Boundary-value Problems in Half-space

Abstract

Boundary-value problems in the half-space ℝ + n ≔ {x ∈ ∝n: x > 0} serve as models for general boundary-value problems in domains with smooth boundary. For equations with constant coefficients the solutions can be found in an explicit form. By analysing such solutions we can understand the basic features of the theory of boundary-value problems. In particular, we can determine the number of boundary conditions to be prescribed on the boundary xn = 0 so that the boundary-value problem has a solution, and indeed a unique solution, for any boundary data from the class of functions in question. This number, roughly speaking, is equal to the number of well-defined roots of the corresponding equation in the sense of Petrovskij. For example, the Cauchy problem, where the number of boundary conditions is the order of the equation in \(\frac{{{\partial ^{2}}u}}{{\partial {t^{2}}}} - {k^{2}}\sum {\frac{{{\partial ^{2}}u}}{{\partial z_{i}^{2}}}} = 0,\) is wellposed only if all its roots are well defined in the sense of Petrovskij. Such equations are known as Petrovskij well-defined equations. To this type belong all parabolic and hyperbolic equations as well as the Schrodinger equation. On the other hand, elliptic equations of order m in ℝn, with n ≥ 3, have exactly m/2 Petrovskij well-defined roots (when n≥ 3, m is even, by Proposition 5.1, Chap. 3), and hence m/2 boundary conditions must be prescribed for such equations.