# 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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### 3D Homogenous Laplace equation with integral boundary conditions

I have the 3D Laplace equation:
$$\nabla^{3} T_w = 0$$
where $\nabla^{(3)}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ defined on $x \in [...

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### Iterative method for $p$-Laplacian

Consider the following iterative procedure for solving the $p$-Laplace equation $\nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$ with fixed Dirichlet boundary data:
$u_0$ is our initial guess, for ...

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### A priori estimate of an inhomogeneous p-Laplace equation with Dirichlet boundary condition

I'm currently working on this Dirichlet problem:
\begin{cases}
div(\sigma |\nabla u|^{p-2} \nabla u) = f &\quad {in }~ \Omega\\
u = g &\quad in~\partial\Omega
\end{cases}
with $\sigma \in L^...

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### When 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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### Euler-Lagrange equations for $p$-Harmonic vector fields

Harmonic vector fields are critical points of Dirichlet energy function on the set of all unit vector fields on $M$, which is defined as follows:
$$E(X):=\frac{1}{2}\int_M\|dX\|^2\mathrm{dVol_g}\...

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### Is the vanishing on boundary condition for the eigenvalue problem of the $p$-Laplacian important?

Consider the eigenvalue problem of the $p$-Laplacian, $$-\Delta _p u=\lambda |u|^{p-2}u,\ u\in W_0^{1,p}(\Omega)$$
In most of the literature I saw, an extra condition is mentioned that $u$ vanish on ...

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### Backgrounds of the p-Laplacian Operator

Motivation
I encountered the following partial differential equation (PDE) in a mathematical paper
$$\begin{array}{}
u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)
\\\qquad\quad-\...

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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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### Connection between the p and q Laplacians

I'm just looking for some quick and dirty intuition(and/or reading material) about the following:
I read that Hodge duality provides a way to interchange the p-Laplacian $ \Delta_p = \nabla\cdot( |\...

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### Discrete p-Laplacian

One of the definitions of the discrete (weighted) $p$-Laplacian is the following:
$$\Delta_{p,w}u(x):=\sum_y |u(y)-u(x)|^{p-2}(u(y)-u(x))w(x,y).$$
Consider the one dimensional case. Then the free ...

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### Space of p-harmonic functions

Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful.
I am interested in the space of p-harmonic functions on $\Omega$; that is, the ...

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### Density of restrictions of $p$-harmonic functions on a hypersurface

Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$.
Let $1<p<\infty$.
Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...

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### $\lVert u\rVert_{W^{2,p}}$ is bounded above by $\lVert \Delta_p u\rVert_{L^2}$ for $u \in W^{1,p}_0 \cap W^{2,p}$?

I work on a bounded domain in $\mathbb{R}^n$ and let $p \geq 2$ and the operator $\Delta_p u = \nabla \cdot (|\nabla u |^{p-2}\nabla u)$.
Does the following inequality (or something similar hold) for ...

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### Strong solution to $u_t - \Delta_p u = f$

For $p > 1$, consider the equation
$$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$
$$u(0) = u_0$$
$$u|_{\partial\Omega} =0$$
for all $v \in W^{1,p}(...

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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 $2$?

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### On the physics background of p-Laplacian equation

Could you tell me the physics background of p-Laplacian equation? Thank you!
Actually, I know nothing about this. But I am curious about the original of these PDEs or where they come from. Could you ...