Geometric constraints on the domain for a class of minimum problems

Crasta, Graziano; Malusa, Annalisa

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 125-133

We consider minimization problems of the form ${\min }_{u\in \varphi +W_0^{1,1}\left(\Omega \right)}{\int }_{\Omega }[f\left(Du\left(x\right)\right)-u\left(x\right)]\mathrm{d}x$ where $\Omega \subseteq {ℝ}^N$ is a bounded convex open set, and the Borel function $f:{ℝ}^N\to [0,+\infty ]$ is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of $\Omega$ 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.

Classification : 49J10, 49L25
Keywords: calculus of variations, existence, non-convex problems, non-coercive problems, viscosity solutions

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

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

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

[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. | Zbl | MR | Numdam

[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. | Zbl | MR

[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. | Zbl | MR

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

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

[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. | Zbl | MR

[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. | Zbl | MR

[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

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

[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. | Zbl | MR

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