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Results tagged with ap.analysis-of-pdes
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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
3
votes
Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains
This Kato square root property is indeed true from abstract principles and without any regularity on the domain since $-\Delta_D$ is selfadjoint. (For $H^1_0(\Omega)$ being the closure of test functio …
- 2,670
5
votes
Reference Request for global Hölder continuity of solutions to elliptic PDEs
Such a global Hoelder regularity result requires some minimal assumptions on the geometry of $\Omega$ but no smoothness assumptions on the coefficients.
The classical assumption is a measure density c …
- 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
6
votes
Accepted
Question about Bochner measurability
Good resources for the mentioned Bochner spaces (with an emphasis on their use in abstract PDEs and the likes) could be for example
Arendt et al: Vector-valued Laplace Transforms and Cauchy Problems …
- 2,670
10
votes
Accepted
Sobolev spaces are smooth? Their dual is strictly convex?
There are a bunch of great books on Banach space geometry but sadly they often do not care very much about Sobolev spaces. There is Example 2.47 in Schuster at al: Regularization Methods in Banach Spa …
- 2,670
2
votes
Accepted
Integrability in the product space can follow from a property of the Nemytskii operator?
Nemytskii operators on Lebesgue spaces are funny objects with a lot of implicit structure. In fact, if $\mathcal N_f$ maps $L^p(\Omega)$ into $L^q(\Omega)$, then $f$ must necessarily satisfy the growt …
- 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
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
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
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
3
votes
Understanding a family of Sobolev-type inequalities
It is indeed elementary with some slight maneuvering:
Since $s' < r' \leq r$, there exists $\alpha \in (0,1]$ such that \begin{equation}\|f\|_{r'} \lesssim \|f\|_{s'}^{1-\alpha} \, \|f\|_r^\alpha.\la …
- 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
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
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
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 …
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