Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators

By Gerd Grubb. 2013

301 citations recorded by the source corpus.

Abstract Let P be a classical pseudodifferential operator of order m ∈ C on an n-dimensional C ∞ manifold Ω 1 . For the truncation P Ω to a smooth subset Ω there is a well-known theory of boundary value problems when P Ω has the transmission property (preserves C ∞ ( Ω ¯ ) ) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian ( − Δ ) μ with μ ∉ Z , are not covered. They have instead the μ-transmission property defined in Hormander's books, mapping x n μ C ∞ ( Ω ¯ ) into C ∞ ( Ω ¯ ) . In an unpublished lecture note from 1965, Hormander described an L 2 -solvability theory for μ-transmission operators, departing from Vishik and Eskin's results. We here develop the theory in L p Sobolev spaces ( 1 p ∞ ) in a modern setting. It leads to not only Fredholm solvability statements but also regularity results in full scales of Sobolev spaces ( s → ∞ ). The solution spaces have a singularity at the boundary that we describe in detail. We moreover obtain results in Holder spaces, which radically improve recent regularity results for fractional Laplacians.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eb74084fefd787bb9ef30968b3bb135ebd582a83

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators (2014, methodology): “Here, (3-9) is obtained by a generalization of [Grubb 2015a, Theorem 4.4] and its proof to the current spaces.”
Nonlocal elliptic equations in bounded domains: a survey (2015, unknown): “…[25], Hoh-Jacob [31], and Barles-Imbert [3]; • Interior regularity of solutions: Bass-Levin [6], Kassmann [33], CaffarelliSilvestre [13, 14, 15], Barles-Chasseigne-Imbert [5], Kassmann-Mimica [34, 35], Schwab-Silvestre [51], and Serra [55]; • Boundary regularity of solutions: Bogdan…”