Existence of homeomorphic minimizers via mappings of finite distortion in compressible magnetoelasticity

Abstract

We establish the existence of an energy minimizer for a variational model of compressible magnetoelastic solids. The analysis is carried out in a new admissible class of deformations consisting of mappings of finite distortion, which extends previously available existence frameworks. A key ingredient is a compactness result under the critical integrability assumption on the outer distortion coefficient, which significantly weakens the regularity requirements imposed in earlier works. To obtain this result, we prove a diameter estimate for finite-distortion mappings satisfying the Ciarlet-Ne\v{c}as condition and derive an open mapping theorem under the optimal integrability assumption, that is, the outer distortion is in $L^{n-1}$. This provides a partial positive result in the direction of the Iwaniec-\v{S}ver\'ak conjecture and implies that admissible deformations are homeomorphisms. These topological and compactness properties allow us to apply the direct method of the calculus of variations and establish the existence of minimizers for compressible magnetoelastic solids within the admissible class of deformations consisting of mappings of finite distortions.