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Mathematics of the USSR-Sbornik, 1971, Volume 14, Issue 1, Pages 131–139
DOI: https://doi.org/10.1070/SM1971v014n01ABEH002608
(Mi sm3181)
 

Density of Cauchy initial data for solutions of elliptic equations

V. I. Voitinskii
References:
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
Bibliographic databases:
UDC: 517.946.82
MSC: 35J40
Language: English
Original paper language: Russian
Citation: V. I. Voitinskii, “Density of Cauchy initial data for solutions of elliptic equations”, 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 Math. USSR-Sb.
\yr 1971
\vol 14
\issue 1
\pages 131--139
\mathnet{http://mi.mathnet.ru//eng/sm3181}
\crossref{https://doi.org/10.1070/SM1971v014n01ABEH002608}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=282049}
\zmath{https://zbmath.org/?q=an:0217.41302}
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  • https://www.mathnet.ru/eng/sm/v127/i1/p132
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