The reflection principle, the Schwarz potential and quadrature

The Schwarz reflection principle for harmonic functions in R 2 can be stated as follows. Let Gamma ⊂ R 2 be a non-singular real analytic curve and P ' ∈ Gamma. Then, there exists a neighborhood U of P' and an anticonformal mapping R : U → U which is identity on Gamma, permutes the components U 1, U2 of UGamma and relative to which any harmonic function u(x,y) defined near Gamma and vanishing on Gamma satisfies the reflection law u(x0, y0) = -u( R(x0, y0)) for any point (x0,y0) sufficiently close to Gamma, where the mapping R is given by R(x0,y0) = R(z0) = Sz0 , and S is the Schwarz function of the curve Gamma.;In Chapter 2, a more general point-to-point reflection law of the form u(P) = a≤ N calphaDalpha u(Q), where N, c alpha ( a ≤ N) are constants depending only on P and Q, is investigated for the solutions of the more general Helmholtz equation in two independent variables, and partial negative answers to the point to compact set reflection conjecture suggested by Garabedian and others are obtained.;In Chapter 3, a reflection formula for polyharmonic functions in R2 is obtained. The formula generalizes the celebrated Schwarz reflection principle for harmonic functions to polyharmonic functions. Modification of the obtained formula to the case of nonhomogeneous data on a reflecting curve is also discussed.;A natural generalization of the Schwarz function to higher dimensions is called the Schwarz potential or the modified Schwarz potential of the surface.;In an effort to prove "the Schwarz potential conjecture" and to study quadrature for harmonic functions, the Schwarz potential of the first few nontrivial surfaces (spheres, cylinders, cones and ellipsoids) has been studied by Khavinson and Shapiro.;In the last Chapter 4, a method to explicitly calculate the modified Schwarz potential and the corresponding quadrature distribution of axially symmetric solid tori in even dimensional spaces is suggested. The quadrature formula for functions harmonic inside the torus and integrable over the torus is also established and examples are provided.