Global Lorentz and Lorentz–Morrey estimates below the natural exponent for quasilinear equations

Abstract

Lorentz and Lorentz–Morrey estimates are obtained for gradients of very weak solutions to quasilinear equations of the form

$$\begin{aligned} \text {div}\,\mathcal {A}(x, \nabla u)=\text {div}\, |\mathbf{f}|^{p-2}\mathbf{f}, \end{aligned}$$

where \(\text {div}\,\mathcal {A}(x, \nabla u)\) is modelled after the p-Laplacian, \(p>1\). The estimates are global over bounded domains that satisfy a mild exterior uniform thickness condition that involves the p-capacity. The vector field datum \(\mathbf{f}\) is allowed to have low degrees of integrability and thus solutions may not have finite \(L^p\) energy. A higher integrability result at the boundary of the ground domain is also obtained for infinite energy solutions to the associated homogeneous equations.

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Correspondence to Nguyen Cong Phuc.

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Communicated by L. Caffarelli.

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Adimurthi, K., Phuc, N.C. Global Lorentz and Lorentz–Morrey estimates below the natural exponent for quasilinear equations. Calc. Var. 54, 3107–3139 (2015). https://doi.org/10.1007/s00526-015-0895-1

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Mathematics Subject Classification

  • 35J92
  • 35B65
  • 42B20