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633 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 $K$. ...
Gil Kalai's user avatar
  • 24.7k
11 votes
0 answers
2k views

A question on trig series

Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$ $$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...
Jacob Lu's user avatar
  • 903
10 votes
1 answer
518 views

Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions

Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
Tian LAN's user avatar
  • 435
8 votes
0 answers
103 views

Sobolev embedding theorems in vector bundles on non-compact manifolds

Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
G. Blaickner's user avatar
  • 1,429
7 votes
0 answers
619 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\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
Carlo Mantegazza's user avatar
7 votes
0 answers
394 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)...
Snoop Catt's user avatar
6 votes
0 answers
220 views

Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized

Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
Feng's user avatar
  • 517
6 votes
0 answers
151 views

Gap between consecutive Dirichlet eigenvalues

Suppose $\Omega \subset \mathbb R^2$ is a domain with a Lipschitz boundary and let $\{\lambda_k\}_{k=0}^n$ be the eigenvalues for the Laplacian operator on $\Omega$, that is to say $$ -\Delta \phi_k = ...
Ali's user avatar
  • 4,145
5 votes
0 answers
608 views

What is the correct $L^\infty$ limit of this strange variational problem, and what does it encode?

1. On the $L^\infty$ calculus of variations: The field known as the $L^\infty$ calculus of variations is a relatively new field that concerns itself with minimising functionals involving the supremum ...
Nate River's user avatar
  • 6,205
5 votes
0 answers
104 views

Convolution of a bounded function and measures

Given a function $f\in L^\infty(\mathbb{R}^n)$ and a family of Radon measure $\mu_\alpha$, under what condition do we have $f*\mu_\alpha$ equi-continuous? One condition I know is if $\mu_\alpha$ has a ...
Sean's user avatar
  • 375
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
5 votes
0 answers
417 views

All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)

I have asked this question on MSE, but this is a better place. The heat equation and the heat kernel. Consider the heat equation on $\mathbb R$: $$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
Lorenzo Pompili's user avatar
5 votes
0 answers
140 views

Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$

Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where \begin{align}\label{eq:bounded-...
Guy Fsone's user avatar
  • 1,101
5 votes
0 answers
218 views

A differential operator analogy of certain fact in real analysis of smooth functions

Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$. Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$. ...
Ali Taghavi's user avatar
5 votes
0 answers
166 views

global estimate for biharmonic function

My question is inspired by the work of Lamm and Rivière : Conservation Laws for Fourth Order Systems in Four Dimensions Here is the setting of the problem. Let $u\in W^{2,2}(B(0,1),S^n)$, where $B(0,...
Paul's user avatar
  • 914
5 votes
0 answers
252 views

Local version of the Hardy-Littlewood-Sobolev theorem for Riesz potentials: $\|I_\alpha(f)\|_{L^q} \le C \|f\|_{L^p}$?

Recently, I have been studying the properties of the Riesz potential $$ I_\alpha(f)(x) = c_{d,\alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy. $$ The classical Hardy-Littlewood-Sobolev ...
Juhana Siljander's user avatar
5 votes
0 answers
310 views

Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...
Elwood's user avatar
  • 562
5 votes
0 answers
913 views

Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
juan rojo's user avatar
  • 103
5 votes
0 answers
428 views

Is there an appropriate weighted Sobolev space to include exponential map and projection map?

Observe that given a non negative function $\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted $L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions $f: \...
Ritwik's user avatar
  • 3,245
4 votes
0 answers
180 views

Approximation by gaussian mollification in Sobolev spaces

I have been trying to find a good estimate on the constant in the inequality (in dimension $d=3$ to simplify) $$\label{0}\tag{0} \|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
LL 3.14's user avatar
  • 230
4 votes
0 answers
318 views

Integral representation of solution of an elliptic PDE in divergence form

Suppose we have a second order elliptic differential operator $$ L(v) = -\text{div}(A(x) \nabla v) $$ $A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...
Harish's user avatar
  • 261
4 votes
0 answers
135 views

Zygmund class, Schwartz class and Littlewood-Paley projection operators

I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions: Consider the Zygmund class of functions defined as ...
pureorapplied's user avatar
4 votes
0 answers
151 views

Estimating the size of $\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}$

Let $\Omega$ be a bounded domain in $\Bbb R^n$. Define $$ \Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}, $$ i.e. it the ring of thickness $r$ at the boundary of $\Omega$. Intuitively, ...
BigbearZzz's user avatar
  • 1,245
4 votes
0 answers
220 views

A metric $w$ on a Kahler manifold is extremal if and only if the gradient vector field of the scalar curvature is holomorphic

I am trying to understand the calculation in An introduction to Extremal kahler metrics. On the fourth line of page 55 the author calculated that $\int_{M} - 2 S R^{\bar k j} \partial_{j} \partial_{\...
qwe's user avatar
  • 91
4 votes
0 answers
747 views

Maximum Principles in Parabolic PDE with Neumann Condition

I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...
Bogdan's user avatar
  • 1,759
4 votes
0 answers
127 views

Anderson Localization and Homogenization theory

I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here. The question is mostly related to homogenization theory in mathematical physics. $\textbf{...
Sanchit's user avatar
  • 81
4 votes
0 answers
159 views

inverse of sobolev riemannian metric still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth ...
X-Naut PhD's user avatar
4 votes
0 answers
174 views

Constant in trace theorem for balls

Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$ The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
user avatar
4 votes
0 answers
144 views

Asymptotic expansion of a Gaussian integral and heat kernel

When considering the heat kernel of a Schr\"odinger operator $$- \Delta + V(x) $$ where $\Delta$ is the standard Laplacian on ${\mathbb R}^n$ and $V$ is a nonnegative potential function that has ...
Guangbo Xu's user avatar
  • 1,207
4 votes
0 answers
131 views

Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$

Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...
Timothy's user avatar
  • 355
4 votes
0 answers
500 views

Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem ...
JumpJump's user avatar
  • 679
4 votes
0 answers
121 views

The best constant in Poincare-liked inequality in $BV$ and $BD$ space

This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention. Let $\Omega\subset \mathbb R^N$ be open, bounded and with ...
JumpJump's user avatar
  • 679
4 votes
0 answers
340 views

Viscosity solution of the PDE

Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and $$|Du|-f(x,u)=0$$ where $f\ge 0$ and $f$ is strictly monotone for fixed $x.$ I am looking for the reference to show that it has unique ...
nick's user avatar
  • 61
3 votes
0 answers
14 views

On compact embeddings in weighted Riesz potential spaces

I wonder if there is any references for the study of the following type of spaces $$ X_{\delta,\alpha}=\{ u\in L^2_\delta(\mathbb{R}^n):\, u= (-\Delta)^\alpha f \quad\text{for some}\quad f\in L^2_{\...
Ali's user avatar
  • 4,145
3 votes
0 answers
167 views

Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate

If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
Cauchy's Sequence's user avatar
3 votes
0 answers
219 views

Strictly contracting solutions to the Eikonal equation on Riemannian manifolds

Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$. Question: Does there exist, on every complete ...
Nate River's user avatar
  • 6,205
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
83 views

Embedding theorems for Dini continuous functions

Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...
Delio Mugnolo's user avatar
3 votes
0 answers
118 views

If $\frac{\partial f}{\partial t}(x,t)$ exists a.e and $\frac{\partial^2 f}{\partial t \,\partial x }$ is continuous, can we improve a.e existence?

The question is as in the title. Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r....
Isaac's user avatar
  • 3,477
3 votes
0 answers
92 views

Questions about article "Ordinary differential equations, transport theory and Sobolev spaces" by DiPerna-Lions

I am reading the article, and I am more or less halfway through it. I have some questions though on some parts I am not understanding, so I wanted to ask about these here. I apologize for listing the ...
tommy1996q's user avatar
3 votes
0 answers
146 views

A uniqueness result for the Neumann problem for the Laplace equation

Let $\Omega \subset \mathbb{R}^{3}$ be a $C^{1}$-domain, not necessarily bounded. Consider solutions $\phi : \overline{\Omega} \to \mathbb{R}$, $\phi \in C^{\infty}(\Omega) \cap C^{1}(\overline{\Omega}...
node's user avatar
  • 351
3 votes
0 answers
84 views

A weighted $W^{2,p}$ estimates

Let $\Omega$ be a bounded smooth domain and $u\in W^{2,p}(\Omega)\cap H^1_0(\Omega)$. By the classical $L^p$ theory of second order elliptic equation, we have $$ \|\nabla^2u\|_{L^p(\Omega)}\leq C\|\...
W.J.'s user avatar
  • 379
3 votes
0 answers
96 views

A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$

Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball. Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves $$\begin{cases} -\Delta ...
Laithy's user avatar
  • 969
3 votes
0 answers
322 views

Heat equation damps backward heat equation?

In a previous question on mathoverflow, I was wondering about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. ...
Sascha's user avatar
  • 536
3 votes
0 answers
46 views

Partial hypoellipticity

The question is posed specifically about the wave equation and is related to partial hypoellipticity of the wave equation. Let $\Omega \subset \mathbb R^n$, $M=(0,T)\times \Omega$ and $\Gamma=(0,T)\...
Ali's user avatar
  • 4,145
3 votes
0 answers
467 views

Opposite of the curl operator and Biot-Savart kernel

Note: I just realized that using $\omega$ and $w$ might not have been the smartest choice of notation -- Sorry about that. Let $\renewcommand{\div}{\operatorname{\div}}Q_0, Q_1$ be two real numbers, $...
Maximilian Janisch's user avatar
3 votes
0 answers
117 views

Are continuous harmonic maps between Riemannian manifolds smooth up to the boundary?

Let $M,N$ be smooth, connected, compact, oriented, two-dimensional Riemannian manifolds, with $C^k$ boundaries. Let $f:M \to N$ be a Lipschitz continuous weakly harmonic map**, and assume that $f(\...
Asaf Shachar's user avatar
  • 6,741
3 votes
0 answers
65 views

Elliptic equations in semi-infinite strips

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good ...
Ali's user avatar
  • 4,145
3 votes
0 answers
125 views

Green operator of elliptic differential operator and radius of convergence

Let $E \to X$ be a hermitian vector bundle over a compact Kähler manifold and let $L$ be a self-adjoint elliptic linear differential operator on $E$. Suppose that $E \to X$ and $L$ are real-analytic. ...
Simon Parker's user avatar
  • 1,383
3 votes
0 answers
119 views

Second derivative estimates

I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates. In one of his papers, Lin proves the following result: Let's consider a ...
Jason's user avatar
  • 59

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