Tagged Questions

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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0
votes
2answers
571 views

Interior regularity for elliptic equations

The monograph "Non-Homogeneous Boundary Value Problems and Applications" by Lions and Magenes is infamous for developing a truly extensive regularity theory for elliptic problems on domains, and for ...
3
votes
1answer
256 views

why is it called the diamagnetic inequality?

How is the Diamagnetic inequality born? Why is it call this name? Diamagnetic inequality: $\big|\nabla|u|(x)\big|\leq \big|(\nabla+iA)u(x)\big|$.
4
votes
1answer
603 views

Nash's paper on parabolic equations.

I am currently studying the paper "CONTINUITY OF SOLUTIONS OF PARABOLIC AND ELLIPTIC EQUATIONS" by John Nash (cf. American Journal of Mathematics, Vol. 80, 1958). The author there establishes some a ...
1
vote
1answer
94 views

Boundedness of a given boundary value problem.

I've been given the following BVP: \begin{align*} -\Delta u = u- u^3,\: x\in \Omega \end{align*}\begin{align} u = 0,\: x\in \partial \Omega \end{align} where $\Omega\subset \mathbb{R}^N$ is bounded. ...
0
votes
1answer
192 views

Quantitative Global Schauder Estimates/Hölder Regularity

Consider the linear second order elliptic Dirichlet problem $$-\nabla\cdot (a\nabla u)\quad u=0 \text{ on }\partial\Omega$$ Condtion 1:$\Lambda |\xi|^2\geq\sum a_{i,j}(x) \xi_i \xi_j \geq \lambda ...
3
votes
2answers
322 views

Schauder estimates for higher order linear elliptic operator on manifold

Hi! Let $(M,g)$ be a smooth compact riemannian manifold without boundary. Let $L$ be a linear elliptic operator on $M$ of order $2k$ with smooth coefficients. Suppose i have $u\in W^{2k,2}(M)$ and ...
5
votes
0answers
644 views

elliptic regularity on manifolds

Hello! I'm reading the book of T. Aubin Some nonlinear problems in Riemannian geometry. In chapter 3 he introduces elliptic operators on manifolds, but then he gives regularity results for elliptic ...
0
votes
0answers
113 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 ...
5
votes
3answers
370 views

Divergence form Elliptic PDE Removable Singularity/Regularity Question

Idea Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) ...
4
votes
2answers
236 views

T. Carleman's method on eigenvalues asymptotics

What is the best available less or more modern introduction to the subject?
2
votes
1answer
265 views

Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation

The standard weak formulation of the Neumann problem for the Poisson equation is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$: $$ \int_{\Omega} \nabla u \nabla v d x = ...
4
votes
1answer
187 views

Simplicity of eigenvalues of an elliptic operator under a symmetry assumption

A striking difference in the spectral analysis of 2nd order elliptic boundary-value problems between one and several space dimensions is the following. In one space dimension, the eigenvalues are ...
1
vote
2answers
299 views

Variational problems whose lagrangian density depends on derivatives higher than 1.

The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...
1
vote
1answer
333 views

Elliptic Differential Equations with rough boundary data

Question stated roughly: Consider the Dirichlet problem for an elliptic equation on a ball. How much can we say about regularity at the boundary of non-linear elliptic equations? Further, how can one ...
2
votes
1answer
196 views

Solvability of a nonlinear elliptic equation

Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation \begin{equation} -\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0. ...
3
votes
1answer
283 views

Upper bounds for the solution of an elliptic PDE depending on a parameter.

Suppose I have the following PDE on $[0,1]^n$ $$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$ with periodic boundary conditions and $\int f(x) dx =0$ . ...
0
votes
1answer
129 views

What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases ?

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an oreintation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data ...
1
vote
1answer
440 views

Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

I asked the question before, but didn't get any reply, so I took the liberty to ask again. Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, ...
2
votes
1answer
220 views

Boundary Value Problem in the space of Distributions

I will not be rigirous cause my question does not require this. It is well known how to treat the solution (for example) of the problem $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial \Omega$ by the ...
3
votes
2answers
529 views

Elliptic regularity on bad domain

Let $\Omega \subset R^n$ be an open bounded set. Consider the Dirichlet problem $$ -\triangle u = 0, x \in \Omega, \quad u = \varphi, x \in \partial \Omega. $$ If $\varphi$ is a continuous function, ...
1
vote
0answers
130 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
1answer
184 views

Reference to the Existence and Uniqueness of the PDE system

Hi all I've the following Problem on systems of Partial Differential Equations.I have " N " Physical variables. and Finally I form the equation on a bounded domain having regular boundary in R^d. ...
2
votes
1answer
219 views

Uniform upper estimate for derivatives of solution of an elliptic PDE

Let for every $i=1,\dots,n$ and $j=1,\dots,n$ be given functions $a^{ij}\colon\mathbb{R}^n\to \mathbb{R}$, $b^{i}\colon\mathbb{R}^n\to \mathbb{R}$, $c\colon\mathbb{R}^n\to \mathbb{R}$ from space ...
3
votes
1answer
347 views

Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?

Consider a Riemannian manifold and let $\mathrm{id}$ be the identity operator, let $\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let $t > 0$ be a parameter. Does ...
1
vote
1answer
211 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
2answers
372 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
1answer
649 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
1answer
208 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 ...
2
votes
1answer
322 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
3answers
172 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
2answers
286 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
1answer
500 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 ...
2
votes
2answers
429 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
839 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
561 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
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
0answers
181 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
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
2answers
527 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
268 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
463 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
542 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
358 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
2answers
769 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 $$\Delta u=f$$ ...
5
votes
2answers
852 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
942 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
328 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
317 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
196 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 + ...