W^1,γ(·)-estimate to non-uniformly elliptic obstacle problems with borderline growth

ABSTRACT

In this paper, we are devoted to a global $W^{1,\gamma (\cdot )}$-estimate for elliptic obstacle problem with double phase growth for the borderline setting in nonsmooth domain. More precisely, we consider the variational inequality ${\int }_{\mathrm{\Omega }}〈A(x,Du),Dv-Du〉\mathrm{d}x\ge {\int }_{\mathrm{\Omega }}〈B(x,F),Dv-Du〉\mathrm{d}x\mathrm{\forall }v\in {\mathcal{A}}_0\left(\mathrm{\Omega }\right),$ whose characteristics of ellipticity and growth switch between a type of polynomial and its logarithm, then we prove a variable exponent Calderón-Zygmund estimate with an implication that $H(x,F),H(x,D\psi )\in L^{\gamma \left(x\right)}\left(\mathrm{\Omega }\right)\Rightarrow H(x,Du)\in L^{\gamma \left(x\right)}\left(\mathrm{\Omega }\right)$ provided that the nonlinearity satisfies a small BMO condition while the boundary of the underlying domain is flat in the sense of Reifenberg.

KEYWORDS:

• Double phase growth
• obstacle problem
• Reifenberg flat domain
• $W^{1\gamma (\cdot )}$-estimate
• the large-M-inequality

AMS SUBJECT CLASSIFICATIONS:

• 35D30
• 35B65
• 35J60

Acknowledgments

We would like to thank the anonymous referee for his/her very valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This paper is supported by the National Science Foundation of China [grant number 12071021].