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Math Help - Calculus of Variations - proof of Semicontinuity

  1. #1
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    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:
    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

    " alt=" 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]
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  2. #2
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    This is the best that I can do to make the question readable. Have I edited it correctly?

    Quote Originally Posted by nylinka View Post
    Dear mathematicians,
    I would like to ask you if anybody of you could explain me the proof of the theorem of semicontinuity:

    Let \Omega be an open set in R^n, 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,

    \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.
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  3. #3
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    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...
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