Optimal rearrangement problem and normalized obstacle problem in the fractional setting
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Abstract We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (–Δ)s, 0 < s < 1, and the Gagliardo seminorm |u|s. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satisfies $$\begin{array}{} \displaystyle -(-{\it\Delta})^s U-\chi_{\{U\leq 0\}}\min\{-(-{\it\Delta})^s U^+;1\}=\chi_{\{U \gt 0\}}, \end{array}$$ which happens to be the fractional analogue of the normalized obstacle problem Δu = χ{u>0}.
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2021 ◽
pp. 175682772110155
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2002 ◽
Vol 66
(6)
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pp. 1103-1130
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2011 ◽
Vol 22
(1)
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pp. 1-38
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1987 ◽
Vol 108
(4)
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pp. 631-646
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