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, p. 511-530
We study the problem of minimizing Ω F(Du(x))dx over the functions uW 1,1 (Ω) that assume given boundary values φ on Γ:=Ω. The lagrangian F and the domain Ω are assumed convex. A new type of hypothesis on the boundary function φ 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 C 2 ). We prove in particular that the solution is locally Lipschitz in Ω. In certain cases, as when Γ is a polyhedron or else of class C 1,1 , we obtain in addition a global Hölder condition on Ω ¯.
Classification:  49J10,  35J20
@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 4},
     number = {3},
     year = {2005},
     pages = {511-530},
     zbl = {1127.49001},
     mrnumber = {2185867},
     language = {en},
     url = {http://www.numdam.org/item/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] P. Bousquet, On the lower bounded slope condition, to appear. | MR 2310433 | Zbl 1132.49031

[2] P. Bousquet and F. Clarke, Local Lipschitz continuity of solutions to a basic problem in the calculus of variations, to appear. | Zbl 1141.49033

[3] G. Buttazzo and M. Belloni, 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 1351738 | Zbl 0852.49001

[4] P. Cannarsa and C. Sinestrari, “Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control”, Birkhäuser, Boston, 2004. | MR 2041617 | Zbl 1095.49003

[5] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, “Nonsmooth Analysis and Control Theory”, Graduate Texts in Mathematics, vol. 178. Springer-Verlag, New York, 1998. | MR 1488695 | Zbl 1047.49500

[6] R. De Arcangelis, Some remarks on the identity between a variational integral and its relaxed functional, Ann. Univ. Ferrara 35 (1989), 135-145. | MR 1079583 | Zbl 0715.49002

[7] L. C. Evans and R. F. Gariepy, “Measure Theorey and Fine Properties of Functions”, CRC Press, Boca Raton, FL, 1992. | MR 1158660 | Zbl 0804.28001

[8] M. Giaquinta, “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems”, Princeton University Press, Princeton, N.J., 1983. | MR 717034 | Zbl 0516.49003

[9] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer-Verlag, Berlin, 1998. (3rd ed). | Zbl 0562.35001

[10] E. Giusti, “Direct Methods in the Calculus of Variations” World Scientific, Singapore, 2003. | MR 1962933 | Zbl 1028.49001

[11] P. Hartman, On the bounded slope condition, Pacific J. Math. 18 (1966), 495-511. | MR 197640 | Zbl 0149.32001

[12] P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901-920. | MR 126812 | Zbl 0094.16303

[13] P. Marcellini, Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl. 90 (1996), 161-181. | MR 1397651 | Zbl 0901.49030

[14] C. Mariconda and G. Treu, Existence and Lipschitz regularity for minima, Proc. Amer. Math. Soc. 130 (2001), 395-404. | MR 1862118 | Zbl 0987.49020

[15] C. Mariconda and G. Treu, Gradient maximum principle for minima, J. Optim. Theory Appl. 112 (2002), 167-186. | MR 1881695 | Zbl 1019.49029

[16] M. Miranda, 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. | Numdam | MR 181918 | Zbl 0137.08201

[17] C. B. Morrey, “Multiple Integrals in the Calculus of Variations”, Springer-Verlag, New York, 1966. | MR 202511 | Zbl 0142.38701

[18] G. Stampacchia, On some regular multiple integral problems in the calculus of variations, Comm. Pure Appl. Math. 16 (1963), 383-421. | MR 155209 | Zbl 0138.36903

[19] W. P. Ziemer, “Weakly Differentiable Functions”, Springer-Verlag, Berlin, 1989. | MR 1014685 | Zbl 0692.46022