## A free boundary problem inspired by a conjecture of De Giorgi

##### Author(s)

Kamburov, Nikola (Nikola Angelov)
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##### Other Contributors

Massachusetts Institute of Technology. Dept. of Mathematics.

##### Advisor

David Jerison.

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We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional I(u) = f lVul2 + V(U), where V(u) is the characteristic function of the interval (-1, 1). This functional is a close relative of the scalar Ginzburg-Landau functional J(u) = f lVul2 + W(u), where W(u) = (1 - u2 )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 levell 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 thesis, 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 {lul < 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.

##### Description

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. Cataloged from PDF version of thesis. Includes bibliographical references (p. 97-99).

##### Date issued

2012##### Department

Massachusetts Institute of Technology. Dept. of Mathematics.##### Publisher

Massachusetts Institute of Technology

##### Keywords

Mathematics.