Mathematical Physics, An Advanced Course

Theoretical Numerical Analysis

“…Chapters 7-9 of the latter contain a more complete and extensive introduction to boundary integral equation reformulations and their numerical solution, again for only Laplace's equation; and a very large set of references are given there. More complete introductions to boundary integral equations and their analysis can be found in Kress [149], Mikhlin [172], and Pogorzelski [187]. From the perspective of applications of BIE, see Jaswon and Symm [130] and Pozrikidis [188].…”

Potential Theory

“…Integral representations of solutions to the Neumann problem for half-spaces, quadrants, circular annuli, etc., can be found in [15,44].…”

“…This method of solution will be sketched only briefly. For further details, see [16,35,44,62]. Consider two points x, y ∈ B 0,ρ ⊂ R n , y = 0, n ≥ 3, and the fundamental harmonic function r −n+2 = |x − y| −n+2 with pole y.…”

“…where u 0 is an arbitrary constant (for further details see [44]). For additional information about spherical harmonics, see the book by Axler, Bourdon, and Ramey [2].…”

Potential Theory

“…This problem has a unique solution v E C 2 (Q c ) n C 1 +<X(Q c )' In fact, following Mikhlin (1970), the solution can be constructed by taking it to be of the fonn…”