Geometric constraints on the domain for a class of minimum problems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 9  (2003), p. 125-133

We consider minimization problems of the form min uϕ+W 0 1,1 (Ω) Ω [f(Du(x))-u(x)]dx where Ω N is a bounded convex open set, and the Borel function f: N [0,+] is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of Ω and the zero level set of f, we prove that the viscosity solution of a related Hamilton-Jacobi equation provides a minimizer for the integral functional.

DOI : https://doi.org/10.1051/cocv:2003003
Classification:  49J10,  49L25
Keywords: calculus of variations, existence, non-convex problems, non-coercive problems, viscosity solutions
@article{COCV_2003__9__125_0,
     author = {Crasta, Graziano and Malusa, Annalisa},
     title = {Geometric constraints on the domain for a class of minimum problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     pages = {125-133},
     doi = {10.1051/cocv:2003003},
     zbl = {1066.49003},
     mrnumber = {1957093},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2003__9__125_0}
}
Crasta, Graziano; Malusa, Annalisa. Geometric constraints on the domain for a class of minimum problems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 9 (2003) , pp. 125-133. doi : 10.1051/cocv:2003003. http://www.numdam.org/item/COCV_2003__9__125_0/

[1] M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997). | Zbl 0890.49011

[2] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer Verlag, Berlin (1994). | Zbl 0819.35002

[3] P. Bauman and D. Phillips, A non-convex variational problem related to change of phase. Appl. Math. Optim. 21 (1990) 113-138. | MR 1019397 | Zbl 0686.73018

[4] P. Cardaliaguet, B. Dacorogna, W. Gangbo and N. Georgy, Geometric restrictions for the existence of viscosity solutions. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 189-220. | Numdam | MR 1674769 | Zbl 0927.35021

[5] P. Celada, Some scalar and vectorial problems in the Calculus of Variations, Ph.D. Thesis. SISSA, Trieste (1997).

[6] P. Celada and A. Cellina, Existence and non existence of solutions to a variational problem on a square. Houston J. Math. 24 (1998) 345-375. | MR 1690397 | Zbl 0980.49020

[7] P. Celada, S. Perrotta and G. Treu, Existence of solutions for a class of non convex minimum problems. Math. Z. 228 (1998) 177-199. | MR 1617955 | Zbl 0936.49010

[8] A. Cellina, Minimizing a functional depending on u and on u. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 339-352. | Numdam | MR 1450952 | Zbl 0876.49001

[9] A. Cellina and S. Perrotta, On minima of radially symmetric functionals of the gradient. Nonlinear Anal. 23 (1994) 239-249. | MR 1289130 | Zbl 0819.49013

[10] G. Crasta, On the minimum problem for a class of non-coercive non-convex variational problems. SIAM J. Control Optim. 38 (1999) 237-253. | MR 1740598 | Zbl 0942.49012

[11] G. Crasta, Existence, uniqueness and qualitative properties of minima to radially symmetric non-coercive non-convex variational problems. Math. Z. 235 (2000) 569-589. | MR 1800213 | Zbl 0965.49003

[12] G. Crasta and A. Malusa, Euler-Lagrange inclusions and existence of minimizers for a class of non-coercive variational problems. J. Convex Anal. 7 (2000) 167-181. | Zbl 0956.49008

[13] G. Crasta and A. Malusa, Non-convex minimization problems for functionals defined on vector valued functions. J. Math. Anal. Appl. 254 (2001) 538-557. | MR 1805523 | Zbl 1093.49501

[14] B. Dacorogna and P. Marcellini, Existence of minimizers for non-quasiconvex integrals. Arch. Rational Mech. Anal. 131 (1995) 359-399. | MR 1354700 | Zbl 0837.49002

[15] B. Kawohl, J. Stara and G. Wittum, Analysis and numerical studies of a problem of shape design. Arch. Rational Mech. Anal. 114 (1991) 349-363. | MR 1100800 | Zbl 0726.65071

[16] R. Kohn and G. Strang, Optimal design and relaxation of variational problems, I, II and III. Comm. Pure Appl. Math. 39 (1976) 113-137, 139-182, 353-377. | MR 820342 | Zbl 0621.49008

[17] P.L. Lions, Generalized solutions of Hamilton-Jacobi equations. Pitman, London, Pitman Res. Notes Math. Ser. 69 (1982). | Zbl 0497.35001

[18] E. Mascolo and R. Schianchi, Existence theorems for nonconvex problems J. Math. Pures Appl. 62 (1983) 349-359. | MR 718948 | Zbl 0522.49001

[19] R.T. Rockafellar, Convex Analysis. Princeton Univ. Press, Princeton (1970). | Zbl 0193.18401

[20] G. Treu, An existence result for a class of non convex problems of the Calculus of Variations. J. Convex Anal. 5 (1998) 31-44. | MR 1649421 | Zbl 0908.49013

[21] M. Vornicescu, A variational problem on subsets of n . Proc. Roy. Soc. Edinburg Sect. A 127 (1997) 1089-1101. | MR 1475648 | Zbl 0920.49002