Related papers: Lipschitz regularity of fractional $p$-Laplacian

We study the fractional $p$-Laplace equation $$ (-\Delta_p)^s u = 0 $$ for $0<s<1$ and in the subquadratic case $1<p<2$. We provide H\"older estimates with an explicit H\"older exponent. The inhomogeneous equation is also treated and there…

Higher Sobolev and H\"older regularity is studied for local weak solutions of the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$. Depending on the regime considered, i.e. $$0<s\le\tfrac{p-2}{p}\quad \text{or}…

In this article, we study nonlinear nonlocal equations with coercive gradient nonlinearity of the form \[ (-\Delta_p)^s u(x) + H(x, \nabla u) = f, \] where $f$ is Lipschitz continuous. We show that any viscosity solution $u$ is locally…

We prove the local gradient H\"older regularity of viscosity solutions to the inhomogeneous normalized $p(x)$-Laplace equation $$ -\Delta u-(p(x)-2)\frac{\left\langle D^{2}uDu,Du\right\rangle }{\left|Du\right|^{2}} = f(x), $$ where $p$ is…

To our knowledge, this paper is the first attempt to consider the existence issue for fractional $p$-Laplacian equation: $(-\Delta)_p^s u= \lambda f(u),\; u> 0 ~\text{in}~\Omega;\; u=0\;\text{in}~ \mathbb{R}^N\setminus\Omega$, where $p>1$,…

We prove that local weak solutions to nonlocal parabolic $p$-Laplace equations are locally Lipschitz continuous in space, uniformly in time for every $1<p<\infty$ and $s \in (0,1)$ whenever $sp > p-1$. Our results hold for symmetric,…

We establish interior $C^{1,\alpha}$ regularity estimates for some $\alpha > 0$, for solutions of the fractional $p$-Laplace equation $(-\Delta_p)^s u = 0$ when $p$ is in the range $p \in [2,2/(1-s))$.

In this article, first we address the regularity of weak solution for a class of $p$-fractional Choquard equations: \begin{equation*} \;\;\; \left.\begin{array}{rl}…

We study the parabolic fractional $p-$Laplace equation $$\p_t u+(-\Delta_p)^su = 0$$ in the degenerate range \(2 \leq p < 2/(1-s)\). We show that weak solutions are Lipschitz continuous in space and, if \(p > 1/(1-s)\), also in time. We…

We study the fractional $(p,q)$-Laplace equation $$ (-\Delta_p)^s u +(-\Delta_q)^t u= 0 $$ for $s,t\in(0,1)$ and $p,q\in(1,\infty)$. We establish H\"older estimates with an explicit exponent. As a consequence, we derive a Liouville-type…

We present an alternative proof for local H\"older regularity of the solutions of the fractional p-Laplace equations, based on clustering and expansion (more precisely, recentering) of positivity.

In this note we consider regularity theory for a fractional $p$-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the $H^{s,p}$-Laplacian. We obtain the natural analogue to the classical $p$-Laplacian…

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 prove the local Lipschitz continuity of viscosity solutions for two-phase free boundary problems for the $p$-Laplacian with non-zero right hand side, where $p\in (1,\infty)$. This is the optimal regularity for the problem. We also obtain…

In this paper, we study qualitative properties of the fractional $p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $p-$Laplacian equation with Dirichlet condition. Moreover, an…

We prove higher H\"older regularity for solutions of equations involving the fractional $p-$Laplacian of order $s$, when $p\ge 2$ and $0<s<1$. In particular, we provide an explicit H\"older exponent for solutions of the non-homogeneous…

In this article, the problems to be studied are the following \leqnomode \begin{equation*} \label{p} \left\{\begin{array}{ll} (-\Delta )_p^s u \pm \dfrac{|u|^{p-2}u}{|x|^{sp}} = \lambda f(x,u) & \quad \mbox{in }\ \Omega\\[0.3cm] u= 0 &…

We study a singular or degenerate equation in non-divergence form modeled by the $p$-Laplacian, $$-|Du|^\gamma\left(\Delta u+(p-2)\Delta_\infty^N u\right)=f\ \ \ \ \text{in}\ \ \ \Omega.$$ We investigate local $C^{1,\alpha}$ regularity of…

Local H\"older regularity is established for certain weak solutions to a class of parabolic fractional $p$-Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling…

We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their…