a_k-v(x)\rvert^2}{\lvert k-x\rvert^{1+2\alpha}}\,dx \le \frac{1}{(k-1)^{1+2\alpha}}\int_0^1 \lvert a_k-v(x)\rvert^2\,dx.\] Summing with respect to \(k\) yields 5 with \[c_1=(N+1)^{-(1+2\alpha)}, \qquad c_2=1.\] ◻
Lemma 3 (Algebraic decomposition around the mean). Let \[\bar v:=\int_0^1 v(x)\,dx.\] Then, for every \(k\in\{2,\dots,N+1\}\), \[\label{eq:single-mean-decomposition} \int_0^1 \lvert a_k-v(x)\rvert^2\,dx =\lvert a_k-\bar v\rvert^2+\lVert v-\bar v\rVert_{L^2(0,1)}^2.\tag{6}\] Consequently, \[\label{eq:sum-mean-decomposition} \sum_{k=2}^{N+1}\int_0^1 \lvert a_k-v(x)\rvert^2\,dx = \sum_{k=2}^{N+1}\lvert a_k-\bar v\rvert^2 +N\lVert v-\bar v\rVert_{L^2(0,1)}^2.\tag{7}\]
Proof. Fix \(k\in\{2,\dots,N+1\}\). We write \[a_k-v(x)=(a_k-\bar v)+(\bar v-v(x)).\] Squaring and integrating over \((0,1)\), we obtain \[\begin{align} \int_0^1\lvert a_k-v(x)\rvert^2\,dx &=\int_0^1 \lvert a_k-\bar v\rvert^2\,dx +\int_0^1\lvert v(x)-\bar v\rvert^2\,dx \\ &\quad +2(a_k-\bar v)\int_0^1(\bar v-v(x))\,dx. \end{align}\] Since \[\int_0^1(\bar v-v(x))\,dx=\bar v-\int_0^1v(x)\,dx=0,\] the mixed term vanishes. This proves 6 . Summing over \(k\) gives 7 . ◻
We now derive the main coercivity statement.
Theorem 2 (Interface coercivity). There exist constants \(C_1,C_2>0\), depending only on \(\alpha\) and \(N\), such that for every \(u=(v,a_2,\dots,a_{N+1})\in\mathcal{H}_\alpha(\mathbb{T})\), \[\begin{align} C_1\Big( [v]_{H^\alpha(0,1)}^2 +\lVert v-\bar v\rVert_{L^2(0,1)}^2 +\sum_{k=2}^{N+1}\lvert a_k-\bar v\rvert^2 \Big) &\le \mathcal{E}_\alpha(u) \tag{8}\\ &\le C_2\Big( [v]_{H^\alpha(0,1)}^2 +\lVert v-\bar v\rVert_{L^2(0,1)}^2 +\sum_{k=2}^{N+1}\lvert a_k-\bar v\rvert^2 \Big). \tag{9} \end{align}\]
Proof. By Proposition 1, \[\mathcal{E}_\alpha(u)=\mathcal{E}_{\mathrm{cc}}(v)+2\mathcal{E}_{\mathrm{cd}}(v,a)+\mathcal{E}_{\mathrm{dd}}(a).\] Since all three terms on the right-hand side are nonnegative, Lemmas 2 and 3 imply \[\begin{align} \mathcal{E}_\alpha(u) &\ge \mathcal{E}_{\mathrm{cc}}(v)+2\mathcal{E}_{\mathrm{cd}}(v,a) \\ &\ge [v]_{H^\alpha(0,1)}^2+2c_1\sum_{k=2}^{N+1}\int_0^1\lvert a_k-v(x)\rvert^2\,dx \\ &= [v]_{H^\alpha(0,1)}^2 +2c_1\sum_{k=2}^{N+1}\lvert a_k-\bar v\rvert^2 +2c_1N\lVert v-\bar v\rVert_{L^2(0,1)}^2. \end{align}\] This proves 8 .
For the reverse estimate, we use again Proposition 1. The term \(\mathcal{E}_{\mathrm{cc}}(v)\) is exactly \([v]_{H^\alpha(0,1)}^2\). By Lemmas 2 and 3, \[\mathcal{E}_{\mathrm{cd}}(v,a) \le c_2\Big( \sum_{k=2}^{N+1}\lvert a_k-\bar v\rvert^2+N\lVert v-\bar v\rVert_{L^2(0,1)}^2 \Big).\] It remains to estimate the discrete–discrete contribution. For \(i\neq j\), \[\lvert a_i-a_j\rvert^2 \le 2\lvert a_i-\bar v\rvert^2+2\lvert a_j-\bar v\rvert^2.\] Hence, \[\begin{align} \mathcal{E}_{\mathrm{dd}}(a) &\le 2\sum_{\substack{i,j=2\\ i\neq j}}^{N+1} \frac{\lvert a_i-\bar v\rvert^2+\lvert a_j-\bar v\rvert^2}{\lvert i-j\rvert^{1+2\alpha}} \\ &\le 4\Big(\sum_{m=1}^{N-1}\frac{1}{m^{1+2\alpha}}\Big) \sum_{k=2}^{N+1}\lvert a_k-\bar v\rvert^2. \end{align}\] Combining these estimates gives 9 . ◻
Remark 3. Theorem 2 shows that the interface term is not a negligible correction. It controls the \(L^2\) oscillation of the continuous component and pins the discrete variables to the average value of the continuous phase. In the variational problem below, this interface term acts as a coercive transmission mechanism between the two phases.
Corollary 1 (Hybrid Poincaré-type inequality). There exists a constant \(C_P>0\), depending only on \(\alpha\) and \(N\), such that for every \(u=(v,a_2,\dots,a_{N+1})\in \mathcal{H}_\alpha(\mathbb{T})\), \[\lVert v-\bar v\rVert_{L^2(0,1)}^2+\sum_{k=2}^{N+1}\lvert a_k-\bar v\rvert^2 \le C_P\,\mathcal{E}_\alpha(u).\]
A direct consequence is that the quadratic form \[\|u\|_{\ast}^2:=\|u\|_{L^2(\mathbb{T},\mu)}^2+ [v]_{H^\alpha(0,1)}^2+\lVert v-\bar v\rVert_{L^2(0,1)}^2+\sum_{k=2}^{N+1}\lvert a_k-\bar v\rvert^2\] defines a norm equivalent to 3 on \(\mathcal{H}_\alpha(\mathbb{T})\).
In line with the variational treatment of nonlocal problems in weak form and prove its well-posedness; see, for instance, [14]–[16]. We now formulate the associated hybrid transmission problem in weak form and prove its well-posedness. The interface coercivity obtained in the previous section is the key tool in this step.
Let \(\lambda>0\) and let \(f\in L^2(\mathbb{T},\mu)\). We consider the quadratic functional \[\label{eq:def-functional} \mathcal{J}(u):=\frac{1}{2}\mathcal{E}_\alpha(u)+\frac{\lambda}{2}\lVert u\rVert_{L^2(\mathbb{T},\mu)}^2-\int_{\mathbb{T}}f u\,d\mu, \qquad u\in\mathcal{H}_\alpha(\mathbb{T}).\tag{10}\]
The bilinear form naturally associated with the energy is \[\label{eq:def-bilinear} \mathfrak a(u,\varphi):= \iint_{\mathbb{T}\times\mathbb{T}} \frac{(u(x)-u(y))(\varphi(x)-\varphi(y))}{\lvert x-y\rvert^{1+2\alpha}} \,d\mu(x)\,d\mu(y).\tag{11}\]
Definition 2. We say that \(u\in\mathcal{H}_\alpha(\mathbb{T})\) is a weak solution of the hybrid nonlocal problem associated with \((\lambda,f)\) if \[\label{eq:weak-solution} \mathfrak a(u,\varphi)+\lambda\int_{\mathbb{T}}u\varphi\,d\mu=\int_{\mathbb{T}}f\varphi\,d\mu \qquad\text{for every }\varphi\in\mathcal{H}_\alpha(\mathbb{T}).\tag{12}\]
Equation 12 may be regarded as the weak form of a nonlocal transmission problem on the hybrid domain: the continuous phase is driven by a nonlocal integral operator, the discrete phase by a finite algebraic system, and the two parts are coupled through the interface term.
Theorem 4 (Existence and uniqueness of the minimizer). For every \(\lambda>0\) and every \(f\in L^2(\mathbb{T},\mu)\), the functional \(\mathcal{J}\) admits a unique minimizer in \(\mathcal{H}_\alpha(\mathbb{T})\).
Proof. The mapping \(u\mapsto \mathcal{E}_\alpha(u)\) is quadratic and nonnegative, and the term \(u\mapsto \lambda\lVert u\rVert_{L^2(\mathbb{T},\mu)}^2\) is strictly convex because \(\lambda>0\). Therefore \(\mathcal{J}\) is strictly convex on the Hilbert space \(\mathcal{H}_\alpha(\mathbb{T})\).
We next prove coercivity. By Theorem 2, \[\mathcal{E}_\alpha(u)\ge C_1\Big( [v]_{H^\alpha(0,1)}^2 +\lVert v-\bar v\rVert_{L^2(0,1)}^2 +\sum_{k=2}^{N+1}\lvert a_k-\bar v\rvert^2 \Big).\] It remains to control the common mean level \(\bar v\). Since \(\bar v=\int_0^1 v(x)\,dx\), Jensen’s inequality gives \[\lvert\bar v\rvert^2\le \int_0^1 \lvert v(x)\rvert^2\,dx\le \lVert u\rVert_{L^2(\mathbb{T},\mu)}^2.\] Hence \[\lVert v\rVert_{L^2(0,1)}^2 \le 2\lVert v-\bar v\rVert_{L^2(0,1)}^2+2\lvert\bar v\rvert^2 \le 2\lVert v-\bar v\rVert_{L^2(0,1)}^2+2\lVert u\rVert_{L^2(\mathbb{T},\mu)}^2,\] and, for each \(k\in\{2,\dots,N+1\}\), \[\lvert a_k\rvert^2\le 2\lvert a_k-\bar v\rvert^2+2\lvert\bar v\rvert^2 \le 2\lvert a_k-\bar v\rvert^2+2\lVert u\rVert_{L^2(\mathbb{T},\mu)}^2.\] Combining these inequalities with the lower bound above, we infer that there exists \(c>0\) such that \[\mathcal{E}_\alpha(u)+\lambda\lVert u\rVert_{L^2(\mathbb{T},\mu)}^2 \ge c\Bigl(\lVert v\rVert_{H^\alpha(0,1)}^2+\sum_{k=2}^{N+1}\lvert a_k\rvert^2\Bigr).\] By Lemma 1, the right-hand side controls \(\lVert u\rVert_{\mathcal{H}_\alpha(\mathbb{T})}^2\). Therefore, \[\mathcal{E}_\alpha(u)+\lambda\lVert u\rVert_{L^2(\mathbb{T},\mu)}^2 \ge c\lVert u\rVert_{\mathcal{H}_\alpha(\mathbb{T})}^2.\] By Cauchy–Schwarz and Young’s inequality, \[\left|\int_{\mathbb{T}}fu\,d\mu\right| \le \lVert f\rVert_{L^2(\mathbb{T},\mu)}\lVert u\rVert_{L^2(\mathbb{T},\mu)} \le \frac{c}{4}\lVert u\rVert_{\mathcal{H}_\alpha(\mathbb{T})}^2+C\lVert f\rVert_{L^2(\mathbb{T},\mu)}^2.\] Consequently, \[\mathcal{J}(u)\to +\infty\qquad\text{as }\lVert u\rVert_{\mathcal{H}_\alpha(\mathbb{T})}\to\infty.\] Thus \(\mathcal{J}\) is coercive and bounded from below.
Let \((u_n)\) be a minimizing sequence. Coercivity implies that \((u_n)\) is bounded in \(\mathcal{H}_\alpha(\mathbb{T})\). Since \(\mathcal{H}_\alpha(\mathbb{T})\) is a Hilbert space, there exist a subsequence, still denoted by \((u_n)\), and some \(u\in\mathcal{H}_\alpha(\mathbb{T})\) such that \(u_n\rightharpoonup u\) weakly in \(\mathcal{H}_\alpha(\mathbb{T})\). The quadratic form \(u\mapsto \mathcal{E}_\alpha(u)+\lambda\lVert u\rVert_{L^2(\mathbb{T},\mu)}^2\) is weakly lower semicontinuous, whereas the linear functional \(u\mapsto \int_{\mathbb{T}}fu\,d\mu\) is weakly continuous. Therefore, \[\mathcal{J}(u)\le \liminf_{n\to\infty}\mathcal{J}(u_n)=\inf_{w\in\mathcal{H}_\alpha(\mathbb{T})}\mathcal{J}(w).\] Hence \(u\) is a minimizer. Uniqueness follows from strict convexity. ◻
Theorem 5 (Variational characterization of weak solutions). A function \(u\in\mathcal{H}_\alpha(\mathbb{T})\) is the unique minimizer of \(\mathcal{J}\) if and only if it is the unique weak solution of 12 .
Proof. Let \(u\in\mathcal{H}_\alpha(\mathbb{T})\) and \(\varphi\in\mathcal{H}_\alpha(\mathbb{T})\). For \(t\in\mathbb{R}\), \[\mathcal{J}(u+t\varphi) =\frac{1}{2}\mathcal{E}_\alpha(u+t\varphi)+\frac{\lambda}{2}\lVert u+t\varphi\rVert_{L^2(\mathbb{T},\mu)}^2-\int_{\mathbb{T}}f(u+t\varphi)\,d\mu.\] Since the energy is quadratic, \[\frac{d}{dt}\Big|_{t=0}\frac{1}{2}\mathcal{E}_\alpha(u+t\varphi)=\mathfrak a(u,\varphi),\] and similarly, \[\frac{d}{dt}\Big|_{t=0}\frac{\lambda}{2}\lVert u+t\varphi\rVert_{L^2(\mathbb{T},\mu)}^2 =\lambda\int_{\mathbb{T}}u\varphi\,d\mu.\] The derivative of the linear term is \[\frac{d}{dt}\Big|_{t=0}\int_{\mathbb{T}}f(u+t\varphi)\,d\mu =\int_{\mathbb{T}}f\varphi\,d\mu.\] Hence \(u\) is a critical point of \(\mathcal{J}\) if and only if 12 holds. Since \(\mathcal{J}\) is strictly convex by Theorem 4, it has at most one critical point. The conclusion follows. ◻
The weak formulation can be split into its continuous and discrete components. This yields a coupled transmission system between the two phases.
Theorem 6 (Weak Euler–Lagrange identity). Let \(u=(v,a_2,\dots,a_{N+1})\in\mathcal{H}_\alpha(\mathbb{T})\) be the minimizer of \(\mathcal{J}\). Then, for every \(\varphi=(\psi,b_2,\dots,b_{N+1})\in\mathcal{H}_\alpha(\mathbb{T})\), \[\begin{align} \label{eq:split-weak-form} &\int_0^1\int_0^1 \frac{(v(x)-v(y))(\psi(x)-\psi(y))}{\lvert x-y\rvert^{1+2\alpha}}\,dx\,dy \notag\\ &\quad +2\sum_{k=2}^{N+1}\int_0^1 \frac{(a_k-v(x))(b_k-\psi(x))}{\lvert k-x\rvert^{1+2\alpha}}\,dx \\ &\quad +\sum_{\substack{i,j=2\\ i\neq j}}^{N+1} \frac{(a_i-a_j)(b_i-b_j)}{\lvert i-j\rvert^{1+2\alpha}} +\lambda\int_{\mathbb{T}}u\varphi\,d\mu =\int_{\mathbb{T}}f\varphi\,d\mu.\notag \end{align}\tag{13}\]
Proof. Let \(u=(v,a_2,\dots,a_{N+1})\in\mathcal{H}_\alpha(\mathbb{T})\) be the minimizer of \(\mathcal{J}\). By Theorem 5, \(u\) is the unique weak solution of 12 . Hence, for every test function \[\varphi=(\psi,b_2,\dots,b_{N+1})\in\mathcal{H}_\alpha(\mathbb{T}),\] one has \[\label{eq:proof-thm71-weak} \mathfrak a(u,\varphi)+\lambda\int_{\mathbb{T}}u\varphi\,\,d\mu=\int_{\mathbb{T}}f\varphi\,\,d\mu.\tag{14}\] It therefore remains to compute the bilinear form \(\mathfrak a(u,\varphi)\) in terms of the continuous and discrete components.
By definition, \[\mathfrak a(u,\varphi)=\iint_{\mathbb{T}\times\mathbb{T}} \frac{(u(x)-u(y))(\varphi(x)-\varphi(y))}{\lvert x-y\rvert^{1+2\alpha}}\,d\mu(x)\,d\mu(y).\] Using the decomposition of the measure \[d\mu(x)=\mathbf{1}_{[0,1]}(x)\,dx+\sum_{k=2}^{N+1}\delta_k(x),\] the product measure \(\mu\otimes\mu\) splits into four contributions: continuous–continuous, continuous–discrete, discrete–continuous, and discrete–discrete. We examine them separately.
For the continuous–continuous part, since \(u|_{[0,1]}=v\) and \(\varphi|_{[0,1]}=\psi\), we obtain \[I_{cc}:=\int_0^1\int_0^1 \frac{(v(x)-v(y))(\psi(x)-\psi(y))}{\lvert x-y\rvert^{1+2\alpha}}\,dx\,dy.\] For the continuous–discrete part, using \(u(k)=a_k\) and \(\varphi(k)=b_k\), we get \[I_{cd}:=\sum_{k=2}^{N+1}\int_0^1 \frac{(v(x)-a_k)(\psi(x)-b_k)}{\lvert x-k\rvert^{1+2\alpha}}\,dx.\] For the discrete–continuous part, by the symmetry of the kernel and the identity \(\lvert k-x\rvert=\lvert x-k\rvert\), we have \[I_{dc}:=\sum_{k=2}^{N+1}\int_0^1 \frac{(a_k-v(x))(b_k-\psi(x))}{\lvert k-x\rvert^{1+2\alpha}}\,dx.\] Since \[(v(x)-a_k)(\psi(x)-b_k)=(a_k-v(x))(b_k-\psi(x)),\] it follows that \(I_{cd}=I_{dc}\). Therefore the mixed contribution equals \[I_{cd}+I_{dc}=2\sum_{k=2}^{N+1}\int_0^1 \frac{(a_k-v(x))(b_k-\psi(x))}{\lvert k-x\rvert^{1+2\alpha}}\,dx.\] Finally, for the discrete–discrete part we obtain \[I_{dd}:=\sum_{i=2}^{N+1}\sum_{j=2}^{N+1} \frac{(a_i-a_j)(b_i-b_j)}{\lvert i-j\rvert^{1+2\alpha}}.\] The terms with \(i=j\) vanish because \((a_i-a_i)(b_i-b_i)=0\), hence \[I_{dd}=\sum_{\substack{i,j=2\\ i\neq j}}^{N+1} \frac{(a_i-a_j)(b_i-b_j)}{\lvert i-j\rvert^{1+2\alpha}}.\] Combining the four pieces yields \[\begin{align} \mathfrak a(u,\varphi) &=\int_0^1\int_0^1 \frac{(v(x)-v(y))(\psi(x)-\psi(y))}{\lvert x-y\rvert^{1+2\alpha}}\,dx\,dy\\ &\quad+2\sum_{k=2}^{N+1}\int_0^1 \frac{(a_k-v(x))(b_k-\psi(x))}{\lvert k-x\rvert^{1+2\alpha}}\,dx\\ &\quad+\sum_{\substack{i,j=2\\ i\neq j}}^{N+1} \frac{(a_i-a_j)(b_i-b_j)}{\lvert i-j\rvert^{1+2\alpha}}. \end{align}\] Substituting this identity into 14 gives exactly 13 . This completes the proof. ◻
Remark 7 (Formal pointwise system). Assume that the weak solution \(u=(v,a_2,\dots,a_{N+1})\) has sufficient additional regularity and integrability for the pointwise expressions below to make sense. Then 13 corresponds formally to \[\label{eq:strong-continuous} 2\,\operatorname{P.V.}\int_0^1\frac{v(x)-v(y)}{\lvert x-y\rvert^{1+2\alpha}}\,dy +2\sum_{k=2}^{N+1}\frac{v(x)-a_k}{\lvert k-x\rvert^{1+2\alpha}} +\lambda v(x)=f(x)\tag{15}\] for \(x\in(0,1)\), and to \[\label{eq:strong-discrete} 2\int_0^1\frac{a_k-v(x)}{\lvert k-x\rvert^{1+2\alpha}}\,dx +2\sum_{\substack{j=2\\ j\neq k}}^{N+1}\frac{a_k-a_j}{\lvert k-j\rvert^{1+2\alpha}} +\lambda a_k=f(k)\tag{16}\] for each \(k\in\{2,\dots,N+1\}\).
The first equation is a nonlocal integral equation on the continuous phase, whereas the second one is a finite nonlocal algebraic system on the discrete phase. The interface contribution acts as a transmission term: it inserts the discrete variables into the continuous equation and, conversely, the continuous profile into each discrete balance law.
In this section, we study a concrete variational problem on a hybrid time scale and derive both its weak formulation and its coupled continuous-discrete structure. Unlike the finite-dimensional reduction introduced later, this problem is formulated on the full energy space and therefore retains the full variational structure of the model.
We consider the hybrid time scale \[\mathbb{T}=[0,1]\cup\{2,3\},\] endowed with the measure \[d\mu(x)=\mathbf{1}_{[0,1]}(x)\,dx+\delta_2+\delta_3.\] A function \(u:\mathbb{T}\to\mathbb{R}\) is identified with a triple \[u=(v,a,b),\] where \[v=u|_{[0,1]},\qquad a=u(2),\qquad b=u(3).\]
For \(\alpha\in(0,1)\), we define the hybrid energy \[\begin{align} \mathcal{E}_\alpha(u) &= \int_0^1\int_0^1 \frac{|v(x)-v(y)|^2}{|x-y|^{1+2\alpha}}\,dx\,dy \notag\\ &\quad +2\int_0^1\frac{|a-v(x)|^2}{|2-x|^{1+2\alpha}}\,dx +2\int_0^1\frac{|b-v(x)|^2}{|3-x|^{1+2\alpha}}\,dx +2|a-b|^2. \label{eq:example-energy} \end{align}\tag{17}\] Accordingly, we introduce the Hilbert space \[\mathcal{H}_\alpha(\mathbb{T}) = \bigl\{u=(v,a,b):\;v\in H^\alpha(0,1),\;a,b\in\mathbb{R}\bigr\},\] endowed with the norm \[\|u\|_{\mathcal{H}_\alpha(\mathbb{T})}^2 = \|u\|_{L^2(\mu)}^2+\mathcal{E}_\alpha(u),\] where \[\|u\|_{L^2(\mu)}^2=\int_0^1 |v(x)|^2\,dx+a^2+b^2.\]
Let \(\lambda>0\), let \(f\in L^2(0,1)\), and let \(F,G\in\mathbb{R}\). We consider the functional \[\label{eq:example-functional} \mathcal{J}(u) = \frac{1}{2}\mathcal{E}_\alpha(u) +\frac{\lambda}{2}\|u\|_{L^2(\mu)}^2 - \left( \int_0^1 f(x)v(x)\,dx+Fa+Gb \right),\tag{18}\] for \(u=(v,a,b)\in \mathcal{H}_\alpha(\mathbb{T})\).
This is the nonlocal variational problem associated with the hybrid time scale \(\mathbb{T}\): \[\label{eq:example-minimization} \min_{u\in \mathcal{H}_\alpha(\mathbb{T})} \mathcal{J}(u).\tag{19}\]
Proposition 8. The minimization problem 19 admits a unique minimizer in \(\mathcal{H}_\alpha(\mathbb{T})\).
Proof. The functional \(\mathcal{J}\) is the sum of the nonnegative quadratic form \(\frac{1}{2}\mathcal{E}_\alpha(u)\), the strictly positive quadratic term \(\frac{\lambda}{2}\|u\|_{L^2(\mu)}^2\), and the continuous linear functional \[u=(v,a,b)\mapsto \int_0^1 f(x)v(x)\,dx+Fa+Gb.\] Hence \(\mathcal{J}\) is strictly convex on \(\mathcal{H}_\alpha(\mathbb{T})\).
Moreover, by Cauchy–Schwarz, \[\left|\int_0^1 f(x)v(x)\,dx+Fa+Gb\right| \le \|f\|_{L^2(0,1)}\|v\|_{L^2(0,1)}+|F||a|+|G||b| \le C\|u\|_{L^2(\mu)}\] for some constant \(C>0\). Therefore \[\mathcal{J}(u) \ge \frac{1}{2}\mathcal{E}_\alpha(u)+\frac{\lambda}{2}\|u\|_{L^2(\mu)}^2-C\|u\|_{L^2(\mu)},\] so \(\mathcal{J}\) is coercive on \(\mathcal{H}_\alpha(\mathbb{T})\). Since its quadratic part is convex and continuous and its linear part is weakly continuous, \(\mathcal{J}\) is weakly lower semicontinuous. The direct method of the calculus of variations therefore yields the existence of a minimizer, and strict convexity implies uniqueness. ◻
For \[u=(v,a,b)\in\mathcal{H}_\alpha(\mathbb{T}), \qquad \varphi=(\psi,\eta,\zeta)\in\mathcal{H}_\alpha(\mathbb{T}),\] we define the bilinear form \[\begin{align} \mathcal{B}_\alpha(u,\varphi) &= \int_0^1\int_0^1 \frac{(v(x)-v(y))(\psi(x)-\psi(y))}{|x-y|^{1+2\alpha}}\,dx\,dy \notag\\ &\quad +2\int_0^1 \frac{(a-v(x))(\eta-\psi(x))}{|2-x|^{1+2\alpha}}\,dx \notag\\ &\quad +2\int_0^1 \frac{(b-v(x))(\zeta-\psi(x))}{|3-x|^{1+2\alpha}}\,dx +2(a-b)(\eta-\zeta). \label{eq:example-bilinear} \end{align}\tag{20}\]
Proposition 9. A function \(u=(v,a,b)\in\mathcal{H}_\alpha(\mathbb{T})\) is the unique minimizer of \(\mathcal{J}\) if and only if \[\label{eq:example-weak} \mathcal{B}_\alpha(u,\varphi) +\lambda\left(\int_0^1 v(x)\psi(x)\,dx+a\eta+b\zeta\right) = \int_0^1 f(x)\psi(x)\,dx+F\eta+G\zeta\qquad{(2)}\] for every \(\varphi=(\psi,\eta,\zeta)\in\mathcal{H}_\alpha(\mathbb{T})\).
Proof. Let \(u=(v,a,b)\in\mathcal{H}_\alpha(\mathbb{T})\) and \(\varphi=(\psi,\eta,\zeta)\in\mathcal{H}_\alpha(\mathbb{T})\). For \(t\in\mathbb{R}\), \[\mathcal{J}(u+t\varphi) = \frac{1}{2}\mathcal{E}_\alpha(u+t\varphi) +\frac{\lambda}{2}\|u+t\varphi\|_{L^2(\mu)}^2 - \left( \int_0^1 f(v+t\psi)\,dx+F(a+t\eta)+G(b+t\zeta) \right).\] Differentiating with respect to \(t\) at \(t=0\), we obtain \[\frac{d}{dt}\mathcal{J}(u+t\varphi)\Big|_{t=0} = \mathcal{B}_\alpha(u,\varphi) +\lambda\left(\int_0^1 v\psi\,dx+a\eta+b\zeta\right) -\left(\int_0^1 f\psi\,dx+F\eta+G\zeta\right).\] Hence \(u\) is a critical point of \(\mathcal{J}\) if and only if ?? holds for every \(\varphi\). By Proposition 8, the critical point is unique and coincides with the unique minimizer. ◻
Equation ?? is the weak formulation of a hybrid nonlocal transmission problem on the time scale \(\mathbb{T}\).
Proposition 10. Let \(u=(v,a,b)\in\mathcal{H}_\alpha(\mathbb{T})\). Then \(u\) satisfies ?? for every \(\varphi=(\psi,\eta,\zeta)\in\mathcal{H}_\alpha(\mathbb{T})\) if and only if the following three conditions hold:
(i) for every \(\psi\in H^\alpha(0,1)\), \[\begin{align} &\int_0^1\int_0^1 \frac{(v(x)-v(y))(\psi(x)-\psi(y))}{|x-y|^{1+2\alpha}}\,dx\,dy \notag\\ &\qquad -2\int_0^1\frac{(a-v(x))\psi(x)}{|2-x|^{1+2\alpha}}\,dx -2\int_0^1\frac{(b-v(x))\psi(x)}{|3-x|^{1+2\alpha}}\,dx \notag\\ &\qquad +\lambda\int_0^1 v(x)\psi(x)\,dx = \int_0^1 f(x)\psi(x)\,dx; \label{eq:example-continuous} \end{align}\qquad{(3)}\]
(ii) \[\label{eq:example-discrete-a} 2\int_0^1\frac{a-v(x)}{|2-x|^{1+2\alpha}}\,dx +2(a-b)+\lambda a = F;\qquad{(4)}\]
(iii) \[\label{eq:example-discrete-b} 2\int_0^1\frac{b-v(x)}{|3-x|^{1+2\alpha}}\,dx +2(b-a)+\lambda b = G.\qquad{(5)}\]
Proof. Assume first that \(u\) satisfies ?? for every \((\psi,\eta,\zeta)\in\mathcal{H}_\alpha(\mathbb{T})\). Choosing \((\psi,\eta,\zeta)=(\psi,0,0)\) gives ?? . Choosing \((\psi,\eta,\zeta)=(0,1,0)\) gives ?? , and choosing \((\psi,\eta,\zeta)=(0,0,1)\) gives ?? .
Conversely, assume that ?? , ?? , and ?? hold. Let \((\psi,\eta,\zeta)\in\mathcal{H}_\alpha(\mathbb{T})\) be arbitrary. Multiplying ?? by \(\eta\), multiplying ?? by \(\zeta\), and adding the resulting identities to ?? , we recover exactly ?? . This proves the equivalence. ◻
The previous proposition shows that the hybrid weak problem consists of a nonlocal equation on the continuous phase, coupled with two scalar nonlocal balance relations on the discrete phase. The interface terms transmit information in both directions between the interval \([0,1]\) and the discrete nodes \(2\) and \(3\).
The full variational problem above can be approximated on finite-dimensional subspaces of \(\mathcal{H}_\alpha(\mathbb{T})\). For instance, one may restrict the continuous component to affine functions, \[v(x)=mx+c,\] and consider the reduced space \[\mathcal{X}_{\mathrm{red}} = \{(mx+c,a,b):\;m,c,a,b\in\mathbb{R}\}.\] The restriction of \(\mathcal{J}\) to \(\mathcal{X}_{\mathrm{red}}\) then yields a finite-dimensional quadratic minimization problem, which may be viewed as a Galerkin approximation of the full hybrid transmission problem. This reduction is useful for explicit computations, but it should be regarded as a derived approximation of the full problem rather than as a substitute for it.
We have studied a hybrid nonlocal transmission problem on a continuous-discrete domain, formulated through a quadratic Gagliardo-type energy. The analysis shows that the interface contribution plays a central role: it yields a coercive coupling between the continuous and discrete phases and provides the key estimate underlying the whole variational theory. In particular, this leads to a Poincaré-type inequality adapted to the hybrid setting, as well as to the existence and uniqueness of a weak solution for the associated variational problem.
The weak formulation obtained in this way naturally splits into a nonlocal equation on the continuous component and discrete balance relations on the nodal part. This confirms that the model should be understood as a transmission problem rather than as a mere juxtaposition of continuous and discrete energies. The example discussed in the last section further illustrates that the hybrid framework remains genuinely variational and gives rise to explicit coupled systems.
A natural continuation of this work would be to consider more general finite discrete sets, to study reduced Galerkin-type approximations of the hybrid problem, and to extend the analysis to nonlinear energies with similar transmission structure. These directions remain close to the present setting and seem realistic within the same general approach.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
During the preparation of this work, the authors used ChatGPT to improve the language, readability, and presentation of the manuscript. The authors edited the content and take full responsibility for the final version of the manuscript.