Perturbations of the Laplacian with variable coefficients in exterior domains and differentiability of the resolvent
Perturbations of the Laplacian with variable coefficients in exterior domains and differentiability of the resolvent
复制标题
外域可变系数拉普拉斯算子的扰动和解的可微性
作者：
Charlotte Kerler
我们考虑变系数拉普拉斯算子的自伴、严格椭圆、短程扰动，其作用于定义在外部域中的平方可积函数。根据极限吸收原理，将再溶剂推广到正的真实的谱线，即这些算子的连续谱。在适当的加权空间，其中的权重也取决于微分的等级，我们证明了扩展的预解式是强可微的w.r.t.光谱参数。此外，我们得到了这些加权空间中的扩展预解式的导数的估计，这是局部一致的谱参数。
We consider self-adjoint, strictly elliptic, short-range perturbations of the Laplacian with variable coefficients, that operate on square integrable functions defined in an exterior domain. According to the limiting absorption principle, the re- solvent is extended to the positive real line, which is the continuous spectrum of these operators. In suitable weighted spaces, where the weight also depends on the grade of differentiation, we show that the extended resolvent is strongly differentiable w.r.t. the spectral parameter. Moreover, we obtain estimates for the derivatives of the extended resolvent in these weighted spaces, which are locally uniform in the spectral parameter.