Related papers: Oscillations and concentrations up to the boundary

Let $\Omega\subset \mathbb{R}^{n+1}$ be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that $\partial\Omega$ may be approximated in a "Big Pieces" sense by boundaries of chord-arc…

Let $K$ be a p.c.f. self-similar set equipped with a strongly recurrent Dirichlet form. Under a homogeneity assumption, for an open set $\Omega\subset K$ whose boundary $\partial \Omega$ is a graph-directed self-similar set, we prove that…

We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent $p$-Laplacian on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$, where the boundary datum belongs to $W^{1,p}(\Omega)$.…

We describe the asymptotic behaviour of the minimal heterogeneous $d$-capacity of a small set, which we assume to be a ball for simplicity, in a fixed bounded open set $\Omega\subseteq \mathbb{R}^d$, with $d\geq2$. Two parameters are…

We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,\infty}(\Omega;\mathbb R^d) \ni u \mapsto\supess_{ x \in \Omega}f(\nabla u(x))$ in the vectorial case, where $\Omega\subset \mathbb R^N$ is a…

We consider the clustering concentration on curves for solutions to the problem $$ \varepsilon^2 {\mathrm {div}}\big( \nabla_{{\mathfrak a}(y)} u\big)- V(y)u+u^p\, =\, 0, \quad u>0 \quad\mbox{in }\Omega, \qquad \nabla_{{\mathfrak a}(y)}…

In this paper we prove that if $\Omega\in\mathbb{R}^n$ is a bounded John domain, the following weighted Poincare-type inequality holds: $$ \inf_{a\in \mathbb{R}}\| (f(x)-a) w_1(x) \|_{L^q(\Omega)} \le C \|\nabla f(x) d(x)^\alpha w_2(x)…

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible…

In this paper, we consider the elliptic operators $\mathcal{L}_\varepsilon = -\nabla\cdot (A(X/\varepsilon) \nabla )$ with periodic coefficients in a bounded domain $\Omega$ without any local smoothness assumption on $A = A(Y)$, where…

We establish sufficient conditions for the local boundedness of weak solutions to a broad class of nonlinear elliptic equations in divergence form, under unbalanced growth conditions on the stress field. Our analysis is carried out in a…

We establish the $\Gamma$-convergence of some energy functionals describing nonlocal attractive interactions in bounded domains. The interaction potential solves an elliptic equation (local or nonlocal) in the bounded domain and the primary…

For $p\in (1,+\infty)$ and $b \in (0, +\infty]$ the $p$-torsion function with Robin boundary conditions associated to an arbitrary open set $\Om \subset \R^m$ satisfies formally the equation $-\Delta_p =1$ in $\Om$ and $|\nabla u|^{p-2}…

We assume that $\Omega_1, \Omega_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their boundaries $F$ has positive harmonic measure. Then we…

We develop a new technique for studying the boundary limiting behavior of a holomorphic function on a domain $\Omega$ -- both in one and several complex variables. The approach involves two new localized maximal functions. As a result of…

We characterize Young measures generated by gradients of bi-Lipschitz orientation-preserving maps in the plane. This question is motivated by variational problems in nonlinear elasticity where the orientation preservation and injectivity of…

We obtain a refinement of the degrees of freedom estimate of Landau and Pollak. More precisely, we estimate, in terms of $\epsilon$, the increase in the degrees of freedom resulting upon allowing the functions to contain a certain…

This paper is mainly concerned with the existence of extremals for the Adams inequality. We first establish an upper bound for the classical Adams functional along of all concentrated sequences in $W^{m,\frac{n}{m}}_{\mathcal{N}}(\Omega)$,…

We consider a class of variational problems for densities that repel each other at distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional \[ D(\mathbf{u}) = \sum_{i=1}^k \int_{\Omega} |\nabla u_i|^2…

We prove that any weakly differentiable function with square integrable gradient can be extended to a capacitary boundary of any simply connected plane domain $\Omega\ne\mathbb R^2$ except a set of a conformal capacity zero. For locally…

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $\Omega $ in ${\mathbb{R}}^N$. We consider deformations $\phi (\Omega)$ of $\Omega $ obtained by means of a locally Lipschitz…