In this work we study a minimization problem with two-phases where in each phase region the problem is ruled by a quasi-linear elliptic operator of Laplacian type. The problem in its variational form is as follows:
Here we minimize among all admissible functions in an appropriate Sobolev space with a prescribed boundary datum on . First, we show existence of a minimizer, prove some properties, and provide an example for non-uniqueness. Moreover, we analyze the limit case where and go to infinity, obtaining a limiting free boundary problem governed by the -Laplacian operator. Consequently, Lipschitz regularity for any limiting solution is obtained. Finally, we establish some weak geometric properties for solutions.
Cite this article
João Vítor da Silva, Julio D. Rossi, A limit case in non-isotropic two-phase minimization problems driven by -Laplacians. Interfaces Free Bound. 20 (2018), no. 3, pp. 379–406DOI 10.4171/IFB/406