Related papers: Convergence of Dirichlet Eigenvalues for Elliptic …
Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric…
One of the reasons for the success of the finite element method is its versatility to deal with different types of geometries. This is particularly true of problems posed in curved domains of arbitrary shape. In the case of second order…
We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, maximum principle refers to the…
After introducing the concept of functional dissipativity of the Dirichlet problem in a domain $\Omega\subset {\mathbb R}^N$ for systems of partial differential operators of the form $\partial_{h}({\mathscr A}^{hk}(x)\partial_{k})$…
We consider the Dirichlet problem for a class of quasilinear elliptic systems in domain with irregular boundary. The principal part satisfies componentwise coercivity condition and the nonlinear terms are Carath\'eodory maps having Morrey…
We first formulate an inverse problem for a linear fractional Lam\'e system. We determine the Lam\'e parameters from exterior partial measurements of the Dirichlet-to-Neumann map. We further study an inverse obstacle problem as well as an…
We prove lower bounds for the Dirichlet Laplacian on possibly unbounded domains in terms of natural geometric conditions. This is used to derive uncertainty principles for low energy functions of general elliptic second order divergence…
In this paper we study the rate of convergence of the eigenvalues of 1-dimensional rapidly oscillating $p-$laplacian type problems and find explicit order of convergence both in $k$ and in $\ve$. Moreover, explicit bounds on the constant…
We study second-order hyperbolic equations with degenerate elliptic operators and non-homogeneous Dirichlet boundary inputs. We establish existence and regularity of weak solutions in weighted Sobolev spaces under mild assumptions on the…
The goal of this paper is to study the Dirichlet eigenvalues of bounded domains $\Omega\subset \Omega'$. With a local spectral stability requirement on $\Omega$, we show that the difference of the Dirichlet eigenvalues of $\Omega'$ and…
In this paper we study some relationships between the first Dirichlet eigenvalue $\Lambda(\Omega)$ and the torsional rigidity $T(\Omega)$ of a domain $\Omega$. We consider the problem of optimizing the product $\Lambda(\Omega)T(\Omega)$…
We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrodinger operators with mixed assumptions on the…
A method of estimating all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in T. Gergelits, K.A. Mardal, B.F. Nielsen, Z. Strako\v{s}: Laplacian…
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic…
We consider the inverse boundary value problem of determining the Lam\'e moduli of an isotropic, static elasticity equations of system at the boundary from the localized Dirichlet-to-Neumann map. Assuming appropriate local regularity…
This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet…
We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures…
This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold $(M,g,\partial M)$ under the singular perturbation. This perturbation involves the imposition of vanishing Dirichlet boundary conditions…
Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the…
We consider the parabolic Lam\'{e} system on a bounded domain. We focus on two types of inequalities for higher-order derivatives of solutions. The first is related to an $L^p$-$L^p$ estimate locally in time in the Lebesgue space setting,…