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Equivalence of solutions for non-homogeneous p(x)-Laplace equations

  • Received: 06 December 2021 Revised: 23 June 2022 Accepted: 23 June 2022 Published: 12 July 2022
  • We establish the equivalence between weak and viscosity solutions for non-homogeneous p(x)-Laplace equations with a right-hand side term depending on the spatial variable, the unknown, and its gradient. We employ inf- and sup-convolution techniques to state that viscosity solutions are also weak solutions, and comparison principles to prove the converse. The new aspects of the p(x)-Laplacian compared to the constant case are the presence of log-terms and the lack of the invariance under translations.

    Citation: María Medina, Pablo Ochoa. Equivalence of solutions for non-homogeneous p(x)-Laplace equations[J]. Mathematics in Engineering, 2023, 5(2): 1-19. doi: 10.3934/mine.2023044

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  • We establish the equivalence between weak and viscosity solutions for non-homogeneous p(x)-Laplace equations with a right-hand side term depending on the spatial variable, the unknown, and its gradient. We employ inf- and sup-convolution techniques to state that viscosity solutions are also weak solutions, and comparison principles to prove the converse. The new aspects of the p(x)-Laplacian compared to the constant case are the presence of log-terms and the lack of the invariance under translations.





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