arXiv 1111.27282011-11math.AP

完全nonlinear elliptic方程解的泛连续模

Universal moduli of continuity for solutions to fully nonlinear elliptic equations

Eduardo V. Teixeira

备注：: 16 pages - improvement of C^{1,Log-Lip} regularity theory, comments on BMO estimates on Du and D^2u

摘要：

本文基于$F(X, D^2u) = f(X)$在不同情况下的最弱可积性，为完全非线性椭圆方程$F(X, D^2u) = f(X)$的粘性解提供了通用的最佳连续模。在这项工作中建立的主要结果是一个尖锐的Log-Lipschitz估计$u$的$f$，这对应于最佳的正则性边界的临界阈值的情况下的$L^n$规范的基础上。当$C^{1,α}$.极限上边界情况，$f\in L^\infty$，也有超越的重要性，椭圆正则性理论及其应用。本文证明了在$F$的凸性假设下，若$f$有界平均振动，则$u \in C^{1,\mathrm{Log-Lip}}$.再一次，这样的估计是最佳的。在理论允许的下界可积条件下，基于$f$的$L^{n-ε}$范数，建立了$u$的$C^{0,\frac{n-2ε}{n-ε}}$范数的\textit{a priori}内部估计，其中$ε$是Escauriaza泛常数.指数$\frac{n-2ε}{n-ε}$是最佳的。当源函数$f$位于$L^q$，$n > q > n-ε$中时，我们还得到了精确的改进的Hölder连续性指数.

英文摘要：

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations $F(X, D^2u) = f(X)$, based on weakest integrability properties of $f$ in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on $u$ based on the $L^n$ norm of $f$, which corresponds to optimal regularity bounds for the critical threshold case. Optimal $C^{1,α}$ regularity estimates are delivered when $f\in L^{n+ε}$. The limiting upper borderline case, $f\in L^\infty$, also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under convexity assumption on $F$, that $u \in C^{1,\mathrm{Log-Lip}}$, provided $f$ has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior \textit{a priori} estimates on the $C^{0,\frac{n-2ε}{n-ε}}$ norm of $u$ based on the $L^{n-ε}$ norm of $f$, where $ε$ is the Escauriaza universal constant. The exponent $\frac{n-2ε}{n-ε}$ is optimal. When the source function $f$ lies in $L^q$, $n > q > n-ε$, we also obtain the exact, improved sharp Hölder exponent of continuity.

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摘要：

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