Related papers: Refined regularity at critical points for linear e…

We investigate the regularity of elliptic equations in double divergence form, where the leading coefficients satisfying the Dini mean oscillation condition. We prove that the solutions are differentiable on the zero level set and derive a…

We study interior $C^{2,\alpha}$ regularity estimates for solutions of fully nonlinear uniformly elliptic equations of the general form $F(D^2u)=0$ in two independent variables and without any geometric condition on $F$. By means of the…

We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy $S(u) := \{X : D u(X) = 0 \}$. Our main Theorem reveals that along…

We establish partial regularity result for vector-valued solutions to second order elliptic system in divergence form. The coefficients safisfies Dini condition respect to $(x,u)$ with growth order lager than 2. We prove $C^1$-regularity of…

We consider second-order elliptic equations in non-divergence form with oblique derivative boundary conditions. We show that any strong solutions to such problems are twice continuously differentiable up to the boundary provided that the…

We consider the fully nonlinear equation with variable-exponent double phase type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$ Under some appropriate assumptions, by making use of geometric tangential methods and…

In this paper, we examine the regularity of the solutions to the double-divergence equation. We establish improved H\"older continuity as solutions approach their zero level-sets. In fact, we prove that $\alpha$-H\"older continuous…

We consider weak solutions $u \in u_0 + W^{1,2}_0(\Omega,R^N) \cap L^{\infty}(\Omega,R^N)$ of second order nonlinear elliptic systems of the type $- div a (\cdot, u, Du) = b(\cdot,u,Du)$ in $\Omega$ with an inhomogeneity satisfying a…

We show that weak solutions to conormal derivative problem for elliptic equations in divergence form are continuously differentiable up to the boundary provided that the mean oscillations of the leading coefficients satisfy the Dini…

We prove new results on the existence of positive radial solutions of the elliptic equation $-\Delta u= \lambda h(|x|,u)$ in an annular domain in $\mathbb{R}^{N}, N\geq 2$. Existence of positive radial solutions are determined under the…

We establish the regularity in 2 dimensions of $L^2$ solutions to critical elliptic systems in divergence form involving involution operators of finite $W^{1,2}$-energy.

This paper establishes an explicit $L^2$-estimate for weak solutions $u$ to linear elliptic equations in divergence form with general coefficients and external source term $f$, stating that the $L^2$-norm of $u$ over $U$ is bounded by a…

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^2$. Assuming that the principal coefficients satisfy the Dini mean oscillation condition, we establish the…

We prove $C^1$ regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients

We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that they yield smoothness properties that go beyond the inherent…

A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the…

We consider gradient estimates for $H^1$ solutions of linear elliptic systems in divergence form $\partial_\alpha(A_{ij}^{\alpha\beta} \partial_\beta u^j) = 0$. It is known that the Dini continuity of coefficient matrix $A =…

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with…

In this paper we consider semilinear equations $-\Delta u=f(u)$ with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution $u$ has…

Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to…