Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with ap.analysis-of-pdes
Search options answers only
not deleted
user 85906
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
vote
Accepted
Is $H_0^1$ a redundant assumption in the 2D Agmon inequality?
There is probably a regularity assumption on $\Omega$ in the lecture notes, right?
Zero traces are very convenient in such proofs because then $\Omega$ may be very irregular and one may rely on resu …
- 2,670
4
votes
Accepted
$W^{1,p}$ ($1\le p<2$) uniqueness of elliptic equations
The answer is yes if and only if you have optimal elliptic regularity in $W^{-1,p'}(B)$ for some $p>2$ for the adjoint differential operator, i.e., the one given by the transpose matrix $A^\top$.
Cons …
- 2,670
3
votes
Accepted
Is there any "extra regularity" to the solution to Poisson's equation posed on a 3-dimension...
As discussed in the comments, such a result can be found in Jochmann's "An $H^s$-Regularity Result for the Gradient of Solutions to Elliptic Equations with Mixed Boundary Conditions".
- 2,670
2
votes
Accepted
Parabolic Regularity with Neumann B.C
Let's see how this goes. First, let me say that the continuity estimate you are looking for is contained in [3, Theorem 4.5] under the integrabilities as I had already mentioned, so $q > n$ and $p > 2 …
- 2,670
4
votes
Accepted
Question on Sobolev spaces in domains with boundary
For the sake of completeness, an expansion on the comment by Mike Miller:
In Evans/Gariepy, Thm. 4.3.1, it is proven that if the boundary of $\Omega$ is Lipschitz, then for $1 \leq p < \infty$ there …
- 2,670
3
votes
Higher regularity of solutions for Laplace equation with mixed boundary condition
This is more of an extended comment, but maybe it is helpful.
At least for the case of mixed boundary conditions involving Dirichlet and Neumann conditions where the corresponding boundary parts actu …
- 2,670
1
vote
Accepted
Showing existence of minimisers with single integral constraint on a possibly non-Lipschitz ...
Your domain is a Lipschitz (graph) domain. The 3-dimensional (!) Lebesgue measure of $\mathcal{F}$ is indeed zero, whereas $\mathcal{F}$ has positive measure for the boundary measure $\sigma$ (which c …
- 2,670
2
votes
Trace of a function
The spatial evaluation (or trace) operator $\mathrm{tr}_L$ at $L$ is well defined and continuous on $H^s(0,L)$ for $s>1/2$; a classic reference is [LM, Chapter 1.9]. (Of course the range $s>1/2$ is ex …
- 2,670
4
votes
Accepted
Gagliardo-Nirenberg inequality for bounded domain
If you assume that $\Omega$ is a bounded uniform extension domain, then your desired inequality holds true. By uniform extension domain, I mean that there exists a linear extension operator $E$ which …
- 2,670
1
vote
Accepted
Solution of hyperbolic equations with $V^*$ data
Chapter 9 in Volume 1 of Lions/Magenes [1] treats this case, even for nonautonomous operators. One essentially gets (somewhat as expected?) a regularity shift just in the spatial components, so the so …
- 2,670
1
vote
Accepted
Optimal control theory of PDEs
Maybe most obviously, there is a whole area of research devoted to numerical analysis of optimal control problems where results like $\|\bar f - f_h\| \in O(h^\alpha)$ as $h \searrow 0$ are of interes …
- 2,670
5
votes
Accepted
Reference request: Uniformly elliptic partial differential operator generates positivity pre...
Check Chapter 4, more specifically Chapter 4.2 in Ouhabaz: Analysis of heat equations on domains (2005). ZBL1082.35003. The approach there goes by the form method which I personally perceive to be ver …
- 2,670
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 …
- 2,670
5
votes
Accepted
Question about Lebesgue Bochner spaces
By Pettis' theorem, $u \colon (0,T) \to W^{1,p}(\Omega)$ is strongly [Bochner] measurable if and only if it is weakly measurable due to separability of $W^{1,p}(\Omega)$.
It is thus sufficient to show …
- 2,670
1
vote
Accepted
An inequality from Bessel potential space to Besov space
I think that your idea is completely correct, but the choice of $s$ and $q$ is indeed somewhat curious in the paper. However, the case $q=p$ should be sufficient for the proof to work:
We need $s$ an …
- 2,670