# Tag Info

### 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 ...
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### 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(...
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### 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 ...
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### 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 ...

### 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 ...
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### 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 ...
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### 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 ...

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### 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
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### 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 ...
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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)\$ ...
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