**1**

vote

**1**answer

217 views

### Harmonic/conformal map composition between manifolds in either order?

Suppose $\mathcal{M}$, $\mathcal{N}$, and $\mathcal{P}$ are Riemannian manifolds (compact and of dimension 2, if it matters). It seems well-known that if $\phi:\mathcal{M}\rightarrow\mathcal{N}$ is ...

**2**

votes

**2**answers

396 views

### How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?

To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$.
Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$.
Prove: Suppose $f\in C^\alpha(M)$ satisfies ...

**1**

vote

**1**answer

724 views

### Extension of harmonic function

Suppose $u$ is a harmonic function of a domain $\Omega\subset \mathbb{R}^n$ and $u$ is continuous up to the boundary. If $\partial\Omega$ has an open smooth portion, can $u$ be extended to a harmonic ...

**0**

votes

**1**answer

212 views

### Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question:
Consider an orientation-preserving ...

**3**

votes

**1**answer

336 views

### A property of weakly harmonic functions

Let $u$ be a distribution in $\mathbb{R}^{n}$ and $\Delta u$=0 in the distributional sense. In addition $u\in L^{p}(\mathbb{R}^{n})$, $p>1$, then can we conclude that $u$ is zero almost everywhere ...

**6**

votes

**3**answers

176 views

### Stability of the spectrum for perturbations of the boundary

Consider the Laplace operator on a smooth bounded open set with Dirichlet boundary conditions. I need some result of the following type: if one perturbs the boundary in a suitable sense to be ...

**2**

votes

**2**answers

292 views

### Reference for all solutions of homogeneous elliptic and parabolic equations with Hölder continuous coefficients to be classical

Consider a uniformly elliptic equation
$$
\sum_{i,j=1}^n a_{ij}(x)\partial_{ij}u+\sum_{i=1}^n b_{i}(x)\partial_{i}u+c(x)u=0
$$
say, in an open ball $B\subset \mathbb R^n$, where coefficients are ...

**2**

votes

**1**answer

553 views

### Variational formulation for bilaplacian

I am trying to derive a variational formulation for the following problem $$\left\{ \begin{array}{ll} \Delta^2u=f, & \Omega \\ \Delta u+\rho \partial_{\nu}u=0, & \partial \Omega ...

**3**

votes

**2**answers

480 views

### Smooth Sobolev extension from $W^{1,p}(U)$ to $W^{1,p} (\mathbb{R}^n) $

The question I would be asking is roughly : do the smooth Sobolev functions defined on an open bounded domain extend to smooth Sobolev functions on the Euclidean space ?
For detail :
Fix $p \geq 1. ...

**1**

vote

**2**answers

853 views

### Geometric Mean Value Property

Does anyone know where I could find a proof of a variant of a version of the mean-value property for harmonic functions in Riemannian manifolds? I'm actually more interested in using an elliptic ...

**6**

votes

**3**answers

580 views

### Moser regularity proof avoiding John-Nirenberg lemma

I heard a rumor that there exists a proof by Moser-style iteration of the $C^{0,\alpha}$-regularity for $W^{1,2}$-solutions $u$ to elliptic equations with measurable coefficients which does not rely ...

**1**

vote

**0**answers

152 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 ...

**2**

votes

**0**answers

184 views

### Core of divergence form operator with unbounded coefficient

Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that ...

**2**

votes

**0**answers

162 views

### A first order ODE with Sobolev estimates

Suppose $f: {\mathbb R}\to {\mathbb R}$ is a continuous function satisfying
$$f(s) = c_- >0,\forall s<-T; f(s) = c_+<0, \forall s>T.$$
(For simplicity we may even assume that $f$ is ...

**19**

votes

**3**answers

1k views

### Convergence of finite element method: counterexamples

There are many known results proving convergence of finite element method for elliptic problems under certain assumptions on underlying mesh [e.g., Braess,2007]. Which of these common assumptions are ...

**1**

vote

**2**answers

546 views

### Hölder estimates on solutions of non-linear elliptic PDE.

In his book "Some non-linear problems in Riemannian geometry" T.
Aubin states the following result (Theorem 3.56):
Let $A(u)=F(x,u,\nabla u,\nabla^2u)$ be a non-linear second order
differential ...

**2**

votes

**0**answers

283 views

### A free boundary problem by finite difference method

I wanna discretize the following free boundary problem
Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$.
I apply finite ...

**2**

votes

**3**answers

477 views

### Finding an $H^1$ function given its values on $\partial\Omega$

Background
I've met this problem when I was trying to convert a elliptic PDE problem
into the corresponding variational problem in order to apply finite element method.
The PDE is an elliptic PDE ...

**4**

votes

**1**answer

565 views

### regularity of solutions of fractional laplacian

Hello, I am looking for boundary regularity of solutions of $(-\Delta)^s u= f(x)$ in $\Omega$ with Dirichlet boundary conditions and where $f $ is nice enough say $f\in C^{1,\alpha}(\overline\Omega)$. ...

**5**

votes

**1**answer

363 views

### A moving boundary in rock mechanics

I'm concern a moving boundary problem in rock mechanics.
We consider a problem of unsaturated flow of an in-compressible fluid in a
porous medium(rock) like D. Moreover suppose that support of a ...

**3**

votes

**2**answers

810 views

### The Hölder continuity condition of the Schauder estimates

The classical Schauder estimates (see the link)
http://en.wikipedia.org/wiki/Schauder_estimates
Requires $f\in C^\alpha$ in order to get a solution $u\in C^{2+\alpha}$ of the equation
$$\Delta u=f$$
...

**6**

votes

**2**answers

905 views

### Estimates on the Green function of an elliptic second order differential operator.

Let $D$ be a linear differential elliptic operator of second order
with infinitely smooth coefficients acting on real valued functions
on a compact manifold $M$. Let us assume that $D$ has no free ...

**5**

votes

**3**answers

954 views

### Integration by parts for a general negative-definite self-adjoint operator.

I suspect I am asking a very stupid question.
Suppose you have self-adjoint negative-definite operator $L$ densely defined on a space $L^2(\pi)$, with $Lf = \nabla \cdot ( A(x)\nabla f)$, for some ...

**4**

votes

**2**answers

338 views

### Can the solution of an elliptic operator with smooth coefficients have zeros of infinite order?

I've been told that the answer is no, but I'm having a hard time finding a reference. More precisely, I'm interested in the following. Let $\Omega \subset \mathbb{R}^n$ be a bonded open connected set, ...

**4**

votes

**0**answers

324 views

### Do there exist generalized conformal maps that preserve elliptic measure?

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction ...

**1**

vote

**0**answers

197 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 + ...

**2**

votes

**0**answers

251 views

### Is the complex harmonic extension of a $\mathcal{C^r}$ map from $S^1$ to $\mathbb{C}$ is smooth upto the boundary ?

Suppose we have a map $ h : S^1\to \mathbb{C} $ that we know is a $\mathcal{C^r} $ map ( in the sense of a map between 1-manifold ( or in the sense of a $2\pi$ periodic map from $\mathbb{R}\to ...

**2**

votes

**1**answer

389 views

### orthonormal basis of eigenvectors for laplacian on a concave polygon

I am interested in the Laplace operator $\Delta$ on a concave polygon.
When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$
is boundedly invertible. In addition, ...

**3**

votes

**0**answers

346 views

### problem with non linear pde

I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution?
$$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 ...

**1**

vote

**3**answers

307 views

### another solution to PDE possible?

hi there,
i have the following pde:
$$(\partial_x x^4 \partial_x - \partial_t^2)y(x,t)=0$$ and found the solution $$y=a+t^2-1/x^2$$, with a a constant.
Is this solution unique? Does anyone know of any ...

**2**

votes

**0**answers

293 views

### Vanishing solution of the Poisson equation at infinity

Hi, I am interested in finding some vanish bahavior at infinity of the solutions of this kind of equations:
$-\Delta\phi+a(x)\phi=b(x)$
where $a(x), b(x)\in L^{p}$ with $1\leq p\leq 3$. Besides ...

**2**

votes

**2**answers

448 views

### smoothness of solution for second order elliptic problem

Hello all,
could someone point me to a reference that ties the smoothness of the solution $u$ to the classical elliptic problem
$\nabla \cdot ( q \nabla u ) = f \;,\; x \in \Omega$
$u = g \;,\; x ...

**4**

votes

**1**answer

327 views

### Does Hölder continuity imply smoothness for the CMC equation: $u:D^2\rightarrow\mathbb{R}^n$, $\Delta u = 2H\partial_xu\times\partial_yu$, $H$ constant?

Context: I am currently reading through the freely available lecture notes from Tristan Riviere (here) on the applicability of integration by compensation in the analysis of various geometrically ...

**2**

votes

**0**answers

283 views

### Poisson problem with a “scaled” Laplacian.

Let $d_1$ and $d_2$ be positive constants. I'm considering a 2D Poisson-like problem of the form
$$ d_1\frac{\partial^2 u}{\partial x_1^2} + d_2\frac{\partial^2 u}{\partial x_2^2} = f$$ in the ...

**1**

vote

**1**answer

742 views

### Laplace equation over concentric spheres

Is there a closed formula for the solution of Dirichlet problem ($\Delta u=0$) for annulus $r <|x| < R$, $x \in R^n$ (n>2), with two given boundary value functions, $f$ over $|x|=r$ and $g$ over ...

**6**

votes

**3**answers

1k views

### Betti number and harmonic forms

Dear Experts,
On a compact, boundless, Riemannian manifold, the dimension of the space of harmonic k-forms is equal to the k-th Betti number. Is this correct (by Hodge theory)? For example, on the ...

**2**

votes

**1**answer

885 views

### Poisson equation with special Neumann BC

Hi
Consider Poisson equation with Neumann boundary condition but the right hand side of boundary condition is in term of the unknown function $u$.
How we can solve it?
$\Delta u(x) = f(x)\quad in~ ...

**3**

votes

**2**answers

653 views

### Trace space and Neumann boundary condition

In which sense is it possible to solve $\Delta u=0$, $\partial_\nu u=\phi$, for $\int\phi=0$ on a closed domain, say a ball $B^3\subset\mathbb R^3$?
For example would a $\phi\in L^p(\partial B^3)$, ...

**2**

votes

**0**answers

224 views

### Finer properties of a sequence of harmonic functions

This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer.
Background:
When ...

**3**

votes

**2**answers

679 views

### Maximum Principle fails when u∉C²(Ω)? Can't find example.

I would like an example where the maximum principle fails in a bounded smooth domain $\Omega$ where one has a solution which is not $C^2(\Omega)$ to $Lu=0$ where $L$ is elliptic and linear. This ...

**2**

votes

**1**answer

158 views

### Weakened conditions on the smoothness of the domain in the regularity and a priori estimate of Agmon, Douglis, and Nirenberg for elliptic systems

I have read in a couple of places (e.g. An Introduction to PDEs by Renardy and Rogers, p.309) that the smoothness hypotheses on the domain in the a priori estimate of Agmon, Douglis, and Nirenberg for ...

**6**

votes

**2**answers

2k views

### Moser iteration for elliptic systems

I heard that De Giorgi-Nash-Moser type regularity arguments fail for elliptic systems, but do not know where to start looking for more substantial information. Why does the regularity fail? Is there ...