Criterion for weakly lower semicontinuity in problems of calculus of variations
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Let’s say I have to minimize a certain integral functional $F(u)= int_{Omega} L(x,u,nabla u)$ with $Omega$ regular and $u$ in a certain Sobolev space (don’t really care to be precise, I just want a general idea). To prove there exists a minimum, let’s say I want to use the direct method. No problems for compactness, but when I want the lower semicontinuity for weakly convergent sequences (the derivatives, typically) I am kinda lost. I know the norm is weakly lower semicontinuous, but is there a simple general criterion to have weakly lower semicontinuity, in general?
optimization sobolev-spaces calculus-of-variations
asked Dec 31 '18 at 15:29
tommy1996qtommy1996q
604415
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add a comment |
$begingroup$
Let’s say I have to minimize a certain integral functional $F(u)= int_{Omega} L(x,u,nabla u)$ with $Omega$ regular and $u$ in a certain Sobolev space (don’t really care to be precise, I just want a general idea). To prove there exists a minimum, let’s say I want to use the direct method. No problems for compactness, but when I want the lower semicontinuity for weakly convergent sequences (the derivatives, typically) I am kinda lost. I know the norm is weakly lower semicontinuous, but is there a simple general criterion to have weakly lower semicontinuity, in general?
optimization sobolev-spaces calculus-of-variations
asked Dec 31 '18 at 15:29
tommy1996qtommy1996q
604415
$endgroup$
$begingroup$
If I'm remembering correctly a convexity assumption on the lagrangian $L$ is sufficient to get the w.l.s.c. I think it's in Evans's book? Or maybe Jost's.
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– Jose27
Dec 31 '18 at 19:08
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Convexity and strong semicontinuity is enough, indeed. The problem is that in dimensions greater than 1 you usually only have $L^p$ convergence for the sequence. I mean, if I had uniform convergence then a a continuous Lagrangian would work. But with $L^p$ convergence, how do I have strong lower semicontinuity given strong $L^p$ convergence?
$endgroup$
– tommy1996q
Dec 31 '18 at 19:34
add a comment |
$begingroup$
Let’s say I have to minimize a certain integral functional $F(u)= int_{Omega} L(x,u,nabla u)$ with $Omega$ regular and $u$ in a certain Sobolev space (don’t really care to be precise, I just want a general idea). To prove there exists a minimum, let’s say I want to use the direct method. No problems for compactness, but when I want the lower semicontinuity for weakly convergent sequences (the derivatives, typically) I am kinda lost. I know the norm is weakly lower semicontinuous, but is there a simple general criterion to have weakly lower semicontinuity, in general?
optimization sobolev-spaces calculus-of-variations
asked Dec 31 '18 at 15:29
tommy1996qtommy1996q
604415
$endgroup$
Let’s say I have to minimize a certain integral functional $F(u)= int_{Omega} L(x,u,nabla u)$ with $Omega$ regular and $u$ in a certain Sobolev space (don’t really care to be precise, I just want a general idea). To prove there exists a minimum, let’s say I want to use the direct method. No problems for compactness, but when I want the lower semicontinuity for weakly convergent sequences (the derivatives, typically) I am kinda lost. I know the norm is weakly lower semicontinuous, but is there a simple general criterion to have weakly lower semicontinuity, in general?
optimization sobolev-spaces calculus-of-variations
optimization sobolev-spaces calculus-of-variations
asked Dec 31 '18 at 15:29
tommy1996qtommy1996q
604415
asked Dec 31 '18 at 15:29
tommy1996qtommy1996q
604415
asked Dec 31 '18 at 15:29
tommy1996qtommy1996q
604415
asked Dec 31 '18 at 15:29
tommy1996qtommy1996q
604415
asked Dec 31 '18 at 15:29
tommy1996qtommy1996q
604415
604415
$begingroup$
If I'm remembering correctly a convexity assumption on the lagrangian $L$ is sufficient to get the w.l.s.c. I think it's in Evans's book? Or maybe Jost's.
$endgroup$
– Jose27
Dec 31 '18 at 19:08
$begingroup$
Convexity and strong semicontinuity is enough, indeed. The problem is that in dimensions greater than 1 you usually only have $L^p$ convergence for the sequence. I mean, if I had uniform convergence then a a continuous Lagrangian would work. But with $L^p$ convergence, how do I have strong lower semicontinuity given strong $L^p$ convergence?
$endgroup$
– tommy1996q
Dec 31 '18 at 19:34
add a comment |
$begingroup$
If I'm remembering correctly a convexity assumption on the lagrangian $L$ is sufficient to get the w.l.s.c. I think it's in Evans's book? Or maybe Jost's.
$endgroup$
– Jose27
Dec 31 '18 at 19:08
$begingroup$
Convexity and strong semicontinuity is enough, indeed. The problem is that in dimensions greater than 1 you usually only have $L^p$ convergence for the sequence. I mean, if I had uniform convergence then a a continuous Lagrangian would work. But with $L^p$ convergence, how do I have strong lower semicontinuity given strong $L^p$ convergence?
$endgroup$
– tommy1996q
Dec 31 '18 at 19:34
$begingroup$
If I'm remembering correctly a convexity assumption on the lagrangian $L$ is sufficient to get the w.l.s.c. I think it's in Evans's book? Or maybe Jost's.
$endgroup$
– Jose27
Dec 31 '18 at 19:08
$begingroup$
If I'm remembering correctly a convexity assumption on the lagrangian $L$ is sufficient to get the w.l.s.c. I think it's in Evans's book? Or maybe Jost's.
$endgroup$
– Jose27
Dec 31 '18 at 19:08
$begingroup$
Convexity and strong semicontinuity is enough, indeed. The problem is that in dimensions greater than 1 you usually only have $L^p$ convergence for the sequence. I mean, if I had uniform convergence then a a continuous Lagrangian would work. But with $L^p$ convergence, how do I have strong lower semicontinuity given strong $L^p$ convergence?
$endgroup$
– tommy1996q
Dec 31 '18 at 19:34
$begingroup$
Convexity and strong semicontinuity is enough, indeed. The problem is that in dimensions greater than 1 you usually only have $L^p$ convergence for the sequence. I mean, if I had uniform convergence then a a continuous Lagrangian would work. But with $L^p$ convergence, how do I have strong lower semicontinuity given strong $L^p$ convergence?
$endgroup$
– tommy1996q
Dec 31 '18 at 19:34
add a comment |
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$begingroup$
If I'm remembering correctly a convexity assumption on the lagrangian $L$ is sufficient to get the w.l.s.c. I think it's in Evans's book? Or maybe Jost's.
$endgroup$
– Jose27
Dec 31 '18 at 19:08
$begingroup$
Convexity and strong semicontinuity is enough, indeed. The problem is that in dimensions greater than 1 you usually only have $L^p$ convergence for the sequence. I mean, if I had uniform convergence then a a continuous Lagrangian would work. But with $L^p$ convergence, how do I have strong lower semicontinuity given strong $L^p$ convergence?
$endgroup$
– tommy1996q
Dec 31 '18 at 19:34