Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth
Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth
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次二次增长抛物线系统的非线性 Calderón-Zygmund 理论
DOI：
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发表时间：
2010
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通讯作者：
Christoph Scheven
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文献类型：
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作者：
Christoph Scheven
抛物型方程组的向量值解
$$partial_tu-{
m div}, a(x,t,Du)={
m div}left(|F|^{p-2}F
ight)$$利用速率$${pinBig(frac{2n}{n+2},2Big)}$$的多项式增长，证明了空间梯度的Calderón-Zygmund型估计。为了处理上述系统的各向异性缩放行为，我们采用了DiBenedetto的本征几何概念。遵循Mingione的思想，我们避免了谐波分析中的奇异积分和极大函数等工具。我们的方法适用于相对于空间变量仅仅连续的系统以及具有vmo型正则性的某些系统。对于时间变量，除了可测量性外，我们不强加任何规律性。
AbstractFor vector-valued solutions of parabolic systems
$$partial_tu-{
m div}, a(x,t,Du)={
m div}left(|F|^{p-2}F
ight)$$with polynomial growth of rate $${pinBig(frac{2n}{n+2},2Big)}$$, we prove Calderón–Zygmund type estimates for the spatial gradient. In order to deal with the anisotropic scaling behaviour of the above system, we employ the concept of intrinsic geometry by DiBenedetto. Following ideas of Mingione, we avoid tools from harmonic analysis such as singular integrals and maximal functions. Our methods apply to systems that are merely continuous with respect to the space variable as well as to certain systems with a VMO-type regularity. With respect to the time variable, we do not impose any regularity except measurability.