Convex integrals on sobolev spaces

Convex integrals on sobolev spaces Article

Barbu, V, Guo, Y, Rammaha, MA et al. (2012). Convex integrals on sobolev spaces . JOURNAL OF CONVEX ANALYSIS, 19(3), 837-852.

Let j 0, j 1: ℝ →[0, ∞ denote convex functions vanishing at the origin, and let Ω be a bounded domain in ℝ ³ with sufficiently smooth boundary T. This paper is devoted to the study of the convex functional J(u) = ∫Ω jo(u)dΩ + ∫ Γj1(γu)dΓ on the Sobolev space H ¹(Ω). We describe the convex conjugate J and the subdifferential ∂J. It is shown that the action of ∂J coincides pointwise a.e. in Ω with ∂j 0(u(x)), and a.e on Γ with ∂j 1(u(x)). These conclusions are nontrivial because, although they have been known for the subdifferentials of the functionals J 0(u) = ∫ Ω and J 1 (u) = ∫Γj 1(γu)dΓ the lack of any growth restrictions on j 0 and j 1 makes the sufficient domain condition for the sum of two maximal monotone operators ∂J 0 and ∂J 1 infeasible to verify directly. The presented theorems extend the results in [6] 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. © Heldermann Verlag.