A universal boundary value problem for partial differential equations

Abstract: A new boundary value problem for partial differential equations is discussed. We consider an arbitrary solution of an elliptic or parabolic equation in a given domain and no boundary conditions are assumed. We study which restrictions the boundary values of the solution and its normal derivatives must satisfy. Linear integral equations for the boundary values of the solution and its normal derivatives are obtained, which we call the universal boundary value equations. A universal boundary value problem is defined as a partial differential equation together with the boundary data which specify the values of the solution on the boundary and its normal derivatives and satisfy to the universal boundary value equations. For the equations of mathematical physics such as Laplace's and the heat equation the solution of the universal boundary value problem is presented. Applications to cosmology and quantum mechanics are mentioned.