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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Weighted Poincare inequality for $p$ harmonic functions

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}...
Adi's user avatar
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3 votes
1 answer
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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 ...
Jingeon An-Lacroix's user avatar
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 ...
Adi's user avatar
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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} ...
Trần Quang Minh's user avatar
3 votes
0 answers
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About p-laplacian and variations

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-...
Cézar Bezerr's user avatar
2 votes
0 answers
110 views

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{...
superdave99's user avatar
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205 views

3D Homogenous Laplace equation with integral boundary conditions

I have the 3D Laplace equation: $$\nabla^{2} T_w = 0$$ where $\nabla^{2}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ defined on $x \in [0,...
Avrana's user avatar
  • 47
2 votes
2 answers
193 views

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 ...
Tommi's user avatar
  • 638
3 votes
1 answer
273 views

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^...
ANZM91's user avatar
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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 ...
Tommi's user avatar
  • 638
6 votes
1 answer
1k views

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 ...
Rajesh D's user avatar
  • 704
3 votes
0 answers
327 views

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}\qquad ...
C.F.G's user avatar
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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 ...
Rajesh D's user avatar
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6 votes
3 answers
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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-\...
Hosein Rahnama's user avatar
4 votes
3 answers
222 views

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 ...
math student's user avatar
3 votes
1 answer
187 views

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( |\...
funda's user avatar
  • 244
2 votes
0 answers
455 views

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 ...
scouser's user avatar
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2 votes
1 answer
175 views

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 ...
Tommi's user avatar
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2 votes
0 answers
122 views

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 $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...
Joonas Ilmavirta's user avatar
3 votes
1 answer
255 views

$\lVert u\rVert_{W^{2,p}}$ 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 ...
Marcus_Thel's user avatar
3 votes
0 answers
223 views

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}(...
jamesC's user avatar
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1 vote
0 answers
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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, ...
math student's user avatar
1 vote
0 answers
185 views

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 $2$?
Flux's user avatar
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6 votes
1 answer
2k views

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 ...
jiaoheming's user avatar