相关论文: On the Bernoulli problem with unbounded jumps
We consider a one-phase free boundary problem of the minimizer of the energy \[ J_{\gamma}(u)=\frac{1}{2}\int_{(B_1^{n+1})^+}{y^{1-2s}|\nabla u(x,y)|^2dxdy}+\int_{B_1^{n}\times \{y=0\}}{u^{\gamma}dx}, \] with constants $0<s,\gamma<1$. It is…
We study a general class of elliptic free boundary problems equipped with a Dirichlet boundary condition. Our primary result establishes an optimal $C^{1,1}$-regularity estimate for $L^p$-strong solutions at points where the free and fixed…
In this survey we go through some of the recent results about the regularity of vectorial free boundary problems of Bernoulli type and free boundary systems. The aim is to illustrate the general methodologies as well as to outline a…
In this paper, we consider a vector-valued one-phase Bernoulli-type free boundary problem on a metric measure space $(X,d,\mu)$ with Riemannian curvature-dimension condition $RCD(K,N)$. We first prove the existence and the local Lipschitz…
We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the $p(x)$-Laplacian. Under the assumption of…
We construct an example of blow-up in a flow of min-plus linear operators arising as solution operators for a Hamilton-Jacobi equation with a Hamiltonian of the form |p|^alpha+U(x,t), where alpha>1 and the potential U(x,t) is uniformly…
This paper concerns the characterization of blowup and global radial solutions of a two-free boundaries system read by \begin{align}\label{bs_pr} \tag{1.1} \left\{\begin{array}{rl} u_t(t,r)= \Delta u(t,r) - \lambda(t,x)|\nabla…
We study the free-boundary equation \[ \Delta u=\chi_{\{|\nabla u|>0\}} \] near the origin. We prove that, at a singular point of \(\partial\{|\nabla u|>0\}\), the quadratic blow-up is unique. As noted in \cite[Notes to Chapter 7]{PSU2012},…
We discuss the extent to which solutions to one-phase free boundary problems can be characterized according to their topological complexity. Our questions are motivated by fundamental work of Luis Caffarelli on free boundaries and by…
We study regularity issues for systems of elliptic equations of the type \[ -\Delta u_i=f_{i,\beta}(x)-\beta \sum_{j\neq i} a_{ij} u_i |u_i|^{p-1}|u_j|^{p+1} \] set in domains $\Omega \subset \mathbb{R}^N$, for $N \geq 1$. The paper is…
We consider the exterior free boundary Bernoulli problem in the case of a rough given domain. An asymptotic analysis shows that the solution of the initial problem can be approximated by the solution of a non-rough Bernoulli problem at…
The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump L\'evy processes. In this paper we study the regularity of the free boundary. Our main…
In this paper we study the existence of bound states of the following class of quasilinear problems, \begin{equation*} \left\{ \begin{aligned} &-\varepsilon ^p\Delta_pu+V(x)u^{p-1}=f(u)+u^{p^\ast -1},\ u>0,\ \text{in}\ \mathbb{R}^{N}, &\lim…
We construct nonnegative weak solutions to the singular parabolic free boundary problem \[ \partial_t u - \Delta u = - \frac{\mathrm{d}}{\mathrm{d} u} u_+^\gamma , \] where $\gamma \in (0,1]$, $u_+ := \max\{u,0\}$, and the term in the…
In this paper we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found variationally as global minimizers of the…
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like $(-\Delta)^s +b\cdot\nabla$, in the subcritical regime $s>\frac{1}{2}$. Our main result states that once the free…
Motivated by the study of branching particle systems with selection, we establish global existence for the solution $(u,\mu)$ of the free boundary problem \[ \begin{cases} \partial_t u =\partial^2_{x} u +u & \text{for $t>0$ and…
This paper studies the regularity of the free boundary for viscosity solutions to a parabolic Bernoulli-type free boundary problem with variable coefficients. The main result is that Lipschitz free boundaries are $C^1$ with a normal vector…
In the unit ball B(0,1), let $u$ and $\Omega$ (a domain in $\R$) solve the following overdetermined problem: $$\Delta u =\chi_\Omega\quad \hbox{in} B(0,1), \qquad 0 \in \partial \Omega, \qquad u=|\nabla u |=0 \quad \hbox{in} B(0,1)\setminus…
We provide an integral estimate for a non-divergence (non-variational) form second order elliptic equation $a_{ij}u_{ij}=u^p$, $u\ge 0$, $p\in[0, 1)$, with bounded discontinuous coefficients $a_{ij}$ having small BMO norm. We consider the…