• Corpus ID: 245424683

Syzygies, constant rank, and beyond

@inproceedings{Harkonen2021SyzygiesCR,
  title={Syzygies, constant rank, and beyond},
  author={Marc Harkonen and Lisa Nicklasson and Bogdan Raiță},
  year={2021}
}
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… 
2 Citations

Natural annihilators and operators of constant rank over $\mathbb{C}$

Even if the Fourier symbols of two constant rank differential operators have the same nullspace for each non-trivial phase space variable, the nullspaces of those differential operators might differ

Oscillation and Concentration in Sequences of PDE Constrained Measures

We show that for constant rank partial differential operators A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}

References

SHOWING 1-10 OF 26 REFERENCES

On the necessity of the constant rank condition forLpestimates

We consider a generalization of the elliptic $L^p$-estimate suited for linear operators with non-trivial kernels. A classical result of Schulenberger and Wilcox (Ann. Mat. Pura Appl. (4) 88: 229-305,

Quasiconvexity, Null Lagrangians, and Hardy Space Integrability Under Constant Rank Constraints

We present a systematic treatment of the theory of Compensated Compactness under Murat’s constant rank assumption. We give a short proof of a sharp weak lower semicontinuity result for signed

Rank-one convexity implies quasiconvexity on diagonal matrices

We prove a conjecture of Tartar regarding weak lower semiconti nuity of functionals on sequences uj vj R R which satisfy uj vj in H This is the simplest example in the theory of compensated

On Rank One Convex Functions that are Homogeneous of Degree One

We show that positively 1-homogeneous rank one convex functions are convex at 0 and at matrices of rank one. The result is a special case of an abstract convexity result that we establish for

Linear PDE with Constant Coefficients

We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the

A simple construction of potential operators for compensated compactness

We give a short proof of the fact that each homogeneous linear differential operator A of constant rank admits a homogeneous potential operator B, meaning that kerA (ξ) = imB(ξ) for ξ ∈ R \ {0}. We

The Euler equations as a differential inclusion

We propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in R n with n 2. We give a reformulation of the Euler equations as a

Controllability and Vector Potential: Six Lectures at Steklov

It turns out, that for systems defined in several important spaces of distributions, controllability is now identical to the notion of vector potential in physics, or of vanishing homology in mathematics.

Compensated Compactness, Separately Convex Functions and Interpolatory Estimates between Riesz Transforms and Haar Projections

We prove sharp interpolatory estimates between directional Haar projections and Riesz Transforms. We apply those to prove a conjecture of L. Tartar that arose within the theory of compensated

A -Quasiconvexity. lower semicontinuity, and young measures

The notion of ${\cal A}$-quasiconvexity is introduced as a necessary and sufficient condition for (sequential) lower semicontinuity of $$ (u,v) \mapsto \int_{\Omega} f(x,u(x), v(x))\, dx $$ whenever