In this work we develop an Almgren type monotonicity formula for a class of
elliptic equations in a domain with a crack, in the presence of potentials
satisfying either a negligibility condition with respect to the inverse-square
weight or some suitable integrability properties. The study of the Almgren
frequency function around a point on the edge of the crack, where the domain is
highly non-smooth, requires the use of an approximation argument, based on the
construction of a sequence of regular sets which approximate the cracked
domain. Once a finite limit of the Almgren frequency is shown to exist, a
blow-up analysis for scaled solutions allows us to prove asymptotic expansions
and strong unique continuation from the edge of the crack.Comment: 32 pages, 2 figure
