G-convergence,Dirichlet to Neumann maps and invisibility

We establish optimal conditions under which the G-convergence of linear elliptic operators implies the convergence of the corresponding Dirichlet to Neumann maps. As an application we show that the approximate cloaking isotropic materials from [19] are independent of the source.     

Spectral stability of the Neumann Laplacian

We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category, then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount.     

An Inequality Between Dirichlet and Neumann Eigenvalues of the Heisenberg Laplacian

Let λ _k and μ _k be the eigenvalues of the Dirichlet and Neumann problems, respectively, in a domain of finite measure in ? ^d , d > 1. Filonov has proved in a simple way that the inequality μ _k+1 < λ _k holds for the Laplacian. We extend his result to the Heisenberg Laplacian in three-dimensional domains which fulfill certain geometric conditions.     

On Dirichlet and Neumann problems for harmonic functions

The aim of the paper is to examine some aspects of the boundary value problems for harmonic functions in half-spaces related to approximation theory. M. V. Keldyshmentioned curious fact on richness in some sense of the solutions of Dirichlet problem in upper half-plane for a fixed continuous boundary data on the real axis. This can be considered as a model version for the Dirichlet problem with continuous boundary data, defined except a single boundary point, with no restrictions imposed on solutions near that point.Some extensions and multi-dimensional versions of Keldysh’s richness are obtained and related questions on existence, representation and richness of solutions for the Dirichlet and Neumann problems discussed.     

Two new Weyl-type bounds for the Dirichlet Laplacian

In this paper, we prove two new Weyl-type upper estimates for the eigenvalues of the Dirichlet Laplacian. As a consequence, we obtain the following lower bounds for its counting function. For , one has

and

where

is a constant which depends on , the dimension of the underlying space, and Bessel functions and their zeros.

     

Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces

The problem of the boundedness of the fractional maximal operator _Mα$M_{\alpha }$, 0<α<n$0<\alpha <n$, in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted _Lp$L_p$-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.     

Regularity of spectral fractional Dirichlet and Neumann problems

Consider the fractional powers and of the Dirichlet and Neumann realizations of a second‐order strongly elliptic differential operator A on a smooth bounded subset Ω of . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of ‐Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on ‐calculus. We also include an overview of the various Dirichlet‐ and Neumann‐type boundary problems associated with the fractional Laplacian.     

Conformal spectral stability estimates for the Neumann Laplacian

We study the eigenvalue problem for the Neumann–Laplace operator in conformal regular planar domains . Conformal regular domains support the Poincaré–Sobolev inequality and this allows us to estimate the variation of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy type integrals. Boundaries of such domains can have any Hausdorff dimension between one and two.     

Conformal spectral stability estimates for the Dirichlet Laplacian

We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains by reducing it, using conformal transformations, to the weighted eigenvalue problem for the Dirichlet Laplacian in the unit disc . This allows us to estimate the variation of the eigenvalues of the Dirichlet Laplacian upon domain perturbation via energy type integrals for a large class of “conformal regular” domains which includes all quasidiscs, i.e. images of the unit disc under quasiconformal homeomorphisms of the plane onto itself. Boundaries of such domains can have any Hausdorff dimension between one and two.     

Selfadjoint commutators and invariant subspaces on the torus II

In a previous paper, the authors determined the invariant subspaces ofL ²(T ²) on which a certain commutator is selfadjoint. In this paper, we give its generalization.Dedicated to Professor Kazuyuki Tsurumi on his sixtieth birthday     

Lower Bound Estimate of Blow Up Time for the Porous Medium Equations under Dirichlet and Neumann Boundary Conditions

In this paper, we establish the lower bounds estimate of the blow up time for solutions to the nonlocal cross-coupled porous medium equations with nonlocal source terms under Dirichlet and Neumann boundary conditions. The results are obtained by using some differential inequality technique.     

Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals

We present an extrapolation theory that allows us to obtain, from weighted L^p inequalities on pairs of functions for p fixed and all A_∞ weights, estimates for the same pairs on very general rearrangement invariant quasi-Banach function spaces with A_∞ weights and also modular inequalities with A_∞ weights. Vector-valued inequalities are obtained automatically, without the need of a Banach-valued theory. This provides a method to prove very fine estimates for a variety of operators which include singular and fractional integrals and their commutators. In particular, we obtain weighted, and vector-valued, extensions of the classical theorems of Boyd and Lorentz-Shimogaki. The key is to develop appropriate versions of Rubio de Francia's algorithm.     

On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

We construct a bounded C¹ domain Ω in for which the regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in such that the solution of in Ω and either on or on is contained in but not in for any . An analogous result holds for Sobolev spaces with .     

Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annulus

Here we study Dirichlet and Neumann problems for a special Helmholtz equation on an annulus. Our main aim is to measure smoothness of solutions for the boundary datum in Besov spaces. We shall use operator theory to solve this problem. The most important advantage of this technique is that it enables to consider equations in vector-valued settings. It is interesting to note that optimal regularity of this problem will be a special case of our main result.     

Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón-Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.     

Regularity in Capacity and the Dirichlet Laplacian

Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective, where denotes the Perron solution of the Dirichlet problem. Let be the set of all open subsets of which are regular in capacity. Then one can define metrics and on only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function . We prove that the spaces and are complete and contain the set of all open sets which are regular in the sense of Wiener (or Dirichlet regular) as a closed subset.     

Extrapolation methods and Rubio de Francia's extrapolation theorem

We develop a general framework to study extrapolation of inequalities.     

Reconstruction of convection coefficients of an elliptic equation in the plane by Dirichlet to Neumann map

The inverse problem of determining two convection coefficients of an ellipticpartial differential equation by Dirichlet to Neumann map is discussed.It is well knownthat this is a severely ill-posed problem with high nonlinearity.By the inverse scatteringtechnique for first order elliptic system in the plane and the theory of generalized analyticfunctions,we give a constructive method for this inverse problem.