Related papers: Higher-order boundary regularity estimates for non…
We study solutions to $Lu=f$ in $\Omega\subset\mathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable L\'evy process. On the one hand, we study the regularity of solutions to $Lu=f$ in $\Omega$, $u=0$ in $\Omega^c$, in…
As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(\rho^aA\nabla w\right)=\rho^af+\mathrm{div}\left(\rho^aF\right) \quad\textrm{in}\; \Omega \]…
This paper is a continuation of our previous work [21], where we have established that, for the second-order degenerate hyperbolic equation (\p_t^2-t^m\Delta_x)u=f(t,x,u), locally bounded, piecewise smooth solutions u(t,x) exist when the…
This paper is devoted to the investigation of the boundary regularity for the Poisson equation $${{cc} -\Delta u = f & \text{in} \Omega u= 0 & \text{on} \partial \Omega$$ where $f$ belongs to some $L^p(\Omega)$ and $\Omega$ is a…
In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $\Omega=\mathcal O\times\mathbb R$…
Let $\Omega$ be a bounded $C^2$ domain in $\mathbb{R}^n$ and $u\in C(\mathbb{R}^n)$ solves \begin{equation*} \begin{aligned} \Delta u + a Iu + C_0|Du| \geq -K\quad \text{in}\; \Omega, \quad \Delta u + a Iu - C_0|Du|\leq K \quad \text{in}\;…
In this paper, we establish refined regularity estimates for nonnegative solutions to the fractional Poisson equation $$ (-\Delta)^s u(x) =f(x),\,\, x\in B_1(0). $$ Specifically, we have derived H\"{o}lder, Schauder, and Ln-Lipschitz…
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are $C^{1,\alpha}$, then they are $C^\infty$. This completes the study of…
A by now classical result due to DiBenedetto states that the spatial gradient of solutions to the parabolic $p$-Laplacian system is locally H\"older continuous in the interior. However, the boundary regularity is not yet well understood. In…
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…
This article establishes the boundary H\"{o}lder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions $n \leq 9$, for $C^{1,1}$ domains. We consider equations $- L u = f(u)$ in a bounded…
We consider the parabolic Lam\'{e} system on a bounded domain. We focus on two types of inequalities for higher-order derivatives of solutions. The first is related to an $L^p$-$L^p$ estimate locally in time in the Lebesgue space setting,…
This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional $1$-Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the…
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…
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 article we establish fine results on the boundary behavior of solutions to nonlocal equations in $C^{k,\gamma}$ domains which satisfy local Neumann conditions on the boundary. Such solutions typically blow up at the boundary like $v…
This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_t u + {\mathcal L}u^m=0$, $m>1$, where the operator ${\mathcal L}$ belongs to a general class of…
We study the boundary weighted regularity of weak solutions $u$ to a $s$-fractional $p$-Laplacian equation in a bounded smooth domain $\Omega$ with bounded reaction and nonlocal Dirichlet type boundary condition, in the singular case…
In this manuscript, we establish the existence and sharp geometric regularity estimates for bounded solutions of a class of quasilinear parabolic equations in non-divergence form with non-homogeneous degeneracy. The model equation in this…
We prove sharp boundary regularity of solutions to nonlocal elliptic equations arising from operators comparable to the fractional Laplacian over Reifenberg flat sets and with null exterior condition. More precisely, if the operator has…