Conjectura de De Giorgi em dimensões 2 e 3

• Autor(a) principal: Sousa, Ivaldo Tributino de
• Data de Publicação: 2012
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: Biblioteca Digital de Teses e Dissertações da UFPB
• Texto Completo: https://repositorio.ufpb.br/jspui/handle/tede/7409
• topic: Conjectura de De Giorgi; Equações elípticas semilineares; Hiperplanos; De Giorgi s conjecture; Semilinear elliptic equations; Hyperplanos; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA

This word is concerned with the study of bounded solutions of semilinear elliptic equations u − F0(u) = 0 in the whole space Rn, under the assumption that u is monotone in one direction, say, @u/@xn > 0 em Rn. The goal is to establish the one-dimensional character or symmetry of u, namely, that u only depends on one variable or, equivalently, that the level sets of u are hyperplanos. This type of symmetry question was raised by de Giorgi in 1978 (see [6]), who made the folowing conjecture: Conjecture Suppose that u 2 C2(Rn) is solution of the equation u + u − u3 = 0 satisfying |u(x)| 1 and @u @xn > 0 in the whole Rn. Then the level sets of u must be hyperplanes. We show a stronger version of De Giorgi s conjecture is indeed true in dimension 2 and 3 using some techniques in the linear theory developed by Berestychi, Caffarelli and Nirenberg [5] in one of their papers on qualitative properties of solutions of semilinear elliptic equations.