# Questions tagged [p-laplace]

Questions involving the $p$-Laplace operator $\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$.

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Suppose $u$ is $p$-harmonic, i.e., it solve $-\operatorname{div} |\nabla u|^{p-2} \,\nabla u = 0$ where $1<p<\infty$. Then is the following inequality true? $$\int_{S_1} (u-k)^2|\nabla u|^{p-2}... • 473 3 votes 1 answer 372 views ### Geometric flow by the level sets of a harmonic function Let u be an harmonic function in a cylindrical domain B_2^{n-1}\times(-1,1)\subset\mathbb{R}^n, and suppose its level sets \Gamma_t=\{u=t\} are graphs of functions on B_2^{n-1}. Consider a ... 2 votes 0 answers 55 views ### Stability of weak solutions to quasilinear parabolic equations Consider the quasilinear operator A(x,t,\nabla u) satisfying$$A(x,t,\nabla u).\nabla u \geq C_0 |\nabla u|^p$$and$$|A(x,t,\nabla u)| \leq C_1 |\nabla u|^{p-1}$$where 1<p<\infty. Note ... • 473 2 votes 0 answers 54 views ### Solution of \vec{p}-Laplace equation Let \Omega \subset {\mathbb{R}^n} is bounded domain with smooth boundary. We consider the bvp$$ - \sum\limits_{I = 1}^n {{\partial _{{x_i}}}\left( {{{\left| {{\partial _{{x_i}}}u} \right|}^{{p_i} ...
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Let $\Omega \subset \mathbb{R^{n}}$ be a domain (open and connected set), for $p\geq 2$, the $p$-laplacian is defined by: $\Delta_p u= \operatorname{div} (|\nabla u|^{p-2} \nabla u)$, in non-...
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### Using weak maximum principle to prove continuous dependence of the boundary data?

I am currently looking at the following ingomogenous Dirichlet problem over an open, bounded domain $\Omega \subset \mathbb{R}^2$ with continuous boundary: \begin{align} \begin{cases} -\operatorname{...
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### How often does the gradient of a solution to elliptic equation vanish on the boundary?

This question is motivated by an inverse coefficient problem, for which it is useful to find solutions to a particular PDE so that the gradient of the solution does not vanish at all, or at least too ...
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### First eigenfunction of $p=3$-Laplacian of a square domain in $\Bbb R^2$ : reference for any work on this?

In the last few decades, lots of work on first eigenfunction of $p$-Laplace with Dirichlet and other boundary conditions. But I couldn't find much on periodic boundary conditions. I have computed the ...
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### Trick in a inequality of a paper of free boundary problem that involves the p-laplacian with 1<p<2

I tried to ask this in mathstack, but no one answered me. Let $B = B(x_0,R) \subset \subset \Omega$ a ball in $R^n$ with $\Omega$ a domain in $R^n$ with smooth boundary and consider two functions ...
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### Basic doubt in a free boundary problem for the Laplacian

I am studying the following article : http://hal.archives-ouvertes.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf In this article the authors considers $K \subset \{ x \in R^n ; x_1 = 0 \}$ a smooth, ...
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### Degeneracy and singularity of the $p$-laplace equation
In what sense is the $p$-Laplacian degenerate for $p$ greater than $2$ and singular for $p$ less than $2$?