Convex integrals on Sobolev spaces. (English) Zbl 1254.49006

Barbu, Viorel; Guo, Yanqiu; Rammaha, Mohammad A.; Toundykov, Daniel

J. Convex Anal. 19, No. 3, 837-852 (2012).

Summary: Let \(j_0, j_1: \mathbb {R}\to [0,\infty)\) 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 domain conditio 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. Brézis [Intégrales convexes dans les espaces de Sobolev. Proc. int. Symp. partial diff. Equ. Geometry normed lin. Spaces I. (French), Isr. J. Math. 13, 9–23, (1972; Zbl 0249.46017)] 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.

MSC:

49J52

Nonsmooth analysis

Keywords:

convex integral; Sobolev space; subdifferential; convex conjugate

Citations:

Zbl 0249.46017