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Comparison of polynomial matrix differential operators

Eduard Curcă, Bogdan Raiţă

Abstract

We characterize matrix polynomials $P,Q$ such that the inequality $$ \left\Vert Q(D)u\right\Vert _{L^{2}}\leq C\left\Vert P(D)u\right\Vert _{L^{2}}\quad\text{for all }u\in C_c^\infty(Ω), $$ holds on bounded open sets $Ω$. We also characterize the operators $P,Q$ for which the linear continuous embedding above is compact, i.e., if $u_n\in C_c^\infty(Ω)$ are such that $(P(D)u_n)_{n\geq 1}$ is bounded in $L^2$, then $(Q(D)u_n)_{n\geq 1}$ is strongly compact in $L^2$.

Comparison of polynomial matrix differential operators

Abstract

We characterize matrix polynomials such that the inequality holds on bounded open sets . We also characterize the operators for which the linear continuous embedding above is compact, i.e., if are such that is bounded in , then is strongly compact in .
Paper Structure (9 sections, 12 theorems, 106 equations)

This paper contains 9 sections, 12 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. $L^{2}$-estimates
  3. Compactness
  4. Application to lower semicontinuity of variational integrals
  5. Proofs of estimates and compactness
  6. Definitions and old results
  7. $L^{2}$-estimates
  8. Compactness
  9. Application to lower semicontinuity

Key Result

Theorem 1

Let $P : \mathbf{R}^d\to \mathcal{M}_{M\times N}(\mathbf{C})$, $Q : \mathbf{R}^d\to \mathcal{M}_{L\times N}(\mathbf{C})$ be matrix polynomials. The following are equivalent:

Theorems & Definitions (24)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Theorem 7
  • Remark 8
  • ...and 14 more