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 Ω + {\Omega ^ + } whose boundary ∂ Ω + {\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 p for which the system is not Fredholm on L p ( ∂ Ω + ) {L^p}(\partial {\Omega ^ + }) . Using the method of layer potentials we study the singularity types of the system of elastostatics \[ L u = μ Δ u + ( λ + μ ) ∇ div ⁡ u = 0. L{\mathbf {u}} = \mu \Delta {\mathbf {u}} + (\lambda + \mu )\nabla \operatorname {div} {\mathbf {u}} = 0. \] in a plane domain Ω + {\Omega ^ + } whose boundary ∂ Ω + {\partial \Omega ^ + } is a curvilinear polygon. Here μ > 0 \mu > 0 and − μ ≤ λ ≤ + ∞ -\mu \le \lambda \le +\infty . When λ = + ∞ \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 > Re ⁡ z > 1 0 > \operatorname {Re} z > 1 lie in the interval ( 1 2 , 1 ) \left ( {\frac {1} {2},1} \right ) so that the system of integral equations is a Fredholm operator of index 0 0 on L p ( ∂ Ω + ) {L^p}(\partial {\Omega ^ + }) for all p p , 2 ≤ p > ∞ 2 \le p > \infty . The explicit dependence of the singularity types on λ \lambda and the interior angles θ \theta of ∂ Ω + {\partial \Omega ^ + } is calculated; the singularity type of each corner is independent of λ \lambda iff the corner is nonconvex.