【作者】 李晓燕；

【导师】 严国政；

【作者基本信息】 华中师范大学 ， 应用数学， 2008， 硕士

【摘要】 我们考虑部分边界被覆盖的散射体上的散射问题．主要利用第一类边界积分方程和变分方法来解决在散射体D的Lipschitz边界上满足Neumann-Robin混合边界条件的Helmholtz方程的散射问题．考虑如下：混合边值内问题显然，如果我们能知道解在整个边界Γ上的Dirichlet Cauchy数据或NeumannCauchy数据，解的存在性问题便可由[1]得到．基于此，参考[8]，我们利用如下方法证明：利用单双层位势理论以及格林公式，先将混合边值内问题(*)转化为2×2的第一类边界积分方程组．在某种意义下，所得积分方程组等价于最开始的混合边值内问题(*)(参见文献[6])．一旦未知的Cauchy数据由此边界积分方程组确定，则(*)有唯一的弱解．我们的证明可分为两部分．第一部分利用边界积分方程理论证明内问题(*)解的存在性．第二部分介绍第一部分中用到的引理并对该引理进行证明．更多还原

【Abstract】 We consider the direct and inverse scattering problems for partially coated obsta-cles.To this end,we first use the method of integral equations of the first kind together with variational methods to solve a scattering problem for the Helmholtz equation where the scattered field satisfies mixed Neumann-Robin boundary conditions. We consider it like this:Interior mixed boundary value problemObviously,if we can know the whole Dirichlet Cauchy data or Neumann Cauchy data of the solution on the whole boundaryΓ,the existence of the solution can be gotten from[1].For this,we use the following method which referred to [8]:By single- and double- layer potential theories and Green’s formula,we first reformulate the interior mixed boundary problem(*)as a 2×2 system of boundary integral equation of the first kind which is equivalent to our original interior mixed boundary value problem in some senses.(see[6]).0nce the unknown Cauchy data are determined from the 2×2 system of boundary integral equation of the first kind, the representation of formula(8) determines the unique weak solution.Our proof can be divided into two parts. In the first part we use the boundary integral equation theory to prove the existence of the solution for problem (*). In the second part we introduce and prove the lemma which was used in the first part .更多还原