Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains

Abstract:

The symbolic calculus of pseudodifferential operators of Mellin type is applied to study layer potentials on a plane domain ${\Omega ^ + }$ whose boundary ${\partial \Omega ^ + }$ is a curvilinear polygon. A "singularity type" is a zero of the determinant of the matrix of symbols of the Mellin operators and can be used to calculate the "bad values" of $p$ for which the system is not Fredholm on ${L^p}(\partial {\Omega ^ + })$. Using the method of layer potentials we study the singularity types of the system of elastostatics \[ L{\mathbf {u}} = \mu \Delta {\mathbf {u}} + (\lambda + \mu )\nabla \operatorname {div} {\mathbf {u}} = 0.\] in a plane domain ${\Omega ^ + }$ whose boundary ${\partial \Omega ^ + }$ is a curvilinear polygon. Here $\mu > 0$ and $-\mu \le \lambda \le +\infty$. When $\lambda = +\infty$, the system is the Stokes system of hydrostatics. For the traction double layer potential, we show that all singularity types in the strip $0 < \operatorname {Re} z < 1$ lie in the interval $\left ( {\frac {1} {2},1} \right )$ so that the system of integral equations is a Fredholm operator of index $0$ on ${L^p}(\partial {\Omega ^ + })$ for all $p$, $2 \le p < \infty$. The explicit dependence of the singularity types on $\lambda$ and the interior angles $\theta$ of ${\partial \Omega ^ + }$ is calculated; the singularity type of each corner is independent of $\lambda$ iff the corner is nonconvex.