The Boutet de Monvel Conjecture.
Navn på bevillingshaver
David Scott Winterrose
Institution
University of California Irvine
Beløb
DKK 820,000
År
2021
Bevillingstype
Internationalisation Fellowships
Hvad?
The aim of this project is to prove, in full generality, an old conjecture due to the famous mathematician Boutet de Monvel. If true in general, it would imply a supply of pseudo-differential operators on closed real-analytic manifolds preserving real-analytic functions with holomorphic extension belonging to holomorphic analogues of the classical Sobolev spaces. Ultimately, my hope is to establish a calculus analogous to the pseudo-differential boundary operators, but adapted to real-analytic boundary value problems, such that one can make precise statements about the domain of holomorphic extendibility of solutions or (missing) Cauchy boundary data. The conjecture is known to be true in a prototypical special case.
Hvorfor?
A proof of the conjecture would be an important contribution to analysis on real-analytic manifolds. It would be a step towards a deeper understanding of the properties of solutions to real-analytic boundary value problems. These appear, for example, in models of electromagnetic wave scattering. Moreover, certain numerical methods for approximating solutions to these types of problems rely directly on the real-analyticity, and their stability and convergence rates are determined by how far the solutions extend as holomorphic functions into a complexification of the problem domain.
Hvordan?
This research is of a pure mathematical nature. It is conducted by reading literature, thinking, and having conversations with other mathematicians. The work will be carried out at the University of California, Irvine.