Gaussian Estimates for Second Order Elliptic Divergence Operators on Lipschitz and C 1 Domains

We study the heat kernel of elliptic second order divergence operators defined on Lipschitz or C ¹ domains subject to Dirichlet or Neumann boundary condition. Our purpose is to obtain gaussian upper bounds and Hölder regularity of these kernels when we allow the coefficients to be complex. We obtain a criterion to decide on whether such estimates hold and apply it in various situations such as uniformly continuous or υmo(Ω) coefficients on C ¹ domains or Lipschitz domains with small Lipschitz constant. We also prove an analyticity result for the heat kernels as functions of the coefficients. Although not treated here the strategy works for second order systems subject to Garding inequality.