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Gerassimos Barbatis

Publications and source records attributed to Gerassimos Barbatis.

17 recordsLinked to original sources
arXiv2025

Geometric Hardy inequalities on the Heisenberg groups via convexity

We prove $L^p$-Hardy inequalities with distance to the boundary for domains in the Heisenberg group ${\mathbb{H}}^n$, $n\geq 1$. Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is then implemented for the distance defined by the gauge quasi-norm related to the fundamental solution of the horizontal Laplacian when the domain is a half-space or a convex polytope. Finally it is implemented for the Carnot-Carath\'eodory distance on half-spaces and arbitrary bounded convex domains of ${\mathbb{H}}^n$. In all cases the constant $((p-1)/p)^p$ is obtained. In the more general context of a stratified Lie group of step two we study the superharmonicity and the weak $H$-concavity of the Euclidean distance to the boundary, thus obtaining a proof of the $L^p$-Hardy inequality on convex domains.

math.AP↗
arXiv2024

Sobolev improvements on sharp Rellich inequalities

There are two Rellich inequalities for the bilaplacian, that is for $\int (Δu)^2dx$, the one involving $|\nabla u|$ and the other involving $|u|$ at the RHS. In this article we consider these inequalities with sharp constants and obtain sharp Sobolev-type improvements. More precisely, in our first result we improve the Rellich inequality with $|\nabla u|$ obtained recently by Cazacu in dimensions $n=3,4$ by a sharp Sobolev term thus complementing existing results for the case $n\geq 5$. In the second theorem the sharp constant of the Sobolev improvement for the Rellich inequality with $|u|$ is obtained.

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arXiv2023

The Hardy constant: a review

We present a review of results that have been obtained in the past twenty-five years concerning the $L^p$-Hardy inequality with distance to the boundary. We concentrate on results where the best Hardy constant is either computed exactly or estimated from below.

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arXiv2022

Heat kernel estimates for fourth order non-uniformly elliptic operators with non strongly convex symbols

We obtain heat kernel estimates for a class of fourth order non-uniformly elliptic operators in two dimensions. Contrary to existing results, the operators considered have symbols that are not strongly convex. This rises certain difficulties as it is known that, as opposed to the strongly convex case, there is no absolute exponential constant. Our estimates involve sharp constants and Finsler-type distances that are induced by the operator symbol. The main result is based on two general hypotheses, a weighted Sobolev inequalitry and an interpolation inequality, which are related to the singularity or degeneracy of the coefficients.

math.AP↗
arXiv2022

Heat and Martin kernel estimates for Schrödinger operators with critical Hardy potentials

Let $Ω$ be a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary and let $K\subset\partialΩ$ be either a $C^2$ submanifold of the boundary of codimension $k<N$ or a point. In this article we study various problems related to the Schrödinger operator $L_μ =-Δ- μd_K^{-2}$ where $d_K$ denotes the distance to $K$ and $μ\leq k^2/4$. We establish parabolic boundary Harnack inequalities as well as related two-sided heat kernel and Green function estimates. We construct the associated Martin kernel and prove existence and uniqueness for the corresponding boundary value problem with data given by measures. Next we apply the results to the study of $L_μu+g(u) = 0$ and establish existence and uniqueness under suitable assumptions on the function $g$. To prove our results we introduce among other things a suitable notion of boundary trace. This trace is different from the one used by Marcus and Nguyen \cite{MT} thus allowing us to cover the whole range $μ\leq k^2/4$.

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arXiv2021

Finsler-Rellich inequalities involving the distance to the boundary

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 half-spaces 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.

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arXiv2018

On the heat kernel of a class of fourth order operators in two dimensions: sharp Gaussian estimates and short time asymptotics

We consider a class of fourth order uniformly elliptic operators in planar Euclidean domains and study the associated heat kernel. For operators with $L^{\infty}$ 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.

math.AP↗
arXiv2018

Sharp Hardy and Hardy--Sobolev inequalities with point singularities on the boundary

We study the Hardy inequality when the singularity is placed on the boundary of a bounded domain in $\mathbb{R}^n$ that satisfies both an interior and exterior ball condition at the singularity. We obtain the sharp Hardy constant $n^2/4$ 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.

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arXiv2015

Monotonicity, continuity and differentiability results for the $L^p$ Hardy constant

We consider the $L^p$ Hardy 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 constant is in general not explicitly known.

math.AP↗
arXiv2014

On the Hardy constant of some non-convex planar domains

The Hardy constant of a simply connected domain $Ω\subset\mathbf{R}^2$ is the best constant for the inequality \[ \int_Ω|\nabla u|^2dx \geq c\int_Ω \frac{u^2}{{\rm dist}(x,\partialΩ)^2}\, dx \; , \;\;\quad u\in C^{\infty}_c(Ω). \] 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 \cite{BT} 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.

math.AP↗
arXiv2013

Shape sensitivity analysis of the Hardy constant

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.

math.AP↗
arXiv2013

On the Hardy constant of non-convex planar domains: the case of the quadrilateral

The Hardy constant of a simply connected domain $Ω\subset\R^2$ is the best constant for the inequality \[ \int_Ω|\nabla u|^2dx \geq c\int_Ω \frac{u^2}{{\rm dist}(x,\partialΩ)^2}\, dx \;, u\in C^{\infty}_c(Ω). \] 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.

math.AP↗
arXiv2012

Higher order linear parabolic equations

We first highlight the main differences between second order and higher order linear parabolic equations. Then we survey existing results for the latter, in particular by analyzing the behavior of the convolution kernels. We illustrate the updated state of art and we suggest several open problems.

math.AP↗
arXiv2012

Stability estimates in $H^1_0$ for solutions of elliptic equations in varying domains

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 $\tilde u$ of the corresponding elliptic equations with the same right-hand side $f\in L^2(Ω\cupϕ(Ω))$. Under certain assumptions we estimate the difference $\|\nabla\tilde u-\nabla u\|_{L^2(Ω\cupϕ(Ω))}$ 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 $|ϕ(Ω)\triangle Ω|$. We provide an example which shows that the estimates obtained are in a certain sense sharp.

math.AP↗
arXiv2011

Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $Ω$ in ${\mathbb{R}}^N$. We consider deformations $ϕ(Ω)$ of $Ω$ obtained by means of a locally Lipschitz homeomorphism $ϕ$ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of $ϕ$. We prove general stability estimates without using uniform upper bounds for the gradients of the maps $ϕ$. 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.

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