We study the problem of minimizing over the functions that assume given boundary values on . The lagrangian and the domain are assumed convex. A new type of hypothesis on the boundary function is introduced: the lower (or upper) 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 (or semiconcave) boundary data (instead of ). We prove in particular that the solution is locally Lipschitz in . In certain cases, as when is a polyhedron or else of class , we obtain in addition a global Hölder condition on .
@article{ASNSP_2005_5_4_3_511_0, author = {Clarke, Francis}, title = {Continuity of solutions to a basic problem in the calculus of variations}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {511--530}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 4}, number = {3}, year = {2005}, mrnumber = {2185867}, zbl = {1127.49001}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2005_5_4_3_511_0/} }
TY - JOUR AU - Clarke, Francis TI - Continuity of solutions to a basic problem in the calculus of variations JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2005 SP - 511 EP - 530 VL - 4 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2005_5_4_3_511_0/ LA - en ID - ASNSP_2005_5_4_3_511_0 ER -
%0 Journal Article %A Clarke, Francis %T Continuity of solutions to a basic problem in the calculus of variations %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2005 %P 511-530 %V 4 %N 3 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2005_5_4_3_511_0/ %G en %F ASNSP_2005_5_4_3_511_0
Clarke, Francis. Continuity of solutions to a basic problem in the calculus of variations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 4 (2005) no. 3, pp. 511-530. http://archive.numdam.org/item/ASNSP_2005_5_4_3_511_0/
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