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
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References
SHOWING 1-10 OF 26 REFERENCES
On the necessity of the constant rank condition forLpestimates
- Mathematics
- 2020
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
- MathematicsArchive for Rational Mechanics and Analysis
- 2022
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
- Mathematics, Philosophy
- 1999
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
- Mathematics
- 2015
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
- MathematicsGlasgow Mathematical Journal
- 2021
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
- Mathematics
- 2021
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
- Mathematics
- 2007
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
- MathematicsArXiv
- 2019
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
- Mathematics
- 2009
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
- Mathematics
- 1999
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…