Journal of the Mathematical Society of Japan

Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains

Kentaro HIRATA

Abstract

In a uniform domain Ω, we present a certain reverse mean value inequality and a Harnack type inequality for positive superharmonic functions satisfying a nonlinear inequality -Δu(x) ≤ cδΩ(x)u(x)p for x ∈ Ω, where c > 0, α ≥ 0 and p > 1 and δΩ(x) is the distance from a point x to the boundary of Ω. These are established by refining a boundary growth estimate obtained in our previous paper (2008). Also, we apply them to show the existence of nontangential limits of quotients of such functions and to give an extension of a certain minimum principle studied by Dahlberg (1976).

Article information

Source
J. Math. Soc. Japan, Volume 62, Number 4 (2010), 1043-1068.

Dates
First available in Project Euclid: 2 November 2010

https://projecteuclid.org/euclid.jmsj/1288703096

Digital Object Identifier
doi:10.2969/jmsj/06241043

Mathematical Reviews number (MathSciNet)
MR2761913

Zentralblatt MATH identifier
1210.31002

Citation

HIRATA, Kentaro. Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains. J. Math. Soc. Japan 62 (2010), no. 4, 1043--1068. doi:10.2969/jmsj/06241043. https://projecteuclid.org/euclid.jmsj/1288703096

References

• H. Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan, 53 (2001), 119–145.
• H. Aikawa, K. Hirata and T. Lundh, Martin boundary points of a John domain and unions of convex sets, J. Math. Soc. Japan, 58 (2006), 247–274.
• H. Aikawa, T. Kilpeläinen, N. Shanmugalingam and X. Zhong, Boundary Harnack principle for $p$-harmonic functions in smooth Euclidean domains, Potential Anal., 26 (2007), 281–301.
• D. H. Armitage and S. J. Gardiner, Classical potential theory, Springer-Verlag London Ltd., London, 2001.
• A. Beurling, A minimum principle for positive harmonic functions, Ann. Acad. Sci. Fenn. Ser. A I No.,372 (1965), 7 pp.
• B. Dahlberg, A minimum principle for positive harmonic functions, Proc. London Math. Soc. (3), 33 (1976), 238–250.
• J. L. Doob, A non-probabilistic proof of the relative Fatou theorem, Ann. Inst. Fourier, Grenoble, 9 (1959), 293–300.
• K. Hirata, Sharp estimates for the Green function, 3G inequalities, and nonlinear Schrödinger problems in uniform cones, J. Anal. Math., 99 (2006), 309–332.
• K. Hirata, Estimates for the products of the Green function and the Martin kernel, Nagoya Math. J., 188 (2007), 1–18.
• K. Hirata, The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations, Math. Ann., 340 (2008), 625–645.
• K. Hirata, Global estimates for non-symmetric Green type functions with applications to the $p$-Laplace equation, Potential Anal., 29 (2008), 221–239.
• K. Hirata, Boundary behavior of superharmonic functions satisfying nonlinear inequalities in a planar smooth domain, J. Aust. Math. Soc., 87 (2009), 253–261.
• K. Hirata, Boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains, to appear in Trans. Amer. Math. Soc.
• Ü. Kuran, On NTA-conical domains, J. London Math. Soc. (2), 40 (1989), 467–475.
• L. Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier, Grenoble, 7 (1957), 183–281.
• S. C. Port and C. J. Stone, Brownian motion and classical potential theory, Academic Press, New York, 1978.