Density of Cauchy initial data for solutions of elliptic equations
V. I. Voitinskii
Abstract:
In this paper we examine a problem connected with Cauchy's problem for linear
elliptic equations.
Let
$G$ be a bounded region of
$E_n$, and let
$\Gamma$ be its boundary. In
$G$ we consider the elliptic equation
\begin{gather*}
\mathscr Lu(x)=\sum_{|\mu|\leqslant 2m}a_\mu(x)D^\mu u(x)=0
\tag{1}\\
\biggl(\mu=(\mu_1,\dots,\mu_n);\quad|\mu|=\mu_1+\dots+\mu_n;\quad
D^\mu=D_1^{\mu_1}\cdots D_n^{\mu_n},\quad D_k=-i\frac\partial{\partial x_k}\biggr),
\end{gather*}
where
$\mathscr L$ is a regular elliptic expression with complex coefficients. Let
$\Gamma_1$ be a piece of the surface
$\Gamma$. The coefficients of the expression
$\mathscr L$, the surface
$\Gamma$, and the boundary
$\Gamma_1$ are assumed to be infinitely smooth. We are concerned with Cauchy's problem on
$\Gamma_1$ with the initial conditions $\{\partial^{j-1}u/\partial\nu^{j-1}|_{\Gamma_1}=f_j\}$,
$j=1,\dots,2m$,
where
$\nu$ designates the direction normal to
$\Gamma$. In this paper we prove that under our assumptions the set of Cauchy initial data for solutions of (1) in
$H^l(G)$ is dense in
$\sum_{j=1}^{2m}H^{l-j+1/2}(\Gamma_1)$ for any integer
$l\geqslant2m$
if Cauchy's problem is unique for the formal conjugate operator
$\mathscr L^+$, as is the case, for example, when
$\mathscr L$ has no multiple complex characteristics.
In addition, in this paper we give conditions under which the analogous assertion holds for certain elliptic systems.
Bibliography: 4 titles.
UDC:
517.946.82
MSC: 35J40 Received: 16.06.1970