On the Lipschitz regularity of solutions of a minimum problem with free boundary

  • Aram L. Karakhanyan

    Australian National University, Canberra, Australia

Abstract

In this article under assumption of "small" density for negativity set, we prove local Lipschitz regularity for the one phase minimization problem with free boundary for the functional

Ep(v,Ω)=Ωvp+λ1p\Xu0+λ2p\Xu>0,1<p<,\mathcal E_p(v,\Omega)=\int_\Omega|\nabla v|^p+\lambda^p_1\X{u\leq0}+\lambda^p_2\X{u>0},\hspace{3mm} 1<p<\infty,

where λ1,λ2\lambda_1, \lambda_2 are positive constants so that Λ=λ1pλ2p<0\Lambda=\lambda_1^p-\lambda_2^p<0, χD\chi_{D} is the characteristic function of set DD, Ω\Rn\Omega\subset\Rn is (smooth) domain and minimum is taken over a suitable subspace of W1,p(Ω)W^{1,p}(\Omega).

Cite this article

Aram L. Karakhanyan, On the Lipschitz regularity of solutions of a minimum problem with free boundary. Interfaces Free Bound. 10 (2008), no. 1, pp. 79–86

DOI 10.4171/IFB/180