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Higher integrability for singular doubly nonlinear systems.

Kristian Moring1, Leah Schätzler2, Christoph Scheven1

  • 1Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany.

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Summary
This summary is machine-generated.

This study proves a local higher integrability result for spatial gradients of weak solutions to doubly nonlinear parabolic systems. An intrinsic geometry considering both the solution and its gradient is key.

Keywords:
Doubly nonlinear systemsGradient estimateHigher integrabilityReverse Hölder inequality

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Area of Science:

  • Partial Differential Equations
  • Nonlinear Analysis
  • Mathematical Physics

Background:

  • Doubly nonlinear parabolic systems are crucial in modeling various physical phenomena.
  • Understanding the regularity of weak solutions, particularly their spatial gradients, is essential for analyzing system behavior.

Purpose of the Study:

  • To establish a local higher integrability result for the spatial gradient of weak solutions.
  • To investigate doubly nonlinear parabolic systems within specific parameter ranges.

Main Methods:

  • Development and application of an intrinsic geometry.
  • The intrinsic geometry accounts for both the solution (u) and its spatial gradient (Du).

Main Results:

  • A local higher integrability result for the spatial gradient (Du) of weak solutions is proven.
  • The findings are applicable to doubly nonlinear parabolic systems within the specified parameter ranges.

Conclusions:

  • The study provides a significant advancement in the regularity theory of weak solutions to these complex systems.
  • The introduced intrinsic geometry offers a novel approach for analyzing related problems in nonlinear PDEs.