Minimizing sequence

A minimizing sequence satisfies $E(u_n)\to \inf E$. Coercivity bounds an appropriate sublevel set, Compactness supplies a convergent subsequence, and sequential lower semicontinuity turns its limit into a global minimizer.

A minimizing sequence can converge uniformly while its derivatives keep oscillating between distinct energy wells. For the functional $I(u)={\int }_0^1[u^4+(u^{\prime 4}-16{)}^4]$, with zero endpoint values and Lipschitz admissible functions, the triangular waves displayed above have slopes $\pm 2$ almost everywhere and height $1/N$. Direct integration gives $I(u_N)=1/(5N^4)$. Nonnegativity therefore makes the infimum zero. Attainment would require both $u=0$ and $|u^{\prime }|=2$ almost everywhere, an impossibility. The uniform limit is zero but its energy is $16^4$; thus the energy fails to be lower semicontinuous for this convergence. The example explains why control of function values alone cannot replace suitable convexity in the derivative variable in the direct method in the calculus of variations.