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arXiv · 2208.07456

The functional dissipativity of certain systems of partial differential equations

Alberto Cialdea·Vladimir Maz'ya
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Abstract

In the present paper we consider the functional dissipativity of the Dirichlet problem for systems of partial differential operators of the form $\partial_{h} ({\mathop{\mathscr A}\nolimits}^{hk}(x)\partial_{k})$ (${\mathop{\mathscr A}\nolimits}^{hk}$ being $m\times m$ matrices with complex valued $L^{1}_{\text{loc}}$ entries). In the particular case of the operator $\partial_{h} ({\mathop{\mathscr A}\nolimits}^{h}(x)\partial_{h})$ (where ${\mathop{\mathscr A}\nolimits}^{h}$ are $m\times m$ matrices) we obtain algebraic necessary and sufficient conditions. We give also three different notions of functional ellipticity and investigate the relations between them and the functional dissipativity for the operators in question.

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Alberto Cialdea, Vladimir Maz'ya. 2022-08-15. The functional dissipativity of certain systems of partial differential equations. https://arxiv.org/abs/2208.07456

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