On the Root Functions of General Elliptic Boundary Value Problems
On the Root Functions of General Elliptic Boundary Value Problems
复制标题
一般椭圆边值问题的根函数
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
N. Tarkhanov
中科院分区：
文献类型：
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作者：
N. Tarkhanov
摘要。研究了在$${mathcal{D}} subset {mathbb{R}}^{n}$$域上二阶椭圆微分算子的边值问题。$${mathcal{D}}$$的边界在维数为0≤q < n−1的光滑流形Y外是光滑的，并且$$partial {mathcal{D}}$$沿Y方向具有边型奇点。假设在$$partial {mathcal{D}}$$的光滑部分满足Lopatinskii条件。相应的空间是有权的Sobolev空间$$H^{{s,gamma }} {left( {mathcal{D}}
ight)}$$，这允许我们为问题定义权γ的椭圆性。这个问题的解决方法假定具有最小增长的射线。主要结果表明，如果存在最小生长射线，相邻射线之间的夹角不超过π(γ + 2m)/n，则问题的根函数在$$L^{2} ({mathcal{D}})$$中是完备的。对于二阶椭圆方程，结果对所有有Lipschitz边界的区域都成立。迈克尔·夏皮罗通讯。
Abstract.We consider a boundary value problem for an elliptic differential operator of order 2m in a domain $${mathcal{D}} subset {mathbb{R}}^{n}$$. The boundary of $${mathcal{D}}$$ is smooth outside a smooth manifold Y of dimension 0 ≤ q < n − 1, and $$partial {mathcal{D}}$$ bears edge type singularities along Y . The Lopatinskii condition is assumed to be fulfilled on the smooth part of $$partial {mathcal{D}}$$. The corresponding spaces are weighted Sobolev spaces $$H^{{s,gamma }} {left( {mathcal{D}}
ight)}$$, and this allows one to define ellipticity of weight γ for the problem. The resolvent of the problem is assumed to possess rays of minimal growth. The main result says that if there are rays of minimal growth with angles between neighbouring rays not exceeding π(γ + 2m)/n, then the root functions of the problem are complete in $$L^{2} ({mathcal{D}})$$. In the case of second order elliptic equations the results remain true for all domains with Lipschitz boundary.Communicated by Michael Shapiro.