Nodal count for Dirichlet-to-Neumann operators with potential
Asma Hassannezhad, David Sher
备注：
v.3, 10 pages. Replacement of paper due to a gap in the proof of the result for Dirichlet-to-Neumann eigenfunctions; the bound for the Steklov eigenfunctions is unaffected
We consider Dirichlet-to-Neumann operators associated to $Δ+q$ on a Lipschitz domain in a smooth manifold, where $q$ is an $L^{\infty}$ potential. We prove a Courant-type bound for the nodal count of the extensions $u_k$ of the $k$th Dirichlet-to-Neumann eigenfunctions $ϕ_k$ to the interior satisfying $(Δ+q)u_k=0$. The classical Courant nodal domain theorem is known to hold for Steklov eigenfunctions, which are the harmonic extension of the Dirichlet-to-Neumann eigenfunctions associated to $Δ$. Our result extends it to a larger family of Dirichlet-to-Neumann operators. Our proof makes use of the duality between the Steklov and Robin problems.
Recent work of the authors and their collaborators has uncovered fundamental connections between the Dirichlet-to-Neumann map, the spectral flow of a certain family of self-adjoint operators, and the nodal deficiency of a Laplacian eigenfunction (or an analogous deficiency associated to a non-bipartite equipartition). Using a refined construction of the Dirichlet-to-Neumann map, we strengthen all of these results, in particular getting improved bounds on the nodal deficiency of degenerate eigenfunctions. Our framework is very general, allowing for non-bipartite partitions, non-simple eigenvalu…
We introduce an analogue of Payne's nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of the domain. The assertion here is that any eigenfunction associated with the first nontrivial eigenvalue of the Neumann Laplacian on a domain $Ω$ with rotational symmetry of order two (i.e., $x\inΩ$ iff $-x\inΩ$) "should normally" be rotationally antisymmetric. We give both positive and negative results which highlight the heuristic similarity of this asser…
In this paper, we show that equality in Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in the case of an open, bounded and connected set in R n with a C 1,1 boundary. This result is analogous to Pleijel's nodal domain theorem for the Dirichlet Laplacian (1956). It confirms, in all dimensions, a conjecture formulated by Pleijel, which had already been solved by I. Polterovich for a two-dimensional domain with a piecewise-analytic boundary (2009). We also show that the argument and the result extend to a class of Robin boundary con…
We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser-Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds on its curvature, with uniform estimates up to the two points where it meets the domain at right angles, and show that many of our results hold for relatively small aspect ratios of the side length…
Let ϕ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {ϕ=0}.
Nodal solutions of a parametric (p_1,p_2)-Laplacian system, with Neumann boundary conditions, are obtained by chiefly constructing appropriate sub-super-solution pairs.
We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the interior nodal sets is obtained for Robin eigenfunctions in the smooth domain. For the analytic domain, the sharp upper bounds of the interior nodal sets was shown for Robin eigenfunctions. More importantly, we obtain the sharp upper bounds for the boundary nodal sets of Neumann eigenfunctions with new quantitative global Carleman estimates. Furthermore, the sharp doubling inequality and vanishing order of Robin eigenfunctions on the bo…
We prove an analogue of Pleijel's nodal domain theorem for piecewise analytic planar domains with Neumann boundary conditions. This confirms a conjecture made by Pleijel in 1956. The proof is a combination of Pleijel's original approach and an estimate due to Toth and Zelditch for the number of boundary zeros of Neumann eigenfunctions.
We consider a nonlinear Neumann problem driven by a $p$-Laplacian-type, nonhomogeneous elliptic differential operator and a Carathéodory reaction term. In this paper we prove the existence of two extremal constant sign smooth solutions and a nontrivial nodal smooth solution. In the proof we use variational methods with truncation techniques, critical point theory and Morse theory (critical groups).
We present a new method for locating the nodal line of the second eigenfunction for the Neumann problem in a planar domain. The technique is based on the `mirror coupling' of reflected Brownian motions.
We consider the Dirichlet eigenvalue problem on a simple polytope. We use the Rellich identity to obtain an explicit formula expressing the Dirichlet eigenvalue in terms of the Neumann data on the faces of the polytope of the corresponding eigenfunction. The formula is particular simple for polytopes admitting an inscribed ball tangent to all the faces. Our result could be viewed as a generalization of similar identities for simplices recently found by Christianson [1][2].
Let $Ω$ be a bounded domain in $\mathbb{R}^n$ with $C^{1}$ boundary and let $u_λ$ be a Dirichlet Laplace eigenfunction in $Ω$ with eigenvalue $λ$. We show that the $(n-1)$-dimensional Hausdorff measure of the zero set of $u_λ$ does not exceed $C(Ω)\sqrtλ$. This result is new even for the case of domains with $C^\infty$-smooth boundary.
We introduce the class of quasiconvex Lipschitz domains, which covers both $C^1$ and convex domains, to the study of boundary unique continuation for elliptic operators. In particular, we prove the upper bound of the size of nodal sets for Dirichlet eigenfunctions of general elliptic equations in bounded quasiconvex Lipschitz domains. Our result is new even for Laplace operator in convex domains.
Let $Ω$ be a bounded annular $C^{1,1}$ domain in $\mathbb{R}^2$ which is left invariant under the action of the dihedral group $D_n$ of isometries of $\mathbb{R}^2$ .We show that the nodal line of a second Dirichlet eigenfunction must intersect the boundary of $Ω$, under suitable conditions on $\frac{\partial}{\partial θ}$ .
The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this paper is to present a systematic study of the Dirichlet-to-Neumann map and its inverse, the Neumann-to-Dirichlet map, in the framework of linear relations in Hilbert spaces. Our treatment is inspired by abstract methods from extension theory of symmetric operators, utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value pr…
The Dirichlet problem and Dirichlet to Neumann map are analyzed for elliptic equations on a large collection of infinite quantum graphs. For a dense set of continuous functions on the graph boundary, the Dirichlet to Neumann map has values in the Radon measures on the graph boundary.
According to the Schwarz symmetry principle, every harmonic function vanishing on a real analytic curve has an odd continuation, while a harmonic function satisfying homogeneous Neumann condition has the even continuation. There are different generalizations of the Schwarz symmetry principle. Most of them are dealing with homogeneous conditions and the Dirichlet case. Using a technique of Dirichlet to Neumann and Robin to Neumann operators, we derive reflection formulae for nonhomogeneous Neumann and Robin conditions from a reflection formula subject to a nonhomogeneous Dirichlet condition.
We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
In this paper we consider the inverse problem of determining on a compact Riemannian manifold the metric tensor in the wave equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uniquely determines the metric tensor and we establish logarithm-type stability.
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
We establish two commutator estimates for the Dirichlet-to-Neumann map associated with a second-order elliptic system in divergence form in Lipschitz domains. Our approach is based on Dahlberg's bilinear estimates.
We prove Hölder type stability estimates near generic simple Riemannian metrics for the inverse problem of recovering such metrics from the Dirichlet-to-Neumann map associated to the wave equation for the Laplace-Beltrami operator.
We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -Δu & = &|u|^{q-2}u & \textrm{in }Ω, \\ u_n & = & 0 & \textrm{on }\partialΩ,\end{array}\right. \end {equation*} where $Ω\subset \real^N$ is a bounded domain, and $1 \leq q < 2$. For $q=1$, $|u|^{q-2}u$ will be identified with $\sgn(u)$. We establish a variational principle for least energy nodal solutions, and we investigate their qualitative properties. In particular, we show that they satisfy a unique continuation property (their zero set is Lebesgue-negligible). Moreover, if $Ω$ is radial, then least energy n…
A mixed Dirichlet-Neumann problem is regularized with a family of singularly perturbed Neumann-Robin boundary problems, parametrized by $\varepsilon > 0$. Using an asymptotic development by Gamma-convergence, the asymptotic behavior of the solutions to the perturbed problems is studied as $\varepsilon \to 0^+$, recovering classical results in the literature.
Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann map with respect to movements of a polygonal conductivity inclusion, [11], we extend the results obtained in [8] proving global Lipschitz stability for the determination of a polygonal conductivity inclusion embedded in a layered medium from knowledge of the Dirichlet to Neumann map.
We study the inverse problem of identifying a periodic potential perturbation of the Dirichlet Laplacian acting in an infinite cylindrical domain, whose cross section is assumed to be bounded. We prove log-log stable determination of the potential with respect to the partial Dirichlet-to-Neumann map, where the Neumann data is taken on slightly more than half of the boundary of the domain.
In the framework of Hilbert spaces we shall give necessary and sufficient conditions to define a Dirichlet-to-Neumann operator via Dirichlet principle. For singular Dirichlet-to-Neumann operators we will establish Laurent expansion near singularities as well as Mittag--Leffler expansion for the related quadratic form. The established results will be exploited to solve definitively the problem of positivity of the related semigroup in the $L^2$ setting. The obtained results are supported by some examples on Lipschitz domains. Among other results, we shall demonstrate that regularity of the boun…
Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a \emph{nonlinear} Dirichlet-to-Neumann mapping that is constructed using a solution to a \emph{fully nonlinear} elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about …
We give formulae that yield an information about the location of an unknown polygonal inclusion having unknown constant conductivity inside a known conductive material having known constant conductivity from a partial knowledge of the Neumann -to-Dirichlet operator.
In this paper we study the dependence of the Hölder estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the Hölder regularity of the solutions to the Dirichlet and Neumann problems in the disk (and in the exterior of the disk), from which we get a relation between harmonic extensions and harmonic functions with prescribed Neumann condition on the boundary of the disk (for both interior and exterior problems).
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.
In this article we seek stability estimates in the inverse problem of determining the potential or the velocity in a wave equation in an anisotropic medium from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uniquely determines the electric potential and we prove Hölder-type stability in determining the potential.
In the very influential paper \cite{CS07} Caffarelli and Silvestre studied regularity of $(-Δ)^s$, $0<s<1$, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea \cite{ST10} and Galé, Miana and Stinga \cite{GMS13} gave several more abstract versions of this extension procedure. The purpose of this paper is to study precise regularity properties of the Dirichlet and the Neumann problem in Hilbert spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between interpolation spaces and its part in the underlying Hilbert space is exactly the…
In this paper we derive rigorously the derivative of the Dirichlet to Neumann map and of the Neumann to Dirichlet map of the conductivity equation with respect to movements of vertices of triangular conductivity inclusions. We apply this result to formulate an optimization problem based on a shape derivative approach.
Let $ Ω\subset R^2$ be a bounded piecewise smooth domain and $ϕ_λ$ be a Neumann (or Dirichlet) eigenfunction with eigenvalue $λ^2$ and nodal set ${ N}_{ϕ_λ} = {x \in Ω; ϕ_λ(x) = 0}.$ Let $H \subset Ω$ be an interior $C^ω$ curve. Consider the intersection number $$ n(λ,H):= \# (H \cap N_{ϕ_λ} ).$$ We first prove that for general piecewise-analytic domains, and under an appropriate "goodness" condition on $H$, $$ n(λ,H) = {\mathcal O}_H(λ) (*)$$ as $λ\rightarrow \infty.$ We then prove that the bound in $(*)$ is satisfied in the case of quantum ergodic (QE) sequences of interior eigenfunctions, p…
We prove an analyticity result for the Dirichlet-Neumann operator under space periodic boundary conditions in any dimension in an unbounded domain with infinite depth. We derive an analytic bifurcation result of analytic Stokes waves -- i.e. space periodic traveling solutions -- of the water waves equations in deep water.
We consider the Calderòn problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log-log stability in the determination of the isotropic periodic conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the solution.
Given the Laplacian on a planar, convex domain with piecewise linear boundary subject to mixed Dirichlet-Neumann boundary conditions, we provide a sufficient condition for its lowest eigenvalue to dominate the lowest eigenvalue of the Laplacian with the complementary boundary conditions (i.e. with Dirichlet replaced by Neumann and vice versa). The application of this result to triangles gives an affirmative partial answer to a recent conjecture. Moreover, we prove a further observation of similar flavor for right triangles.
We study the Dirichlet-to-Neumann map for the stationary linear equation of elasticity in a bounded domain in R d , d $\ge$ 2, with smooth boundary. We show that it can be approximated by a pseudodifferential operator on the boundary with a matrix-valued symbol and we compute the principal symbol modulo conjugation by unitary matrices.
Quantum waveguide with the shape of planar infinite straight strip and combined Dirichlet and Neumann boundary conditions on the opposite half-lines of the boundary is considered. The absence of the point as well as of the singular continuous spectrum is proved.
We present an asymptotic study of the Dirichlet to Neumann map of high contrast composite media with perfectly conducting inclusions that are close to touching. The result is an explicit characterization of the map in the asymptotic limit of the distance between the particles tending to zero.
It is well known that both the heat equation with Dirichlet or Neumann boundary conditions are null controlable as soon as the control acts in a non trivial domain (i.e. a set of positive measure, see [10, 11, 12, 1, 6]. In this article, we show that for any couple of initial data (u0, v0) we can achieve the null control for both equations (Dirichlet and Neumann boundary conditions respectively) simultaneously with the same control function for both equations.
We prove Poisson upper bounds for the kernel $K$ of the semigroup generated by the Dirichlet-to-Neumann operator if the underlying domain is bounded and has a $C^\infty$-boundary. We also prove Poisson bounds for $K_z$ for all $z$ in the right half-plane and for all its derivatives.
In this article, we study the boundary inverse problem of determining the aligned magnetic fiaeld appearing in the magnetic Schrödinger equation in a periodic quantum cylindrical waveguide. Provided that the Dirichlet-to-Neumann map of the magnetic Schrödinger equation. We prove a Holder stability estimate with respect to the Dirichlet-to-Neumann map, by means of the geometrical optics solutions of the magnetic Schrodinger equation.
A large class of initial-boundary value problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit formulae are presented for the generalized Dirichlet to Neumann maps. Namely, the determination of the unknown boundary values in terms of an essential set of given data.
