A transmission problem for (p,q)-Laplacian

2021/06/14 by Colombo, Maria, Kim, Sunghan, Shahgholian, Henrik
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.07315

Abstract

In this paper, we consider a double-phase problem characterised by a transmission that takes place across the zero level "surface" of the minimiser of the functional J(v,Ω) = ∫_Ω( |D v⁺|^p + |D v^-|^q ) dx. We prove that a minimiser exists, and is Hölder continuous, whence using an intrinsic variation we prove a weak formulation of the free boundary condition across the zero level surface, formally represented by (q-1)|D u^-|^q = (p-1) |D u⁺|^p, \hboxon ∂ \u gt; 0\. We show that the free boundary is C^1,α a.e. with respect to the measure Δ_p u⁺, whose support is of σ-finite (n-1)-dimensional Hausdorff measure.

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