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
votes
2answers
305 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 ...
3
votes
1answer
628 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
2answers
524 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
2answers
877 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
3answers
615 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
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 ...
2
votes
0answers
189 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
0answers
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
3answers
2k 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
2answers
570 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
0answers
297 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
3answers
491 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
1answer
589 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
1answer
364 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 ...
4
votes
2answers
868 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
2answers
958 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
3answers
979 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
2answers
357 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
0answers
330 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
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 + ...
2
votes
0answers
252 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
1answer
402 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
0answers
354 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
3answers
308 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
0answers
303 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
2answers
453 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 ...
5
votes
1answer
336 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
0answers
291 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
1answer
773 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
3answers
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
1answer
906 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
2answers
680 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
0answers
227 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
2answers
697 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
1answer
160 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
2answers
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 ...