Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-19T10:41:31.003Z Has data issue: false hasContentIssue false

On the existence of variations, possibly with pointwise gradient constraints

Published online by Cambridge University Press:  12 May 2007

Simone Bertone
Affiliation:
Dipartimento di Matematica e Applicazioni, Universitá degli Studi di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano, Italy; simone.bertone@unimib.it; arrigo.cellina@unimib.it
Arrigo Cellina
Affiliation:
Dipartimento di Matematica e Applicazioni, Universitá degli Studi di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano, Italy; simone.bertone@unimib.it; arrigo.cellina@unimib.it
Get access

Abstract

We propose a necessary and sufficient condition about the existence of variations, i.e., of non trivial solutions $\eta\in W^{1,\infty}_0(\Omega)$ to the differential inclusion $\nabla\eta(x)\in-\nabla u(x)+{\bf D}$.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

V.I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Mathematics 60, Springer-Verlag, New York, Heidelber, Berlin.
H. Brezis, Analyse fonctionnelle, théorie et applications. Masson, Paris (1983).
A. Cellina, On minima of a functional of the gradient: necessary conditions. Nonlinear Anal. 20 (1993) 337–341.
A. Cellina, On minima of a functional of the gradient: sufficient conditions. Nonlinear Anal. 20 (1993) 343–347.
Cellina, A. and Perrotta, S., On the validity of the maximum principle and of the Euler-Lagrange equation for a minimum problem depending on the gradient. SIAM J. Control Optim. 36 (1998) 19871998. CrossRef
L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, Rhode Island (1998).
L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999) 653.
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992).
Tsuji, M., Lindelöf's, On theorem in the theory of differential equations. Jap. J. Math. 16 (1939) 149161.