11
votes
Accepted
Does the Poincaré inequality hold on annular domains?
I will prove the stronger result without the subtraction of $\bar{f}$. As we know $\int |f|^2 = \int |f - \bar{f}|^2 + \int |\bar{f}|^2$, the result without subtracting $\bar{f}$ would imply what you ...
- 37.6k
6
votes
Accepted
Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Yes, there is a standard procedure to analyze such systems, essentially, it is Cartan's method of prolongation combined with his theory of involutive systems. There are other approaches as well, but ...
- 106k
6
votes
Accepted
Uniqueness of constructed solutions to the Helmholtz equation
The old Sherlock Holmes adage
When you have eliminated the impossible, whatever remains, however improbable, must be the truth.
applies here. Since nothing else you did was wrong, it must be your ...
- 37.6k
5
votes
Accepted
Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
The question essentially boils down to asking about sufficient conditions under which an Orlicz space $L_\Phi(\mathcal{X},\mu)$ over a nonatomic finite measure space $(\mathcal{X},\mu)$, equipped with ...
- 176
4
votes
Accepted
An asymmetric quadrilinear estimate
OK. Let's first figure out a necessary condition on $a_x>0; a_{xy}, a_{xz}, a_{yt}\in(\frac 1p,1)$ for the quantity
$$
Q=\int_\Omega \frac{f_1(x)f_2(y)f_3(z)f_4(t)}{x^{a_x}(1-x^2-y^2)^{a_{xy}}|x-z|^...
- 59.8k
3
votes
Question about Neumann eigenvalues on manifolds
Let $\Omega$ be a connected domain and denote its the Neumann eigenvalues by $ 0 = \mu_0(\Omega) < \mu_1(\Omega) \leq \cdots $
Let $\mathbb{S}_+^2 = \{(x,y,z)\in\mathbb{S}^2\ |\ y \geq 0\}$ be the ...
- 881
3
votes
Accepted
On the weak derivative of $|u|^{(p-2)/2}u$
Only a partial answer : (2) seems strange. In dimension $1$, if $u$ does not change sign, your setting includes the one of $u^\alpha \in H^1(0,1)$ (boundary values are irrelevant here) for some $\...
- 2,710
3
votes
How to understand the unique continuation result
If I understand things correctly, $u$ is vanishing on some non-empty open subset. Moreover, you have pointwisely on $\mathbb R^3$ the differential inequality
$$
\lvert \Delta u\rvert\le C\lvert u\...
- 15.2k
3
votes
Accepted
A simple bilinear estimate
Since $f$ is only defined for $x\in [0,1]$ and I can perform a change of variables to get $y \mapsto 1-y$, I think you are asking about the inequality
$$ \int_{[0,1]\times [0,\delta]} \frac{f(x)g(y)}{|...
- 37.6k
2
votes
Accepted
How is this interpolating curve well-defined in the minimizing movement scheme?
$\newcommand{\id}{\operatorname{id}}$This is because there is a typo in (8.11). One could have guessed that (8.11) is wrong for homogeneity reasons: if $x\in \Omega$ has units [m], then $\id(x)=x$ has ...
- 5,163
2
votes
$L^\infty$ estimate for elliptic PDE with mixed boundary conditions
Too long for comment. take for instance $f=0, g=0$. Then the mapping $h\mapsto u$ is a pseudo-differential operator which will have some Sobolev continuity properties for spaces $W^{s,p}$ with $p\in (...
- 15.2k
2
votes
Accepted
A bilinear estimate with a simple one-dimensional oscillatory integral kernel
We can make more concrete what I suggested in my comment. Take $f(x)=x^{1/2p'}e^{-ix}$ on $1\le x\le 1/(10\epsilon)$ and $f=0$ otherwise. Then the LHS of (*) equals
$$
\int_1^{1+\epsilon} |G(t)|^2\, \...
2
votes
PDE for the probability of Brownian motion staying in an area (reference request)
The easiest way to show this is to check that if $\hat{u}$ is a bounded solution to your boundary value problem, then $\hat{u}(t-s,B_{s\wedge\tau}+x)$ is a martingale, where $\tau$ is the minimum of ...
- 8,672
2
votes
Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?
This is a somewhat delicate topic and I think what has been proposed so far did not capture fully what is going on. (Probably I am missing something, too, and I also did not fully answer OP's question,...
- 2,195
2
votes
Accepted
Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?
This may be an overkill : you can use the closed graph theorem. If $(u_n)_n$ converges to $u$ in $W^{s,p}_0(\Omega)$ and $(E(u_n))_n$ converges to $v$ in $W^{s,p}(\mathbf{R}^d)$, then both ...
- 2,710
2
votes
Accepted
Well posedness of the Plateau problem under lack of uniqueness
I am far from being an expert, but if I understand your question correctly, the answer is "no", at least for the Plateau problem in the realm of Caccioppoli sets.
Vinti's cited result [2] ...
- 618
2
votes
Well-posedness of PDE with $\partial_{tt}\Delta u$ - like term
Here's what I would try:
Check that for each $t$ and $f \in S$, where $S$ is an appropriately chosen function space, there is a unique solution to the elliptic problem:
$$ L\phi = f \text{ on }\Omega,...
- 27k
1
vote
Derive elliptic maximum principle from weak derivatives
This is not a complete answer.
Since $u-\sup_{\partial U} u$ solves the same inequation, w.l.o.g. you can assume that $ u\leq 0$ on $\partial U$ and you want to prove $u\leq 0$ inside.
By density your ...
- 2,710
1
vote
Feynman–Kac formula for other operators
Some pointers to the (extensive) literature on generalized Feyman-Kac formulas:
Stochastic Solution of Elliptic and Parabolic Boundary Value Problems for the Spectral Fractional Laplacian
Fractional ...
- 178k
1
vote
Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?
This is "trivial" in the sense that $W^{s,p}_0(\Omega)$ can be regarded as a subspace of $W^{s,p}(\mathbb R^n)$ where the functions are vanishing on $\Omega^c$. And you don't need to take ...
- 896
1
vote
PDE for the probability of Brownian motion staying in an area (reference request)
I think your idea ("PS") works fine, at least when $A$ has finite measure and is a moderately reasonable set (let's say open) and probably in general with more effort. It does seem to get a ...
1
vote
Accepted
Linearized operator of higher order $p$ Laplacian
First, there's maybe a more revealing way to write the linearization of the p-Laplacian:
$$L_v(\phi) = -\text{div}(|Dv|^{p - 2}\mathscr{C} D\phi)$$
where we define the rank-4 tensor
$$\mathscr{C} = I +...
- 688
1
vote
An expansion for 2d Euler equation
Let's consider the simplified equation (which is really WLOG, since you are just absorbing constants)
$$ x \sqrt{-\ln x} = y $$
for $x,y > 0$ and $x,y \ll 1$. (Here $x$ is playing the role of $s_\...
- 37.6k
1
vote
Method of characteristics for higher order PDEs in more than two variables
Check Harry Bateman’s book, Partial Differential Equations, section 2.24, pg. 133. The book is on the internet archive web site. He gives an equation for the characteristics for a second order PDE in ...
- 11
1
vote
Hörmander's hypoellipticity theorem for complex coefficients
As Bazin notes, the situation is more complicated for complex vector fields.
For example, Kohn (Annals of Mathematics, 162 (2005), 943–986) gave an example of an $L^2$ sum of squares of complex vector ...
- 458
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