We study the problem of minimizing $\int _\Omega F)\, dx \;$ over the functions $u\in W^{1,1}$ that assume given boundary values $\phi $ on $\Gamma := \partial \Omega $. The lagrangian $F$ and the domain $\Omega $ are assumed convex. A new type of hypothesis on the boundary function $\phi $ is introduced: the lower bounded slope condition. 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 boundary data. We prove in particular that the solution is locally Lipschitz in $\Omega $. In certain cases, as when $\Gamma $ is a polyhedron or else of class $C^{1,1}$, we obtain in addition a global Hölder condition on $\,\overline{\!\,\Omega }$. (shrink)

This book is a volume in the Penn Press Anniversary Collection. To mark its 125th anniversary in 2015, the University of Pennsylvania Press rereleased more than 1,100 titles from Penn Press's distinguished backlist from 1899-1999 that had fallen out of print. Spanning an entire century, the Anniversary Collection offers peer-reviewed scholarship in a wide range of subject areas.

This anthology contains essays by a distinguished group of British and American scholars prominent in the field of medieval philosophy. Setting high standards of clarity and exactness, the papers reflect current analytic, formalistic, and traditional methods of philosophy applied to topics such as logic, linguistics, and the philosophy of the mind.