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21 votes
1 answer
742 views

Non real eigenvalues for elliptic equations

I am looking for an example of a pure second order uniformly elliptic operator $L=\sum_{i,j=1}^da_{ij}(x)D_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a ...
Giorgio Metafune's user avatar
14 votes
2 answers
536 views

Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...
Tobias Diez's user avatar
  • 5,824
11 votes
3 answers
678 views

Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...
badmf's user avatar
  • 532
11 votes
2 answers
1k views

Concentration compactness. Can this concept be stated in a theorem?

I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk. When I approached the speaker ...
Zinkin's user avatar
  • 501
10 votes
1 answer
973 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here: Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...
Svetoslav's user avatar
  • 261
9 votes
2 answers
1k views

Density of restrictions of harmonic functions inside a ball

Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let $$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
Ali's user avatar
  • 4,145
9 votes
2 answers
418 views

Reference request: Parabolic Equations

I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
Falcon's user avatar
  • 452
9 votes
1 answer
1k views

Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
Gio712's user avatar
  • 253
8 votes
3 answers
1k views

Are all positive eigenfunctions principal eigenfunctions?

In a given domain $\Omega$, we have: $\Delta u=-\lambda u$ with $u>0$. Does this mean that $u$ is a principal eigenfunction for $\Delta$ in $\Omega$? Also, more generally, does this also apply for $...
Holden Lyu's user avatar
8 votes
0 answers
251 views

Smoothness of solution map for PDE

I am wondering what sort of results are available for the following sort of problem, or where to look in the literature for work dealing with such problems, especially in the degenerate elliptic ...
Quarto Bendir's user avatar
7 votes
1 answer
342 views

Does the pointwise mean value property imply harmonicity?

Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property: for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that $$ u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(...
Guy Fsone's user avatar
  • 1,101
7 votes
0 answers
80 views

Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator

Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
MyShepherd's user avatar
7 votes
0 answers
351 views

Fractional Laplacian and chain rule

For the classical Laplacian, we have $$\Delta (h(u)) = h'\Delta u + h''(u)|\nabla u|^2$$ for smooth functions $h$ and $u$. Does a similar chain rule hold (up to a reminder term) also for the ...
Zac's user avatar
  • 161
6 votes
1 answer
696 views

Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$

The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to $$ -\Delta u=f\hspace{3cm}(1)? $$ I'm of ...
leo monsaingeon's user avatar
6 votes
1 answer
322 views

finding subharmonic function on the ball with both Dirichlet and Neumann boundaries prescribed

I have a question which looks like some sort of inverse problem. Let $B$ denote the unit ball centered at the origin in $R^N$ (take $N \ge 2$). Given any $h:\partial B \rightarrow (0,\infty)$ (smooth) ...
Math604's user avatar
  • 1,385
5 votes
1 answer
297 views

A scaled fractional ''Sobolev inequality''

Does a fractional interpolation inequality similar to $$ \int_{B_R(0)} |u| dx \le C R^{2} \sqrt{\log(2R)} \Big( \int_{\mathbb R^2}\int_{\mathbb R^2} \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy + \int_{B_1(...
Jay's user avatar
  • 109
5 votes
1 answer
166 views

Strong maximum principle for a PDE with coefficient in $L^1$

Let $U$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$. Set $N = \frac{2n}{n-2}$. I am interested in the following equation: $$ -\Delta \phi + R \phi + \phi^{N-1} = 0 $$ ...
Romain Gicquaud's user avatar
5 votes
0 answers
213 views

Elliptic regularity and Sobolev spaces

Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e. $$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$ where $a$ are ...
G. Blaickner's user avatar
  • 1,429
4 votes
2 answers
1k 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 $\phi\in L^p(\partial B^3)$, $...
Mircea's user avatar
  • 2,041
4 votes
2 answers
505 views

Eigenfunctions of the Laplacian on singular spaces

Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$...
guest's user avatar
  • 43
4 votes
1 answer
698 views

Poincare Inequality for $H^2$ function satisfying homogeneous Robin boundary conditions

Let $\Omega\subset\mathbb{R}^3$ be a bounded smooth domain. In general, for a Poincare inequality of the type $$\|u\|_{L^2}\le C \|\nabla u\|_{L^2}$$ to hold for all $u\in X\subset H^1(\Omega)$ and $C$...
Fozz's user avatar
  • 287
4 votes
3 answers
309 views

Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$

Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
Bogdan's user avatar
  • 1,759
4 votes
1 answer
147 views

Embeddings of the maximal domain for the Laplacian

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function: $$D = \left\{ f \in L^2(\...
MeS's user avatar
  • 41
4 votes
1 answer
2k views

Crandall & Rabinowitz Theorem, bifurcation curves

Crandall & Rabinowitz Theorem states what follows. We have got a Banach Space $(X,||\cdot||)$ and an equation of the following type: $$ F(\lambda,u) = \lambda u - G(u) = 0, $$ where $G \in C^1(X,X)...
Alessio Di Lorenzo's user avatar
4 votes
1 answer
267 views

Scaled Harnack inequality $\sup_{B_r} v \le c\,(1-r)^{-p}\, \inf_{B_r} v$

Where can I find a proof of the following scaled version of Harnack inequality? Let $v$ be a non-negative solution of ${L}u = 0$ in $B_1$, with $L$ a uniformly elliptic operator. Then, for $r<1$,...
Riku's user avatar
  • 839
4 votes
1 answer
562 views

Fundamental solutions for degenerate elliptic equations

I am looking for a paper or a book that says about the existence and some estimates (like these in the non-degenerate case) of the fundamental solutions for degenerate elliptic equations $L = -divA\...
Phi Le's user avatar
  • 51
4 votes
1 answer
343 views

Conditions for the existence of a solution to a semilinear second-order PDE with a-priori bounds

Consider the general semilinear elliptic second-order PDE $$ u_t-\mathcal L u=f\left(t,x,u,\nabla u\right) $$ where $\mathcal L$ is an elliptic linear operator (like minus the Laplace operator), $t \...
FDR's user avatar
  • 91
4 votes
1 answer
370 views

Equivalence of viscosity and weak solutions for the Poisson equation

Suppose $\Omega$ is a bounded smooth domain in $\mathbb{R}^d$. How does one prove that weak solutions are viscosity solutions and vice versa for the problem $$ \begin{cases} -\Delta u = f(x) & \...
user avatar
4 votes
1 answer
586 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 $\int f(x) dx =0$ . ...
RadonNikodym's user avatar
4 votes
0 answers
90 views

Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bF}{\mathbb{F}} \newcommand{\cF}{\mathcal{F}} \newcommand{\eps}{\...
Akira's user avatar
  • 835
4 votes
0 answers
129 views

Trace-class heat semigroups

Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator. Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$ $$T_{\varphi}(u) :=...
user avatar
4 votes
0 answers
103 views

Characteristic of Sobolev space generated by Hörmander vector fields

Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. Suppose that $X=(X_{1},X_{2},\ldots,X_{m})$ are smooth vector fields defined on $\mathbb{R}^{n}$ and ...
pxchg1200's user avatar
  • 287
4 votes
0 answers
174 views

Techniques for showing non-degeneracy results (PDE)

Motivation: Consider the equation, $$-\Delta u = u^p$$ in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
Student's user avatar
  • 537
4 votes
0 answers
143 views

If theorem valid for compactly supported distribution then is it also valid for tempered distribution?

I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution. For instance, Theorem: Any $A \in \Psi^{m}$ ...
Curious student's user avatar
4 votes
0 answers
136 views

Davies' definition of elliptic operators in "Heat Kernels and Spectral Theory"

I am trying to find my way through Davies' book, and one of the difficult points is his choice of what "elliptic operator" means in his text. This is the first time that I encounter some of the ...
Alex M.'s user avatar
  • 5,407
4 votes
0 answers
198 views

Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space

It is known that for $\alpha\in(0,1)$ and $p>1$, the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by $$ W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^...
Wenguang Zhao's user avatar
4 votes
0 answers
89 views

How can I can derive an explicit bound for the solution of the poisson's PDE?

i need some help on this question Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ (say a square) with $\partial{\Omega} =\Gamma_{1} \cup \Gamma_{2} \cup\Gamma_{3} \cup\Gamma_{4}$. A structure ...
user106481's user avatar
3 votes
2 answers
618 views

Interior smooth regularity

I recently read the PDE book of L. Evans, and in its chapter 6 some kinds of regularities of second-order elliptic equations were discussed. My question is about its proof of interior smooth ...
User's user avatar
  • 402
3 votes
1 answer
333 views

References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$

I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a_{ij}= a_{ji}$ are only assumed to ...
Kernel's user avatar
  • 446
3 votes
1 answer
362 views

Solution of the fractional Laplace equation on a ball

What is the expression of the (non $u \equiv 0$) solutions to \begin{align*} (-\Delta)^s u &= 0 && x \in B_r(0) \\ u&=0 && x \in \mathbb R^N \setminus B_r(0), \end{align*} ...
user173196's user avatar
3 votes
2 answers
1k views

Orthogonality to harmonic functions

Let $a_0$ and $b_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define $a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{B}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$ $b_n=f\Delta^{...
A random mathematician's user avatar
3 votes
1 answer
845 views

Moser estimates?

Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
student's user avatar
  • 31
3 votes
1 answer
344 views

Pseudoinverse of Neumann-Laplacian

Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that $$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$ Further assume a solvability condition $$\int_\Omega f ~\mathrm{d}\...
Elias Ka's user avatar
3 votes
1 answer
203 views

What are the subelliptic estimates for the Rockland operator?

Let $X=(X_1, X_2, \dots, X_n)$ be a smooth vector field on $\mathbb{R}^n$. The operator $L=(\sum_{i=1}^{m}X_i^2)^p$, where $p$ is an integer, is a degenerated operator. If $X$ satisfies the Hörmander ...
displayname's user avatar
3 votes
0 answers
84 views

About the naturality of Krasnoselskii genus on Variational Methods

I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
Pitbull's user avatar
  • 131
3 votes
0 answers
41 views

Functional of fully nonlinear equations

Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is ...
Davidi Cone's user avatar
3 votes
0 answers
110 views

On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections

Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
Pelle Steffens's user avatar
3 votes
0 answers
130 views

Is the range of the exterior covariant derivative closed in $L^{2}$?

Let $(M,g)$ be a compact Riemannian manifold. Given a tensor bundle $\mathbb{E}$, let $\nabla:\Gamma(\mathbb{E}) \rightarrow \Gamma(T^{*}M\otimes \mathbb{E})$ be the canonical connection induced by ...
MyShepherd's user avatar
3 votes
0 answers
190 views

$C^1$-regularity of solution of a Dirichlet problem

I've been stuck trying to hunt down a proof/reference for a certain regularity result which I need for my thesis (all the authors I've consulted refer to it as "well-known"). Consider the ...
Othman El Hammouchi's user avatar
3 votes
0 answers
324 views

Would you help me to find this expression?

I'm studying a paper, which I'll include a small piece here. And I'm struggling to calculate $$C_n\|u_{m,n}\|^{\left(\frac{2*}{2}\right)^k\frac{2*-q}{(r_k)^k}}_{L^{2*}(\Omega)}$$ Where $\Omega$ is an ...
Silvinha's user avatar