Variational problems with lipschitzian minimizers

F. H. Clarke; P. D. Loewen

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We study the problem of minimizing $\int_{Ω} F (D u (x)) d x$ over the functions $u \in W^{1, 1} (Ω)$ that assume given boundary values $φ$ on $Γ : = \partial Ω$ . The lagrangian $F$ and the domain $Ω$ are assumed convex. A new type of hypothesis on the boundary function $φ$ is introduced: the (or upper) . This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary data (instead...

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