Transmission problem for the Laplace equation and the integral equation method

We shall study a weak solution in the Sobolev space of the transmission problem for the Laplace equation using the integral equation method. First we use the indirect integral equation method. We look for a solution in the form of the sum of the double layer potential corresponding to the skip of traces on the interface and a single layer potential with an unknown density. We get an integral equation on the boundary. We prove that this equation has a form (I+M)φ=F where M is a contractive operator. So, we can obtain a solution of this equation using the successive approximation method. Moreover, we are able to estimate the norm of the operator M and control how quickly this process converges. Then we study the direct integral equation method. We obtain the same integral equation like for the indirect integral equation method. So, we can again calculate a solution using the successive approximation method.