Abstract
We find the maximum of ¦Du f ¦ L∞ when uf is the solution, which vanishes at infinity, of the Poisson equation Δu =f on ℝn in terms of the decreasing rearrangement off. Hence, we derive sharp estimates for ¦Du f ¦ L∞ in terms of suitable Lorentz orL p norms off. We also solve the problem of maximizing ¦Du B f (0)¦ whenu B f is the solution, vanishing on∂B, to the Poisson equation in a ballB centered at 0 and the decreasing rearrangement off is assigned.
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References
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Cianchi, A. Maximizing the L∞ norm of the gradient of solutions to the Poisson equation. J Geom Anal 2, 499–515 (1992). https://doi.org/10.1007/BF02921575
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DOI: https://doi.org/10.1007/BF02921575