I have the following problem:
The definition if weak lower semicontinuity states
A function $I[\cdot]$ is weakly lower semicontinuous on $W^{1,q}(U)$ provided \begin{equation} I[u] \leq \liminf_{k \to \infty} I[u_k] \end{equation} whenever $u_k \rightharpoonup u$ weakly in $W^{1,q}(U)$. Here $U \subset \mathbb{R}^N$.
Now assume $I[\cdot]$ has the form \begin{equation} I[w] = \int_U L(Dw(x),w(x),x) dx ~~~~~ w \in W^{1,q}(U,\mathbb{R})\end{equation} so that \begin{equation} L: \mathbb{R}^N \times \mathbb{R} \times U \to \mathbb{R}, ~~ (p,z,x) \mapsto L(p,z,x) \end{equation}
Then one can proof
Assume $L$ is smooth, bounded below and in addition the mapping $p \mapsto L(p,z,x)$ is convex for each $z \in \mathbb{R}$, $x \in U$. Then $I[\cdot]$ is lower semicontinuous on $W^{1,q}(U)$
My Question is:
Now consider the specific functional \begin{equation} I[u]=\frac{1}{2} \int \vert Du \vert^2 + \frac{1}{4} \int (1-\vert u \vert^2)^2, \end{equation} so that \begin{equation} L(p,z,x)=\frac{1}{2} \vert p \vert^2+ \frac{1}{4} (1- \vert z \vert^2)^2. \end{equation} One could apply the above theorem, but in my case $u$ is complex valued, so that $u \in W^{1,q}(U,\mathbb{C})$.
$L$ is now still convex and bounded below, but it is not smooth anymore nor is $z \in \mathbb{R}$ as in the theorem.
Can one relax the assumptions in the above theorem, so that I can deduce weak lower semi continuity for this functional? Can one give me a hint to a more general result in literature? Or is there another way to proof the weak lower semi-continuity of this functional?
Any help is much appreciated!
Forget the complex structure: think of $u$ as a Sobolev map $u:U\to \mathbb R^2$. The Lagrangian is $C^\infty$ smooth, since $|z|^2 = u_1^2+u_2^2$ is a polynomial. (I guess that $|z|$ is squared in $L$ specifically to keep $L$ smooth.)
Weak lower semicontinuity of convex functionals still holds, for the same basic reason: a convex functional is the supremum of a family of linear functionals, which are weakly continuous. In fact, one can even replace convexity by weaker assumptions of polyconvexity or quasiconvexity. (In the scalar case, both of those reduce to ordinary convexity). See section 2.2 here and references there.
Vector-valued calculus of variations becomes dramatically different from scalar-valued when the regularity of minimizers is brought into consideration. If you are interested in this aspect, take a look at Regularity of minima: an invitation to the Dark Side of the Calculus of Variations by Mingione.