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arXiv 1909.01873math.AP
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二阶线性抛物型方程解在层中梯度的精确逐点估计

Sharp pointwise estimates for the gradients of solutions to linear parabolic second order equation in the layer

Gershon Kresin, Vladimir Maz'ya

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12 pages
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摘要：

研究了二阶实常系数线性齐次和非齐次抛物型方程在层${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$中的Cauchy问题的解，其中$n\geq 1$和$T<\infty$ .在$L^p({\mathbb R}^n)$、$1\leq p \leq \infty $中考虑齐次方程和初始数据。对于非齐次方程，我们假设初始函数为零，右边的函数属于$f\in L^p({\mathbb R}^{n+1}_T)\cap C^α\big (\bar{{\mathbb R}^{n+1}_T} \big ) $，$p>n+2$和$α\in (0, 1)$ .得到了这些问题解的梯度长度逐点估计中锐系数的显式表达式。

英文摘要：

We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second order equations with real constant coefficients in the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$, where $n\geq 1$ and $T<\infty$. The homogeneous equation is considered with initial data in $L^p({\mathbb R}^n)$, $1\leq p \leq \infty $. For the nonhomogeneous equation we suppose that initial function is equal to zero and the function in the right-hand side belongs to $f\in L^p({\mathbb R}^{n+1}_T)\cap C^α\big (\bar{{\mathbb R}^{n+1}_T} \big ) $ , $p>n+2$ and $α\in (0, 1)$. Explicit formulas for the sharp coefficients in pointwise estimates for the length of the gradient to solutions to these problems are obtained.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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