Local Calderón-Zygmund estimates for parabolic equations in weighted Lebesgue spaces

Mikyoung Lee; Jihoon Ok; Mikyoung Lee; Jihoon Ok

doi:10.3934/mine.2023062

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-20. doi: 10.3934/mine.2023062

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Local Calderón-Zygmund estimates for parabolic equations in weighted Lebesgue spaces

Mikyoung Lee ¹,
Jihoon Ok ^{2
,
,}

1.
Department of Mathematics, Pusan National University, Busan 46241, Republic of Korea
2.
Department of Mathematics, Sogang University, Seoul 04107, Republic of Korea

Received: 24 August 2022 Revised: 21 October 2022 Accepted: 31 October 2022 Published: 24 November 2022

We prove local Calderón-Zygmund type estimates for the gradient of weak solutions to degenerate or singular parabolic equations of $ p $-Laplacian type with $ p > \frac{2n}{n+2} $ in weighted Lebesgue spaces $ L^q_w $. We introduce a new condition on the weight $ w $ which depends on the intrinsic geometry concerned with the parabolic $ p $-Laplace problems. Our condition is weaker than the one in ^[13], where similar estimates were obtained. In particular, in the case $ p = 2 $, it is the same as the condition of the usual parabolic $ A_q $ weight.
- parabolic equation,
- $ p $-Laplacian,
- Calderón-Zygmund estimate,
- weighted Lebesgue space
Citation: Mikyoung Lee, Jihoon Ok. Local Calderón-Zygmund estimates for parabolic equations in weighted Lebesgue spaces[J]. Mathematics in Engineering, 2023, 5(3): 1-20. doi: 10.3934/mine.2023062

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Abstract

We prove local Calderón-Zygmund type estimates for the gradient of weak solutions to degenerate or singular parabolic equations of $ p $-Laplacian type with $ p > \frac{2n}{n+2} $ in weighted Lebesgue spaces $ L^q_w $. We introduce a new condition on the weight $ w $ which depends on the intrinsic geometry concerned with the parabolic $ p $-Laplace problems. Our condition is weaker than the one in ^[13], where similar estimates were obtained. In particular, in the case $ p = 2 $, it is the same as the condition of the usual parabolic $ A_q $ weight.

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