Convex Integrals on Sobolev Spaces

Let

denote convex functions vanishing at the origin, and let

\Omega Ω

be a bounded domain in

\mathbb{R}^3 R 3

with sufficiently smooth boundary

\Gamma Γ

. 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 J ( u ) = ∫ Ω j 0 ( u ) d Ω + ∫ Γ j 1 ( γ u ) d Γ

on the Sobolev space

H^1(\Omega) H 1 ( Ω )

. We describe the convex conjugate

J^* J ∗

and the subdifferential

\partial J ∂ J

. It is shown that the action of

\partial J ∂ J

coincides pointwise a.e. in

\Omega Ω

with

\partial j_0(u(x)) ∂ j 0 ( u ( x ) )

, and a.e on

\Gamma Γ

with

\partial j_1(u(x)) ∂ 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 J 0 ( u ) = ∫ Ω j 0 ( u ) d Ω

and

J_1(u) = \int_\Gamma j_1(\gamma u)d\Gamma J 1 ( u ) = ∫ Γ j 1 ( γ u ) d Γ

, the lack of any growth restrictions on

j_0 j 0

and

j_1 j 1

makes the sufficient domain condition for the sum of two maximal monotone operators

\partial J_0 ∂ J 0

and

\partial J_1 ∂ J 1

infeasible to verify directly. The presented theorems extend the results of H. Brézis [Intégrales 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 combination of supercritical interior and boundary damping feedbacks can be modeled by the subdifferential of a suitable convex functional on the state space