JournalsifbVol. 10, No. 1pp. 79–86

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

  • Aram L. Karakhanyan

    Australian National University, Canberra, Australia
On the Lipschitz regularity of solutions of a minimum problem with free boundary cover
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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)=∫Ω∣∇v∣p+λ1p\Xu≤0+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=λ1p−λ2p<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