A compensated compactness theorem for pseudodifferential operators on vector bundles
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918360726372352 |
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| author | Li, Siran Su, Xiangxiang Zhu, Yuantu |
| author_facet | Li, Siran Su, Xiangxiang Zhu, Yuantu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19078 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A compensated compactness theorem for pseudodifferential operators on vector bundles Li, Siran Su, Xiangxiang Zhu, Yuantu Functional Analysis Analysis of PDEs Differential Geometry 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. |
| title | A compensated compactness theorem for pseudodifferential operators on vector bundles |
| topic | Functional Analysis Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2602.19078 |