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

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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.

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Acknowledgements

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

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Correspondence to Shenzhou Zheng.

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This work was supported by the Fundamental Research Funds for the Central Universities (No. 2018YJS167) and NSFC (No. 11371050).

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Liang, S., Zheng, S. On \(W^{1,\,\gamma (\cdot )}\)-regularity for nonlinear non-uniformly elliptic equations. manuscripta math. 159, 247–268 (2019). https://doi.org/10.1007/s00229-018-1053-9

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