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Results tagged with ap.analysis-of-pdes
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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.
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 …
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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 …
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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 …
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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
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 …
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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 …
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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
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 …
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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".
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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 …
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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
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
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 …
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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
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 …
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