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On the strong maximum principle


Author: Arrigo Cellina
Journal: Proc. Amer. Math. Soc. 130 (2002), 413-418
MSC (1991): Primary 35B50, 49N60
Published electronically: May 23, 2001
MathSciNet review: 1862120
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents a necessary and sufficient condition on the convex function $f$ in order that continuous solutions to

\begin{displaymath}\hbox {minimize} \int _{\Omega } f(\Vert\nabla u(x)\Vert) \, dx \hbox { on } u^{0} + W^{1,1}_{0}(\Omega )\end{displaymath}

satisfy a Strong Maximum Principle on any open connected $\Omega $.


References [Enhancements On Off] (What's this?)

  • 1. A. Cellina, On the Bounded Slope Condition and the validity of the Euler Lagrange equation, SIAM J. Control Optim., to appear.
  • 2. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • 3. C. Mariconda and G. Treu, A comparison principle for minimizers, C.R. Acad. Sci. Paris Sér. I Math. 330 (2000), 681-686. CMP 2000:14
  • 4. Giovanni Maria Troianiello, Elliptic differential equations and obstacle problems, The University Series in Mathematics, Plenum Press, New York, 1987. MR 1094820

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Additional Information

Arrigo Cellina
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Viale Sarca 202, 20126 Milano, Italy
Email: cellina@matapp.unimib.it

DOI: https://doi.org/10.1090/S0002-9939-01-06104-4
Keywords: Strong Maximum Principle, Comparison Theorem.
Received by editor(s): March 2, 2000
Received by editor(s) in revised form: June 13, 2000
Published electronically: May 23, 2001
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society