The convolution method for fourier expansions: The case of a crack (screen) in an inhomogeneous space

Abstract

We consider boundary value problems in the space Rn for the equation $$ \partial _x (K_i \partial _x \varphi _i ) + K_i L[\varphi _i ] = 0 $$ with generalized transmission conditions corresponding to a nonideal contact of two media of the type of a strongly permeable crack or a weakly permeable screen on the surface x = 0. (Here L is an arbitrary linear differential operator in all variables except for x ∈ R.) The desired solution has arbitrary singular points (sources, sinks, etc.) and satisfies a sufficiently weak condition at infinity. The coefficients Ki in the regions separated by the film x = 0 are arbitrary combinations of the functions Ki(x) = pi tanh2αi(x−xi) and Ki(x) = pi coth2αi(x−xi), where i = 1 for x 0. We derive formulas directly expressing the solutions of the above-mentioned problems in the form of simple quadratures via the solutions of the similar problems with Ki ≡ 1 and without the film inclusion x = 0. In particular, the obtained formulas permit one to solve boundary value problems for equations with functional piecewise continuous coefficients in the presence of a crack or a screen in the framework of the theory of harmonic functions.