A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented.
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Vladimir G. Maz'ya, Andrea Cianchi, Gradient regularity via rearrangements for -Laplacian type elliptic boundary value problems. J. Eur. Math. Soc. 16 (2014), no. 3, pp. 571–595DOI 10.4171/JEMS/440