Extremality in solving general quasilinear parabolic inclusions
This paper deals with initial boundary value problems for parabolic inclusions extending previous results by the same authors. More precisely, the multivalued term is a difference between the Clark generalized gradient of some locally Lipschitz function, verifying a unilateral growth condition, and the subdifferential of a convex function; the elliptic part is a quasilinear operator of Leray-Lions type. This type of differential inclusions has applications in solving problems with non-convex and non-smooth functionals in mechanics and in engineering. The authors, assuming the existence of an upper solution \(\overline u\) and a lower solution \(\underline u\) such that \(\underline u\leq\overline u\) and some conditions on the data, prove the existence of an extremal solution in the interval \([\underline u,\overline u]\), and the compactness of the solution set in a sector in adequate function spaces.
Clark generalized gradient of locally Lipschitz functions
scientific article; zbMATH DE number 2154316