Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients

Abstract

ABSTRACT In this contribution, we establish a calculus of pseudodifferential boundary value problems with Hölder continuous coefficients. It is a generalization of the calculus of pseudodifferential boundary value problems introduced by Boutet de Monvel. We discuss their mapping properties in Bessel potential and certain Besov spaces. Although having non-smooth coefficients and the operator classes being not closed under composition, we will prove that the composition of Green operators a 1(x, D x )a 2(x, D x ) coincides with a Green operator a(x, D x ) up to order m 1 + m 2 − Θ, where Θ ∈ (0, τ2) is arbitrary, a j (x, ξ) is in C τ j (ℝ n ) w.r.t. x, and m j is the order of a j (x, D x ), j = 1, 2. Moreover, a(x, D x ) is obtained by the asymptotic expansion formula of the smooth coefficient case leaving out all terms of order less than m 1 + m 2 − Θ. This result is used to construct a parametrix of a uniformly elliptic Green operator a(x, D x ).