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2 votes

A ball with slit at the radius is not $W^{1,1}$-extension domain

If $\Omega$ was a $W^{1,1}$-extension domain, then restrictions of $C_c^\infty(\mathbb{R}^n)$ functions to $\Omega$ would be dense in $W^{1,1}(\Omega)$ since they are dense in $W^{1,1}(\mathbb{R}^n)$. ...
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1 vote
Accepted

On the domain of the Neumann Laplacian

This is a partial (positive) answer for the convex case only but not every detail has been worked out. Let first $U$ be convex and smooth and all functions be in $C^3$ up to the boundary. Integrating ...
0 votes

Regularity of solution to Fokker Planck equation

A bootstrap argument is given in the proof of Lemma 10.7 of a paper of Mei-Montanari-Nguyen: https://web.stanford.edu/~montanar/RESEARCH/FILEPAP/mean_field.pdf The authors only show $C^{1,2}$ ...
  • 1
3 votes

Second order differentiability of convex functions

The following argument is used in my recent paper with D. Azagra and A. Cappello (work in progress). The answer is yes. I asked about whether the proof of Theorem 1 can be modified so to cover Theorem ...
0 votes

Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?

No, the measure density condition is not necessary I would say. Possibly there are more precise arguments, but I would argue as follows, in a nutshell: The desired inequality can be proven using ...
  • 1,808
1 vote
Accepted

Proof that sesquilinear form in is coercive

The first eigenvalue of the second derivative with Dirichlet b.c. on $(-1,1)$ is $\pi^2/4$ (with eigenfunction $\cos \frac{\pi x}{2}$) and then Poincare' inequality with optimal constant is $\|u\|_2^2 ...
0 votes

Function monotony between [0,T] and $L^2$

First, since you have $H^1(0,T)$ imbedds in $\mathscr{C}^0([0,T])$, $z$ can be seen as an element of $\mathscr{C}^0([0,T];L^2(\Omega))$ and you can speak without ambiguity of $z(t_1)$ and $z(t_2)$. ...
  • 2,240
0 votes

Periodic solution for linear parabolic equation - existence, regularity

For 1., if I am not mistaking you're searching for time-periodic functions enjoying Sobolev regularity in the space variable so the Sobolev regularity is not really linked with the periodicity: you're ...
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