WIAS Preprint No. 511, (1999)

Resolvent and heat kernel properties for second order elliptic differential operators with general boundary conditions on L^p


  • Griepentrog, Jens André
  • Kaiser, Hans-Christoph
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 58D25 35B65 35P10


  • Elliptic differential operators on Lipschitz domains in arbitrary space dimension, regular sets, essentially bounded coefficients, mixed boundary conditions, resolvent estimates, heat kernel properties, symmetric Markov semigroups on L^p, ultracontractivity, linear and semilinear parabolic equations, Hölder continuity


Under general (including mixed) boundary conditions, nonsmooth coefficients and weak assumptions on the spatial domain, resolvent estimates for second order elliptic operators in divergence form are proved. The semigroups generated by them are analytic, map into Hölder spaces, are positivity improving, and their heat kernels are Hölder continuous in both arguments. We regard perturbations of the elliptic operator by nonnegative potentials, by first order differential operators and multiplicative perturbations. Finally the results provide that the solutions of the corresponding linear and semilinear parabolic equations are Hölder continuous in space and time.

Appeared in

  • Advances in Mathematical Sciences and Applications, 2001, Vol. 11, num. 1, pp. 87-112

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