1 Introduction↩︎

In this paper we will study a boundary value problem of Neumann type for the Helmholtz equation with a compactly-supported electric potential, of the form \[\label{eq:neumann} \begin{cases} (\Delta+\lambda-V)\,u=f_1 & \textrm{in }\Omega,\\ \partial_\nu u=f_2 & \textrm{on }\partial\Omega, \end{cases}\tag{1}\] where \(\Omega\) is a bounded open domain in \(\mathbb{R}^n\), \(n\geq3\), containing the support of the potential \(V\). Here \(\lambda>0\) is the energy, and \(f_1\) and \(f_2\) are given functions. Note that this problem doesn’t have a solution in general. The goal of this work, however, is to show that 1 can always be solved if one has some freedom in the choice of the domain. Roughly, we aim to answer the following question:

Given \(\lambda\) and \(V\), can one find a domain \(\Omega\) such that the problem 1 has a unique solution \(u\) for any \(f_1\in L^2(\Omega)\) and \(f_2\in L^2(\partial\Omega)\)?

We will show that the answer is affirmative for big enough values of the energy, under low-regularity assumptions on the potential. In particular, we will consider potentials of the form \[\label{V}V=V^0+\gamma^s+\alpha \,\mathrm{d}\sigma,\tag{2}\] where

• \(V^0\in L^{n/2} (\mathbb{R}^n;\mathbb{R}),\) and it’s compactly supported,

• \(\,\mathrm{d}\sigma\) denotes the surface measure of a compact hypersurface \(\Gamma\) which is locally described by the graph of Lipschitz functions and \(\alpha\in L^\infty(\Gamma;\mathbb{R})\), and

• \(\gamma^s\) is of the form \[\gamma^s=\chi^2\, D^sg,\] for some \(s<1\), \(g\in L^\infty(\mathbb{R}^n;\mathbb{R})\), and \(\chi\,\in \mathcal{C}_0^\infty(\mathbb{R}^n;[0,1])\) is a compactly supported cut-off function. Here \(D^s\) denotes the fractional derivative, defined via Fourier transform as \(\widehat{D^s f}(\xi)=|\xi|^s\hat{f}(\xi)\).

In a given domain \(\Omega\), the Neumann problem above can be solved for every choice of \(f_1\) and \(f_2\) as long as \(\lambda\) is not a Neumann eigenvalue (NEV) for the operator \(-\Delta+V\) in \(\Omega\). We say that a number \(\lambda\in\mathbb{R}\) is a NEV for \(-\Delta+V\) in \(\Omega\) if there exists \(\phi\in H^1(\Omega)\) not identically zero solving the homogeneuos Neumann problem. \[\label{eq:homogeneousneumann} \begin{cases} \left(\Delta +\lambda- V\right)\phi= 0 & \textrm{in }\Omega,\\ \partial_{\nu}\phi=0 & \textrm{on }\partial \Omega, \end{cases}\tag{3}\] and the space of such \(\phi\) is its corresponding eigenspace. We will call \(\lambda\) a simple NEV if the corresponding eigenspace has dimension 1.

Whenever \(\lambda\) is not a NEV, one can solve 1 using the method of layer potentials. With this in mind, if we fix \(\lambda>0\), our goal will be to find a domain \(\Omega\) in which \(\lambda\) is not a NEV. We will give a semi-constructive argument to find such a domain, with an approach based in boundary deformation techniques. Indeed, we will first choose an arbitrary domain and then show how to perturb it by means of a diffeomorphism.

In case \(\lambda\) is a simple NEV, we will use the techniques in [1] and [2] to find a formula for its derivative with respect to the perturbation of the domain, and prove that we can choose such a perturbation so that this derivative doesn’t vanish. In particular, for any \(\mathcal{C}^2\) vector field \(\mathbb{X}\) supported away from the potential, we may deform the domain \(\Omega\) through a family of diffeomorphisms of the form \(h_t=i_\Omega+t\mathbb{X}\), for a fixed vector field \(\mathbb{X}\) and \(t\) small enough. We will then show that there is a unique differentiable function \(t\mapsto \lambda(t)\) such that \(\lambda(t)\) is an eigenvalue of \(-\Delta+V\) in \(h_t(\Omega)\), \(\lambda(0)=\lambda\) and its derivative at \(t=0\) is \[\dot{\lambda}(0)=\int_{\partial \Omega} \left(|\nabla_{\partial\Omega}u|^2-\lambda u^2\right)\mathbb{X}\cdot \nu,\] where \(u\) is the normalized eigenvector of \(\lambda\) and \(\nu\) is the normal outward-pointing vector of \(\Omega\). Then, it will be a matter of choosing \(\mathbb{X}\) in a way that the integral above doesn’t vanish

We will also show that the property of having simple NEVs is generic, this is, most of the domains will have simple NEVs. In particular, we will show that the set of perturbations that produce multiple eigenvalues is meager (of first category), in the appropiate space of perturbations. This notion will be defined later. The main ingredient here is a generalization of Smale-Sard theorem proved by Dan Henry in [1]. This theorem controls the size of the critical set of a map between Banach manifolds. The idea is to characterize multiple eigenvalues as critical points of such a map and then use this theorem.

1.1 Application to a Runge approximation↩︎

Our initial motivation to answer this question comes from an inverse problem of electric scattering with local near-field data. In this problem, which was studied by the author in [3], one attempts to identify an electric potential of compact support by placing point sources and measuring time-harmonic waves in points close to the support of the potential. To be precise, one has access to the scattered wave \(u_{sc}\), which is the solution to the equation \[\label{eq:scatt} \begin{cases} \left(\Delta +\lambda- V\right)u_{sc}(\centerdot\,,y) = V \Phi_\lambda(\centerdot-y) & \textrm{in }\mathbb{R}^n,\\ u_{sc}(\centerdot\,,y)\textrm{ satisfying SRC}. \end{cases}\tag{4}\] Here \(\Phi_\lambda\) is the fundamental solution to the free Helmoltz equation at energies \(\lambda>0\), which solves the distributional problem \[\label{eq:homo} \begin{cases} \left(\Delta +\lambda\right)\Phi_\lambda = \delta_0 & \textrm{in }\mathbb{R}^n,\\ \Phi_\lambda\textrm{ satisfying SRC}. \end{cases}\tag{5}\] SRC stands for the Sommerfeld Radiation Condition, which is a condition of decay at infinity. More precisely, a function \(u\) is said to satisfy SRC if \[\label{eq:src} \lim_{|x|\to\infty}|x|^{\frac{d-1}{2}}\left(\frac{x}{|x|}\cdot\nabla u(x)-i\lambda^{1/2}u(x)\right)=0\tag{6}\] uniformly in every direction. This condition is physically meaningful, since solutions that satisfy it represent waves that “radiate energy at infinity” [4]. Note that \(u_{sc}(x,y)\) can be interpreted as placing a point source in \(y\) and measuring the scattered wave at the point \(x\)

The existence of the scattering wave as solution of the problem 4 was given by the author in [3] by adapting an argument by Caro and García [5], who studied a similar inverse problem with potentials of the form \(V=V^0+\alpha\,\mathrm{d}\sigma\). They defined a family of functional-analytic spaces in which one can exploit appropiate resolvent estimates from harmonic analysis [6]–[8]. These estimates give a decay in suitable norms in terms on \(\lambda\), which in turn means that the scattering solution can be constructed for big enough energies. Throughout this work, we will need to assume that 4 has an unique solution, so that an assumption of the form \(\lambda>\lambda^V\) will be present, where \(\lambda^V\) is such that the scattering solution exists.

In our inverse problem, the denomination “local near-field data” refers to the fact that the measurements are taken in arbitrary small sets of codimension \(1\). In particular, we aim to prove that, if \(V_1\) and \(V_2\) are two potentials of the form 2 , then \[{ \left.\kern-\nulldelimiterspace u_{sc,1}(x,y) \right|_{\Sigma_1\times\Sigma_2} }={ \left.\kern-\nulldelimiterspace u_{sc,2}(x,y) \right|_{\Sigma_1\times\Sigma_2} }\implies V_1=V_2,\] where \(u_{sc,j}\) is the scattering solutions associated to \(V_j\), and \(\Sigma_j\) are two open sets of codimension 1 and of class \(C^3\). To prove identifiability with this data, the author made use of a Runge approximation from single-layer potentials with densities supported in \(\Sigma_j\). This relied on the existence of a unique solution to the Neumann problem 1 in a domain \(\Omega\) whose boundary contained \(\Sigma_1\) and \(\Sigma_2\). The set of NEVs is countable and has no accumulation points, which means that, by choosing any such domain arbitrarily, it is possible to prove uniqueness for most energies, this is, for all \(\lambda>\lambda^V\) except for those that turn out to be NEVs of \(-\Delta+V\) in \(\Omega\).

The assumption that \(\lambda\) is not an eigenvalue has been taken in other problems of inverse scattering, such as in [9]. However, in the aforementioned inverse problem, the choice of the domain in which one performs the Runge approximation -and thus that in which 1 has to be solved- is quite arbitrary. Therefore, it makes sense to fix any energy \(\lambda\), and attempt to find a suitable domain \(\Omega\).

A similar problem was considered by Stefanov in [10], where he sought to avoid Dirichlet eigenvalues of the Helmholtz equation in the context of an inverse problem in electric scattering. His argument is relatively straightforward as a consequence of the fact that Dirichlet eigenvalues are strictly monotonically decreasing with respect to domain inclusion [11]. However, the setting of NEVs proves to be more complex. For instance, this monotonicity already does not hold in general for NEVs of the Laplacian [12].

1.2 The result↩︎

The main theorem of this work is stated below. Note that the second part of the theorem is needed in the inverse problem to perform the Runge approximation from the measurement sets \(\Sigma_j\).

1.3 Outline of the paper↩︎

• In Section 2 we give some preliminary results that will be of use throughout the paper. In Section 2.1, we give a series of unique continuation properties for the operator \(\Delta-\lambda-V\). In Section 2.2 we recall the solution of the scattering problem 4 as given in [3], with the definition of Caro and Garcia’s functional-analytical spaces. This will be relevant in the subsequent sections.

• In Section 3, we use the method of layer potentials to characterize the conditions in which the Neumann problem 1 is solvable in terms of the eigenspace associated to \(\lambda\). This will allow us to solve it in the case in which \(\lambda\) is not a NEV and develop the perturbation of eigenvalues later on.

• In Section 4 we will lay out the approach of boundary perturbations, and study the differentiation of differential operators and boundary conditions with respect to these perturbations. We end the section finding a formula for the derivative of a simple eigenvalue in Section 4.2.

• In Section 5, we show that the simplicity of eigenvalues is generic in the set of suitable perturbations. We start by giving the statement of Henry’s genericity theorem [1], and recalling the notions of meager set, Lindelöf space and semi-Fredholm operator in Section 5.1. Then, in Section 5.2, we use this theorem to prove this genericity property.

• Finally, we give the proof of Theorem [thm:nev] in Section 6.

• In Appendix 7, we recall the notion of Banach manifolds and their tangent space, based on the exposition by Lang [13].

2 Preliminaries↩︎

2.1 Unique continuation properties↩︎

We pull now a Carleman estimate from [3], which was obtained by perturbating such an estimate that Caro and Rogers obtained in [14] for the Laplacian. This estimate is done in a family of Bourgain-type spaces that were introduced by Caro and Rogers, inspired by the works of Haberman and Tataru [15], [16]. For \(s\in\mathbb{R}\) and \(\zeta\in\mathbb{C}^n\) we define the inhomogeneous Bourgain-type space \(X_\zeta^s\) as the space of distributions \(u\in\mathscr{S}'(\mathbb{R}^n)\) such that \(\widehat{u}\in L^2_\text{loc}(\mathbb{R}^n)\) and \[\lVert u\rVert_{X_\zeta^s}=\lVert(M|\Re(\zeta)|^2+M^{-1}|p_\zeta|^2)^{s/2}\,\widehat{u}\rVert_{L^2}<\infty,\] endowed with the norm \(\lVert\centerdot\rVert_{X_\zeta^s}\), with \(M>1\), where \(\Re\) denotes the real part and \[\label{eq:pzeta} p_\zeta(\xi)=-|\xi|^2+2i\zeta\cdot\xi+\zeta\cdot\zeta.\tag{7}\] The Carleman estimate can be stated as follows:

Take now \(R_0\) such that \(\Omega\subset B_{R_0}\). We can check that the spaces \(X_\zeta^{1/2}\) and \(H^1(\mathbb{R}^n)\) are equal as sets, and that, for every \(u\in H^1(\mathbb{R}^n)\) such that \(\mathrm{supp}\,u\subset \overline{\Omega}\), we have that \(e^{\varphi_{\zeta}}(\Delta+\lambda-V)\,(e^{-\varphi_{\zeta}} u)\) is in \(X_{-\zeta}^{-{1/2}}\). Therefore, by density, the estimate ?? also holds for every \(u\in H^1(\mathbb{R}^n)\) such that \(\mathrm{supp}\,u\subset \overline{\Omega}\). This will be useful to prove the following proposition:

Another result in the literature for unique continuation of elliptic operators is the following, whose proof can be found in [9]. We say that an open subset \(O\subset \overline{\Omega}\) is connected to \(\Sigma\) if \(O\) is connected and \(\Sigma\cap O \neq \emptyset\).

As an easy consequence of the previous propositions, we have the following:

2.2 The Scattering Problem↩︎

We will briefly give the definition of the suitable spaces that were used in [3] to solve the scattering problem 4 . These were defined in [5] to solve this same problem with potentials of the form \(V=V^0+\alpha\,\mathrm{d}\sigma\).

Actually, the space \(X_\lambda^*\) is isomorphic to the space of \(u\in\mathscr{S}'(\mathbb{R}^n)\) for which the following norm is finite: \[\begin{align} \lVert u\rVert^2_{X^*_\lambda}\mathrel{\vcenter{\baselineskip 0.5ex \lineskiplimit 0pt \scriptsize.\scriptsize.}} =& \lVert m_\lambda^{1/2}\widehat{P_{<I}f}\rVert^2_{L^2}+\sum_{k\in I}\left(\lambda^{1/2}\lVert P_kf\rVert^2_{B^*}+\lambda^{n(\frac{1}{q_n}-\frac{1}{p_n})}\lVert P_kf\rVert^2_{L^{q_n}}\right)+\\&\sum_{k>k_\lambda+1}\lVert m_\lambda^{1/2}\widehat{P_kf}\rVert^2_{L^2}, \end{align}\] where the norm \(\lVert\centerdot\rVert_{B^*}\) is defined by \[\lVert u\rVert_{B^*}=\sup_{j\in\mathbb{N}_0}\,\left(2^{-j/2}\lVert f\rVert_{L^2(D_j)}\right)\] Furthermore, it might interesting to note that \(\mathscr{S}(\mathbb{R}^n)\) is dense in both \(X_\lambda\) and \(X_\lambda^*\). It will later be useful to see the elements of \(X_\lambda^*\) as elements of \(H^1_{\textrm{loc}}(\mathbb{R}^n)\), and in fact we have the following bound:

The following theorem gives the existence of a unique solution to the scattering problem:

3 Layer potentials↩︎

Layer potentials are a classical tool in the study of elliptic boundary value problems. They allow us to turn these problems into integral equations with kernels associated to the fundamental solution. According to [19], the idea was introduced by Gauss in 1839, and then developed by Neumann in the 1870s and 1880s for convex domains. It was then studied by Poincaré in the case of smooth domains. Several generalizations have been made over the years. We can cite for instance the recent works of David and Semmes [20] and of Tolsa [21], where they study general layer potentials defined by weakly singular kernels on rectifiable sets. However, we restrict our attention to layer potentials defined by the fundamental solution for the operator \(\Delta+\lambda-V\) in \(\mathcal{C}^2\) domains. Our main references here are the book by Folland [22], who studies the Laplacian on \(\mathcal{\mathcal{C}}^2\) domains in \(\mathbb{R}^n\), and that of Colton and Kress [23], who focus on the Helmholtz equation in \(\mathcal{C}^2\) domains in \(\mathbb{R}^3\), as well as the book [24] by the same authors. We also point to the books [25] and [26] for classical references. Along this section, let \(V\) be a potential of the type 2 and fix \(\lambda>\lambda^V\), where \(\lambda^V>0\) is the lower bound required by Theorem 6 for the solvability of the forward problem. From now on, fix also a bounded open domain \(\Omega\) of class \(\mathcal{C}^2\) such that \(\mathrm{supp}\,V\subset\Omega\).

Remember that we want to study the following general Neumann boundary: find \(u\in H^1(\Omega)\) such that \[\label{eq:neumanngen} \begin{cases} \left(\Delta +\lambda- V\right)u= f_1 & \textrm{in }\Omega,\\ \partial_{\nu}u=f_2 & \textrm{on }\partial \Omega, \end{cases}\tag{12}\] for \(f_1\in H^{-1}_0(\Omega)\cap L^2(\Omega\setminus\mathrm{supp}\,V)\) and \(f_2\in L^2(\partial\Omega)\). In this section, we are going to characterize the pairs \((f_1,f_2)\) that admit a solution in terms of the eigenspace associated to \(\lambda\) and, in particular, we will show that one can solve the problem above whenever \(\lambda\) is not a NEV for \(-\Delta+V\) in \(\Omega\).

Let’s start by doing a transformation on the equation. Let \(w\in X_\lambda^*\) be the unique solution to \[\label{eq:w} \begin{cases} \left(\Delta +\lambda- V\right)w = f_1 & \textrm{in }\mathbb{R}^n,\\ w\textrm{ satisfying SRC}, \end{cases}\tag{13}\] which exists by Theorem 6, since the extension by \(0\) of \(f_1\) to \(\mathbb{R}^n\) lives in \(X_\lambda\). Remember that the spaces \(X_\lambda\) and \(X_\lambda^*\) were defined in Section 2.2. Denote now \(g=f_2-{ \left.\kern-\nulldelimiterspace \partial_\nu w \right|_{\partial\Omega} }\), and note that, since \(w\) is a solution to \((\Delta+\lambda)w=f_1\) away from \(\mathrm{supp}\,V\), then \(w\in H^2(\mathbb{R}^n\setminus\mathrm{supp}\,V)\) and therefore \({ \left.\kern-\nulldelimiterspace \partial_\nu w \right|_{\partial\Omega} }\in L^2(\Omega)\). Thus solving 12 is equivalent to solving the problem \[\label{eq:neumann2} \begin{cases} \left(\Delta +\lambda- V\right)v= 0 & \textrm{in }\Omega,\\ \partial_{\nu}v=g & \textrm{on }\partial \Omega. \end{cases}\tag{14}\] with \(g\in L^2(\Omega)\) by setting \(u=w+v\).

We will study this problem using the method of layer potentials. Note also that \({ \left.\kern-\nulldelimiterspace w \right|_{\Omega} }\in H^1(\Omega)\) by Proposition 5, so we will only need to prove that \(v\) belongs to \(H^1(\Omega)\) to conclude that \(u\) belongs too. In this case it will be useful to think of the fundamental solution for the operator \(\Delta+\lambda\) in \(\mathbb{R}^n\) as a Hankel function. In fact, \(\Phi_\lambda\) as defined by 5 will take the form \[\Phi_\lambda(x)=\frac{i}{4}\left(\frac{\lambda^{1/2}}{2\pi|x|}\right)^{n/2-1}H_{n/2-1}^{(1)}\left(\lambda^{1/2}|x|\right),\] where \(H_\nu^{(1)}\) denotes the Hankel function of the first kind (or Bessel function of the third kind). If we define \(u_{in}(x,y)=\Phi_\lambda(x-y)\) and recall the limiting properties of the Hankel functions, it’s relatively easy to check that \[\begin{align} \label{eq:kernel} \begin{aligned} u_{in}(x,y)&= F(x,y)|x-y|^{2-n},\\ \partial_{\nu_x}u_{in}(x,y)&=G(x,y)|x-y|^{2-n},\\ \partial_{\nu_y}u_{in}(x,y)&=\partial_{\nu_x}u_{in}(y,x)=G(y,x)|x-y|^{2-n}, \end{aligned} \end{align}\tag{15}\] with \(F\) and \(G\) being bounded functions on \(\partial \Omega\times\partial \Omega\). Then, \(u_{in}\), \(\partial_{\nu_x}u_{in}\) and \(\partial_{\nu_y}u_{in}\) are, by definition, weakly singular kernels of order \(n-2\) on \(\partial \Omega\times\partial \Omega\). The following lemma is a combination of those in [22], Chapter 3B, and is an important piece to solve the problem 14 .

Take now \(u_{to}=u_{in}+u_{sc}\), where \(u_{sc}\) is the scattering solution of 4 , and define for \(f\) continuous on \(\partial \Omega\) and \(x\in \mathbb{R}^n\setminus \partial \Omega\) the single layer potential with moment \(f\) as \[\left(\mathcal{S}f\right)(x)=\int_{\partial \Omega}u_{to}(x,y)\,f(y)\,\mathrm{d}S(y),\] and the double layer potential as \[\left(\mathcal{D}f\right)(x)=\int_{\partial \Omega}\partial_{\nu_y}u_{to}(x,y)\,f(y)\,\mathrm{d}S(y),.\] Define further the operator \(\mathcal{N}\), which is the adjoint of \(\mathcal{D}\) over \(\partial \Omega\), as \[\left(\mathcal{N}f\right)(x)=\int_{\partial \Omega}\partial_{\nu_x}u_{to}(x,y)\,f(y)\,\mathrm{d}S(y),\quad x\in \partial \Omega,\] which must be understood as an improper integral.

Now, we have the following lemmas, which are modifications of classical results that can be found in [23], [22] and [24] for the Helmholtz and Laplace equations. We denote by \(u_+\) the trace on \(\partial \Omega\) of \({ \left.\kern-\nulldelimiterspace u \right|_{\mathbb{R}^n\setminus \overline{\Omega}} }\), and by \(u_-\) the trace on \(\partial \Omega\) of \({ \left.\kern-\nulldelimiterspace u \right|_{\Omega} }\). As well, we denote by \(\partial_\nu u_+\) and \(\partial_\nu u_-\) the normal derivative of those, always with respect to the outward-pointing normal vector of \(\partial \Omega\) (as seen from inside \(\Omega\)).

Now, we can try to solve the problem 14 by means of Fredholm alternative. This is done in the following lemma.

Finally, we can conclude the following result:

4 Perturbation of domains↩︎

4.1 The approach of \(\mathcal{C}^m\) embeddings↩︎

Let \(V\) be a potential as in 2 , \(\lambda\geq 0\) and take \(\Omega\) and be a \(\mathcal{C}^m\) domain such that and \(\mathrm{supp}\,V\subset\Omega\). From now on, we will denote \(\mathrm{supp}\,V\) by \(\mathcal{V}\). Define then the operator \[\begin{align} L_\Omega : H^1(\Omega)\cap H^m(\Omega\setminus\mathcal{V})&\to H^{-1}_0(\Omega)\cap H^{m-2}(\Omega\setminus\mathcal{V})\\ u & \mapsto \left(\Delta+\lambda-V\right) u. \end{align}\] We will deform the domain \(\Omega\) by a \(\mathcal{C}^m\) embedding \(h:\Omega\to\mathbb{R}^n\). This is, \(h\) is of class \(\mathcal{C}^m\), it is a diffeomorphism onto its image and its inverse \(h^{-1}\) is also of class \(\mathcal{C}^m\). Denote the space of such embeddings as \(\textrm{Diff}^m(\Omega)\). We would like to study the natural operator that acts on functions on the deformed domain \[L_{h(\Omega)} : H^1(h(\Omega))\cap H^m(h(\Omega\setminus\mathcal{V}))\to H^{-1}_0(h(\Omega))\cap H^{m-2}(h(\Omega\setminus\mathcal{V})).\] However, the fact that the function spaces change with \(h\) poses a difficulty. To solve this, we will consider the pullback \(h^*\), defined by \[h^*u(x)=u\, (h(x)),\] and the pushforward \(h_*\), defined by \[h_*u(x)=u \,(h^{-1}(x)).\] Then, the Lagrangian form of the operator \(L_{h(\Omega)}\) is defined as \[h^* L_{h(\Omega)} h_*: H^1(\Omega)\cap H^m(\Omega\setminus\mathcal{V})\to H^{-1}_0(\Omega)\cap H^{m-2}(\Omega\setminus\mathcal{V}).\]

Take now a curve of class \(\mathcal{C}^1\) of such embeddings \(t\mapsto h_t\). In [1], Henry introduces the anti-convective derivative, defined for \(\mathcal{C}^1\) functions \(\varphi:\mathbb{R}\times\mathbb{R}^n\to \mathbb{R}\) as \[\label{eq:anti-convective} D_t\varphi(t,x)=\partial_t \varphi(t,x)-U(t,x)\cdot\nabla \varphi(t,x),\tag{19}\] where \[U(t,x)=\left((Jh_t)^{-1}\,\partial_t h_t\right)(x)\] and \(Jh\) denotes the Jacobian matrix of \(h\) with respect to \(x\). This object will allow us to differentiate operators and boundary conditions in the Lagrangian form. It satisfies the following rule, which can be proved as consequence of the chain rule:

Although stated pointwise, this lemma and the results below will be later interpreted as an equality between \(\mathcal{C}^1\) curves with values on Sobolev spaces. This interpretation is inmediate for curves valued in \(\mathcal{C}^1\), and can be extended to Sobolev spaces by density.

We will now use this to differentiate the differential operator \(\Delta+\lambda-V\) with respect to boundary deformation. The reader can be pointed to Chapter 2 of [1] for an expression of the derivative of a general class of differential operators.

Consider that \(t\mapsto \lambda(t)\), \(t\mapsto u(t,\centerdot)\) and \(t\mapsto h_t\) are curves of class \(\mathcal{C}^1\). Suppose that \(h_0=i_\Omega\), the trivial embedding of \(\Omega\) into \(\mathbb{R}^n\). We want to find now \[{ \left.\kern-\nulldelimiterspace \partial_t\big(h_t^*L_{\Omega_t}h_{t*}u(t,\centerdot)\big) \right|_{t=0} },\] where \(\Omega_t=h_t(\Omega)\). Define then \(v(t,x)=h_{t*}u(t,x)\) and \(\psi(t,x)=L_{\Omega_t}v(t,x)\). Then, by the definition of the anticonvective derivative 19 and Lemma 5, we have formally that \[\begin{align} \partial_t\big(h_t^*\psi(t,\centerdot)\big)=&D_t\left(h_t^*\psi(t,\centerdot)\right) + U(t,\centerdot)\cdot\nabla (h_t^*\psi(t,\centerdot))\\ =&h_t^*\partial_t\psi(t,\centerdot) + U(t,\centerdot)\cdot\nabla (h_t^*\psi(t,\centerdot))\\ =&h_t^*\partial_tL_{\Omega_t}v(t,\centerdot)+h_t^*L_{\Omega_t}\,\partial_tv(t,\centerdot) + U(t,\centerdot)\cdot\nabla (h_t^*\psi(t,\centerdot)). \end{align}\] Note now that, again by Lemma 5 \[h_{t*}D_t u = h_{t*}D_t\left(h_t^*v(t,\centerdot)\right)= h_{t*}h_t^*\partial_t v(t,\centerdot)=\partial_t v(t,\centerdot).\] Therefore, if we observe that \(\partial_tL_{\Omega_t}=\lambda'(t)\), and using again the definition of the anticonvective derivative for \(u\), we obtain \[\label{eq:derivativeL} \begin{align} { \left.\kern-\nulldelimiterspace \partial_t\big(h_t^*L_{\Omega_t}h_{t*}u(t,\centerdot)\big) \right|_{t=0} }=&L_\Omega(\dot{u}-\dot{h} \cdot\nabla u)+\dot{\lambda} u+\dot{h} \cdot \nabla (L_\Omega u)\\ =&L_\Omega \dot{u} + \dot{\lambda} u + \big[\dot{h} \cdot \nabla,L_\Omega\big]. \end{align}\tag{20}\] Here, we have denoted with \(\dot{u}\), \(\dot{\lambda}\) and \(\dot{h}\) the respectives partial derivaties of \(u\), \(\lambda\) and \(h\) with respect to \(t\) at \(t=0\), and we have used the fact that \(Jh_0=Ji_\Omega=1\), so that \(U(0,\centerdot)=\dot{h}\).

Note that this expression may not make sense in general, as the quantity \[\big[\dot{h} \cdot \nabla,L_\Omega\big]=\big[\dot{h} \cdot \nabla,V\big]\] poses technical problems due to the low regularity of the potential \(V\). However, we will later restrict our attention to curves of deformations of the form \[h_t(x)=x+t\mathbb{X}(x),\] for vector fields \(\mathbb{X}\in \mathcal{C}^k(\Omega;\mathbb{R}^n)\), \(k\geq 1\), such that \(\mathrm{supp}\,\mathbb{X}\cap\mathrm{supp}\,V = \emptyset\). Note that \(h_t\in\mathrm{Diff}^k(\Omega)\) as long as \(Jh_t\neq0\), i.e. for \(t\) small enough.

We also need to be able to differentiate the Neumann boundary conditions. For this purpose, we choose an extension of \(\nu_\Omega\) near \(\partial\Omega\), where \(\nu_\Omega\) is the unit outward-pointing normal vector to \(\partial\Omega\), and then define \(\nu_{\Omega_t}=\nu_{h_t(\Omega)}\) by the expression \[\label{eq:hnormal} h^*\nu_{\Omega_t}(x)=\frac{((Jh_t)^{-1})^\top\nu_\Omega(x)}{|((Jh_t)^{-1})^\top\nu_\Omega(x)|}\tag{21}\] for \(x\) near \(\partial\Omega\). Here \(((Jh)^{-1})^\top\) is the inverse-transpose of the Jacobian matrix of \(h\) with respect to \(x\). An expression for the derivative of this extension of the normal vector is given in the following lemma from [1].

This allows us to do a similar computation as before for the Neumann boundary conditions. Indeed, we want to find an expression for \[{ \left.\kern-\nulldelimiterspace \partial_t\big(h_t^*\nu_{\Omega_t}\cdot\nabla (h_{t*}u)\big) \right|_{t=0} }\] in points of \(\partial\Omega\). Consider a curve of embeddings \(t\mapsto h_t\) of the form
\(h_t(x)=x+t\mathbb{X}(x)\), with \(\mathbb{X}\in\mathcal{C}^2(\Omega;\mathbb{R}^n)\). Define again \(v(t,x)=h_{t*}u(t,x)\) and \(\phi(t,x)=\nu_{\Omega_t}\cdot\nabla v(t,x)\). Then, by the definition of the anticonvective derivative 19 and Lemma 5, we have that \[\begin{align} \partial_t \psi(t,\centerdot)&=\nu_{\Omega_t}\cdot\nabla\big(\partial_t v(t,\centerdot)\big)+\partial_t\nu_{\Omega_t}\cdot\nabla v(t,\centerdot)\\ &=\nu_{\Omega_t}\cdot\nabla(h_{t*}D_t u)-\nabla_{\partial\Omega_t}\big(\mathbb{X}\cdot\nu_{\Omega_t}\big)\cdot\nabla v, \end{align}\] and thus \[\begin{align} \partial_t\big(h_t^*\psi(t,\centerdot)\big)=&D_t\left(h_t^*\psi(t,\centerdot)\right) + U(t,\centerdot)\cdot\nabla (h_t^*\psi(t,\centerdot))\\ =& h_t^*\partial_t\psi(t,\centerdot) + U(t,\centerdot)\cdot\nabla (h_t^*\psi(t,\centerdot)) \end{align}\] means that \[\label{eq:derivativeN} { \left.\kern-\nulldelimiterspace \partial_t\big(h_t^*\nu_{\Omega_t}\cdot\nabla (h_{t*}u)\big) \right|_{t=0} }= \partial_\nu\big(\dot{u}-\dot{h} \nabla u\big)-\nabla_{\partial\Omega_t}\big(\dot{h}\cdot\nu\big)\cdot\nabla u+\dot{h}\cdot\nabla\big(\partial\nu u\big)\tag{22}\] Here we have again used the fact that \(Jh_0=Ji_\Omega=1\), so that \(U(0,\centerdot)=\dot{h}\), and the dot notation as before, observing that \(\partial_t h_t=\mathbb{X}\) for all \(t\).

4.2 Perturbation of a simple eigenvalue↩︎

We will end this section by finding a formula for the derivative of a simple NEV with respect to a boundary perturbation, in the case that \(\lambda\) is a simple NEV for \(-\Delta+V\) in \(\Omega\). The formula is contained in the following proposition.

5 Simple eigenvalues are generic↩︎

In this section, we will prove that most perturbations produce domains in which Neumann eigenvalues are simple, i.e., that the property of having simple eigenvalues is generic in the appropiate set of perturbations. This will allow us to be in the position to use Proposition 10 to avoid Neumann eigenvalues. To prove it, we will use Henry’s genericity theorem which we introduce below. The reader might want to recall some notions on Banach manifolds in Appendix 7.

5.1 Henry’s genericity theorem↩︎

Henry gave a generalization of Sard-Smale’s theorem in [1]. We use it in Section 5.2 to prove that “most perturbations” on the domain produce simple NEVs.

Sard proved in 1942 [27] that the set of critical values of a \(\mathcal{C}^k\) map between differentiable manifolds of finite dimension has measure zero, when \(k\) is big enough. Smale generalized this result in 1965 [28] to maps between more general Banach manifolds. In particular, he proved that the set of critical values of a \(\mathcal{C}^k\) Fredholm map between Banach manifolds is meager.

The original version of Smale-Sard theorem can be found in [28]. Other classical references for the genericity of eigenvalues are [29]–[31]. We also used the references [32], [33] for better understanding of Henry’s work on boundary perturbation.

Henry’s generalization can be stated as follows:

Let’s clarify some notions that appear in the theorem.

The notion of Lindelöf set is a generalization of the notion of compactness:

In particular, any separable metric space is Lindelöf, as well as any of its closed subsets. We should also recall the notion of semi-Fredholm operator:

5.2 Generic simplicity of eigenvalues↩︎

Now, we have to carefully define the appropiate spaces of perturbations. In [1], Henry chooses the space \(\mathrm{Diff}^3(\Omega)\), the set of embeddings \(h:\Omega\to\mathbb{R}^n\) of class \(\mathcal{C}^3\) with \(\mathcal{C}^3\) inverse.

However, we have to note that, to avoid regularity problems, the potentials \(V\) should not be affected by the deformation. We would like to restrict ourselves to perturbations that equal the identity around \(\mathrm{supp}\,V\). To achieve this we fix from now a cut-off \(\chi_1\) in \(\mathcal{C}^\infty(\mathbb{R}^n;[0,1])\) such that \(\chi_1=0\) in \(\mathrm{supp}\,V\) and \(\chi_1=1\) in a neighborhood of \(\partial\Omega\).

Also, we would like that an open set of the observation set \(\Sigma\) is contained in the perturbed domain \(h(\Omega)\). Therefore, we will later fix another cut-off \(\chi_2\) in \(\mathcal{C}^\infty(\mathbb{R}^n;[0,1])\) such that \(\chi_2=0\) on a neighborhood of some point \(x_0\in\Sigma\).

Then, we may call \(\eta=\chi_1\chi_2\). We will consider vector fields \(\mathbb{X}\in\mathcal{C}^3\big(\overline{\Omega};\mathbb{R}^n\big)\) such that \(\lVert\eta\mathbb{X}\rVert_{\mathcal{C}^1}<1,\) so that any perturbation of the form \(h(x)=x+\eta(x)\mathbb{X}(x)\) belongs to \(\mathrm{Diff}^3(\Omega)\). We denote the set of such perturbations as \(\mathbb{D}\) and, if we set \(M=\lVert\eta\rVert_{\mathcal{C}^3}\), it can be written as follows \[\label{eq:mathbbd} \mathbb{D}=\left\{h:\Omega\to\mathbb{R}^n: h=i_\Omega+\eta\mathbb{X},\,\mathbb{X}\in\mathcal{C}^3\big(\overline{\Omega};\mathbb{R}^n\big),\,\lVert\mathbb{X}\rVert_{\mathcal{C}^3}<M^{-1}\right\},\tag{25}\] where \(i_\Omega\) is the identity in \(\Omega\).

Note that \(\mathbb{D}\) is a Banach manifold of class \(\mathcal{C}^3\), since it is isomorphic to an open ball in the Banach space \(\mathcal{C}^3\big(\overline{\Omega};\mathbb{R}^n\big)\).

It will be useful to characterize the tangent space of \(\mathbb{D}\) at the identity \(i_\Omega(x)=x\). Observe that any curve in \(t\mapsto h_t \in \mathbb{D}\) such that \(h_0(x)=x\) can be written as \(h_t(x)=x+t\eta(x)\mathbb{X}(x)\) for some vector field \(\mathbb{X}\in\mathcal{C}^3\big(\overline{\Omega};\mathbb{R}^n\big)\), and \(t\in(-T,T)\), where \[T=\lVert\eta(x)\mathbb{X}(x)\rVert_{\mathcal{C}^1}^{-1}.\] Differentiating with respect to \(t\) at \(t=0\) gives the shape of the tangent space at \(i_\Omega\): \[T_{i_\Omega}\mathbb{D}=\left\{\dot{h} = \eta\mathbb{X}:\mathbb{X}\in\mathcal{C}^3\big(\overline{\Omega};\mathbb{R}^n\big)\right\}.\]

We are now ready to state the main result of this section:

We will divide the proof of this proposition in a series of lemmas, to hopefully make it more clear. We again denote by \(\mathcal{V}\) the support of the potential \(V\).

We are now ready to prove the proposition.

6 Proof of Theorem [thm:nev]↩︎

Let’s observe first that eigenvalues in fact form a countable set. To see this, define the unbounded operator \(\left(T_N,D(T_N)\right)\) over \(L^2(\Omega)\) as \(T_Nu=\left(-\Delta+V\right)u\), with domain \[D(T_N)=\{u\in L^2(\Omega): \left(-\Delta+V\right)u\in L^2(\Omega),\; \partial_\nu { \left.\kern-\nulldelimiterspace u \right|_{\partial \Omega} }=0\}.\] Observe now that a Neumann eigenvalue for \(-\Delta+V\) on \(\Omega\) will be an eigenvalue for \(\left(T_N,D(T_N)\right)\). The domain \(D(T_N)\) is a separable Hilbert space, since it is a subspace of \(L^2(\Omega)\), and \(T_N\) is symmetric over \(D(T_N)\), which ensures that the set of its eigenvalues must be countable.

Indeed, suppose that there is an uncountable set of such eigenvalues. Let \(\lambda\) and \(\mu\) be any two distinct eigenvalues, and \(u\) and \(v\) be corresponding distinct eigenfunctions. Then, \[\lambda\braket{u,v}=\braket{T_N u, v}=\braket{u,T_N v}= \mu\braket{u,v},\] and thus \(u\perp v\), which contradicts the fact that \(D(T_N)\) is separable. We recall once again the statement of the theorem we want to prove:

7 Banach manifolds↩︎

Banach are a generalization of the usual finite dimensional manifolds, where the charts are defined on general Banach spaces instead of on \(\mathbb{R}^n\). A general reference for this topic is [13]. We give here just a brief introduction to suit our needs.

We can easily check that there is a unique topology in \(M\) such that each \(U_i\) is open and each \(\varphi_i\) is an homeomorphism. For \(M\) to be Haussdorff, we have to place a separation condition on the covering.

We have not made any assumption on the relationship between the \(E_is\), but if they are all equal to the same space \(E\), we say that we are in possesion of an \(E\)-atlas. Furthermore, we can show that, whenever \(U_i\cap U_j\neq\emptyset\), \(E_i\) and \(E_j\) are isomorphic as topological vector spaces, and therefore, we will have an \(E\)-atlas in any connected component on \(M\).

Now suppose that we are given an open subset \(U\subset M\) and a topological isomorphism \(\varphi\) onto an open subset \(\varphi(U)\) of a Banach space \(E\). We say that \((U,\varphi)\) is compatible with the atlas \(\{(U_i,\varphi_i)\}_{i\in I}\) if \[\varphi\varphi_i^{-1}:\varphi(U_i\cap U)\to\varphi_i(U_i\cap U_j)\] is an isomorphism of class \(\mathcal{C}^k\) for all \(i\in I\). We say that two atlas are compatible if any chart in one of them is compatible with the other atlas. The relation of compatibility is a equivalence relation on the set of atlases of class \(\mathcal{C}^k\).

If all the spaces \(E_i\) in an atlas are isomophormic as topological vector spaces, then we can find an equivalent atlas for which all spaces are all equal, say to \(E\). In this case, we say that \(M\) is a \(E\)-manifold, or that \(M\) is modeled on \(E\). If \(E\) is isomorphic to \(\mathbb{R}^m\), we say that \(M\) is a finite dimensional manifold of dimension \(m\).

We will also need to define tangent vectors to endow the manifold with a differentiable structure. Let \(M\) be a \(E\)-manifold of class \(\mathcal{C}^k\), \(k\geq 1\), and let \(x\) a point in \(M\). Consider triples \((U,\varphi,v)\), where \((U,\varphi)\) is an atlas such that \(x\in U\), and \(v\) is a vector in \(E\). We say that two such triples \((U,\varphi,v)\) and \((V,\psi,w)\) are equivalent if \[{ \left.\kern-\nulldelimiterspace D(\psi\varphi^{-1}) \right|_{\varphi(x)} }v = w.\]

Each chart determines a bijection between the tangent space and a Banach manifold, and allows to transport the structure of topological vector space to it. This structure will be independent of the chart.

We can equivalently define tangent vectors as equivalence classes of curves that pass through \(x\) with the same velocity. Indeed, let \(\gamma_1,\gamma_2:(-\varepsilon,\varepsilon)\to M\) be two curves such that \(\gamma_1(0)=\gamma_2(0)=x\). We say that \(\gamma_1\) and \(\gamma_2\) are equivalent at \(x\) if \[(\varphi\gamma_1)'(0)=(\varphi\gamma_2)'(0),\] for \((U,\varphi)\) a chart containing \(x\). Then, we can define the tangent vector \(v\) at \(x\) as \(v=(\varphi\gamma)'(0)\) for any \(\gamma\) in the equivalence class.

References↩︎