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Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$? Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
user avatar
8 votes
1 answer
380 views

Lavrentiev phenomenon between $C^1$ and Lipschitz

Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere) $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt
 $$ such that $$ \inf_{y\in Lip([a,b])}F(y)<\inf_{...
Carlo Mantegazza's user avatar
9 votes
1 answer
1k views

Traces of Sobolev spaces

Is there a simple proof of the following fact? Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset W^{1-\frac{1}{n},n}(\...
Piotr Hajlasz's user avatar
5 votes
2 answers
459 views

Backward heat equation and forward perturbed heat equation well posed?

I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
Sascha's user avatar
  • 536
4 votes
1 answer
213 views

Mapping properties of backward and forward heat equation

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

Bounding integral expression with total variation of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$ for $\epsilon>0$, $f \in L^\infty(\mathbb R)$,...
user avatar
2 votes
1 answer
997 views

Derivative and Jacobian determinant of solution of ODE [closed]

Let $\Phi$ be the unique solution of $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{cases}$$ where we have assumed $f$ smooth. ...
user avatar
2 votes
0 answers
117 views

Bounding integral expression with BV norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
user avatar
2 votes
0 answers
252 views

Dense property of intersection of Sobolev space

I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim: Pick an arbitrary real number $s$, we have that the ...
geooranalysis's user avatar
14 votes
0 answers
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
12 votes
1 answer
1k views

Proof of Green's formula for rectifiable Jordan curves

$\newcommand{\Ga}{\Gamma}$ I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
Iosif Pinelis's user avatar
11 votes
1 answer
1k views

Proof of the "Neo-classical Inequality", a fractional extension of the binomial theorem

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1$ and $n\in\mathbb N$: $$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{\frac{j}p}b^{\frac{n-j}p}}{\...
Alex R.'s user avatar
  • 4,952
10 votes
1 answer
700 views

Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?

Let us denote the Riesz potential in $\mathbb R^d$ by $$ I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.$$ By the classical Hardy-Littlewood-Sobolev theorem ...
Juhana Siljander's user avatar
9 votes
2 answers
553 views

Asymptotic behavior of Sturm-Liouville eigenvalues

I have two questions. Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$. Let $(\lambda_n)$ denote the sequence of eigenvalues of $A$....
M.S.'s user avatar
  • 369
7 votes
2 answers
682 views

Hölder continuity for operators

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$ we obtain $1/...
user avatar
7 votes
2 answers
2k views

Method of characteristics for higher order PDEs in more than two variables

I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
Puzzled's user avatar
  • 8,998
7 votes
3 answers
2k views

Gross's log Sobolev inequality proof with variational calculus?

For $f\in C^{1}(\mathbb{R}^{n})$, Gross's logarithmic Sobolev inequality says that $$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla f|^{2}...
Thomas Kojar's user avatar
  • 5,474
7 votes
1 answer
609 views

$H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao. They study the ...
Guo's user avatar
  • 71
6 votes
3 answers
1k views

Orthonormal basis in $W^{1,2}([0,1])$

Consider the Hilbertspace $W^{1,2}([0,1])$ (i.e. Sobolev space) with the standard inner product which is defined by: $(f,g) = (f,g)_{L^{2}([0,1])} + (f',g')_{L^{2}([0,1])}$. Here $[0,1]$ is not ...
Pablo's user avatar
  • 63
5 votes
2 answers
358 views

Linear transport equation with unbounded coefficients

Consider the PDE $$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$ with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$ I am wondering then if $q$ and all its ...
Pritam Bemis's user avatar
5 votes
1 answer
564 views

Convergence of discrete Laplacian to continuous one

I make the following observation: Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.) This one has eigenvalues ...
Sascha's user avatar
  • 536
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
4 votes
1 answer
379 views

A constant ratio of integrals? Part I

Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$. For $0<r\leq1$, consider the average of its Dirichlet integral $$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
T. Amdeberhan's user avatar
4 votes
1 answer
377 views

Differential inequalities under which a flat function must be identically zero

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
Ali Taghavi'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
1 answer
266 views

Prove $\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx dist(x,\partial \Omega)^{-s}$, $s \in (0,2)$

Let $\Omega \subset \mathbb R^N$ and $s \in (0,2)$. Under what assumptions on $\partial \Omega$ do we have $$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx \mathrm{dist}(x,\partial \...
Jun's user avatar
  • 303
4 votes
1 answer
279 views

Schroedinger operator in 2 dimensions with singular potential

Consider the Schroedinger operator $$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$ This operator has a self-adjoint realization, since it is a positive symmetric operator ...
António Borges Santos's user avatar
4 votes
2 answers
3k views

Chain rule for fractional laplacian

Does anyone know a formula of chain rule for fractional laplacian? say we take the fractional laplacian of order a on function $g(U(x))$ $x\in \mathbb{R}^2$, $U \in \mathbb{R}$, $g \colon \mathbb{R} \...
Grant's user avatar
  • 41
4 votes
1 answer
222 views

A continuous bi-Lipschitz shrinking of a domain into a compact subset

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. My main problem/question is: (1) Show there exist a sequence of bi-Lipschitz (i.e injective Lipschitz function with Lipschitz inverse) maps $F_n ...
Ben Ciotti's user avatar
3 votes
1 answer
274 views

Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$ where $x_0$ is an ...
Andrea Tauber's user avatar
2 votes
1 answer
211 views

Hölder continuity in time of heat semigroup for regular initial distribution

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e., $$ p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
Akira's user avatar
  • 835
2 votes
2 answers
952 views

Differentiability of Nemytskii operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator $$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$ is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
D G's user avatar
  • 201
2 votes
0 answers
144 views

Does this geometric PDE have a solution?

Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes. Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$,...
Asaf Shachar's user avatar
  • 6,741
1 vote
1 answer
239 views

Reference request for weak solutions of an Elliptic PDE

Edit : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one. I want to find weak, non trivial, continuous, solutions of $$\...
Rajesh D's user avatar
  • 698
1 vote
1 answer
93 views

Is there $f$ such that $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(Ct, x)$ for all $t>0$ and $x \in \mathbb R^d$?

We consider the heat kernel $$ g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ). $$ Then $$ \partial_t g(t, x)...
Analyst's user avatar
  • 657
1 vote
0 answers
92 views

Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case

By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user avatar
1 vote
1 answer
193 views

Quantitative finite speed of propagation property for ODE (cone of dependence)

Consider the following ODE initial value problem \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\ &\Phi(0,x) = x, & x \in \...
Riku's user avatar
  • 839
1 vote
0 answers
922 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
Rajesh D's user avatar
  • 698
1 vote
1 answer
518 views

Interpolation between Schatten classes

I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the ...
Kinzlin's user avatar
  • 305
1 vote
1 answer
236 views

Continuity of the solution of a Pde system

Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$ both continuous and bounded. I have the following system of PDE's \begin{align} \begin{cases} \frac{\partial}{\partial t} u_0(t,r)=- J* ...
user268193's user avatar
0 votes
0 answers
173 views

Is this has anything to do with Riesz representation?

The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49. Here I come up with a question which has similar ...
JumpJump's user avatar
  • 679
0 votes
1 answer
143 views

An estimate of the integral of the higher order derivative of a bump function

Let $\kappa_1>0$, $\beta\in [0, 1]$ and $b: \mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ such that for all $t\ge0$ and $x,y \in \mathbb R^d$ we have $|b(t, 0)| \le \kappa_1$ and $|b(t, x) - b(t, ...
Akira's user avatar
  • 835
0 votes
2 answers
387 views

Derivative of fractional Laplacian is the fractional Laplacian of the derivative

Is it true that $$\partial_x ((-\Delta)^s u(x)) = ((-\Delta)^s \partial_x u(x))?$$
user avatar
0 votes
1 answer
125 views

Bounding integral expression with Sobolev norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
user avatar
0 votes
1 answer
491 views

Is this set of function belongs to $L^\infty$?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write $$ Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal H^{N-...
JumpJump's user avatar
  • 679
0 votes
1 answer
139 views

Singular integral bounded by Dirichlet form?

We define for some fixed $L$ $$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$ in particular $x_1,x_2 \in \mathbb R^2.$ Let $f \in C_c^{\infty}(\Omega)$, then I am ...
António Borges Santos's user avatar