Journal of Convex Analysis 27 (2020), No. 3, 1003--1013
Copyright Heldermann Verlag 2020
Characterizations of a Slow Growth Assumption in Variational Calculus and Gale-Klee Continuity of Epigraphs
Farhad Husseinov
Dept. of Economics, ADA University, 1008 Baku, Azerbaijan
fhusseinov@ada.edu.az
The paper is devoted to the geometric and topological characterizations of a slow growth assumption in variational calculus introduced by Cellina. This assumption plays an important role in studying relaxation, Lipschitzianity of solutions, and nonoccurrence of the Lavrentiev phenomenon in variational problems. It is shown that the slow growth assumption is equivalent to the continuity of the epigraph (as defined by Gale and Klee, 1959) of the second Legendre-Fenchel conjugate f** of the integrand with respect to the derivative variables. It is also shown that this assumption is equivalent to the openness of the domain of the Legendre-Fenchel conjugate with respect to the derivative variables plus continuity of f* on the whole space. As a spin-off of the tools developed in the paper we obtain a criterion for the openness of the domain of the conjugate of a convex function in terms of the continuity of the epigraph of this function. We use these tools also to establish a necessary and sufficient condition for the inf-compactness of lower semicontinuous functions and some related results.
Keywords: Variational calculus, slow growth assumption, Legendre-Fenchel conjugate, continuous set, subdifferential.
MSC: 49N99, 26B25.
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