Continuity of solutions to a basic problem in the calculus of variations

Clarke, Francis

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 3, pp. 511-530.

We study the problem of minimizing ${\int }_{\Omega }F\left(Du\left(x\right)\right)dx$ over the functions $u\in W^{1,1}\left(\Omega \right)$ 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 (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 $\Omega$. 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 $\overline{\Omega }$.

[1] P. Bousquet, On the lower bounded slope condition, to appear. | MR | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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

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

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

[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. | EuDML | Numdam | MR | Zbl

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

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

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