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!!!

  • $\begingroup$ 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'? $\endgroup$ – Uskebasi Apr 17 '17 at 10:55
  • $\begingroup$ @QWERTZ I' m sorry.. I meant $\;\mathcal M\;$ You're right.. I'll edit my post right now! Thanks $\endgroup$ – kaithkolesidou Apr 17 '17 at 10:58
  • $\begingroup$ 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 $W^{1,2}_{loc}(l^-,l^+)$. $\endgroup$ – Uskebasi Apr 17 '17 at 11:18
  • $\begingroup$ 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). 2) 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 $\endgroup$ – Uskebasi Apr 17 '17 at 11:23
  • $\begingroup$ 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. $\endgroup$ – Uskebasi Apr 17 '17 at 11:29

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