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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questions
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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?
$$
\int_{S_1} (u-k)^2|\nabla u|^{p-2}...
- 473
3
votes
1
answer
372
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Geometric flow by the level sets of a harmonic function
Let $u$ be an harmonic function in a cylindrical domain $B_2^{n-1}\times(-1,1)\subset\mathbb{R}^n$, and suppose its level sets $\Gamma_t=\{u=t\}$ are graphs of functions on $B_2^{n-1}$.
Consider a ...
2
votes
0
answers
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Stability of weak solutions to quasilinear parabolic equations
Consider the quasilinear operator $A(x,t,\nabla u)$ satisfying $$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}$$ where $1<p<\infty$. Note ...
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Solution of $\vec{p}-$Laplace equation
Let $\Omega \subset {\mathbb{R}^n}$ is bounded domain with smooth boundary. We consider the bvp
$$ - \sum\limits_{I = 1}^n {{\partial _{{x_i}}}\left( {{{\left| {{\partial _{{x_i}}}u} \right|}^{{p_i} ...
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0
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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-...
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2
votes
0
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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{...
- 31
0
votes
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answers
203
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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,...
- 47
2
votes
2
answers
177
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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 ...
- 628
3
votes
1
answer
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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^...
- 31
2
votes
0
answers
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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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6
votes
1
answer
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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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3
votes
0
answers
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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:
$$E(X):=\frac{1}{2}\int_M\|dX\|^2\mathrm{dVol_g}\qquad ...
- 4,145
1
vote
0
answers
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Is the vanishing on boundary condition for the eigenvalue problem of the $p$-Laplacian important?
Consider the eigenvalue problem of the $p$-Laplacian, $$-\Delta _p u=\lambda |u|^{p-2}u,\ u\in W_0^{1,p}(\Omega)$$
In most of the literature I saw, an extra condition is mentioned that $u$ vanish on ...
- 704
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votes
3
answers
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Backgrounds of the p-Laplacian Operator
Motivation
I encountered the following partial differential equation (PDE) in a mathematical paper
$$\begin{array}{}
u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)
\\\qquad\quad-\...
- 271
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 ...
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3
votes
1
answer
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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( |\...
- 244
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votes
0
answers
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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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votes
1
answer
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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 ...
- 628
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votes
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answers
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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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1
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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 ...
- 31
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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}(...
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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, ...
- 259
1
vote
0
answers
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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 $2$?
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votes
1
answer
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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 ...
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