= Solution
Test the <weak formulation> with $u$ itself. Uniform ellipticity and the <Cauchy-Schwarz inequality> give
$$
\lambda\|\nabla u\|_2^2\le\int_\Omega A\nabla u\cdot\nabla u=-\int_\Omega fu\le\|f\|_2\|u\|_2.
$$
The <Poincare inequality> for the bounded shell with zero trace gives $\|u\|_2\le C_P\|\nabla u\|_2$. If the <gradient> is nonzero, divide to obtain $\|\nabla u\|_2\le C_P\lambda^{-1}\|f\|_2$; if it is zero the estimate is immediate and the zero trace forces $u=0$. Thus
$$
\boxed{\|u\|_{H^1(\Omega)}\le C_P\sqrt{1+C_P^2}\,\lambda^{-1}\|f\|_2.}
$$
The constant depends on the shell and $g$, not on $u$ or $f$. Subtracting two solutions with the same forcing gives a zero-forcing solution and hence proves uniqueness. This is a coercive energy argument, not an appeal to <real analytic> Cauchy solvability.
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