ISSN : 1226-0657
In this paper, we study some characterizations of unbounded domains. Among these, so-called G-domain is introduced by Cabre for the Aleksandrov-Bakelman-Pucci maximum principle of second order linear elliptic operator in a non-divergence form. This domain is generalized to wG-domain by Vitolo for the maximum principle of an unbounded domain, which contains G-domain. We study the properties of these domains and compare some other characterizations. We prove that sA-domain is wG-domain, but using the Cantor set, we are able to construct a example which is wG-domain but not sA-domain.
Berestycki, H.;Nirenberg, L.;Varadahn, S.R.S.;. (1994). The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. Math., 47, 47-92. 10.1002/cpa.3160470105.
Cabre, X.;. (1995). On the Alexandroff-Bekelman-Pucci estimate and reversed Hölder inequality for solutions of elliptic and parabolic equations. Comm. Pure Appl. Math., 48, 539-570. 10.1002/cpa.3160480504.
Cafagna, V.;Vitolo, A.;. (2002). On the maximum principle for second-order elliptic operators in unbounded domains. C. R. Acad. Sci. Paris. Ser. I, 334, 1-5. 10.1016/S1631-073X(02)02204-5.
Ladyzhenskaya, O.A.;Uraltseva, N.N.;. Linear and Quasilinear Elliptic Equations.
Vitolo, A.;. (2003). On the maximum principle for complete second-order elliptic operators in general domains. J. Diff. equations, 194(1), 166-184. 10.1016/S0022-0396(03)00193-1.