Related papers: The free boundary for the singular obstacle proble…

We study convex solutions to the Monge-Amp\`ere obstacle problem \[ \operatorname{det} D^2 v=g v^q\chi_{\{v>0\}}, \quad v \geq 0, \] where $q \in [0,n)$ is a constant and $g$ is a bounded positive function. This problem emerges from the…

The boundary value problem is examined for the system of elliptic equations of from $-\Delta u + A(x)u = 0 \quad\text{in} \Omega,$ where $A(x)$ is positive semidefinite matrix on $\mathbb{R}^{{k}\times{k}},$ and $\frac{\partial u}{\partial…

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…

The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem $$ \partial_t u-\Delta u=a u-b(x) u^p \text{in} \Omega\times \R^+, u(0)=u_0, u(t)|_{\partial \Omega}=0 $$ as $p\to +\infty$, where…

In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equations \begin{equation} \begin{cases} u_{t}\left(x,t\right)=\mbox{div}(|\nabla u\left(x,t\right)|^{p-2}\nabla u(x,t))+f(u(x,t)), &…

We consider the homogeneous equation ${\mathcal A} u=0$, where ${\mathcal A}$ is a symmetric and coercive elliptic operator in $H^1(\Omega)$ with $\Omega$ bounded domain in ${{\mathbb R}}^d$. The boundary conditions involve fractional power…

We study the regularity of the viscosity solution $u$ of the $\sigma_k$-Loewner-Nirenberg problem on a bounded smooth domain $\Omega \subset \mathbb{R}^n$ for $k \geq 2$. It was known that $u$ is locally Lipschitz in $\Omega$. We prove…

In a bounded domain $\Omega$, we consider a positive solution of the problem $\Delta u+f(u)=0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions…

We construct a monotonicity formula for a class of free boundary problems associated with the stationary points of the functional \[ J(u)=\int_\Omega F(|\nabla u|^2)+\mbox{meas}(\{u>0\}\cap \Omega), \] where $F$ is a density function…

For any $\Omega\subset \mathbb{R}^N$ smooth and bounded domain, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on $\Omega$ in a neat interval depending only by the best constant of the Sobolev…

Using the epiperimetric inequalities approach, we study the obstacle problem $\min\{(-\Delta)^su,u-\varphi\}=0,$ for the fractional Laplacian $(-\Delta)^s$ with obstacle $\varphi\in C^{k,\gamma}(\mathbb{R}^n)$, $k\ge2$ and $\gamma\in(0,1)$.…

For $n\ge 3$ and $0<\epsilon\le 1$, let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $u_\epsilon:\Omega \subset\R^n\to \mathbb R^2$ solve the Ginzburg-Landau equation under the weak anchoring boundary condition: $$\begin{cases}…

The parabolic problem $u_t-\Delta u=\frac{\lambda f(x)}{(1-u)^2}+P$ on a bounded domain $\Omega$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this…

This paper focuses on the following Keller-Segel system with singular sensitivity and logistic source $$ \left\{\begin{array}{ll} u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ au-\mu u^2,\quad x\in \Omega, t>0, \disp{ v_t=\Delta v-…

We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when…

We consider the boundary value problem $-\Delta_p u = \lambda c(x) |u|^{p-2}u + \mu(x) |\grad u|^p + h(x)$, $u \in W^{1,p}_0(\Omega) \cap L^{\infty}(\Omega)$, where $\Omega \subset \mathbb R^N$, $N \geq 2$, is a bounded domain with smooth…

In this article we investigate the higher regularity properties of the regular free boundary in the fractional thin obstacle problem. Relying on a Hodograph-Legendre transform, we show that for smooth or analytic obstacles the regular free…

A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form $$\sup_{\int_D\theta\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+\theta}{2}|\nabla…

We introduce the nonlocal analogue of the classical free boundary minimal hypersurfaces in an open domain $\Omega$ of $\mathbb{R}^n$ as the (boundaries of) critical points of the fractional perimeter $\operatorname{Per}_s(\cdot,\,\Omega )$…

In this paper, we prove the regularity of the free boundary in the Monge-Amp\`ere obstacle problem $\det D^2 v= f(y)\chi_{\{v>0\}}. $ By duality, the regularity of the free boundary is equivalent to that of the asymptotic cone of the…