具小平均振荡系数的椭圆方程的正则性
Regularity for Elliptic Equations with Coefficients of Small Mean Oscillation
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中文总结 AI 辅助
针对非散度型具小平均振荡系数的一致椭圆方程，通过实变量论证等方法证明其内部\\(W^{2,p}\\)正则性，得到\\(\VMO_{\rm loc}\\)系数下的正则性结论。
中文摘要 AI 辅助
我们给出非散度型一致椭圆方程\\(a^{ij}(x)D_{ij}u+b^i(x)D_i u=f\\)的内部\\(W^{2,p}\\)正则性的详细证明。假设主导矩阵在所考虑的球上具有足够小的平均振荡，证明采用现代实变量论证：常系数调和替换产生 Hessian 的尖锐函数估计，Fefferman--Stein 定理与 Hardy--Littlewood 定理可吸收系数误差；通过 Agmon 的辅助变量论证得到的参数估计在所有\\(L^p\\)空间上提供一致预解式，使后续可积性提升非循环。一阶项全程保留，由尺度不变量\\(R^{1-n/q}\\|b\\|_{L^q(B_R)}\\)控制，其中\\(q>\max\{n,p\}\\)。因此，局部平均振荡空间\\(\VMO_{\rm loc}\\)中的系数对每个有限\\(p\\)都给出通常的局部\\(W^{2,p}\\)正则性。
英文摘要
We give a detailed proof of interior $W^{2,p}$ regularity for uniformly elliptic equations in nondivergence form \[ a^{ij}(x)D_{ij}u+b^i(x)D_i u=f. \] The leading matrix is assumed to have sufficiently small mean oscillation on the balls under consideration. The proof uses a modern real-variable argument: a constant-coefficient harmonic replacement yields a sharp-function estimate for the Hessian, and the Fefferman--Stein and Hardy--Littlewood theorems permit the coefficient error to be absorbed. A parameter estimate, obtained by Agmon's auxiliary-variable argument, supplies a consistent resolvent on all $L^p$ spaces and makes the subsequent gain of integrability non-circular. The first-order term is retained throughout and is controlled by the scale-invariant quantity $R^{1-n/q}\|b\|_{L^q(B_R)}$, with $q>\max\{n,p\}$. As a consequence, coefficients in $\VMO_{\rm loc}$ give the usual local $W^{2,p}$ regularity for every finite $p$.
