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-Nečas condition and derive an open mapping theorem under the optimal integrability assumption, that is, the outer distortion is in $L<sup>{n-1}$. 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.

• The paper proves existence of energy minimizers for compressible magnetoelastic solids by combining polyconvex elasticity, magnetic exchange and stray-field energies with homeomorphic mappings of finite distortion.
• It establishes openness at the critical outer-distortion threshold under the Ciarlet–Nečas condition, using inverse-image diameter estimates, area formulas and inverse Sobolev regularity.
• The compactness framework preserves homeomorphism, magnetic saturation and lower semicontinuity under weak deformation convergence, while also identifying limits for magnetization and Maxwell stray fields.

The variational model

The paper studies existence of energy minimizers for compressible magnetoelastic solids in the mixed Eulerian–Lagrangian formulation. The total energy comprises three coupled contributions: a magnetostrictive term $$\int_\Omega \mathcal{W}(\nabla \bm{y}, \bm{M}\circ\bm{y})\,dX$$ coupling the deformation gradient with the magnetization defined on the deformed configuration; an exchange energy $\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$ penalizing spatial variation of the magnetization; and the nonlocal magnetostatic (stray-field) energy $\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$, where $H_{\bm{M}}$ solves the stationary Maxwell equations with sources $\chi_{\bm{y}(\Omega)}\bm{M}$. The magnetostrictive density is assumed frame-indifferent, polyconvex in $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$ for fixed magnetization, coercive with growth $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$ for $p>3$, $q\ge 2$, $s>1$, and to blow up under extreme compression ($\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$0). Admissible states satisfy the magnetic saturation constraint of Brown–James–Kinderlehrer,

$\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$1

which reduces to the classical unit-length constraint in the incompressible case.

The principal novelty lies in the choice of admissible deformations: rather than requiring $\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$2 regularity as in Rybka–Luskin's three-dimensional theory, or $\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$3, $\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$4 homeomorphisms as in the incompressible analysis of Kružík–Stefanelli–Zeman, the authors work within a class of mappings of finite distortion — deformations $\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$5 satisfying $\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$6 pointwise, with quantitative control on the outer distortion coefficient. This enlarges the admissible class while retaining geometric information about local stretching.

Openness at critical distortion integrability

The analytical core of the paper is an open mapping theorem in the borderline regime $\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$7, which is precisely the integrability threshold appearing in the Iwaniec–Šverák conjecture. That conjecture asks whether every nonconstant mapping of finite distortion in $\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$8, $\alpha\int_{\bm{y}(\Omega)}|\nabla\bm{M}|^2dx$9, with outer distortion in $\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$0, is open; it is settled affirmatively in dimension two but remains largely open for $\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$1. The paper contributes a partial positive result: openness holds under the additional Ciarlet–Nečas condition $\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$2.

The proof proceeds through three steps. First, a diameter estimate is established for inverse images of balls under continuous finite-distortion mappings satisfying the Ciarlet–Nečas condition:

$\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$3

valid for balls $\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$4 centered at images of points where $\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$5 is differentiable with positive Jacobian. Compared to earlier estimates that presuppose homeomorphism, this result derives the bound purely from the finite-distortion structure plus the Ciarlet–Nečas condition; the argument combines Morrey-type oscillation estimates on $\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$6-dimensional slices with Fubini's theorem and Hölder interpolation. A preliminary step shows that small balls around suitable image points lie inside $\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$7, using topological degree arguments together with the integrability consequence $\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$8 implied by $\frac{\mu_0}{2}\int_{\mathbb{R}^3}|H_{\bm{M}}|^2dx$9.

Second, the diameter estimate feeds into the Rademacher–Stepanov criterion via the area formula, yielding almost-everywhere differentiability of $H_{\bm{M}}$0 and the gradient bound $H_{\bm{M}}$1, so that the inverse is itself of finite distortion. Third, continuity of the inverse follows from the Sobolev embedding applied to $H_{\bm{M}}$2 away from $H_{\bm{M}}$3; the key integral $H_{\bm{M}}$4 is finite by the pointwise comparison between inner and outer distortion coefficients combined with $H_{\bm{M}}$5. Notably, continuity of the inverse is obtained under weaker Sobolev regularity than Morrey's embedding would directly provide — the finite-distortion structure of $H_{\bm{M}}$6 is essential here.

Homeomorphic admissible class and compactness

For $H_{\bm{M}}$7, $H_{\bm{M}}$8, the admissible deformation class consists of $H_{\bm{M}}$9 with $\chi_{\bm{y}(\Omega)}\bm{M}$0 a.e., $\chi_{\bm{y}(\Omega)}\bm{M}$1, the Ciarlet–Nečas condition, and prescribed trace on $\chi_{\bm{y}(\Omega)}\bm{M}$2. Since the coercivity assumption enforces $\chi_{\bm{y}(\Omega)}\bm{M}$3 in dimension three, every such deformation falls under the openness theorem. Combining openness with almost-everywhere injectivity from the Ciarlet–Nečas condition (via Grandi–Kružík–Mainini–Stefanelli), the Lusin (N) condition, and the invariance of domain theorem, the authors show that every admissible deformation is a homeomorphism onto its image. This is the property that makes the Eulerian–Lagrangian formulation tractable: compositions involving $\chi_{\bm{y}(\Omega)}\bm{M}$4 are well behaved, avoiding technicalities present in prior existence theories.

A central structural feature of the compactness theorem deserves emphasis: weak convergence in $\chi_{\bm{y}(\Omega)}\bm{M}$5 does not in general preserve injectivity — the sequence $\chi_{\bm{y}(\Omega)}\bm{M}$6 of homeomorphisms converging to the constant map is the standard counterexample due to Molchanova and Vodopyanov. Here injectivity survives in the limit because the energy controls $\chi_{\bm{y}(\Omega)}\bm{M}$7 in $\chi_{\bm{y}(\Omega)}\bm{M}$8, $\chi_{\bm{y}(\Omega)}\bm{M}$9, which passes to the limit via biting convergence, keeping the limit within the finite-distortion framework where the openness and homeomorphism results apply. This is a genuine advantage over frameworks built solely on Sobolev homeomorphisms.

The full compactness result states that any sequence of admissible states with uniformly bounded energy admits a subsequence converging to an admissible state, in the sense of weak $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$0 convergence of deformations, strong $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$1 convergence of $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$2, and weak $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$3 convergence of $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$4. The magnetization component is handled by localizing on inner domains $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$5 compactly contained in $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$6, whose uniform preimage convergence follows from continuity of the inverses; the saturation constraint converts $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$7 bounds on $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$8 into bounds on $$(\nabla\bm{y}, \mathrm{Cof}\,\nabla\bm{y}, \det\nabla\bm{y})$$9 supplied by coercivity. Preservation of the saturation constraint in the limit is verified via a covering argument and the Lebesgue–Besicovitch differentiation theorem. A companion result establishes strong $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$0 convergence (and a.e. convergence up to subsequence) of the compositions $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$1 toward $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$2, obtained by combining norm convergence of $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$3 in every $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$4 with strong $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$5 convergence of $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$6 derived from Vitali's theorem.

Lower semicontinuity and existence

Lower semicontinuity is established componentwise. For the magnetostrictive term, polyconvexity together with weak continuity of Jacobian minors and the a.e. convergence of $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$7 yields the standard Ball-type inequality. For the exchange energy, the localization argument on $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$8 combined with absolute continuity of the Lebesgue integral gives the required inequality despite the moving domain $$|\nabla\bm{y}|^p + |K^O_{\bm{y}}|^q + (\det\nabla\bm{y})^{-s}$$9. For the stray-field term, uniqueness of weak solutions to the Maxwell system (via Lax–Milgram on the potential formulation in $p>3$0) identifies the weak limit of $p>3$1 with $p>3$2, and weak lower semicontinuity of the $p>3$3 norm completes the argument. The direct method then delivers the main theorem: if the admissible set is nonempty and $p>3$4 satisfies polyconvexity, coercivity, and the compression blow-up condition, the magnetoelastic energy attains a minimum on the admissible class. The authors also note that a Zeeman term $p>3$5 for a prescribed external field can be appended without modifying the compactness argument, since Young's inequality and Fatou's lemma handle the lower bound and semicontinuity respectively.

Limitations and open questions

Several restrictions qualify the results. The openness theorem requires $p>3$6 and continuity of the mapping in addition to $p>3$7 and the Ciarlet–Nečas condition; whether openness holds without the Ciarlet–Nečas condition — the full Iwaniec–Šverák conjecture for $p>3$8 — remains open, and the present result addresses only this special case. The coercivity assumption couples the elastic density to the distortion coefficient itself, i.e., $p>3$9 must dominate $q\ge 2$0; this is stronger than purely polyconvex coercivity in terms of $q\ge 2$1 alone, and it is what forces the limit deformation to remain of finite distortion. Nonemptiness of the admissible class $q\ge 2$2 is assumed rather than proved, so compatibility of the boundary datum with the saturation constraint is not addressed. Finally, the analysis is static; no quasistatic evolution or time-dependent counterpart is treated.

Conclusion

The paper establishes existence of minimizers for a compressible magnetoelastic energy in a new admissible class of homeomorphic mappings of finite distortion, thereby weakening the regularity requirements of earlier three-dimensional theories. Its two methodological contributions — a diameter estimate for inverse images without a priori homeomorphism assumptions, and an open mapping theorem at the critical integrability $q\ge 2$3 under the Ciarlet–Nečas condition — have independent interest in geometric function theory as partial progress toward the Iwaniec–Šverák conjecture. Within variational analysis, the demonstration that homeomorphic character survives weak limits when the distortion coefficient is energy-controlled provides a template for existence problems formulated in mixed Eulerian–Lagrangian coordinates.