Related papers: Obstacle and Dirichlet problems on arbitrary nonop…

We prove the existence and uniqueness of solution of quasilinear stochastic partial differential equations with obstacle (OSPDEs in short) in degenerate case. Using De Giorgi's iteration, we deduce the $L^p-$estimates for the time-space…

We prove optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.

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…

A generic model for the shape optimization problems we consider in this paper is the optimization of the Dirichlet eigenvalues of the Laplace operator with a volume constraint. We deal with an obstacle placement problem which can be…

We study a particular system of partial differential equations in which the harmonic, the divergence and the gradient operators of the unknown functions appear (harmonic-divgrad system). Using the Killing Hopf theorem and leveraging the…

Obstacles to integrability in perturbed evolution equations are overcome by allowing higher-order terms in the expansion of the solution to depend explicitly on time and position. With a special expansion algorithm, obstacles vanish…

We consider a two obstacle problem for the parabolic biharmonic equation in a bounded domain. We prove long time existence of solutions via an implicit time discretization scheme, and we investigate the regularity properties of solutions.

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.

We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we…

Herein, we study an inverse problem for detecting unknown obstacles by the enclosure method using the Dirichlet--to--Neumann map for measurements. We justify the method for an penetrable obstacle case involving a biharmonic equation. We use…

We consider the one and the two obstacles problems for the nonlocal nonlinear anisotropic $g$-Laplacian $\mathcal{L}_g^s$, with $0<s<1$. We prove the strict T-monotonicity of $\mathcal{L}_g^s$ and we obtain the Lewy-Stampacchia…

We study singular perturbations of eigenvalues of the polyharmonic operator on bounded domains under removal of small interior compact sets. We consider both homogeneous Dirichlet and Navier conditions on the external boundary, while we…

We give a definition for Obstacle Problems with measure data and general obstacles. For such problems we prove existence and uniqueness of solutions and consistency with the classical theory of Variational Inequalities. Continuous…

Using a conjecture that allows to approach separable-variables conductivity functions, the elements of the Modern Pseudoanalytic Function Theory are used, for the first time, to numerically solve the Dirichlet boundary value problem of the…

In this article we consider the Dirichlet problem on a bounded domain $\Omega \subset {\bf R}^d$ with respect to a second-order elliptic differential operator in divergence form. We do not assume a divergence condition as in the pioneering…

In this paper, we present a new framework how a PDE with constraints can be formulated into a sequence of PDEs with no constraints, whose solutions are convergent to the solution of the PDE with constraints. This framework is then used to…

In this paper, we consider the obstacle problem for the fractional Laplace operator $(-\Delta)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper…

We study a nonlinear, nonlocal Dirichlet problem driven by the degenerate fractional p-Laplacian via a combination of topological methods (degree theory for operators of monotone type) and variational methods (critical point theory). We…

Dirichlet problem in an $n$-dimensional billiard space is investigated. In particular, the system of ODEs $\ddot x(t) = f(t,x(t))$ together with Dirichlet boundary conditions $x(0) = A$, $x(T) = B$ in an $n$-dimensional interval $K$ with…

We study a superlinear elliptic boundary value problem involving the $p$-laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the…