We consider a class of convex integral functionals with lagrangeans depending only on the gradient and satisfying a generalized symmetry assumption, which includes as a particular case the rotational symmetry. Adapting the method by A. Cellina we obtain a kind of local estimates for minimizers in the respective variational problems, which is applied then to deduce some versions of the Strong Maximum Principle in the variational setting.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Cellina, A.: On the Strong Maximum Principle. Proc. Am. Math. Soc.130, 413–418 (2002)
Cellina, A.: On minima of a functional of the gradient: sufficient conditions. Nonlinear Anal.: Theory Meth. Appl. 20, 343–347 (1993)
Cellina, A., Mariconda, C., Treu, G.: Comparison results without strict convexity. Disc. Cont. Dynam. Syst. Ser. B 11, 57–65 (2009)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Colombo, G., Wolenski, P.: Variational analysis for a class of minimal time functions in Hilbert spaces. J. Convex Anal. 11, 335–361 (2004)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1998)
Kindelehrer, D., Stampachia, G.: An Introduction to Variational Inequalities and their Applications. Academic Press, New York (1980)
Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Math., vol. 1364. Springer, New York (1989)
Pucci, P., Serrin, J.: The strong maximum principle revisited. J. Differ. Equ. 196, 1–66 (2004)
Pucci, P., Serrin, J.: The Strong Maximum Principle. In: Progress in Nonlinear Differential Equations and their Applications, vol. 73. Birkhauser, Switzerland (2007)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, New York (1972)
Work supported by: projecto POCI/MAT/56727/2004 and Financiamento Programático Especial do CIMA-UE (Centro de Investigação em Matemática e Aplicações da Universidade de Évora, Portugal) both of FCT (Fundação para Ciência e Tecnologia, Portugal).
About this article
Cite this article
Goncharov, V.V., Santos, T.J. Local Estimates for Minimizers of Some Convex Integral Functional of the Gradient and the Strong Maximum Principle. Set-Valued Anal 19, 179–202 (2011). https://doi.org/10.1007/s11228-011-0176-x
- Strong Maximum Principle
- Comparison theorems
- Convex variational problems
- Minkowski functional
Mathematical Subject Classifications (2010)