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$\mathcal{A}$-quasiconvexity, Gårding inequalities, and applications in PDE constrained problems in dynamics and statics

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posted on 2023-06-10, 00:20 authored by Konstantinos KoumatosKonstantinos Koumatos, Andreas Panagiotis Vikelis
A Gårding-type inequality is proved for a quadratic form associated to $\mathcal{A}$-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimizers. Using this new Gårding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under $\mathcal{A}$-quasiconvexity assumptions. The application in statics improves existing results by proving uniqueness of $L^p$-local minimizers in the classical $\mathcal{A}={curl}$ case.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

SIAM Journal on Mathematical Analysis

ISSN

0036-1410

Publisher

Society for Industrial and Applied Mathematics

Issue

4

Volume

53

Page range

4178-4211

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2021-07-15

First Open Access (FOA) Date

2021-07-15

First Compliant Deposit (FCD) Date

2021-07-15

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