**2**

votes

**1**answer

175 views

### Does this linear elliptic equation have a weak solution?

Let $Q = \Omega \times (0,C)$ where $\Omega$ is a bounded domain, write $(x,y) \in Q$ for $x \in \Omega$ and $y \in (0,C)$. Is the problem
$$\Delta_{(x,y)}v = 0\quad\text{in $Q$}$$
$$\frac{\partial ...

**1**

vote

**1**answer

65 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 $f : \mathbb R \times \mathbb R/\mathbb Z \to \mathbb R$, $f = f(s,t)$, is the unknown ...

**-1**

votes

**1**answer

98 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

**1**answer

91 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

**0**answers

23 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**

vote

**0**answers

149 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**

vote

**0**answers

76 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**

vote

**0**answers

90 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**

vote

**0**answers

83 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**

vote

**0**answers

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

**0**answers

150 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

**0**answers

140 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

**0**answers

137 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

**0**answers

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**

vote

**0**answers

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

**0**answers

24 views

### Linearization of An Eikonal Equation

I would appreciate any help with the following problem:
Suppose $(\Omega,g)$ is a three dimensional Riemannian manifold with smooth boundary where $\Omega \subset \mathbb{R}^3$ and assume that there ...

**0**

votes

**0**answers

40 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

**0**answers

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

**0**answers

82 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

**0**answers

52 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

**0**answers

73 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

**0**answers

63 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

**0**answers

44 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

**0**answers

81 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

**0**answers

66 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

**0**answers

88 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

**0**answers

77 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

**0**answers

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

**0**answers

112 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

**0**answers

105 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

**0**answers

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

**0**answers

59 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

**0**answers

68 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

**0**answers

263 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

**0**answers

124 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 ...