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A strong maximum principle for globally hypoelliptic operators

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Abstract

In this paper we study the strong maximum principle for globally hypoelliptic differential operators of second-order, and reveal the underlying analytical mechanism of propagation of maximums in terms of the Lie algebra generated by diffusion vector fields and the Fichera function. Our formulation of the strong maximum principle is coordinate-free. The results here may be applied to questions of uniqueness for degenerate elliptic boundary value problems on a manifold. Furthermore, the mechanism of propagation of maximums plays an important role in the interpretation and study of Markov processes from the viewpoint of functional analysis.

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References

  1. Amano, K.: Maximum principles for degenerate elliptic–parabolic operators. Indiana Univ. Math. J. 29, 545–557 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bony, J.-M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19, 277–304 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bony, J.-M., Courrège, P., Priouret, P.: Semigroupes de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum. Ann. Inst. Fourier (Grenoble) 18, 369–521 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chazarain, J., Piriou, A.: Introduction à la théorie des équations aux dérivées partielles linéaires. Gauthier-Villars, Paris (1981)

    MATH  Google Scholar 

  5. Chow, W.-L.: Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117, 98–105 (1940)

    Article  MATH  Google Scholar 

  6. Fefferman, C., Phong, D.H.: Subelliptic eigenvalue problems. Conference on Harmonic Analysis (1981: Chicago, Ill), pp. 590–606. Wadsworth, Belmont, CA (1983)

    Google Scholar 

  7. Fichera, G.: Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine. Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I. 8(5), 1–30 (1956)

    MathSciNet  MATH  Google Scholar 

  8. Friedman, A.: Remarks on the maximum principle for parabolic equations and its applications. Pac. J. Math. 8, 201–211 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fujiwara, D., Omori, H.: An example of a globally hypo-elliptic operator. Hokkaido Math. J. 12, 293–297 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Reprint of the Classics in Mathematics, 1998th edn. Springer, Berlin (2001)

    MATH  Google Scholar 

  11. Hill, C.D.: A sharp maximum principle for degenerate elliptic–parabolic equations. Indiana Univ. Math. J. 20, 213–229 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hopf, E.: Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter ordnung vom elliptischen Typus, Sitz. Ber. Preuss. Akad. Wissensch. Berl. Math. Phys. Kl 19, 147–152 (1927)

    MATH  Google Scholar 

  13. Hörmander, L.: Pseudodifferential operators and non-elliptic boundary problems. Ann. Math. 2(83), 129–209 (1966)

    Article  MATH  Google Scholar 

  14. Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hörmander, L.: A class of hypoeiliptic pseudodifferential operators with double characteristics. Math. Ann. 217, 165–188 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hörmander, L.: The Analysis of Linear Partial Differential Operators III, Pseudo-differential Operators, Reprint of the 1994 Edition, Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2007)

    Google Scholar 

  17. Nicolaescu, L.I.: Lectures on the Geometry of Manifolds, 2nd edn. World Scientific Publishing, Hackensack, NJ (2007)

    Book  MATH  Google Scholar 

  18. Nirenberg, L.: A strong maximum principle for parabolic equations. Commun. Pure Appl. Math. 6, 167–177 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  19. Oleĭnik, O.A.: On properties of solutions of certain boundary problems for equations of elliptic type. Mat. Sb. 30, 595–702 (1952). (in Russian)

    MathSciNet  Google Scholar 

  20. Oleĭnik, O.A., Radkevič, E.V.: Second Order Equations with Nonnegative Characteristic Form. Itogi Nauki, Moscow (1971). ((in Russian); English translation: Amer. Providence, Rhode Island and Plenum Press, New York, Math. Soc. (1973))

    Google Scholar 

  21. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations, Corrected Second Printing. Springer, New York (1984)

    Book  Google Scholar 

  22. Radkevič, E.V.: A priori estimates and hypoelliptic operators with multiple characteristics. Dokl. Akad. Nauk SSSR 187, 274–277 (1969). ((in Russian); English translation in Soviet Math. Dokl. 10, 849–853 (1969))

    MathSciNet  Google Scholar 

  23. Redheffer, R.M.: The sharp maximum principle for nonlinear inequalities. Indiana Univ. Math. J. 21, 227–248 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings of the Sixth Berkeley Symposium of Probability and Mathematical Statistics, vol. 3, pp. 333–359 (1972)

  25. Stroock, D.W., Varadhan, S.R.S.: On degenerate elliptic–parabolic operators of second order and their associated diffusions. Commun. Pure Appl. Math. 25, 651–713 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  26. Taira, K.: A strong maximum principle for degenerate elliptic operators. Commun. Partial Differ. Equ. 4, 1201–1212 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  27. Taira, K.: Semigroups and boundary value problems. Duke Math. J. 49, 287–320 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. Taira, K.: Le principe du maximum et l’hypoellipticité globale. Séminaire Bony–Sjöstrand–Meyer 1984–1985, Exposé No. I, Ecole Polytechnique, Palaiseau (1985)

  29. Taira, K.: Diffusion Processes and Partial Differential Equations. Academic Press Inc., Boston, MA (1988)

    MATH  Google Scholar 

  30. Taira, K.: Semigroups, Boundary Value Problems and Markov Processes. Springer Monographs in Mathematics, 2nd edn. Springer, Heidelberg (2014)

    MATH  Google Scholar 

  31. Yoshino, M.: A class of globally hypoelliptic operators on the torus. Math. Z. 201, 1–11 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is grateful to the referee for many valuable suggestions and comments, which have improved substantially the presentation of the present paper. He is also indebted to Professors Hajime Sato and Koichi Uchiyama for formulating the mapping \(\varPsi \) in terms of differential geometry.

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  1. Institute of Mathematics, University of Tsukuba, Tsukuba, 305–8571, Japan

    Kazuaki Taira

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  1. Kazuaki Taira
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Correspondence to Kazuaki Taira.

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Dedicated to Professor Izumi Kubo on the occasion of his 80th birthday.

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Taira, K. A strong maximum principle for globally hypoelliptic operators. Rend. Circ. Mat. Palermo, II. Ser 68, 193–217 (2019). https://doi.org/10.1007/s12215-018-0351-0

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