Electron. J. Differential Equations, Vol. 2022 (2022), No. 62, pp. 1-28.

Higher differentiability for solutions to nonhomogeneous obstacle problems with 1<p<2

Zhenqiang Wang

Abstract:
In this article, we establish integer and fractional higher-order differentiability of weak solutions to non-homogeneous obstacle problems that satisfy the variational inequality

where 1<p<2, $\varphi \in \mathcal{K}_{\psi } (\Omega ) =\{ v\in u_0+W_0^{1,p}(\Omega ,\mathbb{R} ):v\ge \psi \text{ a.e.\ in } \Omega\} $, $u_0\in W^{1,p}(\Omega)$ is a fixed boundary datum. We show that the higher differentiability of integer or fractional order of the gradient of the obstacle ψ and the nonhomogeneous term F can transfer to the gradient of the weak solution, provided the partial map $ x\mapsto A(x,\xi)$ belongs to a suitable Sobolev or Besov-Lipschitz space.

Submitted May 23, 2022. Published August 22, 2022.
Math Subject Classifications: 35J87, 49J40, 47J20.
Key Words: Nonhomogeneous elliptic obstacle problems; higher differentiability; Sobolev coefficients; Besov-Lipschitz coefficients.
DOI: https://doi.org/10.58997/ejde.2022.62

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Zhenqiang Wang
School of Mathematical Sciences
Nankai University
Tianjin 300071, China
email: 2120190058@mail.nankai.edu.cn

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