In this paper I propose to develop a Laplace transform calculus for the
solution of partial differential equations of the form
D^u(O) = h. j m - 1
(I) P(D+,A)u = EA.D111^ = f
Q-CD^,B.)u = IB..D* un = g.
where the operators A. are partial differential operators acting on a
bounded region H C R with smooth boundary 8ft, and the B.. are boundary
partial differential operators defined on 8ft. I believe that problems of
this kind will have application in control theory.
I shall give two simple examples of problems of type I with which the
Laplace transform calculus of this paper can deal.
(I1) Given f e L [R x ft), find u such that
+ AD u - Au = f
u(0) = 0 u(tO|3fi =0 Vt
Dt(0) = 0
(I") Given f e L2(R+ x ft), find u such that
u(0) = 0 D^u(t)|an =0 0 j 2, Vt
D u(0) = 0
The calculus developed in this paper will allow the treatment of equations
of type (I) whose operator coefficients Aj have orders in D essentially
independent of the corresponding order m - j of D . The major point of
technical difference with previous work on equations of type (I) will be that
I will not impose any homogeneity restrictions on the polynomial forms P(A,i£)