Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems. (English) Zbl 1480.35075
Summary: In this note, we study the mean curvature flow and the prescribed mean curvature type equation with general capillary-type boundary condition, which is \(u_{\nu} = -\phi (x)(1+|Du|^2)^{\frac{1-q}{2}}\) for any parameter \(q>0\). Using the maximum principle, we prove the gradient estimates for the solutions of such a class of boundary value problems. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations. In addition, we study the related additive eigenvalue problem for general boundary value problems and describe the asymptotic behavior of the solution at infinity time. The originality of the paper lies in the range \(0<q<1\), since there are no any related results before. For parabolic case, we generalize the result of X.-N. Ma, P.-H. Wang and the second author [J. Funct. Anal. 274, No. 1, 252–277 (2018; Zbl 1376.53087)] to any \(q>0\). And in elliptic case, we generalize the results in [the third author, Commun. Pure Appl. Anal. 15, No. 5, 1719–1742 (2016; Zbl 1348.35043)] to any \(q\geq 0\) and to any bounded smooth domain.
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 35B50 | Maximum principles in context of PDEs |
| 35J93 | Quasilinear elliptic equations with mean curvature operator |
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
| 35K93 | Quasilinear parabolic equations with mean curvature operator |
Keywords:
capillary-type boundary value; gradient bound; mean curvature flow; additive eigenvalue problem; maximum principleReferences:
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