Potential type operators and transmission problems for strongly elliptic second-order systems in Lipschitz domains
Potential type operators and transmission problems for strongly elliptic second-order systems in Lipschitz domains
复制标题
Lipschitz 域中强椭圆二阶系统的潜在类型算子和传输问题
DOI：
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发表时间：
2009
期刊：
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通讯作者：
M. Agranovich
中科院分区：
文献类型：
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作者：
M. Agranovich
考虑了n维有界区域Ω~+上的强椭圆型二阶方程组，其Lipschitz边界为Γ，n ≥ 2。对系数的平滑假设被最小化。为了方便起见，我们假设域包含在标准环面% MathType! MTEF!二号! 1!+-
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mathbb {T}^n
$$.在以前的文章中，我们得到了在Hp σ和Bp σ空间中Dirichlet和Neumann问题在不使用表面势的情况下唯一可解的结果.本文利用Costabel和McLean提出的方法，在Ω +和补域Ω −上的Dirichlet和Neumann问题唯一可解的前提下，定义了表面势并讨论了它们的性质.特别地，我们证明了积分单层算子和超奇异算子在Γ上的Besov空间中的可逆性.我们描述了它们的一些光谱特性以及相应的传输问题。
AbstractWe consider a strongly elliptic second-order system in a bounded n-dimensional domain Ω+ with Lipschitz boundary Γ, n ≥ 2. The smoothness assumptions on the coefficients are minimized. For convenience, we assume that the domain is contained in the standard torus % MathType!MTEF!2!1!+-
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mathbb{T}^n
$$. In previous papers, we obtained results on the unique solvability of the Dirichlet and Neumann problems in the spaces Hpσ and Bpσ without use of surface potentials. In the present paper, using the approach proposed by Costabel and McLean, we define surface potentials and discuss their properties assuming that the Dirichlet and Neumann problems in Ω+ and the complementing domain Ω− are uniquely solvable. In particular, we prove the invertibility of the integral single layer operator and the hypersingular operator in Besov spaces on Γ. We describe some of their spectral properties as well as those of the corresponding transmission problems.