Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients

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} (? 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 ).