The Solution of the Dirichlet Problem for Lapalce's Equation when the Boundary Data is Discontinuous and the Domain has a Boundary which is of Bounded Rotation by Means of the Lebesgue-Stieltjes Integral Equation for the Double Layer Potential.
Abstract
The Dirichlet problem (u sub xx) + (u sub yy) = 0, (x,y) epsilon R; u = g, (x,y) epsilon C; (1) is considered. Here R is a bounded domain in the (x,y)-plane with boundary C. C is of 'bounded rotation' and g is bounded and Borel-measurable. It is shown that if C has no cusps then the solution of (1) can be obtained in terms of the 'double-layer potential' phi which satisfies the Lebesgue-Stieltjes integral equation (I+T) phi = g/pi. Here (T phi)(s) = the integral over C of (phi(sigma) (Pi sub s) d(sigma)), where s denotes arc-length on C, and (pi sub s) is a Lebesgue-Stieltjes measure which depends on C. The case when C has cusps is also considered. The report also contains a lengthy survey of th literature on double-layer potentials. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1970
- Accession Number
- AD0726417
Entities
People
- Colin Walker Cryer
Organizations
- University of Wisconsin–Madison