Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

learn more… | top users | synonyms

-1
votes
1answer
96 views

Analytic extension of the exterior Newtonian potential into the domain

I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain. Definition of Newtonian ...
-1
votes
1answer
90 views

Regularity of solutions for a non linear elliptic equation

Let $v_k$ be a radial sequence of function that satisfies in $\Omega\subset\mathbb{R}^4$ $(-\Delta)^2 v_k=e^{v_k}$ $v_k(x)\leq v_k(0)=0$ $\left\Vert (-\Delta)v_k\right\Vert_{L^1(B_R(0))}=O(1)\qquad ...
1
vote
0answers
31 views

Constructing a family of domains for application of method of continuity in optimal transportation

Anyway can help me about this paper? http://arxiv.org/pdf/math/0601086v4.pdf I want to ask page 20, The author want to construct a family of subdomain for using method of continuity. But I can't ...
1
vote
0answers
149 views

Bunimovich stadium bouncing ball

http://terrytao.wordpress.com/2008/07/07/hassells-proof-of-scarring-for-the-bunimovich-stadium/ I cannot relate how the eigenfunctions normalizaed correspond to the probabality distribution of the ...
1
vote
0answers
135 views

Result like Brezis-Kato Lemma for Biharmonic equation.

Is there any result for the 4th order elliptic pde,like we do have for 2nd order pde: Consider the equation,\ $\Delta^2 u = a(x)u$ and $a(x)\in L^{\frac{n}{4}}(\Omega)$, where $\Omega$ is bounded. If ...
1
vote
0answers
134 views

Compactness of solutions of elliptic equation

Consider the following nonlinear elliptic equation $$ -\triangle u + u + u^3 = g, \quad x \in R^3. $$ If $g \in L^2(R^3)$, then the set $Q$ of solutions of above equation is bounded in $H^2(R^3)$, and ...
1
vote
0answers
153 views

On explicit eigenfunctions

Given an algebraic surface $S$ defined by an algebraic equation such as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest eigenvalue $\mu_{3}$ for the differential equation $\Delta ...
1
vote
0answers
198 views

References? Stability of the Cauchy problem for elliptic and backward-parabolic operators.

Dear community. I'm looking for survey articles about recent and old results about stability for the elliptic and backward-parabolic Cauchy problem. For example: Take $\partial_t u + ...
0
votes
0answers
43 views

Control of Hessian by its trace in a bounded domain

We know if $u:\mathbb{R}^n \to \mathbb{R}$ is a smooth function with compact support, then we can show, via integration by parts, that $$ \|\Delta u\|=\|\nabla^2u\| \tag 1 $$ where $||\cdot||$ is ...
0
votes
0answers
79 views

Schauder estimate on a bounded domain

We know if $u:\mathbb{R}^{n}\to \mathbb{R}$ is a smooth function with compact support, then we can show, via integration by part, that $$ ||\Delta u||=||{\nabla}^2u|| $$ where $||\cdot||$ is the ...
0
votes
0answers
50 views

global bi harmonic functions in a riemannian manifold

Any help will be appreciated thanks! Consider $(\mathbb{R^n},g)$ to be a Riemannian manifold. For simplicity we can assume the manifold to be asymptotically euclidean outside a compact domain ...
0
votes
0answers
69 views

Showing existence of positive weak solution of a PDE by CoV

Given the following PDE $$ \begin{cases} -\Delta u+\alpha=u^q &x\in\Omega\\ u=0 &x\in\partial\Omega \end{cases} $$ where $\Omega\subset\mathbb R^3$ is open bounded with smooth boundary, ...
0
votes
0answers
58 views

Linking theorem, elliptic pde

I am trying to solve some linear system of the form $$ \Delta \phi +p v(r)^{p-1} \psi = -f, \qquad \Delta \psi + qu(r)^{q-1} \phi =g $$ in $ \mathbb{R}^N$ where $ f,g$ are given and $ \phi,\psi$ are ...
0
votes
0answers
39 views

Regularity of solutions of strongly elliptic system: how smooth must the boundary be?

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with the boundary of class $C^{1,1}$. Let $A$ be a selfadjoint operator acting in $L^2(\Omega;\mathbb{C}^n)$. The operator $A$ is given by the ...
0
votes
0answers
77 views

Implicit function theorem on boundary points

I have the following examples: (1) $xy-1=0$ with $x\ge 0$. By the implicit function theorem, we can solve when $x\in(0, \infty)$. Here on the boundary we have $y=\frac{1}{x}\rightarrow \infty$ as ...
0
votes
0answers
65 views

How to prove it is uniformly bounded?

Let $\Omega$ be a bounded domain with smooth boundary. Say $\theta\in(0, 1]$. Let $u(x, \theta)$ be a solution to the problem $\Delta u-\theta u=g(x)$ subject to Neumann boundary condition. Suppose ...
0
votes
0answers
85 views

Elliptic PDE-Fredholm PDE(Is there a contradictory situation)

Let $E$ be a smooth vector bundle on a closed manifold $M$. Assume that $D:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a diff. operator which is a fredhoolm operator, in the algebraic ...
0
votes
0answers
74 views

inequality involving gradient of two harmonic functions

My question is about the last inequality in the case i) of the proof of lemma 2.3 of this paper: Existence of classical solutions to a free boundary problem for the p-Laplace operator, (I) by A. ...
0
votes
0answers
85 views

Estimate for an integral of a function of the solution to a PDE

Let $\Omega \in \mathbb{R}^3$ be a bounded smooth domain. Assume that smooth functions $\sigma_1,\sigma_2$ satisfy $\sigma_1-\sigma_2 \in C_0^\infty(\Omega)$ and $\lambda\leq \sigma_1, \sigma_2 \leq ...
0
votes
0answers
110 views

positive eigenfunction on complete Riemannian manifold

Let $(M^n,g)$ be a complete(non-compact) Riemannian manifold. Consider the positive solution to the equation $$ \Delta u = u $$ where $\Delta=\nabla_i \nabla_i$ is negative semi-definite. Is there ...
0
votes
0answers
98 views

Heat equation with heterogeneous heat conduction

I'm trying to discretize and a heat conduction/diffusion problem using finite differences and I was wondering how to use a discrete heat conduction coefficient defined per cell (instead of per ...
0
votes
0answers
77 views

Mean value formula for high order elliptic operators

Weyl's lemma says any distributionally harmonic function $u\in L^1_{\text{loc}}$ is harmonic. One way to prove it is as follows. Take radially symmetric compactly supported and smooth approximation ...
0
votes
0answers
58 views

basic doubt in a free boundary problem for the p-Laplacian

I am studying the following article : http://hal.archives-ouvertes.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf In this article the author consider $K \subset \{ x \in R^n ; x_1 = 0 \}$ a smooth, ...
0
votes
0answers
66 views

A density result for biharmonic functions

Let $U_1, U_2\subset\mathbb R^3$ open and homeomorphic to the open unit ball, with sufficiently smooth boundary, such that $\overline{U}_1\subset U_2$. Is is true that every biharmonic function $u$ ...
0
votes
0answers
259 views

Analytic solution of Poisson's equation

Consider Poisson's equation $\nabla^2 u = 1$ on a square of side-length 1 centered at the origin. Cut out a circle of radius 1/3 at the center of this square. Impose a von Neumann boundary condition ...
0
votes
0answers
122 views

second derivatives of initial conditions for a non-linear PDE

Hello, I have the following problem: Having a non-linear PDE of second order, there are the initial conditions $F(x,0)=g(x)$, $\partial_yF(x,0)=h(x)$. The question is: Can the second derivatives ...