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Mat. Sb. (N.S.), 1971, Volume 85(127), Number 1(5), Pages 132–139 (Mi msb3181)  

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}
(\mu=(\mu_1,…,\mu_n);\quad|\mu|=\mu_1+…+\mu_n;\quad D^\mu=D_1^{\mu_1}\cdots D_n^{\mu_n},\quad D_k=-i\frac\partial{\partial x_k}), \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 $\{\partial^{j-1}u/\partial\nu^{j-1}|_{\Gamma_1}=f_j\}$, $j=1,…,2m$, 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.

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English version:
Mathematics of the USSR-Sbornik, 1971, 14:1, 131–139

Bibliographic databases:

UDC: 517.946.82
MSC: 35J40
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

Citation in format AMSBIB
\Bibitem{Voi71}
\by V.~I.~Voitinskii
\paper Density of Cauchy initial data for solutions of elliptic equations
\jour Mat. Sb. (N.S.)
\yr 1971
\vol 85(127)
\issue 1(5)
\pages 132--139
\mathnet{http://mi.mathnet.ru/msb3181}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=282049}
\zmath{https://zbmath.org/?q=an:0217.41302}
\transl
\jour Math. USSR-Sb.
\yr 1971
\vol 14
\issue 1
\pages 131--139
\crossref{https://doi.org/10.1070/SM1971v014n01ABEH002608}


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