L p -boundedness of a boundary integral operator on a contour with a peak
А. А. Соловьев · Vestnik St Petersburg University Mathematics · 2008
Results on the solvability of boundary integral equations on a plane contour with a peak obtained in collaboration with V.G. Maz’ya are developed. Earlier, it was proved that, on a contour Γ with an outward peak, the operator of the boundary equation of the Dirichlet boundary value problem maps the space ℒ p, β + 1 (Γ) continuously onto $$ \mathcal{N}_{p,\beta } (\Gamma ) $$ . The norm of a function in ℒ p, β (Γ) is defined as , provided that the peak is at the origin. In this case, the norms on the spaces $$ \mathcal{N}_{p,\beta }^ \mp (\Gamma ) $$ are defined by , where q ± are the intersection points of Γ with the circle {z: |z| = |q|} and δ > 0 is a fixed small number. On a contour with an inward peak, the operator of the boundary equation of the Dirichlet problem continuously maps ℒ p, β + 1 (Γ) onto ℳ p, β(Γ), where ℳ p, β(Γ) is the direct sum of $$ \mathcal{N}_{p,\beta }^ + (\Gamma ) $$ (Γ) and the space (Γ) of functions on Γ of the form p(z) = Σ = 0 t (k)Rez k with the parameter m = [μ − β − p −1]. The operator I − 2W of the boundary integral equation of plane elasticity theory, where W is the elastic double-layer potential, is considered. The main result is that the operator I − 2W continuously maps the space ℒ p, β + 1 × ℒ p, β + 1(Γ) to the space $$ \mathcal{N}_{p,\beta }^ - \times \mathcal{N}_{p,\beta }^ - (\Gamma ) $$ . On a contour with an inward peak, the obtained representation of the operator I − 2W and theorems on the boundedness of auxiliary integral operators imply that the images of vector-valued functions from ℒ p, β + 1 × ℒ p, β + 1(Γ) have components representable as sums of functions from the spaces $$ \mathcal{N}_{p,\beta }^ - (\Gamma ) $$ (Γ) and ℳ p, β(Γ).