临界指数下非局部能量均匀化中凸奇异项的出现
Emergence of a convex strange term via homogenization of non-local energies at the critical exponent
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中文总结 AI 辅助
本文在临界指数下推导周期穿孔域上非局部能量的Γ-极限，证明尺度分离，并发现奇异项能量密度具有凸性，该结论适用于向量值情形。
中文摘要 AI 辅助
我们在临界指数下，推导了在周期穿孔区域上满足Dirichlet边界条件的卷积型和离散型能量的Γ-极限。我们假设非局部相互作用的长度尺度远小于立方体穿孔的边长，并证明会发生尺度分离。利用我们所关注的变分框架之间的相似性，我们采用统一论证来克服由能量的尺度不变性引起的技术困难。我们的多尺度分析产生了一个新颖的观察结果，该结果与泛函的非局部性质无关，而是与临界指数下的分析有关：我们证明了奇异项的能量密度是凸的，即使在向量值设定下也是如此。
英文摘要
We derive the $Γ$-limit of convolution-type and discrete energies subject to Dirichlet boundary conditions on periodically perforated domains at the critical exponent. We assume that the length-scale of the non-local interactions is much smaller than the side-length of the cubic perforations and prove that a separation of scales occurs. Exploiting the analogies between the variational frameworks of our interest, we employ a unified argument to overcome the technical difficulties that arise from the scaling invariance of the energies. Our multiscale analysis yields a novel observation that is not related to the non-local nature of the functionals, but rather to the analysis at the critical exponent: we prove that the energy density of the $\textit{strange term}$ is convex, even in the vector-valued setting.
发表机构
- SISSA（的里雅斯特高等研究学院）
- Scuola Superiore Meridionale（南方高等学院）
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