Commutative partial differential operators

Commutative Partial Differential Operators

In one variable, there exists a satisfactory classification of commutative rings of differential operators. In several variables, even the simplest generalizations seem to be unknown and in this report we give examples and pose questions that may suggest a theory to be developed. In particular, we address the existence of a “spectral variety” generalizing the spectral curve of the one dimension...

Commutative Algebras of Ordinary Differential Operators with Matrix Coefficients

A classification of commutative integral domains consisting of ordinary differential operators with matrix coefficients is established in terms of morphisms between algebraic curves.

Partial Differential Equations for Morphological Operators

Two of G. Matheron’s seminal contributions have been his development of size distributions (else called ‘granulometries’) and his kernel representation theory. The first deals with semigroups of multiscale openings and closings of binary images (shapes) by compact convex sets, a basic ingredient of which are the multiscale Minkowski dilations and erosions. The second deals with representing inc...

Linear Operators in the Theory of Partial Differential Equations

Multilinear Operators in Harmonic Analysis and Partial Differential Equations

Grafakos’ two lectures at the Research Institute for Mathematical Sciences (RIMS), Kyoto University Workshop entitled “Harmonic Analysis and Nonlinear Partial Differential Equations”, July 4 July 6, 2011, are based on these notes.