Yau and Souplet-Zhang type gradient estimates on Riemannian manifolds with boundary under Dirichlet boundary condition

Abstract

In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation.

Gradient estimates for a nonlinear parabolic equation with Dirichlet boundary condition

“…Remark 1.2. When a = 0 and L = 0, the theorem returns to linear cases [9] and [6]. Notice that our weighted mean curvature assumption here could be bounded below by a nonnegative constant (not just non-negative) and hence it indicates that our result is suitable to a little more general setting.…”

“…In [24], Souplet and Zhang generalized Yau's gradient estimate to the heat equation by adding a necessary logarithmic correction term. Recently, Kunikawa and Sakurai [9] extended Yau and Souplet-Zhang type gradient estimates to the case of manifolds with the boundary under some Dirichlet boundary condition. Shortly later, H. Dung, N. Dung and Wu [6] generalized Kunikawa-Sakurai results to the f -Laplacian equation and the f -heat equation on smooth metric measure spaces with the compact boundary under some Dirichlet boundary condition; see also N. Dung and Wu [7] for further generalizations in this direction.…”

“…In Theorem 1.1, Ric f ≥ −(n − 1)K means that the infimum of Ric f on the unit tangent bundle in the interior of M is more than −(n − 1)K, while H f ≥ −L means that boundary ∂M has some weak convex property. In previous works, Kunikawa and Sakurai [9] proved Souplet-Zhang type gradient estimates for the heat equations under some Dirichlet boundary condition; H. Dung, N. Dung and Wu [6] generalized their result to the f -heat equation; N. Dung and Wu [7] further studied the case of the f -heat equation. Now we further generalize these linear cases to a nonlinear parabolic equation.…”

“…The proof of Theorem 1.1 employs the arguments of [9], [6] and [29]. The outline of the proof is as follows.…”

“…In the interior of the manifold, as in [29], we essentially use the standard Souplet-Zhang argument for the nonlinear parabolic equation. On the boundary of the manifold, adapting arguments of [9] and [6], we apply a derivative equality (see Proposition 2.3 in Section 2) to prove the desired estimate. Compared with the previous proof, here we need to carefully deal with an extra nonlinear term.…”