🚀 go-pugleaf

Hi.

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%  \input amssym.def  \input amssym

Let $\Omega\in{\Bbb R}^d$ and $\Omega_1\in{\Bbb C}^d$ be simply
connected regions having analytic boundaries, with the closure
of~$\Omega$ contained in the interior of~$\Omega_1$.  Let $L$ be a
strongly elliptic operator of order $2m$, whose coefficients are
analytic in~$\Omega_1$.  Let $B_0,\ldots,B_{m-1}$ be boundary
operators on~$\partial\Omega$, whose coefficients are analytic functions.
Let $(L,B_0,\ldots,B_{m-1})$ be a strongly elliptic boundary value
problem over~$\Omega$.  Let $u$ be the solution of the problem
$$\displaylines{
Lu=f \qquad\hbox{\rm in }\Omega,\cr
B_0 u=\ldots=B_{m-1}u = 0 \qquad\hbox{\rm on }\partial\Omega.
}$$

If $f$ is analytic on~$\Omega_1$, is it true that $u$ is also analytic
on~$\Omega_1$, and that
$$\|u\|_{\sup,\Omega_1} \le C \|f\|_{\sup,\Omega_1}$$
for some constant~$C$, independent of~$f$ and~$u$?

Thanks.

%      Art Werschulz
%      InterNet:  agw@cs.columbia.edu
%      ATTnet:    Columbia University (212) 854-8348 854-2736
%                 Fordham University  (212) 636-6325