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

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2
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2answers
146 views

What can we say about the boundary of the level set of a Sobolev function?

I'm a beginner of the area of free boundary problem. Let me first give some background: $\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph. ...
1
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2answers
72 views

References for non-zero boundary value problem

I studied linear elliptic, parabolic and hyperbolic PDEs (boundary/initial value problem) in terms of existence, uniqueness and regularity. I studied always, following Evans book "PDE", the case with ...
3
votes
1answer
92 views

Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman ...
1
vote
1answer
177 views

Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research: $$\Delta f - e^f \partial_s f = E(s,t)\,,$$ where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the ...
-1
votes
1answer
102 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
92 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
70 views

Regularity result for the elliptic equation with Neumann conditions

I am having troubles to justify well some inequalities related with the classical theory of elliptic equations. Let us consider the problem \begin{align*} -\Delta c & =f,\text{ in }\Omega\\ ...
1
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0answers
90 views

Gilbarg-Trudinger's book Theorem 4.13

I am reading Gilbarg-Trudinger's book "Elliptic Partial Differential Equations of Second Order". I do not understand the proof of Theorem 4.13. Theorem 4.13 is a special case of Kellogg's theorem in ...
1
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0answers
92 views

Asymptotics of “heat” semigroup

Consider a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary. Consider a second order elliptic operator $L$ on $L^2(\Omega)$, defined by either the Dirichlet or Neumann boundary ...
1
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0answers
54 views

Second derivative estimates for gradient dipendent elliptic equations

Caffarelli in the article " Interior a priori estimates for solutions of Fully non linear Equations" Ann. Math 130,No.1, 1989 proved that a continuous viscosity solution $ u $ of a uniformly elliptic ...
1
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0answers
30 views

Eigenvalue overdetermined problem

Consider the following overdetermined eigenvalue problem for $\Omega \subset \Bbb{R}^2$: $$(1) \ \ \ \ \begin{cases} - \Delta u = \lambda u & \text{ in }\Omega \\ u = 0 ...
1
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0answers
125 views

Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...
1
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0answers
108 views

$L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value

Let $u \in H^1((0,T)\times S)$ be the unique solution of $$u_{tt} + \Delta u =0$$ $$u|_{t=0}= u_0$$ $$u|_{t=T}=0$$ where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...
1
vote
0answers
83 views

maximum principle on compact manifolds with boundary

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ ...
1
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0answers
75 views

Method of proving the regularity of the minimizer of geometric variational problems

Recall the proof of the $(\Lambda,r_0)$-perimeter minimizer. We say that $E$ is a $(\Lambda,r_0)$-perimeter minimizer in an open set A provided $sptD\chi_E=\partial E$ and there exist two constants ...
1
vote
0answers
48 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 $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...
1
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0answers
90 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, ...
1
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0answers
60 views

Regularity of Schrödinger Resolvent

The following problem keeps bothering me: Let $H:=-\Delta+V$ be a Schrödinger-Operator in $\mathbb{R}^n$, where $V$ is a Kato-Potential of type $K_n$, which especially yields that $H$ is e.s.a. on ...
1
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0answers
146 views

Existence of solution?

I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here. Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf Let ...
1
vote
0answers
25 views

Differential operator with codimension 2 singularity in the domain

The soft version of the question is as follows: suppose I have a linear operator, and I know it is a 'nice' differential operator on its domain minus a singular set of codimension two. Does the ...
1
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0answers
152 views

Stokes operator without dirichlet boundary condition

Let $\Omega$ be a domain, then the following stokes operator is quite well known : $\mathcal{H} \rightarrow \mathcal{V}_{\sigma} $ $f \rightarrow u$ such that $ - \Delta u = f $ where $ ...
1
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0answers
83 views

mixed Dirichlet Neumann regularity for an elliptic equation

Here is a problem which may be easy for some of you but not for me. Statement of the problem: Denote $\Omega := \{ (x,y) \in (0, \infty) \times (-\infty,\infty) \}$. Let $f \in L^2(\Omega)$ then by a ...
1
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0answers
93 views

On the proof of a $W^{2,p}$ estimate - regularity on eliptic PDE

I see this proof on http://www.math.uiowa.edu/~lwang/cccalderon.pdf and I couldn't understand what he did. If $||f||_{L^p(B_{4})} = \delta$ is small and the measure $|\{ x \in B_1; ...
1
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0answers
86 views

Limit Toward Discontinuous Point of Dirichlet Boundary Value

The question arises from a paper on Schwarz's domain decomposition method (click here). We consider a bounded domain in $\mathbb{R}^2$ and a curve splits it into two, see the figure below. Now we ...
1
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0answers
34 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
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0answers
89 views

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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0answers
155 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
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0answers
152 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
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0answers
143 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 ...
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0answers
154 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
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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 + ...
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0answers
33 views

Is the heat kernel satisfies the heat equation in viscosity sense?

Let us see the heat kernel \begin{equation} k(x,t)= \begin{cases} (4\pi t)^{-\frac{n}{2}}\exp\{-\frac{1}{4}t^{-1}|x|^{2}\},t>0\\ 0,t\leq0. \end{cases} \end{equation} It is easy to see that $k\in ...
0
votes
0answers
74 views

Limit for eigenvalues of the Dirichlet problem

If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem $$-\Delta\,u=\lambda\,u\,\,\mbox{in}\,\,\, \Omega, \, u=0\,\,\,\mbox{on}\,\,\, ...
0
votes
0answers
50 views

regularity of non-local linear elliptic equation

$\alpha\in (0,1)$, $u$ satisfies: \begin{equation*} b\cdot \nabla u(x)+\sum_{i=1}^d \int_{R} \left[u(x+se_i)-u(x)-s\mathbb{I}_{\{|s|\leq ...
0
votes
0answers
57 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
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0answers
101 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
75 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
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0answers
53 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
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0answers
96 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
70 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
98 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 ...
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0answers
87 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
118 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
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0answers
121 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
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0answers
88 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 ...
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0answers
70 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
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0answers
272 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
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0answers
126 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 ...