In the unit ball, radial non-symmetric sources always break the pointwise fractional Talenti inequality near the boundary, but the reverse boundary inequality holds; nonradial sources supported in small balls can satisfy the forward boundary inequality.
Lieb and Michael Loss
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On a fractional boundary version of Talenti's inequality in the unit ball
In the unit ball, radial non-symmetric sources always break the pointwise fractional Talenti inequality near the boundary, but the reverse boundary inequality holds; nonradial sources supported in small balls can satisfy the forward boundary inequality.