Preprint no. 152
Mathematical Institute of the Academy of Sciences of the Czech Republic

Neumann and Robin problems in a cracked domain with
jump conditions on cracks

Dagmar Medkova and Pavel Krutitskii

Dagmar Medkova, Mathematical Institute, Acad. Sci.of the Czech Republic, 115 67 PRAHA 1, Zitna 25, Czech Republic;
Czech Technical University, Faculty of Mechanical Engineering, Department of Technical
Mathematics, Karlovo nam. 13, 121 35 Praha 2, Czech Republic, e-mail: medkova@math.cas.cz

Pavel Krutitskii, Department of Mathematics, Faculty of Physics, Moscow State University, Moscow 117234, Russia, e-mail: krutitsk@math.phys.msu.su

Summary:

The boundary value problem for the Laplace equation is studied on a domain with smooth compact boundary and with smooth internal cracks. The Neumann or the Robin condition is given on the boundary of the domain. The jump of the function and the jump of its normal derivative is prescribed on the cracks. The solution is looked for in the form of the sum of a single layer potential and a double layer potential. The solvability of the corresponding integral equation is determined and the explicit solution of this equation is given in the form of the Neumann series.

Keywords:    Laplace equation; crack; single layer potential; double layer potential.

Mathematics Subject Classification 2000: 35J05; 31B10

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