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Geometric constraints on the domain for a class of minimum problems

Published online by Cambridge University Press:  15 September 2003

Graziano Crasta  and
Annalisa Malusa
Graziano Crasta
Affiliation:
Dip. di Matematica, P.le Aldo Moro 2, 00185 Roma, Italy; crasta@mat.uniroma1.it. malusa@mat.uniroma1.it.
Annalisa Malusa
Affiliation:
Dip. di Matematica, P.le Aldo Moro 2, 00185 Roma, Italy; crasta@mat.uniroma1.it. malusa@mat.uniroma1.it.
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Abstract

We consider minimization problems of the form ${\rm min}_{u\in \varphi +W^{1,1}_0(\Omega)}\int_\Omega [f(Du(x))-u(x)]\, {\rm d}x$ where $\Omega\subseteq \mathbb{R}^N$ is a bounded convex open set, and the Borel function $f\colon \mathbb{R}^N \to [0, +\infty]$ 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.

Keywords

Calculus of Variationsexistencenon-convex problemsnon-coercive problemsviscosity solutions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 125 - 133
Copyright
© EDP Sciences, SMAI, 2003

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References

M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997).
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer Verlag, Berlin (1994).
Bauman, P. and Phillips, D., A non-convex variational problem related to change of phase. Appl. Math. Optim. 21 (1990) 113-138. CrossRef
Cardaliaguet, P., Dacorogna, B., Gangbo, W. and Georgy, N., Geometric restrictions for the existence of viscosity solutions. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 189-220. CrossRef
P. Celada, Some scalar and vectorial problems in the Calculus of Variations, Ph.D. Thesis. SISSA, Trieste (1997).
Celada, P. and Cellina, A., Existence and non existence of solutions to a variational problem on a square. Houston J. Math. 24 (1998) 345-375.
Celada, P., Perrotta, S. and Treu, G., Existence of solutions for a class of non convex minimum problems. Math. Z. 228 (1998) 177-199. CrossRef
Cellina, A., Minimizing a functional depending on ∇u and on u. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 339-352. CrossRef
Cellina, A. and Perrotta, S., On minima of radially symmetric functionals of the gradient. Nonlinear Anal. 23 (1994) 239-249. CrossRef
Crasta, G., On the minimum problem for a class of non-coercive non-convex variational problems. SIAM J. Control Optim. 38 (1999) 237-253. CrossRef
Crasta, G., Existence, uniqueness and qualitative properties of minima to radially symmetric non-coercive non-convex variational problems. Math. Z. 235 (2000) 569-589. CrossRef
Crasta, G. and Malusa, A., Euler-Lagrange inclusions and existence of minimizers for a class of non-coercive variational problems. J. Convex Anal. 7 (2000) 167-181.
Crasta, G. and Malusa, A., Non-convex minimization problems for functionals defined on vector valued functions. J. Math. Anal. Appl. 254 (2001) 538-557. CrossRef
Dacorogna, B. and Marcellini, P., Existence of minimizers for non-quasiconvex integrals. Arch. Rational Mech. Anal. 131 (1995) 359-399. CrossRef
Kawohl, B., Stara, J. and Wittum, G., Analysis and numerical studies of a problem of shape design. Arch. Rational Mech. Anal. 114 (1991) 349-363. CrossRef
Kohn, R. and Strang, G., Optimal design and relaxation of variational problems, I, II and III. Comm. Pure Appl. Math. 39 (1976) 113-137, 139-182, 353-377. CrossRef
P.L. Lions, Generalized solutions of Hamilton-Jacobi equations. Pitman, London, Pitman Res. Notes Math. Ser. 69 (1982).
E. Mascolo and R. Schianchi, Existence theorems for nonconvex problems J. Math. Pures Appl. 62 (1983) 349-359.
R.T. Rockafellar, Convex Analysis. Princeton Univ. Press, Princeton (1970).
Treu, G., An existence result for a class of non convex problems of the Calculus of Variations. J. Convex Anal. 5 (1998) 31-44.
Vornicescu, M., A variational problem on subsets of $\mathbb{R}^n$ . Proc. Roy. Soc. Edinburg Sect. A 127 (1997) 1089-1101. CrossRef