可压缩磁弹性体中通过有限畸变映射实现同胚极小化子的存在性
Existence of homeomorphic minimizers via mappings of finite distortion in compressible magnetoelasticity
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中文总结 AI 辅助
该研究针对可压缩磁弹性固体的变分模型，在有限畸变映射构成的容许形变类中，通过推导关键估计与定理，证明了能量极小化子的存在性，为Iwaniec-Šverák猜想提供了部分正面结果。
中文摘要 AI 辅助
我们建立了可压缩磁弹性固体变分模型的能量极小化子的存在性。分析在由有限畸变映射构成的新型容许形变类中进行，该类扩展了此前可用的存在性框架。一个关键要素是在外畸变系数的临界可积性假设下的紧性结果，这显著弱化了早期工作中施加的正则性要求。为获得该结果，我们证明了满足Ciarlet-Nečas条件的有限畸变映射的直径估计，并在最优可积性假设下推导了一个开映射定理，即外畸变属于$L^{n-1}$。这为Iwaniec-Šverák猜想提供了部分正面结果，并表明容许形变是同胚映射。这些拓扑和紧性性质使我们能够应用变分法的直接方法，在由有限畸变映射构成的容许形变类中建立可压缩磁弹性固体的极小化子的存在性。
英文摘要
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.
