Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

learn more… | top users | synonyms

2
votes
0answers
98 views

Regularity of solution of nonlinear equation

Hi! Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball $B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$. Let $f\in C^{0,\alpha}(B_1)$ ...
7
votes
2answers
331 views

Fredholm alternative result for general elliptic system?

Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
4
votes
1answer
218 views

Please recommend some literature on the systematical theory of the elliptic systems!

Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
1
vote
1answer
259 views

Regularity of the right hand side (the source term) in Evans-Krylov theory

A well-known theorem of Evans and Krylov states that in an equation of the form $F(D^2 u)=g$, provided that the right hand side and $u$ both have Lipschitz gradient, and that $F$ is concave or convex ...
2
votes
1answer
135 views

The maximum in the Poisson problem on the cube with constant source

Question: Let us consider the Poisson problem on the square with constant source $1$ $$ \begin{cases} - \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\ u &= 0, \qquad \text{ on } \partial ...
0
votes
1answer
204 views

Mean value theorem for harmonic functions on ellipsoid

Is there any result like the mean value theorem for harmonic functions on ellipsoids (instead of sphere)?
0
votes
1answer
140 views

H_0^1 and C_infinity on the interior, does that imply classical limit is 0 on the boundary?

The solutions to the Dirichlet problem of elliptic PDE with smooth enough coefficients below to H_0^1 and also belong to C_infinity on the interior. Does that mean the classical limit of the function ...
1
vote
0answers
148 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 ...
15
votes
1answer
490 views

Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood

Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to ...
1
vote
1answer
222 views

In which way is this a linearization of the Gross-Pitaevskii-Equation?

In their paper [1] (full text at [2]) Bethuel et al on page 249 (bottom) linearize the moving frame Gross-Pitaevskii-Equation $0=-ic \partial_{x_1} \widetilde{v} - \Delta \widetilde{v} - ...
0
votes
1answer
136 views

Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following Theorem: ...
0
votes
2answers
626 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
317 views

why is it called the diamagnetic inequality?

How is the Diamagnetic inequality born? Why is it call this name? Diamagnetic inequality: $\big|\nabla|u|(x)\big|\leq \big|(\nabla+iA)u(x)\big|$.
5
votes
1answer
1k 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
98 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
218 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
399 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 ...
6
votes
0answers
726 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
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 ...
5
votes
3answers
417 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
256 views

T. Carleman's method on eigenvalues asymptotics

What is the best available less or more modern introduction to the subject?
2
votes
1answer
293 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
203 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
335 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
367 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
210 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
303 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
144 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 orientation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data ...
1
vote
1answer
443 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, ...
3
votes
2answers
244 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
616 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
140 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
198 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
240 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 ...
2
votes
1answer
387 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
231 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
418 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
809 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
213 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
1answer
353 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
181 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
304 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
615 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
511 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
876 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
605 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 ...