ON SOME UNBOUNDED DOMAINS FOR A MAXIMUM PRINCIPLE
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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.
Abstract
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.
참고문헌 (5)
- H. Berestycki, "The principal eigenvalue and maxi-mum principle for second-order elliptic operators in general domains", Comm. Pure Appl. Math, 47, pp.47~92 (1994)
- X. Cabre, "On the Alexandro®-Bekelman-Pucci estimate and reversed HÄolder inequality for solutions of elliptic and parabolic equations", Comm. Pure Appl. Math, 48, pp.539~570 (1995)
- V. Cafagna, "On the maximum principle for second-order elliptic operators in unbounded domains", C. R. Acad. Sci. Paris. Ser. I, 334, pp.1~5 (2002)
- O.A. Ladyzhenskaya, "Linear and Quasilinear Elliptic Equa-tions" (1964), Nauka
- A. Vitolo, "On the maximum principle for complete second-order elliptic operators in general domains", J. Di®. equations, 194(1), pp.166~184 (2003)