
Journal of Convex Analysis 19 (2012), No. 3, 837852 Copyright Heldermann Verlag 2012 Convex Integrals on Sobolev Spaces Viorel Barbu Dept. of Mathematics, University "Al. J. Cuza", 6600 Iasi, Romania vb41@uaic.ro Yanqiu Guo Dept. of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A. syguo2@math.unl.edu Mohammad A. Rammaha Dept. of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A. mrammaha1@math.unl.edu Daniel Toundykov Dept. of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A. dtoundykov2@unl.edu [Abstractpdf] 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) 923 (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. [ Fulltextpdf (160 KB)] for subscribers only. 