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Results tagged with fa.functional-analysis
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user 51695
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
5
votes
Accepted
Existence of directional heat equation without uniform ellipticity
As you do not have any sort of coupling in any spatial direction other than $x_1$, what you have here is not actually a time-dependent PDE in $d$-dimensions but a $(d-1)$-parameter family of time-depe …
- 2,504
4
votes
Accepted
Regular Lagrangian flow for "square root example": $\frac{d}{dt} X(t,x) = \sqrt{X(t,x)}$
Your intuition is right. The key is in the paper you cite, in that they consider uniqueness in the class $L^1_{\text{loc}}$, which does not allow for concentrations. If you add to this, that the Lagra …
- 2,504
3
votes
Accepted
Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?
Even the modified question does not hold.
Let $u_n$ be a basis such that $\mathcal{L}^d(\operatorname{spt} u_n) \to 0$, e.g. a wavelet basis and let $\phi \in C_0^\infty(\Omega)$ a function such that …
- 2,504
3
votes
Accepted
Arzela-Ascoli for L_p-norm
For your interest in a minimal $f$, you might want to read a beginners textbook on Sobolev-Spaces and the calculus of variations, especially on the direct method, which is all about this. The beginner …
- 2,504
2
votes
Vector measures as metric currents
To me your definition seems to be the right one, you just need to prove that it is well defined when approximating Lipschitz with $C^1$-functions. For that you probably need the distributional diverge …
- 2,504