  Elliptic partial differential equations with diffusion coefficients of
lognormal form, that is $a=exp(b)$, where $b$ is a Gaussian random field, are
considered. We study the $\ell^p$ summability properties of the Hermite
polynomial expansion of the solution in terms of the countably many scalar
parameters appearing in a given representation of $b$. These summability
results have direct consequences on the approximation rates of best $n$-term
truncated Hermite expansions. Our results significantly improve on the state of
the art estimates available for this problem. In particular, they take into
account the support properties of the basis functions involved in the
representation of $b$, in addition to the size of these functions. One
interesting conclusion from our analysis is that in certain relevant cases, the
Karhunen-Lo\`eve representation of $b$ may not be the best choice concerning
the resulting sparsity and approximability of the Hermite expansion.
