The basic ideas concerning the Direct Method as well as lower semicontinuity and its connection to convexity are due to Leonida Tonelli and were established in a series of articles in the early 20th century [275–277]. In the 1960s James Serrin generalized the results to higher dimensions [242].

Most of the material in this chapter is very classical and can be found in a variety of books on the calculus of variations, we refer in particular to [76, 77, 137]. We note that a very general lower semicontinuity theorem for convex integrands can be found in Theorem 3.23 of [76].

All of our abstract results on the Direct Method are formulated using sequences and not using general topology tools like nets. This is justified since the weak topology on a separable, reflexive Banach space and the weak*-topology on a dual space with a separable predual are metrizable on norm-bounded sets. Thus, if the functionals under investigation satisfy suitable coerciveness assumptions, one can work with sequences. The only case where one has to be careful is when one uses the weak topology on a non-reflexive Banach space with a non-separable dual space because then the weak topology might not be metrizable. For instance, in the sequence space \(l^1\) (with non-separable dual space \(l^\infty \)), weak convergence of *sequences* is equivalent to strong convergence, but the weak and strong *topologies* still differ (see Chapter V in [74] for more details on such considerations). For us more relevant is the observation that norm-bounded sets in \(\mathrm {L}^1(\varOmega )\) are not weakly precompact, either sequentially or topologically (these notions turn out to be equivalent by the Eberlein–Šmulian theorem). This corresponds to functionals with linear growth, which indeed require a more involved analysis in the space of functions of bounded variation (BV). We will come back to this topic in Chapters 10– 12.

For the *u*-dependent variational integrals the growth in the *u*-variable can be improved up to *q*-growth, where \(q \in [1,p/(p-d))\) by the Sobolev embedding theorem. Moreover, we can work with the more general growth bounds \(|f(x,v, A)| \le M(h(x) + |v|^q + |A|^p)\), with \(h \in \mathrm {L}^1(\varOmega ;[0,\infty ))\) and \(q \in [1,p/(p-d))\). For reasons of simplicity, we have omitted these generalizations here.

The Lavrentiev gap phenomenon was discovered in [175], our Example 2.14 is due to Manià; we follow the description in [117]. Tonelli’s Regularity Theorem [118, 275] gives regularity and hence the absence of the Lavrentiev gap phenomenon, for some integral functionals with superlinear growth; also see [49, 140–143] for some recent developments in this direction.

Much of the theory of general convex functions was developed by Jean-Jacques Moreau and R. Tyrrell Rockafellar in the 1960s. The books [106, 232] and the more advanced monographs [192, 193, 233] develop these topics in great detail.