论文标题
3D Lipschitz域中Dirac运算符的第一类边界积分方程
First-Kind Boundary Integral Equations for the Dirac Operator in 3D Lipschitz Domains
论文作者
论文摘要
我们为3D Lipschitz域中的欧几里得狄拉克运算符开发了新型的第一类边界积分方程，该域包含正方形的势能，并且仅涉及弱奇异的内核。普遍的花园不等式得出，我们确定获得的边界积分运营商是指数为零的弗雷德·霍尔姆。它们的有限维内核的特征是，我们表明它们的尺寸等于域边界的拓扑不变性数量，换句话说，其betti数字的总和。这是通过基本发现来解释的：相关的双线性形式与H-1/2表面算子诱导的Ham Rham Hilbert综合体诱导的形式一致，其基本的内部产物是通过经典的单层单层边界算子定义的非局部内部产物，用于Laplacian。还介绍了狄拉克系统的自然能量空间中未绑定的外部域中的衰减条件。
We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains comprising square-integrable potentials and involving only weakly singular kernels. Generalized Garding inequalities are derived and we establish that the obtained boundary integral operators are Fredholm of index zero. Their finite dimensional kernels are characterized and we show that their dimension is equal to the number of topological invariants of the domain's boundary, in other words to the sum of its Betti numbers. This is explained by the fundamental discovery that the associated bilinear forms agree with those induced by the 2D surface Dirac operators for H-1/2 surface de Rham Hilbert complexes whose underlying inner-products are the non-local inner products defined through the classical single-layer boundary integral operators for the Laplacian. Decay conditions for well-posedness in natural energy spaces of the Dirac system in unbounded exterior domains are also presented.
