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Existence of homeomorphic minimizers via mappings of finite distortion in compressible magnetoelasticity
Shilpa Dutta, Anja Schlömerkemper
TL;DR
Existing compressible magnetoelasticity theories impose stronger regularity on deformations than the paper seeks to require. The paper uses finite-distortion mappings, proving diameter and openness results at critical integrability to obtain homeomorphic admissible deformations, compactness, and an energy minimizer. These results establish existence in the proposed finite-distortion class while retaining the compressibility constraint.
Problem
Existing existence theories for compressible magnetoelastic solids rely on stronger deformation regularity, motivating geometric conditions based on finite-distortion mappings.
Method
The paper combines a diameter estimate and an open mapping theorem for Ciarlet-Nečas finite-distortion mappings with compactness and lower semicontinuity arguments.
Results
The admissible deformations are homeomorphisms, and the magnetoelastic energy admits a minimum in the proposed class.
Takeaways & Limitations
Finite-distortion mappings provide an admissible framework for compressible magnetoelastic solids in which the direct method yields minimizers.
Takeaways & Limitations
Weak convergence in Sobolev spaces generally does not preserve homeomorphism, so the framework relies on its finite-distortion structure and stated topological conditions.
Abstract
from arXiv · showhide
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-Nečas 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-Šverák 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.
1 Introduction
The paper develops a finite-distortion framework for compressible magnetoelasticity that weakens deformation regularity requirements while preserving homeomorphism and supports minimizer existence. Its analytical core combines diameter estimates, borderline open mapping, compactness, and lower semicontinuity.
- Model: Compressible magnetoelasticity couples deformation and magnetization through a mixed Eulerian-Lagrangian variational model with magnetostrictive, exchange, and magnetostatic energy terms.The deformation is defined on the reference configuration, while magnetization is defined on the deformed configuration.
- Motivation: The proposed admissible class replaces higher-order regularity assumptions with geometric conditions based on mappings of finite distortion.This broadens the deformation framework beyond commonly used Sobolev homeomorphism classes.
- Finite-distortion theory: A diameter estimate for inverse images of balls under Ciarlet-Nečas mappings underpins an open mapping theorem in the borderline regime KO ∈ L^{n−1}.The result is presented as a partial positive result related to the Iwaniec-Šverák conjecture.
- Admissible class: The resulting topological properties imply that admissible deformations are homeomorphisms and satisfy the compressibility constraint.The admissible class consists of homeomorphic finite-distortion mappings together with compatible magnetizations.
- Existence: Compactness and lower semicontinuity allow the direct method to establish minimizers for compressible magnetoelastic solids in the proposed finite-distortion setting.The existence result is stated under the model’s assumptions on the energy density.
2 Preliminary notions
The preliminary framework defines finite-distortion mappings through Sobolev regularity and distortion control, then states structural assumptions on the magnetostrictive energy density. These assumptions include polyconvexity, coerciveness, behavior under extreme compression, and frame indifference.
- Finite-distortion mappings: A finite-distortion mapping belongs locally to W 1,1 and satisfies |∇f|^n ≤ KJ_f almost everywhere for a finite distortion function K.The distortion function takes values in [1,∞] and is finite almost everywhere.
- Distortion coefficients: The outer and inner distortion coefficients K_O and K_I quantify distortion using the Jacobian and adjugate matrix, respectively.The supplied definitions introduce their pointwise constructions through cases involving J_f.
- Energy assumptions: The reference configuration is an open connected subset of R^3, and the magnetostrictive energy density is defined on deformation gradients and magnetization vectors.The energy assumptions are imposed for the subsequent variational analysis.
- Energy assumptions: Polyconvexity requires a continuous representation whose dependence on the relevant deformation variables is convex for each magnetization.This is the first stated structural assumption on W.
- Energy assumptions: The coerciveness assumption introduces constants α1, α2 > 0, p > 3, q ≥ 2, and s > 1 governing the energy bounds.These parameters specify the stated growth conditions on W.
- Energy assumptions: Additional assumptions control the energy as det(∇y) approaches 0+ and require frame indifference.The latter expresses invariance under rotations.
3 Key lemmas and the main theorem
The section develops geometric and topological tools for finite-distortion deformations, beginning with a diameter estimate and culminating in openness under critical distortion integrability. These results establish the inverse-map regularity and topological structure needed for the variational analysis.
- Diameter estimate: The diameter estimate controls inverse images of balls for finite-distortion mappings satisfying the Ciarlet-Nečas condition.The argument uses finite-distortion structure and distortion-integrability assumptions without assuming the deformation is initially a homeomorphism.
- Diameter estimate: The estimate is derived using slices, Morrey-type control, and geometric sections of inverse images of balls.For each slice parameter, an (n −1)-dimensional ball is chosen with radius determined by its distance to the boundary of the inverse image.
- Open mapping theorem: Theorem 3.7 proves openness for mappings in W 1,p with p > n under the Ciarlet-Nečas condition and critical outer-distortion integrability.The proof proceeds through almost-everywhere differentiability and Sobolev estimates for the inverse mapping.
- Main consequence: Combining openness with the admissibility assumptions shows that the deformation is open on the domain and has the topological properties required for the variational analysis.The Ciarlet-Nečas condition supplies the multiplicity control used in the inverse-map estimates.
- Open mapping theorem: The inverse deformation is itself a mapping of finite distortion, and its continuity is obtained using the distortion structure rather than only Morrey’s embedding.This yields the inverse regularity required to complete the openness argument.
4 Existence results and proof
The paper proves existence of minimizers for compressible magnetoelasticity in an admissible class built from finite-distortion deformations. The proof combines homeomorphism and compactness properties with lower semicontinuity and the direct method.
- Admissible configurations: Every admissible deformation is a homeomorphism from Ω onto its image.This follows from the Ciarlet–Nečas condition, positivity of the Jacobian, openness, and the Lusin (N) condition.
- Existence theorem: If the admissible set is nonempty and the energy density satisfies the stated assumptions, Emag(y, M) attains a minimum on A.This is obtained by the direct method of the calculus of variations.
- Compactness: Energy-bounded admissible sequences admit subsequences converging to an admissible limit.The compactness argument preserves the homeomorphic character of the deformation and establishes sequential weak closedness of the deformation class.
- Compactness: The compactness proof combines finite-distortion mapping theory, Sobolev compactness, biting convergence, and trace compactness.These ingredients are used to control the deformation, identify boundary values, and retain admissibility in the limit.
- Existence theorem: The magnetoelastic energy is lower semicontinuous with respect to the admissible convergence.Combining compactness with lower semicontinuity allows every minimizing sequence to yield an admissible minimizer.