9 votes

Propagators and PDEs

You'll find some more info about the fundamental solutions of the wave equation in chapter 5.D of Folland and chapter I.7 of Trèves. The trick you use is the idea that (tempered) distributions, even ...
8 votes
Accepted

Forcing the uniqueness of a solution of an ODE

$\newcommand\ep\varepsilon$First, the conditions that $f_n\in\mathcal{C}^1([0,1],\mathbb{R})$ and $f_n(x)\ge\sqrt{x}$ for $x\in[0,1]$ imply $f_n(0)>0$. Since \begin{equation*} \begin{cases} y_n(...
7 votes
Accepted

Nonsmooth version of Hopf boundary point lemma

I think this is just the comment following Lemma 3.4 of Gilbarg and Trudinger (specifically equation 3.11). I should add that lowering the regularity of the boundary seems like a harder problem (and ...
  • 1,958
6 votes

Forcing the uniqueness of a solution of an ODE

This question would be possibly at a better place on MathStack Exchange. Yet, once the statement of the question is corrected (the functions $y_n$ need to be defined on $\mathbb{R_+}$ and not only on $...
5 votes

A text about Schwartz distributions in vector bundles

This is pretty standard material and I don't know of a specific reference that answers your question and only that. You will find lots of references if you search for "distributions on manifolds&...
4 votes

Equivalence between two fractional Sobolev spaces

Yes, they are. Since $(\phi_j,(-\Delta)^\eta u) = \lambda_j^\eta (\phi_j,u)$, it suffices to consider the case $\alpha = 1$. Since, for any closed operator $A$, the domain of $A$ coincides with that ...
3 votes
Accepted

Why we have $f=0$

Fourier transform of $f(x)=e^{-tx^2}$ is $\hat{f}(\xi)=ce^{-\xi^2/(4t)}$. Multiplication of $f$ by a polynomial results in applying a differential operator with constant coefficients to $\hat{f}$, and ...
3 votes
Accepted

Bott-Chern cohomology for singular complex spaces

closed (1,1)-forms and currents on X are not necessary locally $dd^c$-exact in general What makes it different when X is singular? The obstruction to local $dd^c$-lemma is $R^1\pi_*(O_{X'})$, where $\...
3 votes

What is standard continuity argument for well-posedness?

I understand continuity argument a bit differently than the other post. Suppose you've shown the inequality $$\|u\|_{L^\infty(I,X)} \leq C(\|\varphi\|_{X} + |I|\|u\|_{L^\infty(I,X)}^3) \tag{1},$$ for ...
3 votes

What is standard continuity argument for well-posedness?

The way I understand it is that because of this bound you can derive a solution by a fixed point. Set $u_0=0$ and $$ u_{n+1} = A + \int_0^t B |u_n|^{p-1}u_n ds $$ (A,B represents the various ...
  • 2,272
3 votes
Accepted

Reference or proof of a lemma in PDE

I would call this estimate "the classical Calderon-Zygmund estimate" but indeed it is hard to track down a statement for the right-side in divergence form. Usually it is stated as an ...
3 votes

Sobolev space is spanned by distributions supported on half-lines?

First of all, as explained in my comment, this is the same as asking if the Hilbert transform is bounded on $L^2(\mathbb R, w\, dx)$, with $w(x)=(1+|x|)^{2s}$. Or, to state this one more time, this ...
3 votes

Prove Liouville theorem without using mean value property

Below we will prove, by purely energy methods, the following sharper statement: Let $u$ be a harmonic function which satisfies $$\liminf_{r \to \infty} \frac1{r^2} \frac{1}{|B_r|}\int_{B_r} |u|^2 = 0.$...
2 votes

General solution to a n-dimensional partial differential equation

Denote $x_{ij} = x_i-x_j$, $\ n_{ij} = \frac{c_i+c_j}{1-c_i-c_j}$, and operator $$ D_{ij} = \frac{x_i-x_j}{1-c_i-c_j}(c_i\partial_i-c_j\partial_j). $$ We can write the pde as $$ P_t = \sum_{i<j}...
2 votes

Reference or proof of a lemma in PDE

Suppose $p>1$. For each $i\in \{1,\ldots,n\}$, consider $\varphi_i\in W^{2,p}(B_2)\cap W^{1,p}_0(B_2)$ be the solution of $$ -\Delta \varphi_i = F_i \textrm{ in } B_2, $$ and let $\Phi=(\varphi_i)...
  • 2,272
2 votes

Reference or proof of a lemma in PDE

Here is a sketch of the proof. By assumption $\int_{B_2} f \Delta \phi=-\int_{B_2} F\cdot D\phi$ for every $\phi \in C_c^\infty (B_2)$. Fix $\eta \in C_c^\infty(B_2)$, $\eta=1$ in $B_1$ and $\psi \in ...
2 votes

What are the solutions to this nonlinear equation?

As indicated in the comment, some additional information would help answering the question. Assuming standard hypotheses, the problem you are interested in can be considered as a degenerate case of ...
1 vote
Accepted

Representing solutions of $-\Delta u+au=f$ when $a\leq 0$

Take the eigenbasis $\phi_n$ you exhibited for $a=0$. It is still an eigenbasis for $-\Delta + a$, with shifted eigenvalues $\lambda_n + a$. Write $$ E_a=\{n\in\mathbb N : \lambda_n =- a\}, $$ which ...
  • 2,272
1 vote

$C^2$-solution of Lane-Emden equation with positive frequency

The answer to my questions is indeed negative. In this paper it is shown that solutions to the second equation do exist, which are periodic in $x_1$ (but not constant) and decay to zero when $|x'|\to\...
  • 251
1 vote

Ekeland's standardness-property inheritable?

The principle for getting examples of nonapplicability of Ekeland's theorem is described by the following simple Example. With $\mathbb I=[0,1]$ consider the Frechet space $E=F=C^\infty(\mathbb I)$ ...
  • 2,883

Only top scored, non community-wiki answers of a minimum length are eligible