The hilbert boundary value problem for elliptic differential equation in the plane

Abstract

The solution w to the Hilbert boundary value problem ∂w/∂z = F(z, w,∂w/∂z) in D Re(a+ib) w = g on ∂D has so far been solved in the space of Holder-continuously differentiable functions C₁α(D). It is shown here that theproblem has a unique solution in the more general Sobolev space W_1,p (D), 2 < p < ∞, provided that the boundaryfunction g is allowed to belong to the Slobodecky space W_s,p (∂D), S = 1 − 1/P.

Keywords:

• Hilbert boundary value problem
• elliptic differential equations
• Sobolev space
• Slobodecky space

AMS(MOS):

• 35F15