Related papers: Optimal regularity for the obstacle problem for th…

We consider stable solutions to the equation $ -\Delta_p u =f(u) $ in a smooth bounded domain $\Omega\subset\mathbb{R}^n $ for a $ C^1 $ nonlinearity $f$. Either in the radial case, or for some model nonlinearities $f$ in a general domain,…

We consider the so-called \emph{discrete $p$-Laplacian}, a nonlinear difference operator that acts on functions defined on the nodes of a possibly infinite graph. We study the associated nonlinear Cauchy problem and identify the generator…

We prove that local weak solutions to nonlocal parabolic $p$-Laplace equations are locally Lipschitz continuous in space, uniformly in time for every $1<p<\infty$ and $s \in (0,1)$ whenever $sp > p-1$. Our results hold for symmetric,…

We consider critical points of the geometric obstacle problem on vectorial maps $u: \mathbb{B}^2 \subset \mathbb{R}^2 \to \mathbb{R}^N$ \[ \int_{\mathbb{B}^2} |\nabla u|^2 \quad \mbox{subject to $u \in \mathbb{R}^N \backslash…

Consider the class of optimal partition problems with long range interactions \[ \inf \left\{ \sum_{i=1}^k \lambda_1(\omega_i):\ (\omega_1,\ldots, \omega_k) \in \mathcal{P}_r(\Omega) \right\}, \] where $\lambda_1(\cdot)$ denotes the first…

We consider the well-known following shape optimization problem: $$\lambda_1(\Omega^*)=\min_{\stackrel{|\Omega|=a} {\Omega\subset{D}}} \lambda_1(\Omega), $$ where $\lambda_1$ denotes the first eigenvalue of the Laplace operator with…

Existence of solutions to an obstacle $p$-Laplacian problem exhibiting a singular, discontinuous reaction is proved. The reaction term may be discontinuous in a Lebesgue-negligible set. Moreover, solutions are shown to be locally…

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $\partial_tu - \mbox{div}(A\nabla u)=0$ on the domain $\mathbb R^{n+1}_+\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the…

We consider the homogeneous Dirichlet problem for the parabolic equation \[ u_t- \operatorname{div} \left(|\nabla u|^{p(x,t)-2} \nabla u\right)= f(x,t) + F(x,t, u, \nabla u) \] in the cylinder $Q_T:=\Omega\times (0,T)$, where $\Omega\subset…

The main result of this paper is to prove that viscosity solutions to a parabolic free boundary problem with variable coefficients are Lipschitz continuous under the assumptions that the solution has a Lipschitz free boundary and satisfies…

We study the optimal boundary regularity of solutions to Dirichlet problems involving the logarithmic Laplacian. Our proofs are based on the construction of suitable barriers via the Kelvin transform and direct computations. As applications…

We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire. A variant…

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…

We establish a partial $C^{1,\alpha}$ regularity result for minimizers of the optimal $p$-compliance problem with length penalization in any spatial dimension $N\geq 2$, extending some of the results obtained in…

We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity $C^{1,1/2}$. This improves the known optimal regularity results by allowing the thin obstacle…

We study some regularity issues for solutions of non-autonomous obstacle problems with $(p,q)$-growth. Under suitable assumptions, our analysis covers the main models available in the literature.

In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using the…

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.

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 prove that, given~$p>\max\left\{\frac{2n}{n+2},1\right\}$, the nonnegative almost minimizers of the nonlinear free boundary functional $$ J_p(u,\Omega):=\int_{\Omega}\Big( |\nabla u(x)|^p+\chi_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz…