WIAS Preprint No. 827, (2003)

Solutions for quasilinear nonsmooth evolution systems in $L^p$



Authors

  • Elschner, Johannes
  • Maz'ya, Vladimir
  • Rehberg, Joachim
  • Schmidt, Gunther

2010 Mathematics Subject Classification

  • 35K55 35D10 35R05 35K45 35K50 35J25

Keywords

  • Quasilinear parabolic systems, elliptic boundary value problems, polyhedral domains, piecewise constant coefficients, regularity of solutions

Abstract

We prove that nonsmooth quasilinear parabolic systems admit a local, strongly differentiable (with respect to time) solution in $L^p$ over a bounded three-dimensional polyhedral space domain. The proof rests essentially on new elliptic regularity results for polyhedral Laplace interface problems with anisotropic materials. These results are based on sharp pointwise estimates for Green's function, which are also of independent interest. To treat the nonlinear problem, we then apply a classical theorem of Sobolevskii for abstract parabolic equations and recently obtained resolvent estimates for elliptic operators and interpolation results. As applications we have in mind primarily reaction diffusion systems. The treatment of such equations in an $L^p$ context seems to be new and allows (by Gauss' theorem) to define properly the normal component of currents across the boundary.

Appeared in

  • Arch. Rational Mech. Anal. 171 (2004), pp. 219-262

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