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Solutions for Quasilinear Nonsmooth Evolution Systems in L p

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Abstract.

We prove that nonsmooth quasilinear parabolic systems admit a local solution in L p strongly differentiable with respect to time over a bounded three-dimensional polyhedral space domain. The proof rests essentially on new elliptic regularity results for polyhedral Laplace interface problems for 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) the proper definition of the normal component of currents across the boundary.

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Authors and Affiliations

  1. Department of Mathematics, Linköping University, S-58183, Linköping, Sweden

    V. Maz’ya, J. Elschner & J. Rehberg

  2. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117, Berlin, Germany

    G. Schmidt

Authors
  1. V. Maz’ya
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  2. J. Elschner
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  3. J. Rehberg
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  4. G. Schmidt
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Correspondence to J. Elschner.

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Communicated by S. S. Antman

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Maz’ya, V., Elschner, J., Rehberg, J. et al. Solutions for Quasilinear Nonsmooth Evolution Systems in L p . Arch. Rational Mech. Anal. 171, 219–262 (2004). https://doi.org/10.1007/s00205-003-0297-y

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