Related papers: Regularity for fully nonlinear nonlocal parabolic …
This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely $\partial_t u = { \nabla \cdot} \left(u \nabla(-\Delta)^{\frac{\alpha}{2}-1}u^{m-1} \right)$ where $u:\mathbb{R}_+\times…
We prove the $C^{\alpha}$ regularity for weak solutions to a class of ultraparabolic equation, with measurable coefficients. The results generalized our recent $C^{\alpha}$ regularity results of Prandtl's system to high dimensional cases.
We study existence and regularity of weak solutions to a nonlinear parabolic Dirichlet problem $\partial_{t}u - \rho_{\lambda}(x)u\Delta u = \rho_{\lambda}(x)g_{0}(x)u$ on the half line $(0,\infty)$. We find weak solutions from $L^p\ (p <…
In the present paper, we study sharp C^{1;\alpha} regularity results with boundary Neumann condition for viscosity solutions for a class of degenerate fully non-linear elliptic equations with Neumann boundary conditions.
In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators $$ \partial_{t}u(t,x) = \mathcal{L}^{a}u(t,x) + f(t,x), \quad t>0 $$ in $L_{q}(L_{p})$ spaces. Our spatial…
We study the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla\cdot(u\nabla (-\Delta)^{-1/2}u).$ For definiteness, the problem is posed…
In the present paper we derive Liouville type results and existence of periodic solutions for $\chi^{(2)}$ type systems with non-homogeneous nonlinearities. Moreover, we prove both universal bounds as well as singularity and decay estimates…
We show how a theorem about the solvability in $W^{1,2}_{\infty}$ of special parabolic Isaacs equations can be used to obtain the existence and uniqueness of viscosity solutions of general uniformly nondegenerate parabolic Isaacs equations.…
We discuss the H\"older regularity of solutions to the semilinear equation involving the fractional Laplacian $(-\Delta)^s u=f(u)$ in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the…
In this article we address the regularity of stable solutions to semilinear elliptic equations $-\Delta u = f(u)$ with MEMS type nonlinearities. More precisely, we will have $0\leq u \leq 1$ in a domain $\Omega \subset \mathbb{R}^n$ and…
Investigating for interior regularity of viscosity solutions to the fully nonlinear elliptic equation $$F(x,u,\triangledown u,\triangledown ^2 u)=0,$$ we establish the interior $C^{1+1}$ continuity under the assumptions that $F$ is…
We prove H\"older regularity for a general class of parabolic integro-differential equations, which (strictly) includes many previous results. We present a proof which avoids the use of a convex envelop as well as give a new covering…
We show that locally bounded, local weak solutions to certain nonlocal, nonlinear diffusion equations modeled on the fractional porous media and fast diffusion equations given by \begin{align*} \partial_t u + (-\Delta)^s(|u|^{m-1}u) = 0…
Convergence of Rothe's method for the fully nonlinear parabolic equation u_t + F(D^2 u, Du, u, x, t) = 0 is considered under some continuity assumptions on F. We show that the Rothe solutions are Lipschitz in time, Holder in space, and they…
In this paper we consider the fully nonlinear parabolic free boundary problem $$ \left\{\begin{array}{ll} F(D^2u) -\partial_t u=1 & \text{a.e. in}Q_1 \cap \Omega\\ |D^2 u| + |\partial_t u| \leq K & \text{a.e. in}Q_1\setminus\Omega,…
We study the obstacle problem for parabolic operators of the type $\partial_t + L$, where $L$ is an elliptic integro-differential operator of order $2s$, such as $(-\Delta)^s$, in the supercritical regime $s \in (0,{1/2})$. The best result…
In this paper we study several aspects related with solutions of nonlocal problems whose prototype is $$ u_t =\displaystyle \int_{\mathbb{R}^N} J(x-y) \big( u(y,t) -u(x,t) \big) \mathcal G\big( u(y,t) -u(x,t) \big) dy \qquad \mbox{ in } \,…
This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in [22] to domains of polyhedral type. In particular, we study the smoothness in the specific scale…
In this note, we present the interior $C^{2,\alpha}$ regularity for viscosity solutions of fully nonlinear uniformly elliptic equations in dimension two.
We study a general form of a degenerate or singular parabolic equation $$ u_t-|Du|^{\gamma}\big(\Delta u+(p-2)\Delta_\infty^Nu\big)=0 $$ that generalizes both the standard parabolic $p$-Laplace equation and the normalized version that…