The paper introduces the logarithmic p-Laplacian and proves its first eigenvalue is the s to 0 derivative of the fractional p-Laplacian eigenvalue, with eigenfunction convergence, maximum principles, and a boundary Hardy inequality.
Boundary fractional Hardy's inequality in dimension one: The critical case
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abstract
We prove fractional boundary Hardy's inequality in dimension one for the critical case $sp =1$. Optimality of the inequality is obtained for any $p$. The extra logarithmic correction term appears in usual fashion. We also provide a concrete (workable) example of a sequence of smooth functions that converges to constant function in $W^{s,p}((0,1))$ for $sp=1$ and $p=2$.
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2024 1verdicts
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The Dirichlet Problem For the Logarithmic p-Laplacian
The paper introduces the logarithmic p-Laplacian and proves its first eigenvalue is the s to 0 derivative of the fractional p-Laplacian eigenvalue, with eigenfunction convergence, maximum principles, and a boundary Hardy inequality.