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Jul
19
comment Existence of the solution of a Dirichlet type differential equation
You can show smoothness by local interior regularity: since $f_{\epsilon}$ is bounded you have $u\in W^{2,p}_{loc}$ for any large $p$, thus by Morrey's inequality $u\in C^{1,\alpha}_{loc}$. Bootstrapping on Schauder's estimates you immediately get $u\in \mathcal{C}^{\infty}$ (of course not all the way up to the boundary). As far as I can tell the $u(0)=\epsilon$ is not imposed, so it's not a constraint (cf the "suppose" in their theorem 1.2) but only a normalization (which they explain just below the theorem).
Jun
26
comment continuity of the Boltzmann entropy in the Wasserstein metric
Well, sure, if the relative H is lsc then of course the original one is lsc too. But do you have a reference?
Jun
26
comment continuity of the Boltzmann entropy in the Wasserstein metric
what about lower semi-continuity then? after thinking on it for a while this should be enough for my purpose, contrarily to what I said in my post
Jun
26
revised continuity of the Boltzmann entropy in the Wasserstein metric
fixed typos
Jun
26
accepted continuity of the Boltzmann entropy in the Wasserstein metric
Jun
26
comment continuity of the Boltzmann entropy in the Wasserstein metric
yes you're perfectly right, very nice counterexample. Yet another naive belief goes to the bin...
Jun
26
asked continuity of the Boltzmann entropy in the Wasserstein metric
May
6
awarded  Yearling
Apr
13
comment monotone parabolic systems, convex variational structure and Legendre transform
yep, user5678's answer below answers my question 1 (should have thought of it!), and the reference helps for question 2.
Apr
12
accepted monotone parabolic systems, convex variational structure and Legendre transform
Mar
21
comment if $u_\epsilon \rightarrow u$ weakly in $L^2$ then also $\partial_t u_\epsilon \rightarrow \partial_t u $ weakly in $L^2$
By $u_\epsilon \in L^{\infty}(I,H^1(M))\cap Lip(I,L^2(M))$ do you mean that you have uniform bounds (in $\epsilon$), or that for any fixed $\epsilon$ the function $u_{\epsilon}$ lies in this space? if you have uniform bounds then the statement is trivial, since then $\partial_tu_{\epsilon}$ is bounded in $L^{\infty}(I,L^2(M))\subset L^2(I,L^2(M))$ so in by the Banach-Alaoglu-Bourbaki theorem $\partial_tu_{\epsilon}\rightharpoonup v$ weakly in $L^2(I,L^2(M))$ for some $v$ and then by continuity $v=\partial_t u$. Otherwise I think the statement is false.
Dec
17
awarded  Popular Question
Nov
6
accepted Reference request: Wasserstein metric spaces for non linear weights/mobility?
Nov
6
comment Reference request: Wasserstein metric spaces for non linear weights/mobility?
A quick look indicates that this is exactly what I needed, thank you Nicola.
Nov
6
revised Reference request: Wasserstein metric spaces for non linear weights/mobility?
added 15 characters in body
Nov
6
asked Reference request: Wasserstein metric spaces for non linear weights/mobility?
Aug
12
comment Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
OK, thanks @Christian Remling. Any specific reference I could cite? I was suspecting that, but for example I'm having a hard time figuring out what $L^2(\mathbb R^d,\delta_0)$ looks like (Dirac mass)...
Aug
12
asked Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
Aug
4
comment Spectrum of this ODE
what about the pairwise eigenfunctions, do they have any remarkable "pairwise" behaviour (odd/even, or whatever...)?
Aug
1
comment Weak convergence of a sequence
Indeed, my argument really shows that $\nabla u_k\rightharpoonup \nabla u$ in $L^2L^2$, I just kept using $\nabla \Psi$ as test functions in order to stay consistent with the OP's setting. But I disagree with $\Delta u \in L^2(0,T; H^1_0)$, the above inequality gives $\Delta u \in L^2(0,T; H^{-1})$!