A two-phase free boundary problem involving exponential operator - Pontes, Pedro Fellype; Yang, Minbo (Topological Methods in Nonlinear Analysis, 2025)
In this paper we are interested in the study of a two-phase problem equipped with the $Φ$-Laplacian operator $$ Δ_Φ u \coloneqq div \bigg(φ(| abla u|)\dfrac{ abla u}{| abla u|}\bigg),$$%where $Φ(s)=e^{s^2}-1$ and $φ=Φ'$. We obtain the existence, boundedness, and Log-Lipschitz regularity of the minimizers of the energy functional associated to the two-phase problem. Furthermore, we also prove that the free boundaries of these minimizers have locally finite perimeter and Hausdorff dimension at most $(N-1)$.