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

María Medina; Pablo Ochoa; María Medina; Pablo Ochoa

doi:10.3934/mine.2023044

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-19. doi: 10.3934/mine.2023044

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

María Medina ¹,
Pablo Ochoa ^{2
,
,}

1.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain
2.
Facultad de Ingeniería, Universidad Nacional de Cuyo. CONICET. Parque Gral. San Martín 5500. Mendoza, Argentina

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.

Keywords:

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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Abstract

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