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://www.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://www.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://www.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, Série 5, Tome 4 (2005) no. 3, pp. 511-530. http://www.numdam.org/item/ASNSP_2005_5_4_3_511_0/
[1] On the lower bounded slope condition, to appear. | MR | Zbl
,[2] Local Lipschitz continuity of solutions to a basic problem in the calculus of variations, to appear. | Zbl
and ,[3] A survey on old and recent results about the gap phenomenon, In: “Recent Developments in Well-Posed Variational Problems”, R. Lucchetti and J. Revalski (eds.), Kluwer, Dordrecht, 1995, 1-27. | MR | Zbl
and ,[4] “Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control”, Birkhäuser, Boston, 2004. | MR | Zbl
and ,[5] “Nonsmooth Analysis and Control Theory”, Graduate Texts in Mathematics, vol. 178. Springer-Verlag, New York, 1998. | MR | Zbl
, , and ,[6] Some remarks on the identity between a variational integral and its relaxed functional, Ann. Univ. Ferrara 35 (1989), 135-145. | MR | Zbl
,[7] “Measure Theorey and Fine Properties of Functions”, CRC Press, Boca Raton, FL, 1992. | MR | Zbl
and ,[8] “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems”, Princeton University Press, Princeton, N.J., 1983. | MR | Zbl
,[9] “Elliptic Partial Differential Equations of Second Order”, Springer-Verlag, Berlin, 1998. (3rd ed). | Zbl
and ,[10] “Direct Methods in the Calculus of Variations” World Scientific, Singapore, 2003. | MR | Zbl
,[11] On the bounded slope condition, Pacific J. Math. 18 (1966), 495-511. | MR | Zbl
,[12] On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901-920. | MR | Zbl
and ,[13] Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl. 90 (1996), 161-181. | MR | Zbl
,[14] Existence and Lipschitz regularity for minima, Proc. Amer. Math. Soc. 130 (2001), 395-404. | MR | Zbl
and ,[15] Gradient maximum principle for minima, J. Optim. Theory Appl. 112 (2002), 167-186. | MR | Zbl
and ,[16] Un teorema di esistenza e unicità per il problema dell'area minima in n variabili, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 233-249. | EuDML | Numdam | MR | Zbl
,[17] “Multiple Integrals in the Calculus of Variations”, Springer-Verlag, New York, 1966. | MR | Zbl
,[18] On some regular multiple integral problems in the calculus of variations, Comm. Pure Appl. Math. 16 (1963), 383-421. | MR | Zbl
,[19] “Weakly Differentiable Functions”, Springer-Verlag, Berlin, 1989. | MR | Zbl
,