Singular elliptic equations and free boundary problems / Equações elipticas singulares e problemas de fronteira livre

AUTOR(ES)

Olivaine Santana de Queiroz

DATA DE PUBLICAÇÃO

2008

RESUMO

We study the equation -D.u = X{u>O} (log u+Àf(x, u)) in a smooth bounded domain fl C JRn, with boundary conditions u = O on 8fl. We obtain existence and regularity of the maximal solution. The positivity of such a solution depends on the parameter À and on the domain fl. .If the maximal solution vanishes on a set of positive measure, then we obtain local estimates for the Hausdorff measure of the free boundary. If the singularity logu is replaced by -u-!3, with O <(3 <1, the theory of Alt&Caffarelli and Alt&Phillips implies that the free boundary is regular. We also study the Neumann problem with logarithmic nonlinearity using perturbation techniques and variational methods.

ASSUNTO(S)

singular elliptic equations equações diferenciais parciais não-lineares free boundary problems nonlinear partial differential equations

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