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Let $f:\mathbb R^m \rightarrow [0,+\infty)\;$ be a smooth function that vanishes on a finite set $A\;$ where $\vert A \vert\; \ge 2$ and the maps $v:(l^{-},l^{+}) \rightarrow \mathbb R^m\;$ defined by

$\mathcal M= \{\;v\in W^{1,2}_{loc} (l^{-},l^{+});\;-\infty \le l^{-} \lt l^{+} \le +\infty\;,\;\lim_{x \to l^{-}} v(x)=a_1 \in A\;,\;\lim_{x \to l^{+}} v(x)=a_2 \neq a_1 \in A\;\;,\;v((l^{-},l^{+}))\subseteq \mathbb R \setminus A \}\;$.

Show that a minimizer of the functional $\;I(v)=\int_{l^{-}}^{l^{+}} \frac{{\dot v}^2}{2} + f(v) \;dx$ on $\;\mathcal M\;$ exists.

I found after some research the following theorem:

Let $\;X\;$ be reflexive, $\;M\subset X\;$ nonempty and weakly sequentially closed,$\; F:M\rightarrow \mathbb R \;$ coercive and weakly sequentially lower semi-continuous. Then there exist $\;x_0 \in M\;$ such that ; $\;F(x_0)=\inf_{x \in M} F(x)\;$

I know $\;W^{1,2}\;$ is reflexive and for the coerciveness of $\;I\;$ there is a hint: "Assume $\limsup_{\vert v \vert \to +\infty} f(v) \gt 0\;$" which I don't completely understand why is needed.

My question: How do I prove that $\; \mathcal M\;$ and $\;I\;$ satisfy the above theorem?

I'm really new to Sobolev Spaces and I hadn't seen before the term "weakly sequentially closed". I would appreciate if somebody could help me through this. Furthermore, any suggestions about useful books related to this topic, would be valuable.

Thanks in advance!!!

kaithkolesidou
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  • What do you mean by 'Show that a minimizer of the functional $;I(v)=\int_{l^{-}}^{l^{+}} \frac{{\dot v}^2}{2} + f(v) ;dx$ on A exists'? My doubt is: why on A? Shouldn't the question be something like 'show that the functional has minimum on M'? – Uskebasi Apr 17 '17 at 10:55
  • @QWERTZ I' m sorry.. I meant $;\mathcal M;$ You're right.. I'll edit my post right now! Thanks – kaithkolesidou Apr 17 '17 at 10:58
  • I have a similar problem on my homework sheet. If I have time I will look if I can handle your problem. For the moment two ideas (you have to be really careful when you'll try to formalize them):
    1. You should have a theorem that assures you that a bounded sequence in $W^{1,2}(B)$ admits a weakly convergent subsequence, where B is an open set. You want to apply it to a minimizing sequence, unfortunately you cannot apply it directly since you are working in $W^{1,2}_{loc}(l^-,l^+)$.
     – Uskebasi Apr 17 '17 at 11:18
  • To overcome this problem you have to use a diagonal argument to estract a subsequence weakly convergent in your space M (you have to prove that boundary conditions are satisfied etc etc).
    1. you should have Tonelli's theorem that assures you lower semicontinuity (for functionals convex in $v'$). Again, in works only on bounded sets thus if $l^-=-\infty,\ l^+=\infty$ you have to fix an increasing sequence of compact sets contained in $(l^-,l^+)$, use the theorem in every compact set and try to pass to the limit with standard limit theorems
     – Uskebasi Apr 17 '17 at 11:23
  • You don't need Tonelli, but also the theorem which requires coerciveness and convexity in $v'$, works only on bounded sets (at least I know only that version). In the unbounded case you have to argue in the way I suggested in the second point. – Uskebasi Apr 17 '17 at 11:29
  • @QWERTZ First of all, I'd like to thank you for posting your ideas -even if they are in the comments section-. I 'll try to follow the steps you suggested. Do you know any book I could read in order to get used to handle problems like this? – kaithkolesidou Apr 17 '17 at 11:32
  • @QWERTZ Do you think, the hint might implies something for convexity? – kaithkolesidou Apr 17 '17 at 11:33
  • You need convexity for the lagrangian only in $v'$ and you have it. You know that f takes values in $[0,\infty)$, the hint seems to exclude the case $\lim_{\vert v\vert\rightarrow\infty}f(v)=0$. It is not clear to me why he is suggesting this (maybe there is some result in your notes that needs this assumption). Dacorogna's book 'Introduction to the calculus of variations' is pretty good. There isn't this specific problem but there are many examples and solutions to exercises – Uskebasi Apr 17 '17 at 11:52
  • @QWERTZ I found the book you suggested and I have to admit, it's really helpful! Unfortunately, I was busy the last two days and I saw today the rest of your comments. I would like to have a private conversation with you about the post, but I don't know how to invite you... However, you've helped me a lot. Thanks!!! – kaithkolesidou Apr 19 '17 at 16:32
  • I've created a room. We can continue the discussion there. – Uskebasi Apr 20 '17 at 14:25

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