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Mathematics > Analysis of PDEs
Title: Maximum principles for weakly $1$-coercive operators with applications to capillary and prescribed mean curvature graphs
(Submitted on 22 Jan 2024 (v1), last revised 14 May 2024 (this version, v3))
Abstract: In this paper we establish maximum principles for weakly 1-coercive operators $L$ on complete, non-compact Riemannian manifolds $M$. In particular, we search for conditions under which one can guarantee that solutions $u$ of differential equations of the form $L(u)\geq f(u)$ satisfy $f(u)\leq 0$ on $M$. The case of weakly $p$-coercive operators with $p>1$, including the $p$-Laplacian and in particular the Laplace-Beltrami operator for $p=2$, has been considered in a recent paper of ours. As a consequence of the main results we infer comparison principles for that kind of operators. Furthermore we apply them to geometric situations dealing with the mean curvature operator, which is a typical weakly 1-coercive operator. We first consider the case of $\mathcal C^1$ operators $L$ acting on functions $u$ of class $\mathcal C^2$ and, in the last section of the paper, we show how our results can be extended to the case of less regular operators $L$ acting on functions $u$ which are just continuous and locally $W^{1,1}$ regular.
Submission history
From: Giulio Colombo [view email][v1] Mon, 22 Jan 2024 17:43:29 GMT (28kb)
[v2] Wed, 31 Jan 2024 11:24:33 GMT (24kb)
[v3] Tue, 14 May 2024 17:30:16 GMT (24kb)
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