相关论文: Regularity for nonlocal equations with local Neuma…
We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones $K\subset \mathbb{R}^3$ in the specific scale $\ B^{\alpha}_{\tau,\tau}, \…
The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as shown by Krylov, for any $\alpha>0$ one can find a simple $1$-dimensional constant…
We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class $C_{loc}^{1, \alpha}$,…
In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Companato estimates and Sobolev embedding theorem, we first show the H\"{o}lder continuity (locally in the…
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…
In this paper, we develop systematically the pointwise regularity for viscosity solutions of fully nonlinear elliptic equations in general forms. In particular, the equations with quadratic growth (called natural growth) in the gradient are…
In the present work we establish sharp regularity estimates for the solutions of the porous medium equation, along their zero level-sets. We work under a proximity regime on the exponent governing the nonlinearity of the problem. Then, we…
In this paper we investigate the regularity and solvability of solutions to Dirichlet problem for fully non-linear elliptic equations with gradient terms on Hermitian manifolds, which include among others the Monge-Amp\`ere equation for…
We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without…
On a compact Riemann surface $(\Sigma, g)$ with a smooth boundary $\partial \Sigma$, we consider the following mean field equations with Neumann boundary conditions: $$ -\Delta_g u = \lambda \left(\frac{Ve^u}{\int_{\Sigma} Ve^u \, dv_g} -…
We prove the existence and $C^{1,\alpha}$ regularity of solutions to nonlocal fully nonlinear elliptic double obstacle problems. We also obtain boundary regularity for these problems. The obstacles are assumed to be Lipschitz…
In this survey paper, we study the optimal regularity of solutions to uniformly degenerate elliptic equations in bounded domains and establish the H\"older continuity of solutions and their derivatives up to the boundary.
We present a general blow-up technique to obtain local regularity estimates for solutions, and their derivatives, of second order elliptic equations in divergence form in H\"older spaces with variable exponent. The procedure allows to…
We study the regularity of the free boundary in the parabolic obstacle problem for the fractional Laplacian $(-\Delta)^s$ (and more general integro-differential operators) in the regime $s>\frac{1}{2}$. We prove that once the free boundary…
Let $D$ be an bounded region in ${\bf R}^n$. The regularity of solutions of a family of quasilinear elliptic partial differential equations is studied, one example being $\Delta_nu=Vu^{n-1}$. The coefficients are assumed to be in the space…
In this paper we establish some regularity results concerning the behavior of weak solutions and very weak solutions of the degenerate wave equation near the boundary. For the nondegenerate case, the correponding results were originally…
This is an expository paper on the theory of local regularity for weak solutions to the non-stationary 3D Navier-Stokes equations near the boundary of a domain.
We consider H\"older continuous weak solutions $u\in C^\gamma(\Omega)$, $u\cdot n|_{\partial \Omega}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega\subset\mathbb{R}^d$. If $\Omega$ is of class…
In this paper, we study the boundary H\"older regularity for solutions to the fractional Dirichlet problem in unbounded domains with boundary \begin{equation*} \begin{cases} (-\Delta)^s u(x) = g(x),&\text{in } \Omega, u(x)=0, &\text{in }…
In this paper we survey some results on the Dirichlet problem \[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }\Omega \\ u&=&g&\textrm{in }\mathbb R^n\backslash\Omega \end{array}\right.\] for nonlocal operators of the form…