Extension Problem and Harnack's Inequality for Some Fractional Operators

By Pablo Raúl Stinga, José L. Torrea. 2009

543 citations recorded by the source corpus.

The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential operators in some class. We also get a Poisson formula and a system of Cauchy–Riemann equations for the extension. The method is applied to the fractional harmonic oscillator H σ = (− Δ + |x|2)σ to deduce a Harnack's inequality. A pointwise formula for H σ f(x) and some maximum and comparison principles are derived.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/7802c3362accf69b00ce51c1d1d6c89882a52233

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In Computational Theory and Mathematics

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Citations in the corpus

Weyl and Marchaud Derivatives: A Forgotten History (2017, methodology): “In [58] this approach has been developed and then generalized in [30] to a very large class of operators.”
Some constructions for the fractional Laplacian on noncompact manifolds (2012, methodology): “We apply again the results of [62], this time for the operator L acting on functions on R, and obtain the existence of the solution of (3.”
What is the fractional Laplacian? A comparative review with new results (2018, methodology): “See [36] and [37] for a full discussion.”
Fractional elliptic equations, Caccioppoli estimates and regularity (2014, methodology): “It was shown in [27, 28] that if u = ∑∞ k=0 ukφk then the solution to this problem is given by”
Getting Acquainted with the Fractional Laplacian (2017, methodology): “We refer to [89] for extension methods for this type of operator.”
Fractional Laplacian on the torus (2012, methodology): “2), already found in [23] when 0 < γ < 1, that involves the semigroup e−tL.”
Quantitative approximation properties for the fractional heat equation (2017, background): “Then, following [7], [39], [8], we define (−∆aij )su(x′) := lim xn+1→0 x1−2s n+1 ∂n+1ū(x ′, xn+1),”
Unique Continuation for Fractional Orders of Elliptic Equations (2016, methodology): “The case of bounded domains For a bounded Lipschitz domain Ω ⊂ R, we can still define fractional order of elliptic operators using spectral theory [27][28][29].”
Harnack inequality for fractional sub-Laplacians in Carnot groups (2012, methodology): “A similar formula appears in [35], but, as long as we know, this representation is new for sublaplacians in Carnot groups.”
DENSITY PROPERTIES FOR FRACTIONAL SOBOLEV SPACES (2015, background): “Recently, in the literature a deep interest was shown for nonlocal operators, thanks to their intriguing analytical structure and in view of several applications in a wide range of contexts, such as the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous ”
Nonlocal Diffusion and Applications (2015, methodology): “This technique is classical (see [92]), but it has also been efficiently used in recent research papers (see for instance [34, 90, 22]).”