# Approximation Methods for the Muskhelishvili Equation

Chapter

## Abstract

Let where Δ is the Laplace operator (3.48). We assume that the function

*D*⊂ ℝ^{2}denote a domain bounded by a simple closed piecewise smooth contour Γ and let \( {\bar D} \) :=*D*∪ Γ. It is well known that many problems in plane elasticity, radar imaging, and theory of slow viscous flows can be reduced to the biharmonic problem$$
\Delta ^2 U(x,y) = 0, (x,y) \in D,
$$

**U**is from the space*W*_{ p }^{1}(\( {\bar D} \)) ∪*W*_{ p }^{4}(*D*). The notation*W*_{ p }^{ k }(*X*) is used for the Sobolev space of*k*-times differentiable functions on*X*, the derivatives of which belong to the corresponding space*L*_{ p }(*X*).## Keywords

Integral Operator Boundary Problem Biharmonic Equation Biharmonic Function Fredholm Property
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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