The solution of the third problem for the Laplace equation onplanar domains with smooth boundary and inside cracks and modifiedjump conditions on cracks

摘要

This paper studies the third problem for the Laplace equation on abounded planar domain with inside cracks. The third condition?u/?n+hu=fis given on the boundary of thedomain. The skip of the functionu+?u?=gand the modifiedskip of the normal derivatives(?u/?n)+?(?u/?n)?+hu+=fare given on cracks. Thesolution is looked for in the form of the sum of a modifiedsingle-layer potential and a double-layer potential. The solutionof the corresponding integral equation is constructed.