Regularizing effect in some Mingione’s double phase problems with very singular data

Lucio Boccardo; Giuseppa Rita Cirmi; Lucio Boccardo; Giuseppa Rita Cirmi

doi:10.3934/mine.2023069

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-15. doi: 10.3934/mine.2023069

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Regularizing effect in some Mingione’s double phase problems with very singular data

Lucio Boccardo ^{1
,
,},
Giuseppa Rita Cirmi ²

1.
Istituto Lombardo & Sapienza Università di Roma, Piazzale A. Moro 5, 00185 Roma, Italy
2.
Università di Catania, Dipartimento di Matematica e Informatica, Viale A. Doria 6, 95125 Catania, Italy

Received: 03 October 2022 Revised: 01 December 2022 Accepted: 01 December 2022 Published: 12 December 2022

In this paper we study the existence of solutions of the Dirichlet problem associated to the following nonlinear PDE

$ \begin{equation*} { } -{{{\rm{\;div}}}}\big(a(x)\,\nabla u|\nabla u|^{p-2}\big) -{{{\rm{\;div}}}}\big( |u|^{(r-1)\lambda+1}\nabla u|\nabla u|^{\lambda-2}\big) = f \end{equation*} $

where $ 1 < \lambda \leq p $, $ r > 1 $ and $ f \in L^1(\Omega) $.
- nonlinear elliptic equations,
- weak solutions,
- double phase problems,
- singular data,
- regularity
Citation: Lucio Boccardo, Giuseppa Rita Cirmi. Regularizing effect in some Mingione’s double phase problems with very singular data[J]. Mathematics in Engineering, 2023, 5(3): 1-15. doi: 10.3934/mine.2023069

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Abstract

In this paper we study the existence of solutions of the Dirichlet problem associated to the following nonlinear PDE

$ \begin{equation*} { } -{{{\rm{\;div}}}}\big(a(x)\,\nabla u|\nabla u|^{p-2}\big) -{{{\rm{\;div}}}}\big( |u|^{(r-1)\lambda+1}\nabla u|\nabla u|^{\lambda-2}\big) = f \end{equation*} $

where $ 1 < \lambda \leq p $, $ r > 1 $ and $ f \in L^1(\Omega) $.

References

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