We introduce a new algorithm to construct travel time distances between a point in the interior of a Riemannian manifold and points on the boundary of the manifold, and describe a numerical implementation of the algorithm. It is known that the travel time distances for all interior points determine the Riemannian manifold in a stable manner. We do not assume that there are sources or receivers in the interior, and use the hyperbolic Neumann-to-Dirichlet map, or its restriction, as our data. Our algorithm is a variant of the Boundary Control method, and to our knowledge, this is the first numerical implementation of the method in a geometric setting.
In this paper we prove that a singularly perturbed Neumann problem with potentials admits the existence of interior spikes concentrating in maxima and minima of an auxiliary function depending only on the potentials.
In this paper we study the {\it a priori} gradient estimates for admissible solutions to Neumann boundary value problem of fully nonlinear Hessian equations on Riemannian manifolds. We firstly derive an interior gradient estimates for admissible solutions, then we obtain boundary gradient estimates based on the interior gradient estimates we have got.
We consider a parabolic problem with degeneracy in the interior of the spatial domain and Neumann boundary conditions. In particular, we will focus on the well-posedness of the problem and on Carleman estimates for the associated adjoint problem. The novelty of the present paper is that for the first time it is considered a problem with an interior degeneracy and Neumann boundary conditions so that no previous result can be adapted to this situation. As a consequence new observability inequalities are established.
We study the semi-classical microlocal structure of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a non-empty smooth boundary. We build a new, improved parametrix in the glancing region compaired with that one built in [9], [12]. We also study the way in which the parametrix depends on the refraction index. As a consequence, we improve the transmission eigenvalue-free regions obtained in [12] in the isotropic case when the restrictions of the refraction indices on the boundary coincide.
We study the high-frequency behavior of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a non-empty smooth boundary. We show that far from the real axis it can be approximated by a simpler operator. We use this fact to get new results concerning the location of the transmission eigenvalues on the complex plane. In some cases we obtain optimal transmission eigenvalue-free regions.
We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, enter perpendicularly into a support manifold. For example, harmonic maps, or $H$-surfaces, with a partially free boundary condition. In the interior it is known, by the celebrated work of Riviere, that these maps satisfy a system with an antisymmetric potential, from which one can derive regularity of the solution. We show that these maps satisfy along the boundary a system with a nonlocal antisym…
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.
Bounded smooth solutions of the Dirichlet and Neumann problems for a wide variety of quasilinear parabolic equations, including graphical anisotropic mean curvature flows, have gradient bounded in terms of oscillation and elapsed time.
In this paper, we study the blow-up analysis for a sequence of solutions to the Liouville type equation with exponential Neumann boundary condition. For interior case, i.e. the blow-up point is an interior point, Li \cite{Li} gave a uniform asymptotic estimate. Later, Zhang \cite{Zhang} and Gluck \cite{Gluck} improved Li's estimate in the sense of $C^0$-convergence by using the method of moving planes or classification of solutions of the linearized version of Liouville equation. If the sequence blows up at a boundary point, Bao-Wang-Zhou \cite{Bao-Wang-Zhou} proved a similar asymptotic estima…
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).
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 establish uniform Lipschitz estimates for second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded $C^{1,α}$ domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the …
It was shown by the second author in (CPAA, 2019) for the biharmonic Schrödinger equation and most recently by Himonas and Mantzavinos (Nonlinearity, 2020) for 2D Schrödinger equation that Fokas method based formulas are capable of defining weak solutions of associated nonlinear initial boundary value problems (ibvps) below the Banach algebra threshold. In view of these results, we revisit the theory of interior-boundary Strichartz estimates for the Schrödinger equation posed on the right half line, considering both Dirichlet and Neumann cases. Finally, we apply these estimates to obtain low r…
We study the critical points of Laplace eigenfunctions on polygonal domains with a focus on the second Neumann eigenfunction. We show that if each convex quadrilaterals has no second Neumann eigenfunction with an interior critical point, then there exists a convex quadrilateral with an unstable critical point. We also show that each critical point of a second-Neumann eigenfunction on a Lip-1 polygon with no orthogonal sides is an acute vertex.
In this paper we analyze some properties of a sixth order elliptic operator arising in the framework of the strain gradient linear elasticity theory for nanoplates in flexural deformation. We first rigorously deduce the weak formulation of the underlying Neumann problem as well as its well posedness. Under some suitable smoothness assumptions on the coefficients and on the geometry we derive interior and boundary regularity estimates for the solution of the Neumann problem. Finally, for the case of isotropic materials, we obtain new Strong Unique Continuation results in the interior, in the fo…
We discuss a Steklov-type problem for Maxwell's equations which is related to an interior Calderón operator and an appropriate Dirichlet-to-Neumann type map. The corresponding Neumann-to-Dirichlet map turns out to be compact and this provides a Fourier basis of Steklov eigenfunctions for the associated energy spaces. With an approach similar to that developed by Auchmuty for the Laplace operator, we provide natural spectral representations for the appropriate trace spaces, for the Calderón operator itself and for the solutions of the corresponding boundary value problems subject to electric or…
We build a one-parameter family of S^{1}-invariant metrics on the unit disc with fixed total area for which the second eigenvalue of the Laplace operator in the case of both Neumann and Dirichlet boundary conditions is simple and has an eigenfunction with a closed nodal line. In the case of Neumann boundary conditions, we also prove that this eigenfunction attains its maximum at an interior point, and thus provide a counterexample to the hot spots conjecture on a simply connected surface. This is a consequence of the stronger result that within this family of metrics any given (finite) number …
Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres
Let $Ω$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-Hénon weight $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2α}u^p,\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, Ω,\\[2mm] \frac{\partial u}{\partialν}=0\,\, \qquad\quad\qquad\qquad\qquad \qquad\qquad\qquad\qquad \,\ \ \,\,\,\, \textrm{on}\,\,\, \partialΩ, \end{cases} $$ where $ν$ denotes the outer unit normal vector to $\partialΩ$, $q\in\overlineΩ$, $α\in(-1,+\infty)\setminus\mathbb{N}$, $p>1$ is a large exponent and $a(x)$ is a positive smooth fun…
We consider a second-order hyperbolic equation on an open bounded domain $Ω$ in $\mathbb{R}^n$ for $n\geq2$, with $C^2$-boundary $Γ=\paΩ=\bar{Γ_0\cupΓ_1}$, $Γ_0\capΓ_1=\emptyset$, subject to non-homogeneous Neumann boundary conditions on the entire boundary $Γ$. We then study the inverse problem of determining the interior damping coefficient of the equation by means of an additional measurement of the Dirichlet boundary trace of the solution, in a suitable, explicit sub-portion $Γ_1$ of the boundary $Γ$, and over a computable time interval $T>0$. Under sharp conditions on the complementary pa…
This paper is concerned with a priori $C^\infty$ regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a diophantine condition are automatically $C^\infty…
This work considers properties of the Neumann-to-Dirichlet map for the conductivity equation under the assumption that the conductivity is identically one close to the boundary of the examined smooth, bounded and simply connected domain. It is demonstrated that the so-called bisweep data, i.e., the (relative) potential differences between two boundary points when delta currents of opposite signs are applied at the very same points, uniquely determine the whole Neumann-to-Dirichlet map. In two dimensions, the bisweep data extend as a holomorphic function of two variables to some (interior) neig…
We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $Ω\subset\mathbb{R}^{n}$ when the so-called Neumann-to-Dirichlet map is locally given on a non empty curved portion $Σ$ of the boundary $\partialΩ$. We prove that anisotropic conductivities that are \textit{a-priori} known to be piecewise constant matrices on a given partition of $Ω$ with curved interfaces can be uniquely determined in the interior from the knowledge of the local Neumann-to-Dirichlet map.
The spectrum of a selfadjoint second order elliptic differential operator in $L^2(\mathbb{R}^n)$ is described in terms of the limiting behavior of Dirichlet-to-Neumann maps, which arise in a multi-dimensional Glazman decomposition and correspond to an interior and an exterior boundary value problem. This leads to PDE analogs of renowned facts in spectral theory of ODEs. The main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding Weyl functions, and are applied to the PDE setting afterwards.
We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrödinger operators on compact Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For nonzero potentials, we obtain new geometric invariants determined by the spectrum of what we call the parametric Steklov problem. In particular, for constant potentials parametric Steklov problem, the total geodesic curvature on each connected component of the boundary is a spectral invariant. Under the constant curvatu…
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 t…
Let $e_ł(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $Δ$ on a compact Riemannian manifold $M$ with boundary such that $Δ\, e_ł=ł^2 e_ł$ in the interior of $M$ and the normal derivative of $e_ł$ vanishes on the boundary of $M$. Let $χ_λ$ be the unit band spectral projection operator associated with the Neumann Laplacian and $f$ a square integrable function on $M$. We show the following gradient estimate for $χ_λ\,f$ as $λ\geq 1$: $\|\nabla\ χ_ł f\|_\infty\leq Cł\|χ_ł\f\|_\infty+ł^{-1}\|Δ χ_ł f\|_\infty$, where $C$ is a positive constant depending only on $M$. As a coro…
We consider an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifold with time-like boundary. The time-dependent coefficients of the nonlinear terms can be recovered in the interior from the knowledge of the Neumann-to-Dirichlet map. Either distorted plane waves or Gaussian beams can be used to derive uniqueness.
In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minimal submanifolds in Euclidean space. Lastly we give a sharp estimate of the eigenvalue (with outer Dirichlet and inner Neumann boundaries) in terms of the Dircihlet eigenvalue, and also we give an upper bound of the eigenvalue with inner Dirichlet and outer Neumann problems by the…
On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and boundary of the manifold. The principal angles between these interior subspaces are all acute and are called Poincare duality angles. This paper determines the Poincare duality angles of a collection of interesting manifolds with boundary derived from complex projective spaces and fro…
In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in $\mathbb{R}^n$, we generalize the definition of the Poincaré-Steklov operator to $d$-set boundaries, $n-2< d<n$, and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of $n$-sets. The results are obtained thanks to a generalization of the conti…
We use probabilistic tools based on Brownian motion and Feynman-Kac formulae to investigate the heat profile for the ground state Dirichlet and second Neumann eigenfunctions. Among other topics, we comment on supremum norm bounds for ground state Dirichlet eigenfunctions and look at the corresponding Neumann problem, namely the comparison of maximum temperatures on the interior and the boundary, the latter being partially motivated by the hot spots problem. We also investigate the proximity/distance of level sets of ground state Dirichlet eigenfunctions, some with analogous statements for Neum…
Let $(Ω,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$ and $u_λ:= ϕ_λ |_{\partial Ω}$ the associated Dirichlet data (ie. boundary restriction of $ϕ_λ$). Our first main result (Theorem \ref{T:non-con}) is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary corner points) and any $δ\in [0,1),$ $$ \| ϕ_h \|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$$ Our subsequent results involve applications of the nonconcentration estimate to upper bounds for…
We obtain sharp parabolic interior and global Schauder estimates for solutions to nonlocal space-time master equations $(\partial_t +L)^su = f$ in $\mathbb{R} \times Ω$, where $L$ is an elliptic operator in divergence form, subject to homogeneous Dirichlet and Neumann boundary conditions. In particular, we establish the precise behavior of solutions near the boundary. Along the way, we prove a characterization of the correct intermediate parabolic Hölder spaces in the spirit of Sergio Campanato.
We study the wave equation in a bounded domain or on a compact Riemannian manifold with boundary. Assume that we are given the hyperbolic Neumann-to-Dirichlet map on the boundary corresponding to physical boundary measurements. We consider how to focus waves, that is, how to find Neumann boundary values so that at a given time the corresponding wave converges to a delta distribution $δ_y$ while the time derivative of the wave converges to zero. Such boundary value are generated by an iterative sequence of measurements. In each iteration step we apply time reversal and other simple operators to…
