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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.
Received: 16.06.1970
Citation:
V. I. Voitinskii, “Density of Cauchy initial data for solutions of elliptic equations”, Mat. Sb. (N.S.), 85(127):1(5) (1971), 132–139; Math. USSR-Sb., 14:1 (1971), 131–139
Linking options:
https://www.mathnet.ru/eng/sm3181https://doi.org/10.1070/SM1971v014n01ABEH002608 https://www.mathnet.ru/eng/sm/v127/i1/p132
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Abstract page: | 226 | Russian version PDF: | 67 | English version PDF: | 21 | References: | 42 |
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