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visits | member for | 2 years, 3 months |
seen | 6 hours ago | |
stats | profile views | 331 |
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 $\Delta u \in L^2(0,T; H^{-1})$! |