Small time boundary behavior of solutions of parabolic equations with noncompatible data

We consider the linear parabolic problem: partial derivative(t)u = Lu, u(0) = phi, where L is a uniformly elliptic operator, on a bounded domain Omega of R-N, with Dirichlet boundary conditions. If the initial data phi is not compatible with the Dirichlet condition, i.e., if there exists x(0) epsilo n partial derivative Omega such that phi(x0) not equal 0, then the solution u is not continuous on [0, T] x <(Omega)over bar>. In the present paper, we give a precise description of the discontinuities of the solution occuring from such initial data. We present two kinds of op timal pointwise estimates on u(t, x) in two different regions of the space- time domain ('near' the boundary and 'far' from the boundary). We also prov ide estimates for the solution of the related inhomogeneous problem. The proofs are based on the construction of suitable sub- and supersolution s for auxiliary inhomogeneous problems in balls and annuli and on some mono tonicity and localization arguments. (C) 2000 Editions scientifiques et med icales Elsevier SAS.