Related papers: A Singular Integral approach to a Two Phase Free B…

If $\Omega\subset\R^n$ is a bounded domain, the existence of solutions ${\bf u}\in H^1_0(\Omega)^n$ of ${div} {\bf u} = f$ for $f\in L^2(\Omega)$ with vanishing mean value, is a basic result in the analysis of the Stokes equations. In…

In this paper we consider degenerate Kirchhoff-type equations of the form \[-\phi(\Xi(u)) \left(\mathcal{A}(u)-|u|^{p-2}u\right) = f(x,u)\quad \text{in } \Omega,\] \[\phantom{aaiaaaaaaaaa}\phi (\Xi(u)) \mathcal{B}(u) \cdot \nu = g(x,u)…

The main purpose of this article is to explore the possibility of extending the notion of peripheral Poisson boundary of unital completely positive (UCP) maps to contractive completely positive (CCP) maps and to unital and non-unital…

We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form $$ (u,A) \quad \mapsto \quad…

We consider the inhomogeneous Allen-Cahn equation $$ \epsilon^2\Delta u\,+\,V(y)(1-u^2)\,u\,=\,0\quad \mbox{in}\ \Omega, \qquad \frac {\partial u}{\partial \nu}\,=\,0\quad \mbox{on}\ \partial \Omega, $$ where $\Omega$ is a bounded domain in…

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into…

We investigate, on a bounded domain $\Omega$ of $\R^2$ with fixed $S^1$-valued boundary condition $g$ of degree $d>0$, the asymptotic behaviour of solutions $u_{\varepsilon,\delta}$ of a class of Ginzburg-Landau equations driven by two…

We consider H\"older continuous weak solutions $u\in C^\gamma(\Omega)$, $u\cdot n|_{\partial \Omega}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega\subset\mathbb{R}^d$. If $\Omega$ is of class…

Let $\Omega $ be a smooth bounded domain in $\R^N, N>1$ and let $n\in \N^*$. We are concerned here with the existence of nonnegative solutions $u\_n$ in $BV(\Omega)$, to the problem $$(P\_n) \begin{cases} -{\rm div} \sigma +2n (\int\_…

In this paper we continue the study started in part I (posted). We consider a planar, bounded, $m$-connected region $\Omega$, and let $\bord\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\bord\Omega$,…

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $F \subset \partial \Omega$ be a $C^2$ submanifold of dimension $0 \leq k \leq N-2$. Put $\delta_F(x)=dist(x,F)$, $V=\delta_F^{-2}$ in $\Omega$ and $L_{\gamma…

We introduce a general $L_p$-solvability result for the Poisson equation in non-smooth domains $\Omega\subset \mathbb{R}^d$, with the zero Dirichlet boundary condition. Our sole assumption on the domain $\Omega$ is the Hardy inequality:…

For a linear, strictly elliptic second order differential operator in divergence form with bounded, measurable coefficients on a Lipschitz domain $\Omega$ we show that solutions of the corresponding elliptic problem with Robin and thus in…

We study a 2-phase free boundary problem for harmonic measure first considered by Kenig and Toro and prove a sharp H\"older regularity result. The central difficulty is that there is no a priori non-degeneracy in the free boundary…

We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body $\Omega$. The bound depends only on the perimeter and inradius $r$ of the original body and states that \[|\partial\Omega_t| \geq…

We give a new and self-contained proof of a generic version of the (former) Helgason conjecture. It says that for generic spectral parameters the Poisson transform is a topological isomorphism, with the inverse given by a boundary value…

We present an efficient and accurate algorithm for solving the Poisson equation in spherical polar coordinates with a logarithmic radial grid and open boundary conditions. The method employs a divide-and-conquer strategy, decomposing the…

We address the existence of stationary solutions of the Vlasov-Poisson system on a domain $\Omega\subset\mathbb{R}^3$ describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a…

In this paper, we consider an area minimizing integral $m$-current $T$ within a submanifold $\Sigma$ of $\mathbb{R}^{m+n}$, taking a boundary $\Gamma$ with arbitrary multiplicity $Q \in \mathbb{N} \setminus \{0\}$, where $\Gamma$ and…

In the unit ball B(0,1), let $u$ and $\Omega$ (a domain in $\R$) solve the following overdetermined problem: $$\Delta u =\chi_\Omega\quad \hbox{in} B(0,1), \qquad 0 \in \partial \Omega, \qquad u=|\nabla u |=0 \quad \hbox{in} B(0,1)\setminus…