We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with bound... more We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first involves the volume of the manifold and of its boundary, as well as packing and volume growth constants of the boundary and its distortion. Its proof is based on a metric-measure space technique that was introduced by Colbois and Maerten. The second bound is in terms of the extrinsic diameter of the boundary and its injectivity radius. It is obtained from a concentration inequality, akin to Gromov-Milman concentration for closed manifolds. By applying these bounds to cylinders over closed manifold, we obtain bounds for eigenvalues of the Laplace operator, in the spirit of Grigor'yan-Netrusov-Yau and of Berger-Croke. For a family of manifolds that has uniformly bounded volume and boundary of fixed intrinsic geometry, we deduce that a large first nonzero Steklov eigenvalue implies that each boundary component is contained in a ball of small extrinsic radius.
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary ... more How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander's approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.
The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is ... more The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.
This chapter is a reprint of the article Spectral geometry of the Steklov problem by the same aut... more This chapter is a reprint of the article Spectral geometry of the Steklov problem by the same authors, published in the Journal of Spectral Theory. We would like to thank the Journal of Spectral Theory and the European Mathematical Society for granting permission to reproduce the paper in this book. Spectral geometry of the Steklov problem is a rapidly developing subject, and there have been a number of important advances since the original version of this article has appeared. In the present text, we have added references to some of these new results in the footnotes. In order to make this chapter coherent with the rest of the book, the dimension is denoted by d, and the trivial Steklov eigenvalue is now denoted by σ = , as opposed to σ = in the journal version of this article. The numeration has been also changed.
Given a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, we prove the existence... more Given a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\sigma_k(M_j)$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_j$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.
We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue... more We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point within a conformal class. The second is degeneration of the conformal class to the boundary of the moduli space on the torus and on the Klein bottle. In the latter, we follow the outline proposed by N.
Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the S... more Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev’s upper bound of $$8\pi $$ 8 π for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov ei...
We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian m... more We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for thek-th perimeter-normalized Steklov eigenvalue is$$8\pi k$$8πk, which is the best upper bound for the$$k^{\text {th}}$$ktharea-normalised ei...
Given two compact Riemannian manifolds M 1 and M 2 such that their respective boundaries Σ 1 and ... more Given two compact Riemannian manifolds M 1 and M 2 such that their respective boundaries Σ 1 and Σ 2 admit neighbourhoods Ω 1 and Ω 2 which are isometric, we prove the existence of a constant The constant C depends only on the geometry of Ω 1 ∼ = Ω 2 . This follows from a quantitative relationship between the Steklov eigenvalues σ k of a compact Riemannian manifold M and the eigenvalues λ k of the Laplacian on its boundary. Our main result states that the difference |σ k -√ λ k | is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω 1 ∼ = Ω 2 .
Given a smooth compact hypersurface $M$ with boundary $\unicode[STIX]{x1D6F4}=\unicode[STIX]{x220... more Given a smooth compact hypersurface $M$ with boundary $\unicode[STIX]{x1D6F4}=\unicode[STIX]{x2202}M$, we prove the existence of a sequence $M_{j}$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\unicode[STIX]{x1D70E}_{k}(M_{j})$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_{j}$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.
We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary... more We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is isospectral to the disk.
Proceedings of the American Mathematical Society, 2019
Let ( M , g ) (M,g) be a compact Riemannian manifold with boundary. Let b > 0 b>0 be the nu... more Let ( M , g ) (M,g) be a compact Riemannian manifold with boundary. Let b > 0 b>0 be the number of connected components of its boundary. For manifolds of dimension ≥ 3 \geq 3 , we prove that for j = b + 1 j=b+1 it is possible to obtain an arbitrarily large Steklov eigenvalue σ j ( M , e δ g ) \sigma _j(M,e^\delta g) using a conformal perturbation δ ∈ C ∞ ( M ) \delta \in C^\infty (M) which is supported in a thin neighbourhood of the boundary, with δ = 0 \delta =0 on the boundary. For j ≤ b j\leq b , it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of M M . In fact, when working in a fixed conformal class and for δ = 0 \delta =0 on the boundary, it is known that the volume of ( M , e δ g ) (M,e^\delta g) has to tend to infinity in order for some σ j \sigma _j to become arbitrarily large. This is in stark contrast with the situation for the eigenvalues of the Laplace operator on a closed manifold, where a c...
The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We ... more The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension $d\geqslant 3$ . Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the $(d-2)$ -dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related resu...
In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian ... more In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectral gap arbitrarily large using conformal deformations which are localized on domains of small measure, as long as the support of the deformations contains and connects each component of the boundary.
We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain ... more We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of half this area. The estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Pólya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres.
We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar d... more We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains, for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.
Mathematical Proceedings of the Cambridge Philosophical Society, 2014
We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with bo... more We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neumann operator and on a number–theoretic argument.
The goal of this paper is to study the Dirichlet eigenvalues of bounded domains 0 . With a local ... more The goal of this paper is to study the Dirichlet eigenvalues of bounded domains 0 . With a local spectral stability requirement on , we show that the difference of the Dirichlet eigenvalues of 0 and is explicitly controlled from above in terms of the first eigenvalue of 0 n x and of geometric constants depending on the inner domain . In particular, 0 can be an arbitrary bounded domain.
We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with bound... more We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first involves the volume of the manifold and of its boundary, as well as packing and volume growth constants of the boundary and its distortion. Its proof is based on a metric-measure space technique that was introduced by Colbois and Maerten. The second bound is in terms of the extrinsic diameter of the boundary and its injectivity radius. It is obtained from a concentration inequality, akin to Gromov-Milman concentration for closed manifolds. By applying these bounds to cylinders over closed manifold, we obtain bounds for eigenvalues of the Laplace operator, in the spirit of Grigor'yan-Netrusov-Yau and of Berger-Croke. For a family of manifolds that has uniformly bounded volume and boundary of fixed intrinsic geometry, we deduce that a large first nonzero Steklov eigenvalue implies that each boundary component is contained in a ball of small extrinsic radius.
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary ... more How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander's approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.
The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is ... more The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.
This chapter is a reprint of the article Spectral geometry of the Steklov problem by the same aut... more This chapter is a reprint of the article Spectral geometry of the Steklov problem by the same authors, published in the Journal of Spectral Theory. We would like to thank the Journal of Spectral Theory and the European Mathematical Society for granting permission to reproduce the paper in this book. Spectral geometry of the Steklov problem is a rapidly developing subject, and there have been a number of important advances since the original version of this article has appeared. In the present text, we have added references to some of these new results in the footnotes. In order to make this chapter coherent with the rest of the book, the dimension is denoted by d, and the trivial Steklov eigenvalue is now denoted by σ = , as opposed to σ = in the journal version of this article. The numeration has been also changed.
Given a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, we prove the existence... more Given a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\sigma_k(M_j)$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_j$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.
We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue... more We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point within a conformal class. The second is degeneration of the conformal class to the boundary of the moduli space on the torus and on the Klein bottle. In the latter, we follow the outline proposed by N.
Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the S... more Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev’s upper bound of $$8\pi $$ 8 π for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov ei...
We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian m... more We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for thek-th perimeter-normalized Steklov eigenvalue is$$8\pi k$$8πk, which is the best upper bound for the$$k^{\text {th}}$$ktharea-normalised ei...
Given two compact Riemannian manifolds M 1 and M 2 such that their respective boundaries Σ 1 and ... more Given two compact Riemannian manifolds M 1 and M 2 such that their respective boundaries Σ 1 and Σ 2 admit neighbourhoods Ω 1 and Ω 2 which are isometric, we prove the existence of a constant The constant C depends only on the geometry of Ω 1 ∼ = Ω 2 . This follows from a quantitative relationship between the Steklov eigenvalues σ k of a compact Riemannian manifold M and the eigenvalues λ k of the Laplacian on its boundary. Our main result states that the difference |σ k -√ λ k | is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω 1 ∼ = Ω 2 .
Given a smooth compact hypersurface $M$ with boundary $\unicode[STIX]{x1D6F4}=\unicode[STIX]{x220... more Given a smooth compact hypersurface $M$ with boundary $\unicode[STIX]{x1D6F4}=\unicode[STIX]{x2202}M$, we prove the existence of a sequence $M_{j}$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\unicode[STIX]{x1D70E}_{k}(M_{j})$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_{j}$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.
We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary... more We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is isospectral to the disk.
Proceedings of the American Mathematical Society, 2019
Let ( M , g ) (M,g) be a compact Riemannian manifold with boundary. Let b > 0 b>0 be the nu... more Let ( M , g ) (M,g) be a compact Riemannian manifold with boundary. Let b > 0 b>0 be the number of connected components of its boundary. For manifolds of dimension ≥ 3 \geq 3 , we prove that for j = b + 1 j=b+1 it is possible to obtain an arbitrarily large Steklov eigenvalue σ j ( M , e δ g ) \sigma _j(M,e^\delta g) using a conformal perturbation δ ∈ C ∞ ( M ) \delta \in C^\infty (M) which is supported in a thin neighbourhood of the boundary, with δ = 0 \delta =0 on the boundary. For j ≤ b j\leq b , it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of M M . In fact, when working in a fixed conformal class and for δ = 0 \delta =0 on the boundary, it is known that the volume of ( M , e δ g ) (M,e^\delta g) has to tend to infinity in order for some σ j \sigma _j to become arbitrarily large. This is in stark contrast with the situation for the eigenvalues of the Laplace operator on a closed manifold, where a c...
The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We ... more The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension $d\geqslant 3$ . Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the $(d-2)$ -dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related resu...
In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian ... more In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectral gap arbitrarily large using conformal deformations which are localized on domains of small measure, as long as the support of the deformations contains and connects each component of the boundary.
We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain ... more We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of half this area. The estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Pólya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres.
We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar d... more We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains, for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.
Mathematical Proceedings of the Cambridge Philosophical Society, 2014
We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with bo... more We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neumann operator and on a number–theoretic argument.
The goal of this paper is to study the Dirichlet eigenvalues of bounded domains 0 . With a local ... more The goal of this paper is to study the Dirichlet eigenvalues of bounded domains 0 . With a local spectral stability requirement on , we show that the difference of the Dirichlet eigenvalues of 0 and is explicitly controlled from above in terms of the first eigenvalue of 0 n x and of geometric constants depending on the inner domain . In particular, 0 can be an arbitrary bounded domain.
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