Boundary problems for fractional Laplacians and other mu-transmission operators

Geometry and Analysis seminar. 

Speaker: Gerd Grubb, KU

Title: Boundary problems for fractional Laplacians and other mu-transmission operators

Abstract: The fractional Laplacian (-\Delta)^a , say for 0<a<1, is currently of interest in probability theory, finance, differential geometry and mathematical physics. It is a pseudodifferential operator, in particular nonlocal, so it is nontrivial to study meaningful boundary value problems for it on subsets \Omega  of R^n. The theory of pseudodifferential boundary problems initiated by Boutet de Monvel does not cover it. However, it belongs to a more general class of operators with the so-called mu-transmission property, defined by H?rmander in an old Princeton lecture note and in his 1985 book Ch. 18.2.
  The talk will be concerned with a recently developed solvability theory for such problems in L_p Sobolev spaces and more general function spaces, with results on the well-posedness and the regularity of solutions, the domain and spectrum of L_2-realizations, the regularity of eigenfunctions, and other questions.