Related papers: A brief survey on the obstacle problem

We consider an optimal control problem for the obstacle problem with an elliptic variational inequality. The obstacle function which is the control function is assumed in $H^{2}$. We use an approximate technique to introduce a family of…

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $\Phi$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-\Delta_\Phi u= g(x,u), &…

Recent work by Serfaty and Serra give a formula for the velocity of the free boundary of the obstacle problem at regular points [Serfaty-Serra 2018], and much older work by King, Lacey, and Vazquez gives an example of a singular free…

The thin obstacle problem or $n$-dimensional Signorini problem is a classical variational problem arising in several applications, starting with its first introduction in elasticity theory. The vast literature concerns mostly quadratic…

In this article, we deal with some problems involving a class of singularly perturbed elliptic operator. We prove the asymptotic preserving of a general Galerkin method associated to a semilinear problem. We use a particular Galerkin…

We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2}…

We prove the existence and $C^{1,\alpha}$ regularity of solutions to nonlocal fully nonlinear elliptic equations with gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or…

In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish…

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear stochastic partial differential equations (OSPDEs for short) with Neumann boundary condition. Our method is based on the analytical technics coming…

We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully…

Suppose we are given a pair of points $s, t$ and a set $S$ of $n$ geometric objects in the plane, called obstacles. We show that in polynomial time one can construct an auxiliary (multi-)graph $G$ with vertex set $S$ and every edge labeled…

We study the problem of optimizing the eigenvalues of the Dirichlet Laplace operator under perimeter constraint. We prove that optimal sets are analytic outside a closed singular set of dimension at most $d-8$ by writing a general…

We establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in…

We give a simplified presentation of the obstacle problem approach to stochastic homogenization for elliptic equations in nondivergence form. Our argument also applies to equations which depend on the gradient of the unknown function. In…

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The…

We study the regularity of solutions of elliptic second order boundary value problems on a bounded domain $\Omega$ in $\mathbb R^3$. The coefficients are not necessarily continuous and the boundary conditions may be mixed, i.e. Dirichlet on…

In this paper we consider a class of obstacle problems of the type %\begin{equation*} %\int_{\Omega}\left<A(x, Du), D(\varphi-u)\right> \, \dx\ge0\qquad\forall %\varphi\in W^{1,q}(\Omega) \quad {\mathrm{s.t.}} \quad \varphi \ge \psi…

We study the regularity of solutions to the fully nonlinear thin obstacle problem. We establish local $C^{1,\alpha}$ estimates on each side of the smooth obstacle, for some small $\alpha > 0$. Our results extend those of Milakis-Silvestre…

We provide a brief outlook on recent developments in regularity theory for nonuniformly elliptic problems, with special emphasis on those of variational nature.

We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,\Omega)=\int_\Omega F(x,Dv)dx,$$ under the constraint $v \ge \psi$ a.e. in $\Omega$, where $\psi$ is a…