By J. M. Sanz-Serna
Barbora Benešová and Martin Kružík are the authors of “Weak Lower Semicontinuity of Integral Functionals and Applications,” the Survey and Review paper in the latest issue of SIAM Review.
The notion of weak lower semicontinuity arises in a natural way when studying the existence of solutions of problems seeking to minimize a real-valued function or functional \(I(u)\) as \(u\) ranges in a suitable set Y of admissible candidates. It would be difficult to overemphasize the importance of those problems. Many scientific theories are built upon a minimization principle; Fermat's 1662 least-time principle in geometric optics provides perhaps the earliest example. Algorithms in applied mathematics, from data assimilation to training neural networks, are often based on minimization. What is weak lower semicontinuity and what role does it play in showing the existence of a minimizer of \(I\)? In the simplest case where the set Y of admissible candidates is a bounded, closed interval of the real line (or more generally a bounded, closed subset of a Euclidean space), we may start by considering a minimizing sequence, i.e., a sequence \(u_n \subset Y\) such that \(I(u_n)\) converges to \(I_* = \inf\ I(u): u\in Y \). By the Bolzano--Weierstrass theorem, there exists a subsequence \(u_{n_k}\) converging to a limit \(u_*\in Y\), and then, if \(I\) is continuous,
\[ I(u_*) = lim_k I(u_{n_k}), \]
or \(I(u_*)=I_*\). In this way, the infimum \(I_*\) is finite and \(u_*\) is a minimizer. The argument hinges both on (i) the continuity of \(I\) and (ii) the possibility of extracting convergent subsequences of sequences in \(Y\), or, in other words, the compactness of \(Y\). In the calculus of variations, where \(u\) is a function, \(I\) a functional such as \(\int_a^b f(x,u(x),u^\prime(x))dx\), and \(Y\) a closed, bounded subset of a normed linear space \(Y\), the same procedure will not work, because \(Y\) is not compact. The best we may hope for is that \(Y\) is compact in the weak topology of \(Y\) and extract from the minimizing sequence \(u_n \subset Y\) a weakly convergent subsequence, i.e., a subsequence such that \(lim_k \ell(u_{n_k}) = \ell(u_*)\) for each bounded, linear functional \(\ell\) on \(Y\). However, \(I\) is typically not continuous with respect to the weak topology (see Example 1.1 in the paper), and then the equation displayed above will not be satisfied. Fortunately, in many applications, \(I\) is weakly lower semicontinuous (lower semicontinuous with respect to the weak topology). This means that for sequences \(v_n\) that converge weakly to a limit \(v\), \(I(v) łeq łiminf_n(v_n)\); the prefix semi- refers to the fact that here we have the symbol \(łeq\) rather than \(=\). For weakly lower semicontinuous \(I\),
\[ I(u_*) \leq lim_k inf I(u_{n_k}). \]
Thus \(I(u_*) \leq I_*\), so that the weak limit \(u_*\) is a minimizer. The article, which begins with a very friendly introduction, reviews in detail the literature on weak lower semicontinuity since the 1950s. It also discusses applications, notably to elasticity, including recently studied situations where the energy functional depends on higher derivatives of the deformations.
J. M. Sanz-Serna is a full professor at Universidad Carlos III de Madrid. He is the section of editor of the Survey and Review section of SIAM Review.