Abstract
In this note, we highlight a difference in the conditions of the classical weak maximum principle and the classical strong maximum principle for linear parabolic partial differential inequalities. We demonstrate, by the careful construction of a specific function, that the condition in the classical strong maximum principle on the coefficient of the zeroth-order term in the linear parabolic partial differential inequality cannot be relaxed to the corresponding condition in the classical weak maximum principle. In addition, we demonstrate that results (often referred to as boundary point lemmas) which conclude positivity of the outward directional derivatives of nontrivial solutions to linear parabolic partial differential inequalities at certain points on the boundary where a maxima is obtained cannot be obtained under the same zeroth-order coefficient conditions as in the classical strong maximum principle.
References
Besala P.: An extension of the strong maximum principle for parabolic equations. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 19, 1003–1006 (1971)
Friedman A.: Remarks on the maximum principle for parabolic equations and its applications. Pac. J. Math. 8, 201–211 (1958)
Hopf E.: Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitzungsber. d. Preuss. Akad. d. Wiss. 19, 147–152 (1927)
Nirenberg L.: A strong maximum principle for parabolic equations. Commun. Pure Appl. Math. 6, 167–177 (1953)
Picone M.: Sul problema della propagazione del calore in un mezzo privo di frontiera, conduttore, isotrope e omogeneo. Math. Ann. 101, 701–712 (1929)
Protter M.H., Weinberger H.F.: Maximum Principles in Differential Equations. Springer, New York (1984)
Walter W.: On the strong maximum principle for parabolic differential equations. Proc. Edinb. Math. Soc. 29, 93–96 (1986)
Yoshida N.: Maximum principles for implicit parabolic equations. Proc. Jpn. Acad. 49, 785–788 (1971)
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author acknowledges financial support from EPSRC.
Rights and permissions
About this article
Cite this article
Needham, D.J., Meyer, J.C. A note on the classical weak and strong maximum principles for linear parabolic partial differential inequalities. Z. Angew. Math. Phys. 66, 2081–2086 (2015). https://doi.org/10.1007/s00033-014-0492-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-014-0492-8