Maximum principles for weakly $1$-coercive operators with applications to capillary and prescribed mean curvature graphs
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arXiv
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| Natura: | Preprint |
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2024
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| author | Alías, Luis J. Colombo, Giulio Rigoli, Marco |
| author_facet | Alías, Luis J. Colombo, Giulio Rigoli, Marco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12152 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximum principles for weakly $1$-coercive operators with applications to capillary and prescribed mean curvature graphs Alías, Luis J. Colombo, Giulio Rigoli, Marco Analysis of PDEs Differential Geometry 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. |
| title | Maximum principles for weakly $1$-coercive operators with applications to capillary and prescribed mean curvature graphs |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2401.12152 |