Korean J. Math.  Vol 26, No 4 (2018)  pp.747-756
DOI: https://doi.org/10.11568/kjm.2018.26.4.747

A maximum principle for non-negative zeroth order coefficient in some unbounded domains

Sungwon Cho


We study a maximum principle for a uniformly elliptic second order differential operator in nondivergence form. We allow a bounded positive zeroth order coefficient in a certain type of unbounded domains. The results extend a result by J. Busca in a sense of domains, and we present a simple proof based on local maximum principle by Gilbarg and Trudinger with iterations.  


Maximum principle in unbounded domains, Second-order elliptic equation, Measurable coefficients, A priori estimates, Behavior of subsolution at infinity

Subject classification

35B50; 35B45; 35J15


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H. Berestycki, L. Nirenberg, S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1) (1994), 47–92. (Google Scholar)

J. Busca Existence results for Bellman equations and maximum principles in unbounded domains, Comm. Partial Differential Equations 24 (11-12) (1999), 2023–2042. (Google Scholar)

X. Cabr ́e, Estimates for Solutions of Elliptic and Parabolic Problems, Ph.D. Thesis. (1994), Courant Institute, New York University, U.S. (Google Scholar)

X. Cabr ́e, On the Alexandroff-Bakel’man-Pucci estimate and the reversed Ho ̈lder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 48 (5) (1995), 539–570. (Google Scholar)

L. C. Evans, Partial Differential Equations, Am. Math. Soc., Providence, RI, 1998 (Google Scholar)

D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Or- der, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. (Google Scholar)

O.A. Ladyzhenskaya and N.N. Ural’tseva, Linear and Quasilinear Elliptic Equa- tions, “Nauka”, Moscow, 1964 in Russian; English transl.: Academic Press, New York, 1968; 2nd Russian ed. 1973. (Google Scholar)

E. M. Landis, Second Order Equations of Elliptic and Parabolic Type, “Nauka”, Moscow, 1971 in Russian; English transl.: Amer. Math. Soc., Providence, RI, 1997. (Google Scholar)


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