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




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.


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

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