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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21
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521 views

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 ...
17
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527 views

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 ...
16
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1answer
669 views

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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924 views

Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$

$\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\...
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575 views

Are harmonic mappings non-singular outside a set of measure zero?

Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb D^n$. Let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving immersion, and let $\omega :\...
12
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552 views

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 $K$. ...
10
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109 views

A geometrical problem in terms of a convex function

I wish to know whether the following problem has ever been investigated. Let $D$ be a convex domain in ${\mathbb R}^d$, with smooth boundary $\partial D$. Let $\vec V:\partial D\rightarrow{\mathbb R}^...
10
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194 views

Elliptic regularity of perturbed scalar curvature in Kazdan & Warner

In their paper A Direct Approach to the Determination of Gaussian and Scalar Curvature Functions, Kazdan and Warner claim something along the lines of: if $g$ is a metric in $W^{2,p}$ ($p>n$) whose ...
10
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439 views

Uniqueness of singular Hermitian-Einstein metric along Yau-Donaldson flow?

The following question is related to Singular Yang-Mills theory The Hermitian metric $H$ along the fibers of the holomorphic vector bundle $E$ over a K\"ahler manifold $(M,\omega)$ is Hermitian-...
10
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243 views

Is there a classification of differential equations over the field of fractions of formal power series? (characteristic 0)

Let $k$ be an algebraically closed field of characteristic 0. Consider the field of fractions of formal power series $K:= Frac(k[[T_1,...,T_n]])$. We have the corresponding algebra of differential ...
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467 views

Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks! Definition ...
9
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314 views

Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure. Assume $\partial\Omega=\partial\Omega'$. Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
9
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624 views

Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?

Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...
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224 views

Why is it hard to obtain improved $L^6$ bound of eigenfunction of Laplacian on 2-dimensional compact Riemannian manifold?

Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$. C.D. Sogge proved that we have the ...
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224 views

Measuring the non-commutativity of the codifferential and pullbacks

$\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\N}{\...
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170 views

Lifting a determinant map

This is a kind of a follow-up to Question on Hessian of a function (probability question). Suppose I give you a continuous function $f:\mathbb{R}^n \to \mathbb{R}.$ Is it true that there exists a ($C^...
8
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225 views

Basis for $L^2(\mathbb{R})$ that Solves the Heat Equation

This is a less-than-serious question that I asked on math.SE, but I suspect it is slightly more appropriate to ask it here. Consider the heat equation $$ u_t = \frac12 u_{xx} $$ On $\mathbb{T}$ with ...
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232 views

Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$). Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...
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233 views

Harmonic map heat flow in positive curvature

Suppose I wish to relax/smooth a map $\phi:M\rightarrow N$ between two surfaces $M,N$ embedded in $\mathbb{R}^3$. I could try flowing the map using harmonic heat flow, which (as I understand it) is ...
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598 views

What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?

I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it. Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi \...
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953 views

Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
8
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176 views

Local energy decay for variable-speed, divergence-form wave equation in non-trapping medium without obstacles

I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial ...
8
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329 views

Finding a dimension-free bound for a certain multiplier on Euclidean space

The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
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206 views

Regularity result for the boundary value problem for the heat equation

Let $\Omega$ be an open bounded subset of $\mathbb R^N$. Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$ Consider the following boundary value problem for the heat equation: ...
7
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210 views

(geodesic) smoothness of f-divergence with respect to the Wasserstein metric

We consider the f-divergence, which takes the form $$ D_f(P \| Q) = \int_\Omega f\left(\frac{dP}{dQ}\right) dQ. $$ For example, when $f(t) = t \log t$, we obtain the KL-divergence. My question is ...
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220 views

Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$

I have found the following claim made very clearly at least once in the published literature (see below): Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...
7
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188 views

Lax pairs in an abstract formalism

I am reading Integrals of Nonlinear Equations of Evolution and Solitary Waves by Peter Lax and I'm having a hard time. The methods are pioneering, of course, but Lax does not bother much to provide ...
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207 views

How to interpret heat kernel at unit time on a Riemann surface?

Let $M$ be a compact Riemann Surface. Let $P$ be a fixed point on $M$ and let $\delta_{P}$ be the Dirac point distribution at $M$. Consider the fundamental solution of the heat equation $$ (\partial_{...
7
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152 views

What is nice in gradient flows?

First of all, I am sorry for such naivety. I have not all the intuition in hard analysis as I wish. I am studying Perelman's work and his big first contribution is to prove that the Ricci flow is in ...
7
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252 views

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: ...
7
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205 views

Fixed radius mean value property implies harmonicity?

Let $f$ be a continuous real-valued function on $\mathbb{R}^n$. It is well known that the following are equivalent: $f$ is harmonic. $f$ satisfies the ball mean value property $$ f(x)=\frac{1}{|B(x,r)...
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245 views

Pseudodifferential operators on compact manifolds with boundary

I have heard that the square root of the Dirichlet (or the Neumann) Laplacian is not a pseudodifferential operator on compact manifolds with boundary. The context in which this was said was that ...
7
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1answer
608 views

Lax Pairs for Linear PDEs

I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has ...
7
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0answers
412 views

Regularity of solutions to a linear degenerate parabolic pde

I've encountered the following problem which is causing me some trouble : Let $(M^2,g)$ be a smooth compact Riemannian surface (with constant curvature for instance) and consider a smooth function $u:...
7
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0answers
466 views

planar mappings 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 $f:...
6
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0answers
232 views

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 ...
6
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138 views

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 ...
6
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0answers
161 views

Simple proof for Fefferman-Phong inequality?

The following inequality was proved firstly by Fefferman in the paper: The uncertainty principle, Bulletin of the AMS, 1983. Then it was improved by several authors. The proofs presented in these ...
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174 views

Sobolev space under Mellin transform

The Mellin transform is known to be an isomorphism see wikipedia between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$ where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
6
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0answers
236 views

Is a Sobolev map with invertible smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$ 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 ...
6
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0answers
63 views

Density of squares of radial eigenfunctions

The eigenfunctions of the Laplace operator on the disc can be written in polar coordinates as $f(r,\theta)=R_{nk}(r)e^{ik\theta}$, where $k\in\mathbb Z$ and $n\in\mathbb N$ and the radial function is $...
6
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0answers
214 views

Spectral properties of Non-local Differential operators on real line

I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs. Definition: A ...
6
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0answers
282 views

Lavrentiev Phenomenon

Does there exist a (onedimensional) integral functional of calculus of variations $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt
 $$ such that not only $$ \inf_{y\in Lip([a,b])}F(y)>\inf_{y\in W^{1,1}([a,b])...
6
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0answers
108 views

$L^p$ estimates for solutions of strongly elliptic equations

Let $\Omega\subset \mathbb{R}^d$ be a Lipschitz bounded domain. In $L_2(\Omega;{\mathbb C}^n)$, we consider a strongly elliptic second order operator $A$ with constant coefficients. Namely, it is ...
6
votes
1answer
404 views

Boundary value problems with $L^2$ boundary data

Recently, I read the following result from "A Remark on the Regularity of Solutions of Maxwell’s Equations on Lipschitz Domains" by Martin Costabel: Let $\Omega$ be a bounded Lipschitz domain, $u\in ...
6
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0answers
119 views

Continuous right inverse to the Laplacian operator on $C^\infty$

For each $f\in C^\infty(\mathbb R^n)$, there exists $u\in C^\infty(\mathbb R^n)$ such that $\Delta u=f$. This, I guess, has been known well before the more general Malgrange-Ehrenpreis theorem that ...
6
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0answers
162 views

behaviour of first eigenfunction near the boundary

Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $ \phi_1(x)>0$ is the first eigenfunction of $ -\Delta$ in $H_0^1(\Omega)$ normalized however one chooses. My interest is in how $ \...
6
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0answers
384 views

Laplacians associated to symplectic cohomologies

I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...
6
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0answers
125 views

Lagrangean uniqueness versus Eulerian uniqueness

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE: $$ \dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N. $$ ...
6
votes
0answers
336 views

Compactness of solutions to parabolic equations (parabolic regularity)

I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature. For each $s>0$, I have a ...