Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations (Q2710423)

The paper is concerned with sufficient conditions for minimality in the context of classical integral functionals of the Calculus of Variations NEWLINE\[NEWLINE u\mapsto\int_\Omega f(x,u,\nabla u) dx. NEWLINE\]NEWLINE The main novelty of the paper is the analysis and the characterization of local \(L^r\)-minimality, a concept stronger than the classical \(L^\infty\) minimality. The approach of the paper is not based on field theory; instead, it is based on an extension of Hestenes methods, in turn relying on contradiction arguments and on the continuity and the lower semicontinuity of quadratic forms arising from the expansion of the functional. The treatment is quite complete and includes in the end also some applications to vector-valued problems, where convexity can be achieved using suitable null Lagrangians.