tailieunhanh - Đề tài " On De Giorgi’s conjecture in dimensions 4 and 5 "

In this paper, we develop an approach for establishing in some important cases, a conjecture made by De Giorgi more than 20 years ago. The problem originates in the theory of phase transition and is so closely connected to the theory of minimal hypersurfaces that it is sometimes referred to as “the version of Bernstein’s problem for minimal graphs”. The conjecture has been completely settled in dimension 2 by the authors [15] and in dimension 3 in [2], yet the approach in this paper seems to be the first to use, in an essential way, the solution of. | Annals of Mathematics On De Giorgi s conjecture in dimensions 4 and 5 By Nassif Ghoussoub and Changfeng Gui Annals of Mathematics 157 2003 313 334 On De Giorgi s conjecture in dimensions 4 and 5 By Nassif Ghoussoub and Changfeng Gui 1. Introduction In this paper we develop an approach for establishing in some important cases a conjecture made by De Giorgi more than 20 years ago. The problem originates in the theory of phase transition and is so closely connected to the theory of minimal hypersurfaces that it is sometimes referred to as the eversion of Bernstein s problem for minimal graphs . The conjecture has been completely settled in dimension 2 by the authors 15 and in dimension 3 in 2 yet the approach in this paper seems to be the first to use in an essential way the solution of the Bernstein problem stating that minimal graphs in Euclidean space are necessarily hyperplanes provided the dimension of the ambient space is not greater than 8. We note that the solution of Bernstein s problem was also used in 18 to simplify an argument in 9 . Here is the conjecture as stated by De Giorgi 12 . Conjecture . Suppose that u is an entire solution of the equation Au u u3 0 u 1 X X xn G R satisfying -ễu 0 X G Rn dXn Then at least for n 8 the level sets of u must be hyperplanes. The conjecture may be considered together with the following natural but not always essential condition lim u X xn 1. .Tn x The nonlinear term in the equation is a typical example of a two well potential and the PDE describes the shape of a transitional layer from one N. Ghoussoub was partially supported by a grant from the Natural Science and Engineering Research Council of Canada. C. Gui was partially supported by NSF grant DMS-0140604 and a grant from the Research Foundation of the University of Connecticut. 314 NASSIF GHOUSSOUB AND CHANGFENG GUI phase to another of a fluid or a mixture. The conjecture essentially states that the basic configuration near the interface should .