The harmonic dirichlet problem for a cracked domain with jump conditions on cracks

Abstract

The harmonic problem in a cracked domain is studied in R ^m , m > 2. The boundary of the domain is assumed to be nonsmooth, while cracks are smooth. The Dirichlet condition is specified on the boundary of the domain. Jumps of the unknown function and its normal derivative are specified on the cracks. Uniqueness and solvability results are obtained. The problem is reduced to the uniquely solvable integral equation, its solution is given explicitely in the form of a series. The estimates of the solution of the problem depending on the boundary data are obtained.

Keywords:

• Laplace equation
• Crack
• Single layer potential
• Double layer potential
• AMS Classifications: 31B10
• 35J05

Acknowledgements

Medková was supported by GAČR Grant No. 201/00/1515 and NATO Grant PST.EV.979340. Krutitskii was supported by RFBR grant 02-01-01067.

Notes

E-mail: [email protected]

Additional information

Notes on contributors

Pavel Krutitskii †

E-mail: [email protected]