Optimal rearrangement problem and normalized obstacle problem in the fractional setting
We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (−∆) 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 satis es...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://eprints.nottingham.ac.uk/60945/ |
| Summary: | We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (−∆) 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 satis es −(−∆) sU − χ{U≤0} min{−(−∆) sU + ; 1} = χ{U>0} , which happens to be the fractional analogue of the normalized obstacle problem ∆u = χ{u>0} . |
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