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manuscripta mathematica

, Volume 159, Issue 1–2, pp 247–268 | Cite as

On \(W^{1,\,\gamma (\cdot )}\)-regularity for nonlinear non-uniformly elliptic equations

  • Shuang Liang
  • Shenzhou ZhengEmail author
Article

Abstract

We prove a global \(W^{1,\,\gamma (\cdot )}\)-estimate for nonlinear non-uniformly elliptic problems on bounded smooth domains. More precisely, we consider the following zero-Dirichlet problem of non-uniformly elliptic equations
$$\begin{aligned} \text {div }{} \mathbf A (Du,x)= \text {div } G(\mathbf F ,x)\quad x\in \varOmega \end{aligned}$$
with the model case of \( \mathbf A (Du,x)\approx |Du|^{p-2}Du+a(x)|Du|^{q-2}Du\) and \(G(\mathbf F ,x)\approx |\mathbf F |^{p-2}{} \mathbf F +a(x)|\mathbf F |^{q-2}{} \mathbf F \). Let \(H(\xi ,x)=|\xi |^p+a(x)|\xi |^q\), we obtain its global variable Calderón–Zygmund estimate with that
$$\begin{aligned} H(\mathbf F ,x) \in L^{\gamma (x)}(\varOmega )\Rightarrow H(Du,x) \in L^{\gamma (x)}(\varOmega ) \end{aligned}$$
under the sharp assumptions that \(a(\cdot )\) is \(C^{0,\, \alpha }\)-Hölder continuous, \(1< p< q< p+\frac{\alpha p}{n}\), the boundary of domain is of class \(C^{1,\, \beta }\) with \(\beta \in [\alpha , 1]\), and the variable exponents \(\gamma (x)\ge 1\) satisfy the log-Hölder continuity.

Mathematics Subject Classification

35D30 35K10 

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Notes

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBeijing Jiaotong UniversityBeijingChina

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