二维两相电导率的梯度可积性和刚性结果
Gradient integrability and rigidity results for two-phase conductivities in dimension two
摘要：
本文研究了具有间断系数$σ$的调和函数$u$的高梯度可积性，即$÷(σ\nabla u) = 0$的弱解.我们专注于两相电导率，并研究相应的梯度场$|\nabla u|$的更高的可积性。梯度场及其可积性显然取决于几何形状，即，阶段安排。我们找到了对应于任何一对正定矩阵的$\{σ_1,σ_2\}$的梯度场的最佳可积指数，即，所有可能的微观几何形状中最差的。我们还表明，它是通过所谓的精确解相应的偏微分方程。此外，在所有具有固定椭圆率的两相电导率中，我们描述了那些对应于较差的可积性。
英文摘要：
This paper deals with higher gradient integrability for $σ$-harmonic functions $u$ with discontinuous coefficients $σ$, i.e. weak solutions of $÷(σ\nabla u) = 0$. We focus on two-phase conductivities, and study the higher integrability of the corresponding gradient field $|\nabla u|$. The gradient field and its integrability clearly depend on the geometry, i.e., on the phases arrangement. We find the optimal integrability exponent of the gradient field corresponding to any pair $\{σ_1,σ_2\}$ of positive definite matrices, i.e., the worst among all possible microgeometries. We also show that it is attained by so-called exact solutions of the corresponding PDE. Furthermore, among all two-phase conductivities with fixed ellipticity, we characterize those that correspond to the worse integrability.
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