# Thread: Calculus of Variations - proof of Semicontinuity

1. ## Calculus of Variations - proof of Semicontinuity

Dear mathematicians,
I would like to ask you if anybody of you could explain me the proof of the theorem of semicontinuity:
$\displaystyle Let \Omega be an open set in R^n[IMG]file:///C:/DOCUME%7E1/Jitka/LOCALS%7E1/Temp/msohtml1/01/clip_image002.gif[/IMG], let M be a closed set in R^N , and let F(x, u, z) be a function defined in \Omega \times M \times R^\upsilon and such that (i) F is a CARATHEODORY function, that is measurable in x for every (u,z)\in M \times R^\upsilon and continuous in (u,z) for almost every x\in \Omega (ii) F(x, u, z) is convex in z for almost every x \in \Omega and for every u\in S (iii) F\geq 0 Let u_h, u \in L^1(\Omega,M), z_h,z \in L^1(\Omega, R^\upsilon) and assume that u_h\to u and z_h \to z in L_{loc}^1 (\Omega) Then, \int_{\Omega} F(x, u, z)dx\le \liminf_{h \to \infty}\int_{\Omega}F(x,u_h,z_h)dx$

(I dont know how can I write it here properly, so if anybody of you could help me, you can contact me also through email: jita.holu@seznam.cz and I will send you the file where it is written normally.
Thanks for any kind of your help

[IMG]file:///C:/DOCUME%7E1/Jitka/LOCALS%7E1/Temp/msohtml1/01/clip_image026.gif[/IMG]

2. This is the best that I can do to make the question readable. Have I edited it correctly?

Dear mathematicians,
I would like to ask you if anybody of you could explain me the proof of the theorem of semicontinuity:

Let $\displaystyle \Omega$ be an open set in $\displaystyle R^n$, let M be a closed set in $\displaystyle R^N$, and let $\displaystyle F(x, u, z)$ be a function defined in $\displaystyle \Omega \times M \times R^\upsilon$ and such that

(i) $\displaystyle F$ is a CARATHEODORY function, that is measurable in x for every $\displaystyle (u,z)\in M \times R^\upsilon$ and continuous in $\displaystyle (u,z)$ for almost every $\displaystyle x\in \Omega$,

(ii) $\displaystyle F(x, u, z)$ is convex in $\displaystyle z$ for almost every $\displaystyle x \in \Omega$ and for every $\displaystyle u\in S$,

(iii) $\displaystyle F\geq 0$.

Let $\displaystyle u_h, u \in L^1(\Omega,M),\ z_h,z \in L^1(\Omega, R^\upsilon)$ and assume that $\displaystyle u_h\to u$ and $\displaystyle z_h \to z$ in $\displaystyle L_{loc}^1 (\Omega)$. Then,

$\displaystyle \displaystyle\int_{\Omega} F(x, u, z)dx\le \liminf_{h \to \infty}\int_{\Omega}F(x,u_h,z_h)dx$.
The next question would be: How to prove that result? I haven't thought about it carefully, but my guess would be that Fatou's lemma should be involved.

3. Thanks very much that you made it readable! I have the proof of this but I need if anybody could explain me it in words, not only mathematical symbols. I can write the proof I have, if it is neccessary...