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A compensated compactness theorem for pseudodifferential operators on vector bundles

Siran Li, Xiangxiang Su, Yuantu Zhu

Abstract

We establish a compensated compactness theorem in the microlocal and geometric analytic framework. For a weakly $L^2_{\rm loc}$-convergent sequence of sections of a vector bundle over a semi-Riemannian manifold whose image under a pseudo-differential operator $\mathscr{A}$ of order $s>0$ is precompact in $H^{-s}_{\rm loc}$, we show that a quadratic form $Q$ acting on this sequence converges in the distributional sense, provided that $Q$ vanishes on the operator cone of $\mathscr{A}$. This extends the classical Murat--Tartar theory of compensated compactness from constant-coefficient first-order differential constraints on Euclidean spaces to variable-coefficient pseudo-differential constraints of arbitrary order on semi-Riemannian manifolds.

A compensated compactness theorem for pseudodifferential operators on vector bundles

Abstract

We establish a compensated compactness theorem in the microlocal and geometric analytic framework. For a weakly -convergent sequence of sections of a vector bundle over a semi-Riemannian manifold whose image under a pseudo-differential operator of order is precompact in , we show that a quadratic form acting on this sequence converges in the distributional sense, provided that vanishes on the operator cone of . This extends the classical Murat--Tartar theory of compensated compactness from constant-coefficient first-order differential constraints on Euclidean spaces to variable-coefficient pseudo-differential constraints of arbitrary order on semi-Riemannian manifolds.
Paper Structure (9 sections, 5 theorems, 53 equations)

This paper contains 9 sections, 5 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\Psi {\rm DO}$s on Euclidean domains
  4. $\Psi {\rm DO}$s on manifolds
  5. Semi-Riemannian manifolds
  6. Quadratic forms on a vector bundle
  7. Compensated compactness on Euclidean domains
  8. Compensated Compactness on Manifolds
  9. Proof of Lemma \ref{['lem:quadratic-inequality']}

Key Result

Theorem 1

Let $\{v_\varepsilon\}$, $\{w_\varepsilon\}$ be two sequences of vector fields on $\mathbb{R}^3$, such that $v_\varepsilon \rightharpoonup v$ and $w_\varepsilon \rightharpoonup w$ weakly in $L^2_{\rm loc}(\mathbb{R}^3;\mathbb{R}^3)$. Suppose that $\{{\rm div}(v_\varepsilon)\}$ is precompact in $H^{-

Theorems & Definitions (10)

  • Theorem
  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 2.1
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • proof
  • proof