# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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### What is a Green's function in the language of $\mathcal{D}$-modules?

Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying ...
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### Is there an Infinite dimensional sheaf theory for analysis on manifolds?

I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the ...
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### Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates \...
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### Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $R^d$. (Say, a ball, say a cube...) For which classes $\cal C$ of functions, every function $f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
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### 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 ...
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### a question on the paper of Łaba and Wolff

I'm reading the paper A local smoothing estimate in higher dimensions by Izabella Łaba and Thomas Wolff. The paper can be found at J. Anal. Math. 88 (2002), 149–171, doi: 10.1007/BF02786576, arxiv: ...
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### A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory

The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
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### Probabilistic characterization of first Neumann eigenvalue

In this MO post, a question has been asked (and answered) about the probabilistic interpretation of the first Dirichlet eigenvalue of the Laplacian in terms of boundary hitting times. I wish to ask ...
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### Level Sets of Harmonic Maps

Can anybody point me in the direction of some references in which the level sets of harmonic maps between Riemannian manifolds are studied. (Sorry I am unfamiliar with the area and would like some ...
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### Hints on an expository article about Kardar-Parisi-Zhang (KPZ)

It seems the KPZ is the next big thing in mathematical physics and probability. The skeletal idea is probably that while classical averages are in the Gaussian universality class, lots of other ...
$\newcommand{\Cof}{\text{cof}}$ Let $k,d$ be even integers, such that $d\ge3$ and $2 \le k \le d-1$. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for ...