A free boundary problem with log term singularity

Autor(es)

de Queiroz, Olivaine S. ^[1] ; Shahgholian, Henrik ^[2]

Afiliação do(s) autor(es)

^[1] Univ Estadual Campinas, IMECC, Dept Matemat, Rua Sergio Buarque de Holanda 651, BR-13083859 Campinas, SP - Brazil

^[2] Royal Inst Technol, Dept Math, S-10044 Stockholm - Sweden

• Tipo de documento: Artigo Científico
• Fonte: INTERFACES AND FREE BOUNDARIES; v. 19, n. 3, p. 351-369, 2017.
• Citações Web of Science: 0

Resumo

We study a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional (sic)(v) = integral(Omega) (vertical bar del v vertical bar(2)/2 - v(+)(log v - 1))dx -> min which should be minimized in some natural admissible class of non-negative functions. Here, v(+) = max[0, v]. The Euler-Lagrange equation associated with (sic) is -Delta u = chi([u>0]) log u, which becomes singular along the free boundary partial derivative[u > O]. Therefore, the regularity results do not follow from classical methods. Besides, the logarithmic forcing term does not have scaling properties, which are very important in the study of free boundary theory. Despite these difficulties, we obtain optimal regularity of a minimizer and show that, close to every free boundary point, they exhibit a super-characteristic growth like r(2)vertical bar log r vertical bar. This estimate is crucial in the study of analytic and geometric properties of the free boundary. (AU)

• Processo FAPESP: 12/20197-0 - Aspectos locais de problemas elípticos e parabólicos
• Beneficiário: Olivâine Santana de Queiroz
• Modalidade de apoio: Auxílio à Pesquisa - Regular