arXiv 1503.075782015-03math.AP

随机椭圆算子的高阶大规模正则性理论

A higher-order large-scale regularity theory for random elliptic operators

Julian Fischer, Felix Otto

备注：: 37 pages; revised version, now includes the regularity theory of arbitrary order

摘要：

在随机均匀化的背景下，我们发展了一类具有非均匀系数场的散度型椭圆方程的高阶大规模正则性理论$a$。$a$ -调和函数的大尺度正则性由刘维尔原理编码：最多像$k$次多项式那样增长的$a$ -调和函数的空间具有与常系数情况相同的维数。这个结果可以被看作是大规模$C^{k,α}$ -正则性理论的定性方面，在本工作中，该理论以相应的$C^{k,α}$ -“过量衰变”估计的形式发展：对于球$B_R$上的给定$a$ -调和函数$u$，它在某个球$B_r$上到$a$上的能量距离-最多像次数$k$的多项式那样增长的调和函数在某个最小半径$r_0$以上的半径$r$中具有自然衰减。虽然出于随机均匀化，本文的贡献是纯粹的确定性的性质：我们的工作假设下，对于给定的实现$a$的系数字段，对$(ϕ,σ)$的标量和矢量的潜在的谐波坐标，其中$ϕ$是通常的校正，增长次线性温和量化的方式。然后，我们构建“$k$第三阶校正”，从而$a$ -调和函数的空间，增长最多像一个多项式的次数$k$，建立上述过剩衰减，然后相应的刘维原理。

英文摘要：

We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields $a$ in the context of stochastic homogenization. The large-scale regularity of $a$-harmonic functions is encoded by Liouville principles: The space of $a$-harmonic functions that grow at most like a polynomial of degree $k$ has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale $C^{k,α}$-regularity theory, which in the present work is developed in the form of a corresponding $C^{k,α}$-"excess decay" estimate: For a given $a$-harmonic function $u$ on a ball $B_R$, its energy distance on some ball $B_r$ to the above space of $a$-harmonic functions that grow at most like a polynomial of degree $k$ has the natural decay in the radius $r$ above some minimal radius $r_0$. Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization $a$ of the coefficient field, the couple $(ϕ,σ)$ of scalar and vector potentials of the harmonic coordinates, where $ϕ$ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct "$k$th-order correctors" and thereby the space of $a$-harmonic functions that grow at most like a polynomial of degree $k$, establish the above excess decay and then the corresponding Liouville principle.

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摘要：

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