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.
WISCONSIN UNIV MADISON DEPT OF COMPUTER SCIENCES
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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 IT phi gpi. Here T phis the integral over C of phisigma Pi sub s dsigma, 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
- Theoretical Mathematics