论文标题
完全非线性椭圆方程的周期性均质化统一估计值
Uniform Estimates in Periodic Homogenization of Fully Nonlinear Elliptic Equations
论文作者
论文摘要
本文涉及统一的$ c^{1，α} $和$ c^{1,1} $估计的估计值，以定期同质化完全非线性椭圆方程。分析基于紧凑度方法，该方法涉及在每个近似步骤中操作员的线性化。由于方程的非线性，线性化运算符涉及上一步中出现的校正器的Hessian。校正器的Hessian的参与会恶化线性化操作员的规律性，有时甚至会改变其振荡模式。这些问题是通过新的近似技术解决的，该技术在均质过程的常规部分和不规则部分的精确分解以及对校正器的Hessian统一控制中的确切分解。在线性方程的上下文中，近似技术甚至是新的。我们的论点不仅可以应用于凹操作员，还可以应用于某些类别的非cove操作员。
This article is concerned with uniform $C^{1,α}$ and $C^{1,1}$ estimates in periodic homogenization of fully nonlinear elliptic equations. The analysis is based on the compactness method, which involves linearization of the operator at each approximation step. Due to the nonlinearity of the equations, the linearized operators involve the Hessian of correctors, which appear in the previous step. The involvement of the Hessian of the correctors deteriorates the regularity of the linearized operator, and sometimes even changes its oscillating pattern. These issues are resolved with new approximation techniques, which yield a precise decomposition of the regular part and the irregular part of the homogenization process, along with a uniform control of the Hessian of the correctors in an intermediate level. The approximation techniques are even new in the context of linear equations. Our argument can be applied not only to concave operators, but also to certain class of non-concave operators.
