×
Clarke, Francis

Continuity of solutions to a basic problem in the calculus of variations. (English) Zbl 1127.49001

Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 4, No. 3, 511-530 (2005).
The author studies the problem of minimizing \(\int_\Omega F(Du(x))\,dx\) over the functions \(u\in W^{1,1}(\Omega)\) taking prescribed boundary values \(\phi\), where the function \(F\) and the domain \(\Omega\) are assumed convex. Under a new type of hypothesis on the boundary function \(\phi\) (the lower bounded slope condition), the author extends the classical regularity theory to the case of semiconvex boundary data (instead of \(C^2\)). In particular, he proves that the solution is locally Lipschitz in \(\Omega\). Furthermore, in certain cases (for instance, when the boundary is a polyhedron or of class \(C^{1,1}\)), a global Hölder condition on \(\overline\Omega\) is derived.
Reviewer: Luis Alberto Fernandez (Santander)

Cited in 2 Reviews
Cited in 18 Documents

MSC:

49J10 Existence theories for free problems in two or more independent variables
35J20 Variational methods for second-order elliptic equations
35A15 Variational methods applied to PDEs
49N60 Regularity of solutions in optimal control
Cite Review PDF
Full Text: EuDML

References:

[1] P. Bousquet, On the lower bounded slope condition, to appear. Zbl1132.49031 MR2310433 · 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. Zbl1141.49033 · Zbl 1141.49033 · doi:10.1016/j.jde.2007.05.034
[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. Zbl0852.49001 MR1351738 · Zbl 0852.49001
[4] P. Cannarsa and C. Sinestrari, “Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control”, Birkhäuser, Boston, 2004. Zbl1095.49003 MR2041617 · 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. Zbl1047.49500 MR1488695 · 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. Zbl0715.49002 MR1079583 · Zbl 0715.49002
[7] L. C. Evans and R. F. Gariepy, “Measure Theorey and Fine Properties of Functions”, CRC Press, Boca Raton, FL, 1992. Zbl0804.28001 MR1158660 · Zbl 0804.28001
[8] M. Giaquinta, “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems”, Princeton University Press, Princeton, N.J., 1983. Zbl0516.49003 MR717034 · Zbl 0516.49003
[9] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer-Verlag, Berlin, 1998. (3rd ed). Zbl0562.35001 · Zbl 0562.35001
[10] E. Giusti, “Direct Methods in the Calculus of Variations” World Scientific, Singapore, 2003. Zbl1028.49001 MR1962933 · Zbl 1028.49001
[11] P. Hartman, On the bounded slope condition, Pacific J. Math. 18 (1966), 495-511. Zbl0149.32001 MR197640 · Zbl 0149.32001 · doi:10.2140/pjm.1966.18.495
[12] P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901-920. Zbl0094.16303 MR126812 · Zbl 0094.16303 · doi:10.2307/2372995
[13] P. Marcellini, Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl. 90 (1996), 161-181. Zbl0901.49030 MR1397651 · Zbl 0901.49030 · doi:10.1007/BF02192251
[14] C. Mariconda and G. Treu, Existence and Lipschitz regularity for minima, Proc. Amer. Math. Soc. 130 (2001), 395-404. Zbl0987.49020 MR1862118 · Zbl 0987.49020 · doi:10.1090/S0002-9939-01-06370-5
[15] C. Mariconda and G. Treu, Gradient maximum principle for minima, J. Optim. Theory Appl. 112 (2002), 167-186. Zbl1019.49029 MR1881695 · Zbl 1019.49029 · doi:10.1023/A:1013052830852
[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. Zbl0137.08201 MR181918 · Zbl 0137.08201
[17] C. B. Morrey, “Multiple Integrals in the Calculus of Variations”, Springer-Verlag, New York, 1966. Zbl0142.38701 MR202511 · 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. Zbl0138.36903 MR155209 · Zbl 0138.36903 · doi:10.1002/cpa.3160160403
[19] W. P. Ziemer, “Weakly Differentiable Functions”, Springer-Verlag, Berlin, 1989. Zbl0692.46022 MR1014685 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.