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

In this paper, we prove gradient continuity estimates for viscosity solutions to $\Delta_{p}^N u- u_t= f$ in terms of the scaling critical $L(n+2,1 )$ norm of $f$, where $\Delta_{p}^N$ is the game theoretic normalized $p-$Laplacian operator…

We study a general class of parabolic equations $$ u_t-|Du|^\gamma\big(\Delta u+(p-2) \Delta_\infty^N u\big)=0, $$ which can be highly degenerate or singular. This class contains as special cases the standard parabolic $p$-Laplace equation…

We prove the existence of a weak solution to the problem \begin{equation*} \begin{split} -\Delta_{p}u+V(x)|u|^{p-2}u & =f(u,|\nabla u|^{p-2}\nabla u), \ \ \ \\ u(x) & >0\ \ \forall x\in\mathbb{R}^{N}, \end{split} \end{equation*} where…

We extend the so-called universal potential estimates of the Kuusi-Mingione type (J.Funct. Anal. 2012) to the singular case $1<p\leq 2-1/n$ for the quasilinear equation with measure data \begin{equation*} -\operatorname{div}(A(x,\nabla…

Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \[ \left\{ \begin{array}{c} u_{t} \ - \ (…

We show that p-harmonic functions in the plane satisfy a nonlinear asymptotic mean value property for p>1. This extends previous results of Manfredi and Lindqvist for certain range of p's.

We consider the Laplacian with drift in $\mathbb R^n$ defined by $\Delta_\nu = \sum_{i=1}^n(\frac{\partial^2}{\partial x_i^2} + 2 \nu_i\frac{\partial }{\partial{x_i}})$ where $\nu=(\nu_1,\ldots,\nu_n)\in \mathbb R^n\setminus\{0\}$. The…

In this paper we prove an asymptotic behavior for the radial eigenvalues to the Dirichlet $p$-Laplacian problem $-\Delta_p\,u = \lambda\,|u|^{p-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is an annular domain…

We consider a degenerate parabolic equation associated with the fractional $% p $-Laplace operator $\left( -\Delta \right) _{p}^{s}$\ ($p\geq 2$, $s\in \left( 0,1\right) $) and a monotone perturbation growing like $\left\vert s\right\vert…

We study the positive solutions of the Lane-Emden equation $-\Delta_{p}u=\lambda_{p}|u|^{q-2}u$ in $\Omega$ with homogeneous Dirichlet boundary conditions, where $\Omega\subset\mathbb{R}^{N}$ is a bounded and smooth domain, $N\geq2,$…

We prove existence and uniqueness of viscosity solutions to the degenerate parabolic problem $u_t = \Delta_\infty^h u$ where $\Delta_\infty^h$ is the $h$-homogeneous operator associated with the infinity-Laplacian, $\Delta_\infty^h u =…

In this paper we prove existence and uniqueness results for nonlinear parabolic problems with Dirichlet boundary values whose model is \[ \left\{ \begin{aligned} &b(u)_t-\Delta_{p}u=\mu\;\mbox{in }(0,T)\times\Omega,\\…

We study some basic properties of generalized Morrey spaces $\mathcal{M}^{p,\phi}(\R^{d})$. Also, the problem $-\mbox{div}(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u=0$ in $\Omega$, where $\Omega$ is a bounded open set in $\R^d$, and potential…

We face a rigidity problem for the fractional $p$-Laplace operator to extend to this new framework some tools useful for the linear case. It is known that $(-\Delta)^s(1-|x|^{2})^s_+$ and $-\Delta_p(1-|x|^{\frac{p}{p-1}})$ are constant…

We study the eigenvalue problem for a system of fractional $p-$Laplacians, that is, $$ \begin{cases} (-\Delta_p)^r u = \lambda\dfrac{\alpha}p|u|^{\alpha-2}u|v|^{\beta} &\text{in } \Omega,\vspace{.1cm} (-\Delta_p)^s u =…

We derive a priori bounds for positive supersolutions of $ - \Delta_{p} u = \rho(x) f(u) $, where $p>1$ and $\Delta_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply…

We prove global comparison results for the $p$-Laplacian on a $p$-parabolic manifold. These involve both real-valued and vector-valued maps with finite $p$-energy.

We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau…

In this paper we study nonlinear second-order differential inclusions involving the ordinary vector $p$-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying…

The fractional heat operator $(\partial_t-\Delta_x)^s$ and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value…