Abstract

Let j0,j1:R↦[0,∞)j_0, j_1: \mathbb{R}\mapsto [0,\infty) denote convex functions vanishing at the origin, and let Ω\Omega be a bounded domain in R3\mathbb{R}^3 with sufficiently smooth boundary Γ\Gamma. This paper is devoted to the study of the convex functional J(u)=∫Ωj0(u)dΩ+∫Γj1(γu)dΓJ(u)=\int_{\Omega} j_0(u)d\Omega + \int_{\Gamma} j_1(\gamma u) d\Gamma on the Sobolev space H1(Ω)H^1(\Omega). We describe the convex conjugate J∗J^* and the subdifferential ∂J\partial J. It is shown that the action of ∂J\partial J coincides pointwise a.e. in Ω\Omega with ∂j0(u(x))\partial j_0(u(x)), and a.e on Γ\Gamma with ∂j1(u(x))\partial j_1(u(x)). These conclusions are nontrivial because, although they have been known for the subdifferentials of the functionals J0(u)=∫Ωj0(u)dΩJ_0(u) = \int_\Omega j_0(u)d\Omega and J1(u)=∫Γj1(γu)dΓJ_1(u) = \int_\Gamma j_1(\gamma u)d\Gamma, the lack of any growth restrictions on j0j_0 and j1j_1 makes the sufficient domain condition for the sum of two maximal monotone operators ∂J0\partial J_0 and ∂J1\partial J_1 infeasible to verify directly. The presented theorems extend the results of H. Br{é}zis [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

Author information

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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.

s-yguo2@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

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Suggested citation

V. Barbu, Y. Guo, M. A. Rammaha, D. Toundykov. “Convex Integrals on Sobolev Spaces.” Journal of Convex Analysis 19 (2012), No. 3, 837–852. https://doi.org/10.68381/jca19043

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.