## Convex integrals on Sobolev spaces.(English)Zbl 1254.49006

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

Zbl 0249.46017
Full Text: