Local Lipschitz continuity of solutions to a problem in the calculus of variations

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• 作者：Pierre Bousquet ; Francis Clarke
• 关键词：Pontryagin Maximum Principle ; Set separation ; Transversality ; Lipschitz
• 刊名：Journal of Differential Equations
• 出版年：2007
• 出版时间：15 December 2007
• 年：2007
• 卷：243
• 期：2
• 页码：489-503
• 全文大小：159 K

文摘

This article studies the problem of minimizing ∫_ΩF(Du)+G(x,u) over the functions up://www.sciencedirect.com/scidirimg/entities/2208.gif"" alt=""set membership, variant"" border=0>Wp>1,1p>(Ω) that assume given boundary values p://www.sciencedirect.com/scidirimg/entities/3d5.gif"" alt=""phi"" border=0> on ∂Ω. The function F and the domain Ω are assumed convex. In considering the same problem with e52172df38f4c44e49"" title=""Click to view the MathML source"">G=0, and in the spirit of the classical Hilbert–Haar theory, Clarke has introduced a new type of hypothesis on the boundary function p://www.sciencedirect.com/scidirimg/entities/3d5.gif"" alt=""phi"" border=0>: the lower (or upper) bounded slope condition. This condition, which is less restrictive than the classical bounded slope condition of Hartman, Nirenberg and Stampacchia, is satisfied if p://www.sciencedirect.com/scidirimg/entities/3d5.gif"" alt=""phi"" border=0> is the restriction to ∂Ω of a convex (or concave) function. We show that for a class of problems in which G(x,u) is locally Lipschitz (but not necessarily convex) in u, the lower bounded slope condition implies the local Lipschitz regularity of solutions.