Boundary Value Problems and Propagation of Singularities for Several Partial Differential Equations of Mathematical Physics

This paper deals with several equations of mathematical physics written in explicit form with their solutions. In Theorem 1, an oblique derivative problem for the string equation is studied. More precisely, the initial-boundary value problem for the string equation is investigated. The corresponding vector field on the boundary is non-vanishing and does not have a characteristic direction, but can be tangential to some part of the boundary, and it is allowed to change sign. A classical solution exists with suitable compatibility conditions at the corner points. The picture changes significantly in the case of the wave equation with several (say two: 2D) space variables in a circular cylinder. The initial-boundary value problem turns out to be underdetermined with an infinite-dimensional kernel if the boundary vector field is orthogonal to the time axis. By prescribing extra conditions on the generatrices of the cylinder where the vector field is tangential to the cylinder, we obtain a unique classical solution. In Theorem 2, we consider the Cauchy problem in the interior of the parabola of the Lorentzian-type eikonal equation and find its unique classical solution in {0≤x2≤1/2}∩{x2≥x122}. Propagation of singularities for the D and 3 D hyperbolic (Klein–Gordon) equations in R4, R8 is studied in Theorem 3. In the double characteristic points, the wave front propagates either along the surface of the characteristic cone, or in the solid cone starting from (t0,x0).