非线性弹性力学中的椭圆方程理论

Elasticity is an important branch of solid mechanics, a basic subject in material mechanics, structure mechanics and other subjects. It is widely used in construction, machinery, aerospace and other engineering field. With an increasing demand for new material technology and nonlinear material model, the research on the nonlinear elastic theory is more and more urgent. Professor J. Ball, former president of the International Mathematical Union, member of the Academia Europaea, Professor L. Nirenberg, fellow of USA National Academy of Sciences, Professor I. BabusKa, fellow of USA National Academy of Engineering, and other famous mathematicians have all maken important contributions in this field. ..In recent five years, the applicant and his collaborators have been dedicated in a class of elliptic systems from the study of composite material, and obtained a series of attracted results on the gradient estimates of their solutions. Based on these existing work, this project team will be focus on two issues in the framework of nonlinear elastic mechanics: (a) to buid a unified model of cavitation and fracture, and to study the surface energy variational problem. This will involve the isoperimetric inequality, theoty of harmonic mapping in Ginzburg-Landau model; (b) when two cavities are close, to study the interaction between them; and to investigate the related narrow regional problems: the cone singularity of elastc plate and the Korn-Poincare type inequality in elastic thin plate.

弹性力学是固体力学的重要分支，是材料力学、结构力学等学科的基础，广泛应用于建筑、机械、航天等工程领域。随着新材料技术和非线性材料模型需求的日益增加，对非线性弹性理论的研究也愈加迫切。国际数学联盟前主席、欧洲科学院院士J. Ball，美国科学院院士L. Nirenberg，美国工程院院士I. Babuška等著名数学家都曾在此领域做出重要贡献。.近五年来，申请人与其合作者一直从事一类来自复合材料的椭圆方程组的研究，在解的梯度估计方面得到了一系列引人注目的成果。项目组将在已有的工作基础上，在非线性弹性力学框架下集中研究两个问题：(a)建立腔化和断裂的统一模型，研究带有曲面能量的变分问题，这与等周不等式、Ginzburg-Landau模型中的调和映照理论密切相关；(b) 当空腔靠得很近时，研究空腔之间的相互作用；以及相关的窄区域问题；弹性薄片的锥奇异性与薄板上的Korn-Poincaré不等式。

现代科学技术的发展离不开复合材料，尖端科学技术领域更是对复合材料有着极大需求，如航空航天、核潜艇器件的材料设计。复合材料的研制和应用作为一个新兴的交叉领域已成为21世纪科技发展的主要方向之一。本项目主要研究了复合材料中线性和非线性弹力问题对应的椭圆方程及方程组解的正则性理论，在解的梯度估计、应力爆破分析等方面取得了一系列重要进展。针对Babuška公开问题的原始模型，创造发展了一套能量迭代技术，克服了椭圆型方程组无极值原理的本质困难，系统性建立了带有部分无穷系数 Lamé 方程组解的梯度的最佳爆破估计，回答了Babuška 院士的公开问题，开创性揭示了内含物凸性对爆破速度的决定性作用，找到了决定爆破发生的决定因子；回答了李岩岩教授与Vogelius 院士提出的两个公开问题；相关成果在《Adv. Math.》(2篇)、《Arch. Rational Mech. Anal.》、《SIAM J. Math. Anal.》等杂志发表论文9篇，完善了高对比度复合材料中的断裂产生和空腔引起材料破坏的数学基础理论。