INTRINSIC ULTRACONTRACTIVITY OF DIRICHLET LAPLACIANS IN NONSMOOTH DOMAINS

We give a general criterion for the intrinsic ultracontractivity of Di richlet Laplacians -DELTA(D) on domains D of R(d) d greater-than-or-eq ual-to 3, based on the Lieb's formula. It applies to various classes o f domains (e.g. John, Holder and L(p)-averaging domains) and gives new conditions for intrinsic ultracontractivity in terms of the Minkowski dimension of the boundary partial derivative D. In particular, ''isot ropic self-similar fractals'' and domains satisfying a ''c-covering co ndition'' are considered.