Related papers: Fractional Elliptic Problems on Lipschitz Domains:…
We prove Besov boundary regularity for solutions of the homogeneous Dirichlet problem for fractional-order quasi-linear operators with variable coefficients on Lipschitz domains $\Omega$ of $\mathbb{R}^d$. Our estimates are consistent with…
This paper surveys recent analytical and numerical research on linear problems for the integral fractional Laplacian, fractional obstacle problems, and fractional minimal graphs. The emphasis is on the interplay between regularity,…
We consider the homogeneous Dirichlet problem for the integral fractional Laplacian $(-\Delta)^s$. We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is $C^s$ up to the boundary. We present the…
We prove Besov regularity estimates for the solution of the Dirichlet problem involving the integral fractional Laplacian of order $s$ in bounded Lipschitz domains $\Omega$: \[ \begin{aligned} \|u\|_{\dot{B}^{s+r}_{2,\infty}(\Omega)} \le C…
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition…
We settle the issue of well-posedness for the Dirichlet problem for a higher order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $\Omega$, with boundary data in Besov…
We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-\Delta)^s$ on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in…
The purpose of this work is the study of solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the…
In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the…
We survey some recent regularity results for fractional p-Laplacian elliptic equations, especially focusing on pure and weighted boundary H\"older continuity of the solutions of related Dirichlet problems. Then, we present some applications…
We study the optimal approximation of the solution of an operator equation by certain n-term approximations with respect to specific classes of frames. We study worst case errors and the optimal order of convergence and define suitable…
We study a discretization technique for the parabolic fractional obstacle problem in bounded domains. The fractional Laplacian is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic equation posed on a semi-infinite…
The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is to classify the solutions which have a singularity on the…
We study well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional…
This work is about global H\"older regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the…
This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value…
This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for solutions of an elliptic system with highly oscillating coefficients, over a highly oscillating Lipschitz boundary. The originality of this…
We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic partial differential equation (PDE) as state equation and a control variable…
In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order $s \in (0,1)$. We identify minimal conditions on the nonlinear term and the source which leads to existence of…
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.