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We study Rellich inequalities associated to higher-order elliptic operators in the Euclidean space. The inequalities are expressed in terms of an associated Finsler metric. In the case of a half-space we obtain the sharp constant while for a general convex domain we obtain estimates that are better than those obtained by comparison with the polyharmonic operator. © 2022 American Mathematical Society.

We study the Hardy inequality when the singularity is placed on the boundary of a bounded domain in Rn that satisfies both an interior and exterior ball condition at the singularity. We obtain the sharp Hardy constant n2/4 in case the exterior ball is large enough and show the necessity of the large exterior ball condition. We improve Hardy inequality with the best constant by adding a sharp Sobolev term. We next produce criteria that lead to characterizing maximal potentials that improve Hardy inequality. Breaking the criteria one produces successive improvements with sharp constants. Our approach goes through in less regular domains, like cones. In the case of a cone, contrary to the smooth case, the Sobolev constant does depend on the opening of the cone. © 2018 Elsevier Masson SAS

We consider a class of fourth order uniformly elliptic operators in planar Euclidean domains and study the associated heat kernel. For operators with L∞ coefficients we obtain Gaussian estimates with best constants, while for operators with constant coefficients we obtain short time asymptotic estimates. The novelty of this work is that we do not assume that the associated symbol is strongly convex. The short time asymptotics reveal a behavior which is qualitatively different from that of the strongly convex case. © 2018

We consider the LpHardy inequality involving the distance to the boundary for a domain in the n-dimensional Euclidean space. We study the dependence on p of the corresponding best constant and we prove monotonicity, continuity and differentiability results. The focus is on non-convex domains in which case such a constant is in general not explicitly known. © 2016, Hebrew University of Jerusalem.

We study the homogenization of elliptic systems of equations in divergence form where the coefficients are compositions of periodic functions with a random diffeomorphism with stationary gradient. This is done in the spirit of scalar stochastic homogenization by Blanc, Le Bris and Lions. An application of the abstract result is given for Maxwell's equations in random dissipative bianisotropic media. © 2015 World Scientific Publishing Company.

The Hardy constant of a simply connected domain Ω ⊂ R2 is the best constant for the inequality (Formula presented) After the work of Ancona where the universal lower bound 1/16 was obtained, there has been a substantial interest on computing or estimating the Hardy constant of planar domains. In Barbatis and Tertikas (J Funct Anal 266:3701–3725, 2014) we have determined the Hardy constant of an arbitrary quadrilateral in the plane. In this work we continue our investigation and we compute the Hardy constant for other non-convex planar domains. In all cases the Hardy constant is related to that of a certain infinite sectorial region which has been studied by E.B. Davies. © 2015, Springer International Publishing Switzerland.

The Hardy constant of a simply connected domain Ω⊂R2 is the best constant for the inequality. ∫Ω|∇;u|2dx≥c∫Ωu2dist(x,∂;Ω)2dx,u∈Cc∞(Ω). After the work of Ancona where the universal lower bound 1/16 was obtained, there has been a substantial interest on computing or estimating the Hardy constant of planar domains. In this work we determine the Hardy constant of an arbitrary quadrilateral in the plane. In particular we show that the Hardy constant is the same as that of a certain infinite sectorial region which has been studied by E.B. Davies. © 2013 Elsevier Inc.

We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain Ω and on the domain φ(Ω) resulting from Ω by means of a bi-Lipschitz map φ. We consider the solutions u and ũ of the corresponding elliptic equations with the same right-hand side f L 2(Ω φ(Ω)). Under certain assumptions, we estimate the difference L2(Ω φ(Ω)) in terms of certain measure of vicinity of φ to the identity map. For domains within a certain class, this provides estimates in terms of the Lebesgue measure of the symmetric difference of φ(Ω) and Ω, that is, | φ(Ω) Ω |. We provide an example that shows that the estimates obtained are in a certain sense sharp. © 2013 John Wiley & Sons, Ltd.

We consider the Hardy constant associated with a domain in the n-dimensional Euclidean space and we study its variation upon perturbation of the domain. We prove a Fréchet differentiability result and establish a Hadamard-type formula for the corresponding derivatives. We also prove a stability result for the minimizers of the Hardy quotient. Finally, we prove stability estimates in terms of the Lebesgue measure of the symmetric difference of domains. © 2014 Elsevier B.V. All rights reserved.

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain ω in ℝ N. We consider deformations Ø•(ω) of ω obtained by means of a locally Lipschitz homeomorphism Ø• and we estimate the variation of the eigenfunctions and eigenvalues upon variation of Ø•. We prove general stability estimates without assuming uniform upper bounds for the gradients of the maps Ø•. As an application, we obtain estimates on the rate of convergence for eigenvalues and eigenfunctions when a domain with an outward cusp is approximated by a sequence of Lipschitz domains. © 2012 Copyright University College London.

We consider a periodic pseudodifferential operator H =(- Δ)l + A (l > 0) in Rd which satisfies the following conditions: (i) the symbol of H is smooth in x, and (ii) the perturbation A has order smaller than 2l - 1. Under these assumptions, we prove that the spectrum of H contains a half-line. © Taylor & Francis Group, LLC.

We study the periodic homogenization of Maxwell's equations for dissipative bianisotropic media in the time domain, both in R3 and in a bounded domain with the perfect conductor boundary condition, We consider both local with respect to time (optical response region) and non-local in time (allowing dispersive effects) constitutive laws; in the non-local case the explicit description of the homogenized coefficients is given in terms of the Laplace transform. The principal result of this work is the description of the asymptotic behaviour of the solutions of the considered problems as the period of the electromagnetic parameters tends to zero.Copyright © 2003 John Wiley & Sons, Ltd.