A free boundary problem inspired by a conjecture of De Giorgi
Name
809647693-MIT.pdf
Description
Full printable version
Size
3.32 MB
Format
Adobe PDF
Checksum (MD5)
3411063fa51bb23cf9e3465f3b24b2a5
Author(s)
Kamburov, Nikola (Nikola Angelov)
Advisor(s)
David Jerison.
Date Issued
2012
Publisher
Massachusetts Institute of Technology
Abstract
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).
Subjects
Mathematics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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