Convex Integrals on Sobolev Spaces

V. Barbu ; Y. Guo ; M. A. Rammaha ; D. Toundykov

Journal of convex analysis, Tome 19 (2012) no. 3, pp. 837-852.

Résumé

Let $j_0, j_1: \mathbb{R}\mapsto [0,\infty)$ denote convex functions vanishing at the origin, and let $\Omega$ be a bounded domain in $\mathbb{R}^3$ with sufficiently smooth boundary $\Gamma$. This paper is devoted to the study of the convex functional $$ J(u)=\int_{\Omega} j_0(u)d\Omega + \int_{\Gamma} j_1(\gamma u) d\Gamma $$ on the Sobolev space $H^1(\Omega)$. We describe the convex conjugate $J^*$ and the subdifferential $\partial J$. It is shown that the action of $\partial J$ coincides pointwise a.e. in $\Omega$ with $\partial j_0(u(x))$, and a.e on $\Gamma$ with $\partial j_1(u(x))$. These conclusions are nontrivial because, although they have been known for the subdifferentials of the functionals $J_0(u) = \int_\Omega j_0(u)d\Omega$ and $J_1(u) = \int_\Gamma j_1(\gamma u)d\Gamma$, the lack of any growth restrictions on $j_0$ and $j_1$ makes the sufficient \emph{domain condition} for the sum of two maximal monotone operators $\partial J_0$ and $\partial J_1$ infeasible to verify directly. The presented theorems extend the results of H. Br{\'e}zis [Int\'egrales convexes dans les espaces de Sobolev, Proc. Int. Symp. Partial Diff. Equations and the Geometry of Normed Linear Spaces, Jerusalem 1972, vol. 13 (1972) 9--23 (1973); MR 0341077 (49\#5827)] and fundamentally complement the emerging research literature addressing supercritical damping and sources in hyperbolic PDE's. These findings rigorously confirm that a \emph{combination} of supercritical interior and boundary damping feedbacks can be modeled by the subdifferential of a suitable convex functional on the state space.