Related papers: $C^{1+\alpha}$ regularity for fractional $p$-harmo…

In the seminal paper [Arch. Ration. Mech. Anal. 176 (2005), 351--361], Savin proved the $C^1$-regularity of planar $\infty$-harmonic functions $u$. Here we give a new understanding of it from a capacity viewpoint and drop several high…

Our goal is to establish a hidden regularity result for solutions of time fractional Petrovsky system where the Caputo fractional derivative is of order $\alpha\in(1,2)$. We achieve such result for a suitable class of weak solutions.

In this paper, we introduce and investigate the fractional logarithmic $p$-Laplacian $(-\Delta)_{p}^{s+\log}$, defined as the first-order derivative with respect to the parameter $t$ of the fractional $p$-Laplacian $(-\Delta)_{p}^{t}$…

For $\Omega\subseteq\mathbb{R}^{n}$ an open and bounded region we consider solutions $u\in W_{\text{loc}}^{1,p(x)}\big(\Omega;\mathbb{R}^{N}\big)$, with $N>1$, of the $p(x)$-Laplacian system \begin{equation}…

We study partial regularity for degenerate elliptic systems of double-phase type, where the growth function is given by $H(x,t)=t^p+a(x)t^q$ with $1<p\leq q$ and $a(x)$ a nonnegative $C^{0,\alpha}$-continuous function. Our main result…

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if $u$ is a solution of $(-\Delta)^s u = g$ in $\Omega$, $u \equiv 0$ in $\R^n\setminus\Omega$, for some…

We study the regularity of the extremal solution $u^*$ to the singular reaction-diffusion problem $-\Delta_p u = \lambda f(u)$ in $\Omega$, $u =0$ on $\partial \Omega$, where $1<p<2$, $0 < \lambda < \lambda^*$, $\Omega \subset \mathbb{R}^n$…

Let $(-\Delta_{p})^{s}$, with $0<s<1<p<\infty$, be the fractional $p$-Laplacian operator. We prove that the span of $p$-harmonic functions in $B_{1}$ is dense in $C^{k}(\bar{B}_{1})$.

We establish gradient H\"older continuity for solutions to quasilinear, uniformly elliptic equations, including $p$-Laplace and Orlicz-Laplace type operators. We revisit and improve upon the results existing in the literature, proving…

We examine the interior regularity of solutions to a degenerate normalized $p$-Laplace equation, where the degeneracy is governed by a modulus of continuity whose inverse satisfies a Dini continuity condition. We prove that under very…

We establish pointwise a priori estimates for solutions in $D^{1,p}(\mathbb{R}^n)$ of equations of type $-\Delta_pu=f(x,u)$, where $p\in(1,n)$, $\Delta_p:=\mbox{div}\big(\left|\nabla u\right|^{p-2}\nabla u\big)$ is the $p$-Laplace operator,…

This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small…

It is a classical result that every subharmonic function, defined and ${\mathcal{L}}^p$-integrable for some $p$, $0<p<+\infty$, on the unit disk $\mathbb{D}$ of the complex plane ${\mathbb{C}}$ is for almost all $\theta$ of the form $o((1-|…

The solutions of a kind of second-order homogeneous partial differential equation are called (real kernel) alpha-harmonic functions. The alpha-harmonic functions and their first-order partial derivative functions on unit disk are estimated…

In this paper we study positive solutions to problem involving the fractional Laplacian $(E)$ $(-\Delta)^{\alpha} u(x)+|u|^{p-1}u(x)=0 in x\in\Omega\setminus\mathcal{C}$, subject to the conditions $u(x)=0$ $x\in\Omega^c$ and…

We study the regularity of segregated profiles arising from competition - diffusion models, where the diffusion process is of nonlocal type and is driven by the fractional Laplacian of power $s \in (0,1)$. Among others, our results apply to…

Let $2\le n\le 5$. We establish an apriori interior H\"older regularity of $C^2$-stable solutions to the semilinear equation $-\Delta u=f(u)$ in any domain of $R^n$ for any nonlinearity $f\in C^{0,1}(R) $.If $f $ is nondecreasing and convex…

In this note we are concerned with interior regularity properties of the $p$-Poisson problem $\Delta_p(u)=f$ with $p>2$. For all $0<\lambda\leq 1$ we constuct right-hand sides $f$ of differentiability $-1+\lambda$ such that the (Besov-)…

In this paper we study stable solutions to the fractional equation \begin{align} (-\Delta)^s u =f(u), \quad |u| < 1 \quad \mbox{in $\mathbb{R}^d$}, \end{align}where $0<s<1$ and $f:[-1,1] \rightarrow \mathbb{R}$ is a $C^{1,\alpha}$ function…

In this manuscript, we extend our previous work on the Riemann-Liouville fractional integral of order $\alpha > 0$ in Bochner-Lebesgue spaces. We specifically address the remaining cases concerning its boundedness when $\alpha > 1/p$.…