A Free Boundary Problem Inspired by a Conjecture of De Giorgi

作者：Kamburov Nikola^*

来源：Communications in Partial Differential Equations, 2013, 38(3): 477-528.

DOI：10.1080/03605302.2012.739672

• Barriers
• De Giorgi's conjecture
• Free boundary problem
• Minimal graph
• Monotone minimizer
• 35R35
• 35Q56
• 49Q05
• 35B08
• 35B45
• 35B50

We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional , where V(u) is the characteristic function of the interval (1, 1). This functional is a close relative of the scalar Ginzburg-Landau functional , where W(u)=(1u 2)2/2 is a standard double-well potential. According to a famous conjecture of De Giorgi, global critical points of J that are bounded and monotone in one direction have level sets that are hyperplanes, at least up to dimension 8. Recently, Del Pino, Kowalczyk and Wei gave an intricate fixed-point-argument construction of a counterexample in dimension 9, whose level sets follow the entire minimal non-planar graph, built by Bombieri, De Giorgi and Giusti (BdGG). In this paper we turn to the free boundary variant of the problem and we construct the analogous example; the advantage here is that of geometric transparency as the interphase {|u| <1} will be contained within a unit-width band around the BdGG graph. Furthermore, we avoid the technicalities of Del Pino, Kowalczyk and Wei's fixed-point argument by using barriers only.

• 出版日期2013-3-4