Related papers: Weighted Global Regularity Estimates for Elliptic …
In this paper, we investigate a shape optimization problem for the second Robin eigenvalue of the weighted Laplacian on bounded Lipschitz domains symmetric about the origin. Our main theorem states that the ball centered at the origin…
Recently, their has been development of an abstract approach to the Robin--Robin method, enabling the treatment of linear and nonlinear elliptic and parabolic equations on Lipschitz domains within one framework. However, previously this…
We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique $W^{1,p}$ solvability is obtained with $p$ being in the optimal range $(4/3,4)$. The leading coefficients are assumed…
We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution $u \in H^s$…
We study an inverse boundary value problem associated with $p$-Laplacian which is further perturbed by a linear second order term, defined on a bounded set $\Omega$ in $\R^n, n\geq 2$. We recover the coefficients at the boundary from the…
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…
In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}(\mathcal{A}(x,…
We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on $L^1$ comparison) between solutions to an elliptic partial differential equation on a smooth bounded set $\Omega$ with a…
In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel…
The purpose of this article is to study extrapolation of solvability for boundary value problems of elliptic systems in divergence form on the upper half-space assuming De Giorgi type conditions. We develop a method allowing to treat each…
Using Maz'ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in $R^n$. For $n\ge 8$, combined with a result in \cite{S2}, these estimates lead to the…
This paper continues the development of regularity results for quasilinear measure data problems \begin{align*} \begin{cases} -\mathrm{div}(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ \quad \quad \qquad u &=0 \quad \text{on} \ \…
We study the Robin boundary-value problem for bounded domains with isolated singularities. Because for such domains trace spaces of space $H^1(D)$ on its boundaries are weighted Sobolev spaces $L^{2, \xi}(\partial D)$ existence and…
We consider the problem of determining the unknown boundary values of a solution of an elliptic equation outside a bounded open set $B$ from the knowledge of the values of this solution on a boundary of an arbitrary Lipschitz bounded domain…
We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…
Homogenization of a scalar elliptic equation in a bounded domain with Neuman boundary condition is studied. Coefficients of the operator are oscillating over two different groups of variables with different small periods $\varepsilon$ and…
We study generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schr\"odinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$. We also discuss the case of bounded…
We start presenting an $L^{\infty}$-gradient bound for solutions to non-homogeneous $p$-Laplacean type systems and equations, via suitable non-linear potentials of the right hand side. Such a bound implies a Lorentz space characterization…
We prove existence and up to the boundary regularity estimates in $L^{p}$ and H\"{o}lder spaces for weak solutions of the linear system $$ \delta \left( A d\omega \right) + B^{T}d\delta \left( B\omega \right) = \lambda B\omega + f \text{ in…
In this paper, we study the global regularity for regular Monge-Amp\`ere type equations associated with semilinear Neumann boundary conditions. By establishing a priori estimates for second order derivatives, the classical solvability of…