# 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.

1,041
questions with no upvoted or accepted answers

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555 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 ...

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549 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 ...

**17**

votes

**1**answer

868 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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1k views

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

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586 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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563 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 $K$.
...

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168 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 ...

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119 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}^...

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228 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 ...

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469 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-...

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249 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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481 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 ...

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337 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 $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...

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657 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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205 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 ...

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229 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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273 views

### Measuring the non-commutativity of the codifferential and pullbacks

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172 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^...

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216 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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224 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 ...

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297 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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240 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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248 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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249 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 ...

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1k 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
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...

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334 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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156 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 ...

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238 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:
...

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268 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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209 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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228 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 \...

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381 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])...

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216 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_{...

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253 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: ...

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238 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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639 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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**1**answer

731 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 ...

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413 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:...

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473 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:...

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272 views

### A question in elementary differential geometry

Let $M$ be a finite dimensional manifold of constant curvature $\kappa$. Consider a solution of the Hamilton--Jacobi equation
$$ \partial_t u + |\nabla u|^2 = 0. $$
Can we give a precise estimate of a ...

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321 views

### Lax pair of an integrable non-linear PDE

The following is a fourth-order non-linear PDE that passes the Painleve integrability test
$$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}+ t\left(...

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274 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 ...

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102 views

### 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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147 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 ...

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199 views

### 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 ...

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238 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 ...

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74 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 $...

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230 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 ...

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123 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 ...

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279 views

### Steklov eigenvalue problem for a planar region bounded by ellipse

The Steklov problem for a compact planar region $\Omega$ is
\begin{cases} \Delta u =0 &\text{in $\Omega$}, \\ \frac{\partial u}{\partial n} = \sigma u &\text{on $\partial \Omega$},
\end{...