Timeline for Functionals continuous with respect to weak convergence
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 3, 2011 at 14:25 | comment | added | Phil Isett | Hi Dorian, I think that the theorems you are quoting about lower-semicontinuity etc. regard functionals which include derivatives. Functionals without derivatives make sense at a purely measure-theoretic level, where the notion of quasiconvexity does not make sense. I should clarify the question. | |
| Sep 13, 2011 at 1:41 | history | edited | Dorian | CC BY-SA 3.0 | reversed inclusion for $H^1$ and $L^4$ |
| Sep 13, 2011 at 1:40 | comment | added | Dorian | semi-continuity but not vice versa (the correct notion there is quasiconvexity). | |
| Sep 13, 2011 at 1:40 | comment | added | Dorian | Hi Phil. In fact the reason lower semi-continuity is hard for functionals like $\int f(x,u(x))dx$ is that assuming you have a minimizing sequence, you don't have any coercivity. You don't conclude any bounds in say $H^1$ on your functions $u_n$ from assuming $\limsup_{n} \int f(x,u_n(x)) dx < +\infty$ and so you don't have sufficient compactness. Does that make sense? And do you mean continuity or lower semi-continuity? In the scalar valued case you are considering, lower semi-continuity always implies convexity in that variable. For the vector valued case, convexity always implies | |
| Sep 12, 2011 at 2:39 | comment | added | Phil Isett | I can show that continuity with respect to, say, $L^\infty$ weak-* convergence implies $f(x, \theta u + (1-\theta) v ) = \theta f(x, u) + (1-\theta) f(x, v)$ for all bounded measurable functions $u, v$ and measurable functions $\theta \in [0,1]$; so now I'm wondering: does it follow $f(x,u)$ is convex in $u$ for almost every $x$? | |
| Sep 12, 2011 at 2:37 | comment | added | Phil Isett | Thanks for the reference and all of the tips! I think you meant to say that $H^1 \subseteq \subseteq L^4$. I looked at Dacorogna for a little while. One thing I wanted to understand, which oddly enough does not seem to be in the book but maybe I'll be able to figure it out from the methods, is the lower semicontinuity of functionals $\int f(x, u(x)) dx$ which do not involve the derivatives.... | |
| Sep 10, 2011 at 13:31 | history | answered | Dorian | CC BY-SA 3.0 |