Recent Advances in Regularity of Second-order Hyperbolic Mixed Problems, and Applications

Abstract

The present paper centers on second-order hyperbolic equations in the unknownw(t,x):

$${w_{tt}} + A(x,\partial )w = f{\text{ in }}\Omega =(0,T]x\Omega $$

((1.1))

augmented by initial conditions

$$w(0, \cdot ) = {w_0};{\text{ }}{w_t}(0, \cdot ) = {w_1}{\text{ in }}\Omega $$

((1.2))

and suitable boundary conditions either of Dirichlet type

$$w{|_\Sigma } = u{\text{ in }}\Sigma = (0,T]x\Gamma ,$$

((1.3D))

or else of Neumann type

$$\frac{{\partial w}}{{\partial {\nu _A}}} = u{\text{ in }}\Sigma $$

((1.3N))

where ∂/∂v _A denotes the corresponding co-normal derivative. Here and throughout, Ω is a general open bounded domain inR ⁿ, n typically ≥ 2, with boundary ∂Ω = Γ assumed ‘smooth’ (the ‘degree’ of smoothness depending on the ‘degree’ of regularity of the solutions we wish to consider). Moreover,A(x, ∂) denotes a second-order elliptic operator on Ω:

$$\left\{ {_{\sum\limits_{i,j}^n {aij} (x)\xi {}_i\xi {}_j \geqslant c\sum\limits_i^n {\xi {}_i^2\;cons\tan t\;c\; > \;0,} }^{A(x,\partial )=\; - \mathop \sum \limits_{i,j\;}^n \frac{n}{{\partial x{}_1}}(aij(x)\frac{\partial }{{\partial x{}_1}}),}} \right.$$

((1.4a)(1.4b))

with suitably smooth coefficientsa _ij (x) =a _ij (x).

Submitted August 23, 1990. Research partially supported by the National Science Foundation under Grant DMS-8902811