Introduces the fractional-logarithmic Laplacian via differentiation of the fractional Laplacian, establishes equivalent representations and energy spaces with compact critical embeddings, and derives a Weyl asymptotic combining fractional scaling with logarithmic growth for Dirichlet eigenvalues.
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Existence of an unbounded sequence of Lusternik-Schnirelmann eigenvalues is shown for the logarithmic Laplacian with indefinite weights; the first eigenvalue is simple with constant-sign eigenfunction, higher ones change sign, and nodal inequalities plus monotonicity hold.
A fractional logarithmic p-Laplacian operator is defined by differentiating the fractional p-Laplacian, yielding an integral form with a log term, and applied to prove inequalities and eigenvalue results.
As s approaches 0+, s-harmonic functions u_s have asymptotics and s-derivatives expressible via the logarithmic Laplacian of extensions of the exterior data g, yielding pointwise monotonicity in s for many g.
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The Fractional-Logarithmic Laplacian:Fundamental Properties and Eigenvalues
Introduces the fractional-logarithmic Laplacian via differentiation of the fractional Laplacian, establishes equivalent representations and energy spaces with compact critical embeddings, and derives a Weyl asymptotic combining fractional scaling with logarithmic growth for Dirichlet eigenvalues.
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Spectral Properties of the Logarithmic Laplacian with Indefinite Weights
Existence of an unbounded sequence of Lusternik-Schnirelmann eigenvalues is shown for the logarithmic Laplacian with indefinite weights; the first eigenvalue is simple with constant-sign eigenfunction, higher ones change sign, and nodal inequalities plus monotonicity hold.
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On the fractional logarithmic $p$-Laplacian
A fractional logarithmic p-Laplacian operator is defined by differentiating the fractional p-Laplacian, yielding an integral form with a log term, and applied to prove inequalities and eigenvalue results.
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$s$-harmonic functions in the small order limit
As s approaches 0+, s-harmonic functions u_s have asymptotics and s-derivatives expressible via the logarithmic Laplacian of extensions of the exterior data g, yielding pointwise monotonicity in s for many g.