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Syzygies, constant rank, and beyond

Marc Härkönen, Lisa Nicklasson, Bogdan Raiţă

Abstract

We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs. In this paper we investigate the connection between these two concepts. From the nonlinear analysis point of view, we make some progress in the study of weak lower semicontinuity of integral functionals defined on sequences of PDE constrained fields, when the PDEs do not have constant rank.

Syzygies, constant rank, and beyond

Abstract

We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs. In this paper we investigate the connection between these two concepts. From the nonlinear analysis point of view, we make some progress in the study of weak lower semicontinuity of integral functionals defined on sequences of PDE constrained fields, when the PDEs do not have constant rank.
Paper Structure (7 sections, 23 theorems, 70 equations, 1 table)

This paper contains 7 sections, 23 theorems, 70 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modules over Polynomial Rings
  4. Classes of Operators
  5. Controllable--Uncontrollable Decomposition
  6. Generic Rank and Associated Primes
  7. Applications to the Calculus of Variations

Key Result

Theorem 1.1

Let $A$ have real (resp. complex) constant rank. Then $A$ has a controllable-uncontrollable decomposition as in eq:CUC with real (resp. complex) elliptic $A_u$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 47 more