Related papers: Bernoulli problem for rough domains
We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version…
We establish existence and uniqueness of generalized solutions to the initial-boundary value problem corresponding to an Euler-Bernoulli beam model from mechanics. The governing partial differential equation is of order four and involves…
A singularly perturbed free boundary problem arising from a real problem associated with a Radiographic Integrated Test Stand concerns a solution of the equation $\Delta u = f(u)$ in a domain $\Omega$ subject to constant boundary data,…
We consider a free boundary problem in an exterior domain \begin{cases}\begin{array}{cc} Lu=g(u) & \text{in }\Omega\setminus K, \\ u=1 & \text{on }\partial K,\\ |\nabla u|=0 &\text{on }\partial \Omega, \end{array}\end{cases} where $K$ is a…
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…
We consider the spectral structure of indefinite second order boundary-value problems on graphs. A variational formulation for such boundary-value problems on graphs is given and we obtain both full and half-range completeness results. This…
In this paper we study the local behavior of solutions to some free boundary problems. We relate the theory of quasi-conformal maps to the regularity of the solutions to nonlinear thin-obstacle problems; we prove that the contact set is…
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 construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of…
We give a dimension-free Euler estimation of solution of rough differential equations in term of the driving rough path. In the meanwhile, we prove that, the solution of rough differential equation is close to the exponential of a Lie…
We establish rigorous quantitative inequalities for the first eigenvalue of the generalized $p$-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter $\alpha$ is positive, and the…
We provide a higher order boundary Harnack inequality for harmonic functions in slit domains. As a corollary we obtain the $C^\infty$ regularity of the free boundary in the Signorini problem near non-degenerate points.
In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $Lu=0$ in $\Omega$, $u=g$ in $\mathbb R^N\setminus\Omega$, in non-smooth domains $\Omega$. When $g$ is smooth enough, then it is easy to…
In this paper, we study the Brezis-Nirenberg problem on bounded smooth domains of R3. Using the algebraic topological argument of Bahri-Coron[2] as implemented in [6] combined with the Brendle[4]- Schoen[8]'s bubble construction, we solve…
We give a short and self-contained proof of the Boundary Harnack inequality for a class of domains satisfying some geometric conditions given in terms of a state function that behaves as the distance function to the boundary, is subharmonic…
This paper deals with a priori pointwise error estimates for the finite element solution of boundary value problems with Neumann boundary conditions in polygonal domains. Due to the corners of the domain, the convergence rate of the…
Let $U\subset \mathbb{R}^n$ ($n\geq 3$) be an exterior Euclidean domain with smooth boundary $\partial U$. We consider the Steklov eigenvalue problem on $U$. First we derive a sharp lower bound for the first eigenvalue in terms of the…
We consider the problem of determining the unknown boundary values of a solution of an elliptic equation outside a bounded open set $B$ from the knowledge of the values of this solution on a boundary of an arbitrary Lipschitz bounded domain…
In this paper the first and second domain variation for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions is computed. Minimality and maximality properties of the ball among nearly circular…
We consider the Cauchy-Dirichlet problem for semilinear wave equations in a three space dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical…