For questions about elliptic differential operators.

A differential operator is said to be elliptic if its principal symbol is invertible.

For questions about elliptic differential operators.

A differential operator is said to be elliptic if its principal symbol is invertible.

166 questions

votes

I'm trying to approach the Atiyah-Singer Index Theorem by getting an overview of the area.
One thing that confuses me a lot is that some treatments give (and hence prove) the theorem for Dirac operators, while other sources not even mention Dirac…

EmarJ

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Consider the eigenvalue problem $\Delta u + \lambda u = 0$ on some bounded domain $\Omega \subset \Bbb R^d$ with smooth boundary, with Dirichlet data $u|_{\partial\Omega} =0$. It is known that solutions $u$ are real-analytic inside $\Omega$.
This is…

Yoni Rozenshein

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We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta u dx $$ where $n$ is the unit outward normal ,…

user490539

votes

I'm currently working on my master's thesis, but I'm stuck.
Consider the PDE $\nabla\cdot (\sigma\nabla u)=0$ on $B_1\subset \mathbb{R^3}$, where $\sigma:B_1\rightarrow \mathbb{R}_+$ is constant inbetween the radii $0

Nem49

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I have a question regarding the construction of a barrier frequently used in PDE. The barrier used is the following:
Let $\Omega$ be a uniformly convex domain in $\mathbb{R}^n$ with $C^2$ boundary. Here uniformly convex means there exists some $r>0$…

Cale Rankin

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I'd like to solve the following problem, but I don't know how to approach it.
(Adjoint dynamics) Suppose that $u$ is a smooth solution of
$$\left\{\begin{align}
u_t+Lu&=0\quad \text{in}\quad U_T\\
u&=0\quad \text{on}\quad \partial…

Pedro

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One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ ($C$ probably depends on $s$) such that we have an…

jxnh

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I have seen the cohomological form of the index theorem usually stated in the following form:
$$
\int_X \varphi^{-1}\left(\operatorname{ch}([\sigma(P)])\right).\operatorname{todd}(TX\otimes\mathbb C)
$$
where $\varphi$ is the Thom isomorphism, or…

A Rock and a Hard Place

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Can anyone suggest a reference for elliptic PDE on the all of $\mathbb{R}^d$, as opposed to some bounded domain $\Omega$, covering the standard topics of existence, uniqueness, and regularity. I specifically mean elliptic operators with variable…

user2379888

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$\newcommand{\M}{M}$
Let $\M$ be a smooth oriented Riemannian manifold.
Let $\sigma$ be a differential $k$-form on $\M$ with coefficients in $L^1(\M)$. We say $\sigma$ is weakly harmonic if
$$
\int_{\M} \langle \sigma , \Delta \alpha \rangle=0 \,…

Asaf Shachar

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Let
$$
P:\Gamma(E)\rightarrow\Gamma(F)
$$
be an elliptic partial differential operator, with index zero and closed image of codimension 1, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections of vector bundles $E\rightarrow M$ and…

mdg

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In a paper I am reading there is a line which I don't quite understand. The setup is the following. We have a compact Riemannian manifold $N$, then we consider the cylinder $M=\mathbb{R} \times N$ and we have a nontrivial solution $u:M\rightarrow…

TheManWhoNeverSleeps

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I'm doing some self-study on Hodge theory and elliptic operators right now. I'm trying to come up with an example of a harmonic $p$-form $\omega$ on a compact manifold, i.e. a form such that $d\omega =0$ and $d^* \omega=0$ where…

lvxvl

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I am reading chapter 14 of Michael E. Taylor's book, Partial Differential Equations. I am confused because I can't seem to find what it means for a fully nonlinear PDE to be elliptic.
Here is the setup to my question: suppose $f=f(x,\zeta)$ is…

Andrew

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We are given (weak) Gårding's inequality for elliptic pseudodifferential operators:
Given $a\in S^m$ such that $\operatorname{Op}(a)$ is an elliptic operator, namely $\exists c,R>0$ such that for each $x,\xi\in\mathbb R^n$, we have…

Yai0Phah

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