On the convergence of minimizers of singular perturbation functionals
On the convergence of minimizers of singular perturbation functionals
复制标题
奇异摄动泛函极小值的收敛性
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Rémy Rodiac
中科院分区：
文献类型：
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作者：
Andres A Contreras Marcillo;X. Lamy;Rémy Rodiac
Dirichlet能量奇异微扰的研究是软凝聚物质唯象描述范式的核心。能够越过极限对于理解几何驱动的基态轮廓起着至关重要的作用。在这项工作中，我们研究了在非常一般的假设下，极小值向调和映射的收敛。我们证明了收敛到边界是局部一致的，远离低维奇异集。我们的结果将相关的发现推广到了所有的维度和一般的非线性，尤其是在液晶理论中。我们的证明遵循一个著名的方案，依赖于小能量估计和单调性公式。由于我们不依赖于势的具体形式，它与以前在处理边界上的小能量估计时的研究有很大的不同。特别是，这在3维设置中扩展了现有的结果。在高维中，我们还讨论了与边界单调性公式有关的其他困难。
The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the geometric-driven profile of ground states. In this work we study, under very general assumptions, the convergence of minimizers towards harmonic maps. We show that the convergence is locally uniform up to the boundary, away from the lower dimensional singular set. Our results generalize related findings, most notably in the theory of liquid-crystals, to all dimensions $ngeq 3$, and to general nonlinearities. Our proof follows a well-known scheme, relying on small energy estimate and monotonicity formula. It departs substantially from previous studies in the treatment of the small energy estimate at the boundary, since we do not rely on the specific form of the potential. In particular this extends existing results in 3-dimensional settings. In higher dimensions we also deal with additional difficulties concerning the boundary monotonicity formula.