相关论文: On the A-Obstacle Problem and the Hausdorff Measur…

We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are $C^{1,\alpha}$, then they are $C^\infty$. This completes the study of…

For the two-phase membrane problem $ \Delta u = {\lambda_+\over 2} \chi_{\{u>0\}} - {\lambda_-\over 2} \chi_{\{u<0\}} ,$ where $\lambda_+> 0$ and $\lambda_->0 ,$ we prove in two dimensions that the free boundary is in a neighborhood of each…

We construct an efficient numerical scheme for solving obstacle problems in divergence form. The numerical method is based on a reformulation of the obstacle in terms of an L1-like penalty on the variational problem. The reformulation is an…

This paper concerns the theoretical and numerical analysis of a free boundary problem for the Laplace equation, with a curvature condition on the free boundary. This boundary is described as the graph of a function, and contact angles are…

We investigate a class of n-dimensional free boundary elliptic problems which includes the dam problem, the aluminum problem, and the lubrication problem. We establish that the free boundary in this class is a porous set, which implies its…

We present a finite element method along with its analysis for the optimal control of a model free boundary problem with surface tension effects, formulated and studied in \cite{HAntil_RHNochetto_PSodre_2014a}. The state system couples the…

This study investigates Dirichlet boundary condition related to a class of nonlinear parabolic problem with nonnegative $L^1$-data, which has a variable-order fractional $p$-Laplacian operator. The existence and uniqueness of renormalized…

We study the obstacle problem for integro-differential operators of order $2s$, with $s\in (0,1)$. Our main result establishes that the free boundary is $C^{1,\gamma}$ and $u\in C^{1,s}$ near all regular points. Namely, we prove the…

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a H\"older continuous linear term. With the help of those formulas we are able to…

We prove the -- to the best knowledge of the authors -- first result on the fine asymptotic behavior of the regular part of the free boundary of the obstacle problem close to singularities. The result is motivated by our recent partial…

We provide very mild sufficient conditions for space-time domains (non-necessarily cylindrical) which ensure that the continuous Dirichlet problem and the H\"older Dirichlet problem are well-posed, for any parabolic operator in divergence…

In this paper we prove local interior and boundary Lipschitz continuity of solutions of a free boundary problem involving the $A$-Laplacian. We also show that the free boundary is represented locally by graphs of a family of lower…

We consider obstacle problems for nonlinear stochastic evolution equations. More precisely, the leading operator in our equation is a nonlinear, second order pseudomonotone operator of Leray-Lions type. The multiplicative noise term is…

In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace…

In 2012, F.Leonetti and F.Siepe [1] considered solutions to boundary value problems of some anisotropic elliptic equations of the type $$ \left\{ \begin{array}{llll} \sum\limits _{i=1}\limits^{n} D_i (a_i(x,Du(x)))=0, &x\in \Omega,\\…

We study the obstacle problem with an elliptic operator in divergence form. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the…

It is shown that solutions to the anisotropic least gradient problem for boundary data $f \in L^p(\partial\Omega)$ lie in $L^{\frac{Np}{N-1}}(\Omega)$; the exponent is shown to be optimal. Moreover, the solutions are shown to be locally…

We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^{p(x)}+ \lambda \chi_{\{u>0\}} dx$ in the class of functions $W^{1,p(\cdot)}(\Omega)$ with $u-\phi_0\in W_0^{1,p(\cdot)}(\Omega)$, for a given $\phi_0\geq 0$ and…

In this paper, we consider the properties of a special free boundary point in the following obstacle problem: The Laplacian of u equals f(x) multiplied by the characteristic function of the set where u is positive within the two-dimensional…

In this paper, we consider the analogous of the obtacle problem in $H_0^1(\Omega)$, on the space $W^{1,p}_0(\Omega)$. We prove an existence and uniqueness of the result. In a second time, we define the optimal control problem associated.…