In applications the interesting phenomena frequently occur near the boundary, and consequently the formulation of boundary conditions plays an important r le. In this chapter we treat problems in one space dimension with an interval as spatial domain. After discussion of the heat equation as a specific example, more general parabolic systems will be considered in Section 7.2. The energy method in its discrete and continuous form will be employed to show well-posedness under Dirichlet and Neumann boundary conditions. For more general boundary conditions, the Laplace transform method is the appropriate tool, and we present it in detail in Sections 7.4 and 7.5. If a determinant-condition is satisfied then the problem is strongly well-posed in the generalized sense. Important concepts of well-posedness for initial-boundary value problems are discussed in Section 7.3.

For hyperbolic equations the characteristics play, of course, an eminent role in determining correct boundary conditions: Values for the ingoing characteristic variables must be provided. If this is the case and if the solution is not overspecified at the boundary, then the hyperbolic problem is well-posed. Also, we will derive boundary conditions for mixed hyperbolic-parabolic systems and will apply the results to the linearized (compressible) Navier-Stokes equations. A unified view of all results which can be obtained by the energy method is given in Section 7.8: The energy method applies if the spatial differential operators are semibounded on the space of functions obeying the (homogeneous) boundary conditions.

Throughout this chapter we restrict ourselves to