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Large sets at infinity and maximum principle on unbounded domains for a class of sub-elliptic operators. (English) Zbl 1448.35060

Maximum principles (MP) are among the most powerful and widely used analytic tools in the study of second-order linear and nonlinear elliptic and parabolic equations. In general, MP hold in bounded domains, but it is well known that certain MP for subsolutions (the so called Phragmèn-Lindelöf-type principles) hold true in certain unbounded domains if some restrictions are imposed on the growth of the subsolutions at infinity. The growth condition considered in the present paper is boundedness.
The authors introduce the following notion: An open set \(\Omega\) in \(\mathbb{R}^N\) is called a maximum principle set for the operator \(\mathcal{L}\) if for any bounded \(\mathcal{L}\)-subharmonic \(v\) satisfying \(\limsup_{x \to \partial \Omega}v(x)\leq 0\), we have \(v\leq 0\) in \(\Omega\).
In the paper under review, the authors consider a class of subelliptic operators \(\mathcal{L}\) in \(\mathbb{R}^N\) and establish some criteria for an unbounded open set to be a maximum principle set for \(\mathcal{L}\). The main assumptions of the paper are the existence of a well behaved global fundamental solution \(\Gamma\) such that \(1/\Gamma\) is a quasimetric, and that \(\mathcal{L}\) satisfies the Liouville property in \(\mathbb{R}^N\). We note that quasimetric kernels appears naturally in the study of Green functions, see for example [M. W. Frazier and I. E. Verbitsky, in: Around the research of Vladimir Maz’ya. III. Analysis and applications. Dordrecht: Springer; Novosibirsk: Tamara Rozhkovskaya Publisher. 105–152 (2010; Zbl 1197.35085); W. Hansen, Potential Anal. 21, No. 2, 99–135 (2004; Zbl 1054.31005); Y. Pinchover, Math. Ann. 314, No. 3, 555–590 (1999; Zbl 0928.35010)]. The results of the paper extend celebrated results by Deny, Hayman and Kennedy valid for the Laplacian (see [W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I. London-New York-San Francisco: Academic Press, a subsidiary of Harcourt Brace Jovanovich Publishers (1976; Zbl 0419.31001)]).

MSC:

35B50 Maximum principles in context of PDEs
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
35H20 Subelliptic equations
35J70 Degenerate elliptic equations
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