The initial-Neumann problem for the heat equation in Lipschitz cylinders
The initial-Neumann problem for the heat equation in Lipschitz cylinders
复制标题
Lipschitz 圆柱体热方程的初始诺伊曼问题
DOI：
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发表时间：
1990
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影响因子：
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通讯作者：
Russell M. Brown
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作者：
Russell M. Brown
证明了当横向数据为Lp，i<；p<；2+e时，热传导方程初值Neumann问题解的存在唯一性。为方便起见，我们假定初始数据为零。给出了空间梯度的抛物线极大函数的估计。当数据位于原子Hardy空间HI中时，建立端点结果。当数据位于L P，1<；p<；2+e中具有一个空间导数和一半时间导数的势的空间中时，初始-Dirichlet问题得到了类似的结果，当p=1时，得到了相应的Hardy空间结果。利用这些结果，我们证明了我们的解可以用单层热势表示。通过对偶性，得出了具有L q，2e‘<；q<；00和BMO中数据的初值-狄利克雷问题的解可以表示为双层热势。
We prove existence and uniqueness for solutions of the initialNeumann problem for the heat equation in Lipschitz cylinders when the lateral data is in LP , I < p < 2+e , with respect to surface measure. For convenience, we assume that the initial data is zero. Estimates are given for the parabolic maximal function of the spatial gradient. An endpoint result is established when the data lies in the atomic Hardy space HI . Similar results are obtained for the initial-Dirichlet problem when the data lies in a space of potentials having one spatial derivative and haif of a time derivative in L P , 1 < p < 2 + e , with a corresponding Hardy space result when p = 1 . Using these results, we show that our solutions may be represented as single-layer heat potentials. By duality, it follows that solutions of the initial-Dirichlet problem with data in L q , 2 e' < q < 00 and BMO may be represented as double-layer heat potentials.