Related papers: Gradient estimates for nonlinear elliptic equation…
We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called…
The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a…
We consider nonlocal equations of order larger than one with measure data and prove gradient regularity in Sobolev and H\"older spaces as well as pointwise bounds of the gradient in terms of Riesz potentials, leading to fine regularity…
Under structural conditions which are almost optimal, we derive a quantitative version of boundary estimate then prove existence of solutions to Dirichlet problem for a class of fully nonlinear elliptic equations on Hermitian manifolds.
We develop a new real-variable method for weighted $L^p$ estimates. The method is applied to the study of weighted $W^{1, 2}$ estimates in Lipschitz domains for weak solutions of second-order elliptic systems in divergence form with bounded…
$C^\alpha$ and $W^{1,\infty}$ estimates for the first-order and second-order correctors in the homogenization are presented based on the translation invariant and Li-Vogelius's gradient estimate for the second order linear elliptic equation…
In this paper, we study regularity estimates for a class of degenerate, fully nonlinear elliptic equations with arbitrary nonhomogeneous degeneracy laws. We establish that viscosity solutions are locally continuously differentiable under…
For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the microscopic scale, without smoothness…
We obtain a maximum principle, and "a priori" upper estimates for solutions of a class of non linear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumption of volume growth conditions.…
In this paper, we study the local gradient regularity of non-negative weak solutions to doubly nonlinear parabolic partial differential equations of the type \begin{align*} \partial_t u^q - \mbox{div}\, A(x,t,Du)=0 \qquad\mbox{in…
We study boundary regularity of viscosity solutions to fully nonlinear degenerate or singular parabolic equations. The gradient-dependent degeneracy or singularity, along with the time derivative, introduces significant challenges beyond…
We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…
This paper is concerned with nonlinear elliptic equations in nondivergence form where the operator has a first order drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative…
This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - \mbox{div}[\mathbb{A}(x,t,u,\nabla u)]= \mbox{div}[{\mathbf F}]$ with homogeneous Dirichlet…
In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in $(0,T) \times \R^N$. Our assumptions include the case that the coefficients be both unbounded…
We study a class of Dirichlet boundary value problems whose prototype is \begin{equation}\label{1.2abs} \left\{\begin{array}{ll} -\Delta_p u =h(u)|\nabla u|^p+u^{q-1}+f(x)\, &\quad\hbox{in } \ \Omega\,,\\ u\ge 0\,,&{\quad\hbox{in } \…
We establish sharp geometric $C^{1+\alpha}$ regularity estimates for bounded weak solutions of evolution equations of $p$-Laplacian type. Our approach is based on geometric tangential methods, and makes use of a systematic oscillation…
In this manuscript, we obtain sharp and improved regularity estimates for weak solutions of weighted quasilinear elliptic models of Hardy-H\'{e}non-type, featuring an explicit regularity exponent depending only on universal parameters. Our…
This paper presents an existence result and maximal regularity estimates for distributional solutions to degenerate/singular elliptic systems of $p$-Laplacian type with absorption and (prescribed) locally integrable forcing posed in…
We consider Orlicz-growth generalization to evolutionary $p$-Laplacian and to the evolutionary symmetric $p$-Laplacian. We derive the spatial second-order Caccioppoli estimate for a local weak solution to these systems. The result is new…