Maximum principles for weakly $1$-coercive operators with applications to capillary and prescribed mean curvature graphs

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 $L(u)\ge f(u)$ satisfy $f(u)\le 0$ 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.