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3 votes
2 answers
352 views

General version of Weyl's lemma

The classical Weyl's lemma say, suppose $u \in L^1_{loc}(\Omega­)$ satisfies $$\int_{\Omega}u \Delta \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$$ then $u$ is harmonic in $\Omega.$ What I want ...
3 votes
1 answer
370 views

Is Brascamp-Lieb inequality on the sphere applicable for these functions for some $1\leq p<2$

My question is on Brascamp-Lieb-inequality on the Euclidean sphere (which can be viewed as an analogue of Young's inequality on the sphere) obtained in [1]. (See also this question: Brascamp-Lieb ...
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 ...
2 votes
0 answers
138 views

Sufficient initial conditions for "non-local" PDE

I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
0 votes
0 answers
73 views

Operator globally hypoelliptic

An operateor $T$ is globally hypoelliptic if : $u\in S'(\Bbb R^n),Tu\in S(\Bbb R^n)$ imply $u\in S(\Bbb R^n)$. My question why if $u\in L^2(\Bbb R^n): Tu =\lambda u$. Then $u\in S(\Bbb R^n)$. where $\...
7 votes
1 answer
185 views

Question on ODE involving mollifiers from Taylor's book on PDEs

In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form $$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$ with some initial condition $u(...
3 votes
0 answers
124 views

Estimating a solution to Euler-type ODE #2

This is a similar question to this but with a different ODE. Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
2 votes
1 answer
142 views

Estimating a solution to an Euler-type ODE

Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer and $a$ be a real number. Let $u(r)$ be a function on $[1,\infty)$ ...
1 vote
1 answer
112 views

A bilinear estimate with a simple one-dimensional oscillatory integral kernel

Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$. I am trying to show that $$\int_{\mathbb{R}}\int_{\mathbb{R}} \,K(y,z)\, \frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
12 votes
2 answers
5k views

Where was/is Compensated Compactness used?

This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
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 ...
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 $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
8 votes
3 answers
1k views

Are all positive eigenfunctions principal eigenfunctions?

In a given domain $\Omega$, we have: $\Delta u=-\lambda u$ with $u>0$. Does this mean that $u$ is a principal eigenfunction for $\Delta$ in $\Omega$? Also, more generally, does this also apply for $...
0 votes
2 answers
238 views

Fractional Laplacian of $(a-x)_+^\alpha$ in $(0,1)$

How can I compute the spectral fractional Laplacian of $(a-x)_+^\alpha$ on $\Omega = (0,1)$? Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
6 votes
2 answers
519 views

Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

This question is concerning the paper, particularly the proof of Lemma 2.1 in Section 2.1: Matas, A., Merker, J. Existence of weak solutions to doubly degenerate diffusion equations, Appl Math 57 (...
1 vote
1 answer
160 views

On an integral equation

Let $B: C^{\infty}([0,1]^3)$ satisfy $$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$ Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation: $$ \int_0^1 f(t,x)\,dx + \int_0^t\...
2 votes
0 answers
206 views

Failure of Calderón–Zygmund inequality at the endpoints

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'd like to prove that the famous Calderón–Zygmund elliptic estimate $$ \norm{ \partial_{ij}u }_{L^p} \leq C \norm{\Delta u }_{L^...
1 vote
2 answers
106 views

Green function of symmetric stable process in dimension 1 and 2

Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
3 votes
0 answers
111 views

Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces

The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is, $[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$. Now I ...
0 votes
1 answer
171 views

Looking for English version of a paper of Jean Ginibre

I am in serious need of an English translation for the following paper: Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace, d'après ...
0 votes
1 answer
161 views

Verifying the proof of a bilinear estimate in $L^2$

$\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert_{L^{#2}}}$ $\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert_{L^{...
37 votes
4 answers
4k views

Which differential equations allow for a variational formulation?

Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation $$ \frac{d}{dt}\frac{\...
4 votes
2 answers
315 views

Is this an $L^p-L^{\infty}$ operator?

Let $1\leq p <\infty$ and let $p^{\prime}$ denote its conjugate exponent. Consider the following operator on Schwartz functions: $$Tf(x)=\int_{0}^{\infty}t^{\frac{n}{2 p^{\prime}}-1}e^{-t} \int_{|x-...
2 votes
0 answers
201 views

Green function of a 2D exterior domain

Consider solutions of the laplace equation \begin{equation} \begin{split} -\Delta u=f, \ \ u|_{\partial D}=0, \end{split} \end{equation} where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
1 vote
0 answers
100 views

N-wave solution of conservation law $u_t + (u - u^2)_x = 0$

How can we compute the "N-wave" source-solution of the conservation law $$u_t + (u - u^2)_x = 0, $$ that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
1 vote
0 answers
47 views

Scaling limit of transport equation with double-well potential

Let us consider the transport PDE $$ u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...
2 votes
0 answers
64 views

Scaling limit of ODE with double-well potential

Let us consider the ODE $$ \frac{d}{dt}x_\epsilon(t) = -\frac{1}{\epsilon} W'(x_\epsilon(t)) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the ODE reads $$...
0 votes
1 answer
166 views

Relationship between elliptic and parabolic problems and their discretizations

Let us consider the fully nonlinear problem $$ \begin{cases} F(x,u,Du,D^2 u) = 0 & \text{ in } \Omega \\ u=0 & \text{ in } \partial \Omega \end{cases} $$ Suppose that we know that the ...
1 vote
0 answers
59 views

Identification of a limit point of a sequence of solution of ODE

Let $v^0$ and $v^1$ be the following vector fields over $\big(\mathbb{R}_+^*\big)^3$: for $x\in\big(\mathbb{R}_+^*\big)^3$ and $1\leq i\leq 3$, \begin{align*} & v^0_i(x)=x_i(x_{i-1}-x_{i+1}) \\ &...
3 votes
1 answer
425 views

Regularity of boundary of a level set of a $C^{1,\alpha}$ function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^{1,\alpha}$ function. Denote $S_C=\{x\in\mathbb{R}^2\mid f(x)=C \}$ the level set of $f$ with value $C$. What i want to ask is, if $S_C$ is nonempty for some $...
5 votes
1 answer
170 views

Regular Lagrangian flow for "square root example": $\frac{d}{dt} X(t,x) = \sqrt{X(t,x)}$

Consider the problem $$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \sqrt{X(t,x)}, &t \in [0,T],\\ X(0,x) = x, &x \in \mathbb R \end{cases} $$ This is the prototype of non-uniqueness ...
1 vote
1 answer
158 views

How do I integrate this inequality that appears in a paper of Rabinowitz?

Sorry if this is too easy for MO, but I found it in a research paper, so I thought that it was worth posting here. I was reading a paper by Rabinowitz(this one to be more precise) and I came across ...
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^\...
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^\...
0 votes
1 answer
380 views

How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?

Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
3 votes
1 answer
453 views

Duality argument

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$Throughout my studying for some papers, in particular, the proof of localized Strichartz estimates, I encountered a use of the ...
0 votes
0 answers
171 views

Explanation of a step in a preprinted work

I have been studying this preprinted paper and the references therein. I believe that there are some typos; However, since I am not an expert yet, I would like to make sure I am correct. I do not ...
1 vote
1 answer
203 views

Explanation of a step in a work by C. E. Kenig and A.D. Ionescu

I am studying the work Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. ...
2 votes
1 answer
145 views

Estimate for an oscillatory integral of the first kind

I am confused in finding the right bound for the following oscillatory integral $$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$ Where $\psi(2^{-k} \xi)$ is a smooth ...
2 votes
0 answers
66 views

Existence of saddle points under a $C^0$-perturbation of a continuous function

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and has a strict maximum point $a$ and strict minimum point $b$. Define $g(x,y)=f(x)+f(y)$ and $h_\varepsilon(x,y)$ be a family of continuous ...
3 votes
1 answer
259 views

The continuous dependence of the Green's function on a domain

Let $\Omega\in\mathbb{R}^2$ be a smooth bounded domain and $G(x,y)$ be the Green's function of $-\Delta$ in $\Omega$ with zero Dirichlet condition. Clearly $G(x,y)=-\frac{1}{2\pi}\ln|x-y|-h(x,y)$, ...
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\|\...
2 votes
2 answers
631 views

Decomposition of a positive definite matrix

Let $K(x)_{n\times n}$ be a positive definite matrix defined on $x\in D$ and $K_{i,j}(x)\in C^2(D)$ (or generally $C^k$) for any $1\le i,j\le n$. Of course for any $x$, there exists a invertable ...
11 votes
2 answers
478 views

$x f'$ bounded by $x^2f $ and $f''$?

Consider the Hilbert space of functions $f \in L^2(\mathbb R)$ such that $x^2f \in L^2(\mathbb R) $ and $ f'' \in L^2(\mathbb R).$ I am wondering whether it is true that $xf'\in L^2(\mathbb R)$ as ...
2 votes
1 answer
94 views

Decay rate for a small perturbation of a simple linear ODE

MOTIVATION. Let $f:[0,+\infty)\to \mathbb{R}$. Solutions to $\partial_tf(t) = -\lambda f(t)$, $f(0)\neq 0$ approach zero exactly as $e^{-\lambda t}$. This property is preserved if we apply an ...
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_{...
6 votes
1 answer
376 views

Lavrentiev phenomenon between $C^1$ and $C^2$

Does there exist a (onedimensional) functional that exhibits the Lavrentiev phenomenon between $C^1$ and $C^2$ that is $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \quad\text{or possibly}\quad F(y)=\int_a^b f(...
3 votes
2 answers
129 views

Link between controllability of ODEs and controllability of transport equations

What is the relationship between the controllability of the ODE $$\dot x(t) = v(x) + u(t)$$ using a control $u$ and the controllabilty of the transport equation $$\rho_t(t,x) + \mathrm{div}(v(x) \rho(...
7 votes
3 answers
2k views

Collections of examples and counterexamples in (real, complex, functional) analysis, ODEs and PDEs

What books collect examples and counterexamples (or also "solved exercises", for some suitable definition of "exercise") in real analysis, complex analysis, functional analysis, ODEs, PDEs? The ...
2 votes
0 answers
73 views

Question about Gidas-Ni-Nirenberg result

Background: So I know that the Euler Lagrange equation associated with the Sobolev inequality takes the following form, $$-\Delta u = u^p$$ where $p=2^*-1$ and here we assume that $u>0$ on $\mathbb{...