Related papers: Extensions of divergence-free fields in $\mathrm{L…

We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary…

We study integral functionals constrained to divergence-free vector fields in $L^p$ on a thin domain, under standard $p$-growth and coercivity assumptions, $1<p<\infty$. We prove that as the thickness of the domain goes to zero, the…

We introduce the homogeneous and piecewise multilinear extensions and the eigenvalue problem for locally Lipschitz function pairs, in order to develop a systematic framework for relating discrete and continuous min-max problems. This also…

This is a short survey on the connection between general extension theories and the study of realizations of elliptic operators A on smooth domains in R^n, n > 1. The theory of pseudodifferential boundary problems has turned out to be very…

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two non-equivalent distances. The two distances under consideration are the Euclidean distance and, roughly…

For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in…

We study the exponent of the exponential rate of convergence in terms of the number of degrees of freedom for various non-standard {$p$-version} finite element spaces employing reduced cardinality basis. More specifically, we show that…

In 1995, D. Jerison and C. Kenig in \cite{JK-1995} considered the the inhomogeneous Dirichlet problem $\Delta u= f$ on $\Omega$, $u=0$ on $\partial\Omega$ in Lipschitz domains. One of their main results shows that the $W^{1,p}$ estimate…

Given a bounded domain $\O$ and $f$ of zero integral, the existence of a vector fields $\u$ vanishing on $\partial\O$ and satisfying $\d\u=f$ has been widely studied because of its connection with many important problems. It is known that…

A general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion…

We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show…

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators…

Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from…

We study the existence of uniformly bounded extension and trace operators for W^{1,p}-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. Such extension and trace operators are important for…

We introduce the Lorentz space $\mathcal{L}^{p(\cdot), q(\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The…

For a family of elliptic operators with periodically oscillating coefficients, $-\text{div}( A(\cdot/\varepsilon) \nabla) $ with tiny $\varepsilon>0$, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions…

In this paper we prove two approximation results for divergence free measures. The first is a form of an assertion of J. Bourgain and H. Brezis concerning the approximation of solenoidal charges in the strict topology: Given $F \in…

In this paper, we prove a new continuous embedding theorem for fractional Sobolev spaces with variable exponents into variable exponent Lebesgue spaces on unbounded domains. As an application, we study a class of nonlocal elliptic problems…

The main result of this paper is a proof of the continuity of a family of integral functionals defined on the space of functions of bounded variation with respect to a topology under which smooth functions are dense. These functionals occur…

In this paper we prove the discrete compactness property for a wide class of p-version finite element approximations of non-elliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find…