Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Differential transmutations

Franck Sueur

Abstract

Inspired by Gromov's partial differential relations, we introduce a notion of differential transmutation, which allows to transfer some local properties of solutions of a PDE to solutions of another PDE, in particular local solvability, hypoellipticity, weak and strong unique continuation properties and the Runge property. The latest refers to the possibility to approximate some given local solutions by a global solution, with point force controls in preassigned positions in the holes of the space domain. As examples we prove that 2D Lamé-Navier system and the $3$D steady Stokes system, can be obtained as differential transmutations of appropriate tensorizations of the Laplace operator.

Differential transmutations

Abstract

Inspired by Gromov's partial differential relations, we introduce a notion of differential transmutation, which allows to transfer some local properties of solutions of a PDE to solutions of another PDE, in particular local solvability, hypoellipticity, weak and strong unique continuation properties and the Runge property. The latest refers to the possibility to approximate some given local solutions by a global solution, with point force controls in preassigned positions in the holes of the space domain. As examples we prove that 2D Lamé-Navier system and the D steady Stokes system, can be obtained as differential transmutations of appropriate tensorizations of the Laplace operator.
Paper Structure (28 sections, 49 theorems, 52 equations)

This paper contains 28 sections, 49 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Abstract transmutations
  3. Definitions and first properties
  4. Equivalence class properties
  5. Some examples
  6. Degrees of freedom
  7. Algebraic operations
  8. Abstract adjoints
  9. Differential transmutations
  10. Definitions and first properties
  11. Examples
  12. The transfer theorems
  13. Local solvability
  14. Differential inverses
  15. Hypoellipticity
  16. ...and 13 more sections

Key Result

Proposition 2.3

If $A$ is a silvern transmutation of $B$ by philosophers' stone $(P,Q,R,S,T)$ then $A$ is weakly post-regenerated by $(R,S,T)$. If $A$ is an bronze transmutation of $B$ by $(P,Q,R,S,T)$ then $A$ is weakly pre-regenerated by $(P,Q,T)$.

Theorems & Definitions (93)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Proposition 2.8
  • proof
  • ...and 83 more