Applications of a Local Energy Method to Systems of PDE’s Involving Free Boundaries

  • Gonzalo Galiano
Part of the Progress in Mathematics book series (PM, volume 202)


We present a method of analysis for free boundary problems which is based on local energy estimates. This method allows us to deal with a great variety of problems formulated in a very general form, where generality stands for:
  • No global information, like boundary conditions or boundedness of the domain, is needed.

  • No monotonicity assumption on the nonlinearities is required, as the comparison principle is not invoked.

  • Coefficients may depend on space and time variables and only a weak regularity is required.

  • No restriction on the space dimension is assumed.

In this article we first show how the energy method applies to a simple example, proving the well known property of finite speed of propagation for the porous medium equation. We then give an outline of how the method works in more complicated situations: the occurrence of dead cores for the mangroves’ problem and the finite speed of propagation along the characteristics for an evolution convection-diffusion equation.


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Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Gonzalo Galiano
    • 1
  1. 1.Departamento de MatemàticasUniversidad de OviedoOviedoSpain

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