Boundary Value Problems for Partial Differential Equations in the Plane Revisited

Abstract

AbstractBoth boundary value problems the DIRICHLET and the RIEMANN‐HILBERT problems, were solved by the author in the SOBOLEV space W1,p(D), 2 < p < ∞, for the elliptic differential equation in [8], [11] and [12]. In fact in most of the literature so far it is the case of p > 2 only which has been studied extensively. The results obtained so far relied on the fact that the VEKUA‐type operator TD maps the LEBESGUE space Lp(D) Into the HÖLDER space (D̄) for α= (p–2)/p. By modifying both VEKUA type singular integral operators TD and πD it is shown here that we can then solve both boundary value problems in W1,p(D) for 1 < p < ∞. While enlarging the set of admissible SOBOLEV spaces, the set of partial differential equations that can then be solved is reduced through stricter conditions on the LIPSCHITZ constants.