**3**

votes

**1**answer

231 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 ...

**2**

votes

**1**answer

84 views

### Regularity of Hodge Laplacian on bounded domains

I need a reference for the $W^{s,p}$ regularity of the Hodge boundary value problem on bounded domains. I need estimates
$\lVert \omega \rVert_{W^{s+2,p}} \leq c \lVert f \rVert
_{W^{s,p}}$, for ...

**1**

vote

**1**answer

168 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**

votes

**1**answer

104 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

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

**0**answers

93 views

### Regularity of weak solution

I have also posted the question here. Let me explain what difficulties I have. In fact, one may write
\begin{equation}
\partial_1(f-\partial_1 u)=0
\end{equation}
in $\Omega$. Then one may have the ...

**1**

vote

**0**answers

80 views

### Existence of the solution of a Dirichlet type differential equation

I'm reading the first chapter of the book A geometric approach to free boundary problems by Caffarelli and Salsa, see the PDF here. The question came from Page 14—15. Let me state my question:
...

**1**

vote

**0**answers

52 views

### Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type
$$
-a\Delta u + f(u) = 0,
$$
$$
u|_\Gamma = u_0
$$
by Newton’s method when its convergence is global and monotonic.
...

**1**

vote

**0**answers

78 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**

vote

**0**answers

96 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}\,\,\, ...

**1**

vote

**0**answers

108 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**

vote

**0**answers

94 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**

vote

**0**answers

60 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**

vote

**0**answers

32 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**

vote

**0**answers

109 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

**0**answers

98 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**

vote

**0**answers

86 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

**0**answers

55 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**

vote

**0**answers

102 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**

vote

**0**answers

66 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**

vote

**0**answers

150 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

**0**answers

28 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

161 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

94 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

94 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

89 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

35 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

99 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, ...

**1**

vote

**0**answers

157 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

157 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

146 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

56 views

### How Minimal solution is obtained as limit of approximations

I have encountered a problem in the proof of a Lemma in an article. The image of Lemma and it's proof is this:
I can understand the proof, but I don't know why this solution which is obtained as a ...

**0**

votes

**0**answers

31 views

### Reference request: Weak harnack inequality for biharmonic equation

I have seen a lemma which I do not have any reference and hint for it.
Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and
let $u$ be a positive distributional supersolution to ...

**0**

votes

**0**answers

128 views

### Harmonic function with Dirichlet boundary condition

Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...

**0**

votes

**0**answers

39 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

**0**answers

58 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

108 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

80 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

60 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

71 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

109 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

88 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

119 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

139 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

91 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

71 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

283 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

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 ...