Related papers: A mean value formula for the variational $p$-Lapla…

We extend the classical mean value property for the Laplacian operator to address a nonlinear and non-homogeneous problem related to the $p$-Laplacian operator for $p>2$. Specifically, we characterize viscosity solutions to the $p$-Laplace…

We characterize an asymptotic mean value formula in the viscosity sense for the double phase elliptic equation $$ -{\rm div}(\lvert \nabla u \rvert^{p-2}\nabla u+ a(x)\lvert\nabla u \rvert^{q-2}\nabla u)=0 $$ and the normalized double phase…

Let $1\le p\le\infty$. We show that a function $u\in C(\mathbb R^N)$ is a viscosity solution to the normalized $p$-Laplace equation $\Delta_p^n u(x)=0$ if and only if the asymptotic formula $$ u(x)=\mu_p(\ve,u)(x)+o(\ve^2) $$ holds as…

Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well known equivalence between harmonic functions and mean value properties. In the…

We derive two equivalent definitions of the viscosity solutions to the homogeneous sub-p- Laplace parabolic equations on the Heisenberg group, and characterize the viscosity solutions in terms of an asymptotic mean value formula. Moreover,…

Let $1<p \leq \infty$. We provide an asymptotic characterization of continuous viscosity solutions $u$ of the normalized $p$-Laplacian $\Delta_{p\,\mathbb{G}}^N u=0$ in any Carnot group $\mathbb{G}$.

We introduce a game-theoretical framework for the doubly nonlinear parabolic equation \[ |\partial_t u|^{p-2} \partial_t u - \Delta_p u = 0. \] where $\Delta_p u = \nabla \cdot ( |\nabla u |^{p-2} \nabla u)$ with $p>2$ is the standard…

We propose a new monotone finite difference discretization for the variational $p$-Laplace operator, \[ \Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u), \] and present a convergent numerical scheme for related Dirichlet problems. The…

We study the p-Laplace equation in the plane and prove that the mean value property holds directly for the solutions themselves. This removes the need to interpret the formula in the viscosity sense via test functions. The method is based…

We obtain some regularity results for solutions to vectorial $p$-Laplace equations $$ -{\boldsymbol \Delta}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u})\,\, \mbox{…

We prove that any nonnegative viscosity solution of the inequality $$(-\Delta_p)^s u(x) \geq u^{t} |\nabla u|^{m}\quad \text{ in }\; \mathbb{R}^N,\; N\geq 2,$$ must be constant. This result holds for parameters $p\in (1, \infty), s\in (0,…

We consider the critical $p$-Laplacian system \begin{equation}\label{92} \begin{cases}-\Delta_p u-\frac{\lambda a}{p}|u|^{a-2}u|v|^b =\mu_1|u|^{p^\ast-2}u+\frac{\alpha\gamma}{p^\ast}|u|^{\alpha-2}u|v|^{\beta}, &x\in\Omega,\\ -\Delta_p…

In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge-Amp\`ere equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic…

We consider the (viscosity) solution $u^\varepsilon$ of the elliptic equation $\varepsilon^2\Delta_p^G u= u$ in a domain (not necessarily bounded), satisfying $u=1$ on its boundary. Here, $\Delta_p^G$ is the {\it game-theoretic or…

The $p$-Laplacian operator $\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$ is not uniformly elliptic for any $p\in(1,2)\cup(2,\infty)$ and degenerates even more when $p\to \infty$ or $p\to 1$. In those two cases the Dirichlet…

This paper establishes existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. This operator represents a generalization of the $p(\cdot)$-Laplace…

We consider a number of boundary value problems involving the $p$-Laplacian. The model case is $-\Delta_p u=V|u|^{p-2}u$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R}^n$. We derive necessary conditions for the existence of…

We prove H\"older estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional $p$-Laplace equation $$ \text{PV}…

We find discrete analogs to continuous mean value principles that are used in the numerical analysis of the normalized p-Laplacian for particular values of p, specifically when p is 4.

We obtain necessary and sufficient existence conditions for solutions of the boundary value problem $$ \Delta_p u = f \quad \mbox{on } M, \quad \left. \left| \nabla u \right|^{p - 2} \frac{\partial u}{\partial \nu} \right|_{ \partial M } =…