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
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
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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The authors would like to thank the anonymous referees for their valuable comments and suggestions.
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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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DOI: https://doi.org/10.1007/s00229-018-1053-9