Capillary Orlicz-Minkowski flow in the upper half-space

Abstract: In this paper, we study the long-time existence and asymptotic behavior of an anisotropic capillary Gauss curvature flow. By studying this flow and proving its convergence to a stationary solution, we establish a new existence result for the capillary Orlicz-Minkowski problem without the evenness assumption, and provide a flow approach to the existence of smooth solutions.

• The paper proves long-time existence and smooth subsequential convergence for the capillary Orlicz–Minkowski flow for any smooth strictly convex $f$. It introduces a boundary value problem under the capillary Orlicz-Minkowski equation, extending evenness results to the non-even case.
• The flow method ensures smooth long-time existence and a subsequence convergence of the strictly convex capillary boundary condition, skipping the evenness assumption.
• For $n = 2$ and $p = 1$, the capillary Minkowski problem can be solved without any evenness or symmetry assumption on $f$

Background and problem statement

The classical Minkowski problem asks for a closed convex hypersurface in $\mathbb{R}^{n+1}$ whose Gauss curvature equals a prescribed positive function $f$ on $\mathbb{S}^n$, equivalently the Monge–Ampère equation $\det(\nabla^2 h + h\sigma) = f^{-1}$ for the support function $h$. Its generalizations — the $L_p$-Minkowski problem of Lutwak and the Orlicz–Minkowski problem of Haberl, Lutwak, Yang, and Zhang (2601.14659) — have been studied extensively via both variational methods and geometric flows.

The capillary setting concerns convex hypersurfaces $\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$ meeting the boundary plane at a constant contact angle $\theta \in (0,\pi)$. Mei, Wang, and Weng introduced the capillary Minkowski problem and solved it for $\theta \in (0,\pi/2]$ using the continuity method; subsequent work addressed even and non-even cases of the capillary $L_p$-Minkowski problem, with Hu–Hu–Ivaki resolving the even case for all $f$0 and the non-even case for $f$1 via geometric flows. Independently, Wang and Zhu formulated a capillary Orlicz–Minkowski problem as a Robin boundary value problem of Monge–Ampère type, but their existence theorem requires an even prescribed function $f$2. The paper under review supplies the corresponding non-even existence result by flow methods.

Main results

Let $f$3 denote the unit capillary spherical cap, $f$4 its capillary support function, and $f$5 the capillary Gauss map. The paper studies the boundary value problem

$f$6

where $f$7 is smooth. Taking $f$8 recovers the capillary $f$9-Minkowski equation. The main analytic assumption is

$\mathbb{S}^n$0

the same condition used by Bryan–Ivaki–Scheuer and Liu–Lu for the classical (dual) Orlicz–Minkowski problems. Notably, this condition holds when $\mathbb{S}^n$1 with $\mathbb{S}^n$2, so the theorem subsumes the non-even capillary $\mathbb{S}^n$3 result of Hu–Hu–Ivaki in that range.

Theorem A (existence). For $\mathbb{S}^n$4, any positive $\mathbb{S}^n$5 satisfying the above condition, there exists a smooth strictly convex solution to the capillary Orlicz–Minkowski equation — without any evenness or symmetry assumption on $\mathbb{S}^n$6.

Theorem B (flow convergence). Starting from any smooth strictly convex capillary hypersurface with positive capillary support function, the anisotropic capillary Gauss curvature flow

$\mathbb{S}^n$7

with the capillary boundary condition $\mathbb{S}^n$8, exists smoothly for all time, and a subsequence converges in $\mathbb{S}^n$9 to a smooth strictly convex stationary solution of the above equation.

Method: a priori estimates along the flow

Under the inverse capillary Gauss map parametrization, the flow becomes a parabolic scalar equation for $\det(\nabla^2 h + h\sigma) = f^{-1}$0 with Robin boundary data:

$\det(\nabla^2 h + h\sigma) = f^{-1}$1

The long-time existence argument proceeds through uniform estimates:

• $\det(\nabla^2 h + h\sigma) = f^{-1}$2 bound: applied to $\det(\nabla^2 h + h\sigma) = f^{-1}$3, which satisfies an oblique Neumann-type condition $\det(\nabla^2 h + h\sigma) = f^{-1}$4 on $\det(\nabla^2 h + h\sigma) = f^{-1}$5. At interior minima of $\det(\nabla^2 h + h\sigma) = f^{-1}$6, strict convexity gives $\det(\nabla^2 h + h\sigma) = f^{-1}$7, so the evolution inequality $\det(\nabla^2 h + h\sigma) = f^{-1}$8 combined with the asymptotic condition on $\det(\nabla^2 h + h\sigma) = f^{-1}$9 pins $h$0 between time-independent constants. This is where the structural hypothesis on $h$1 enters decisively.
• $h$2 bound: the auxiliary function $h$3 is controlled by a maximum principle argument; at boundary maxima the relation $h$4 reduces the estimate to the $h$5 bound times $h$6.
• Gauss curvature bounds: an upper bound follows from the auxiliary quantity $h$7, whose evolution satisfies $h$8; the negative $h$9 term forces $L_p$0 bounded above. The lower bound uses $L_p$1 with large $L_p$2; a key claim verified at the boundary is $L_p$3, which relies on the identity $L_p$4 together with $L_p$5.
• Principal curvature bounds: with $L_p$6 bounded both ways, an upper bound on principal radii follows from $L_p$7 for large $L_p$8. The delicate part is the boundary analysis: two cases are distinguished depending on whether $L_p$9 dominates $\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$0, and in the second case either $\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$1 or a direct time-independent bound is obtained from the $\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$2 estimate. This circumvents the extra difficulty that $\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$3 estimates typically pose at capillary boundaries.

With $\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$4 uniformly bounded, standard parabolic regularity theory for fully nonlinear equations with oblique boundary conditions yields $\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$5 bounds for all $\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$6, hence long-time existence.

Convergence via a monotone functional

Convergence is obtained through the functional

$\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$7

where $\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$8 denotes enclosed volume. Using the first variation formula $\Sigma \subset \overline{\mathbb{R}_+^{n+1}}$9 for capillary convex bodies, one computes

$\theta \in (0,\pi)$0

with equality exactly at stationary solutions. Monotonicity plus the uniform estimates imply that along a sequence $\theta \in (0,\pi)$1 the limit $\theta \in (0,\pi)$2 exists and satisfies $\theta \in (0,\pi)$3, hence $\theta \in (0,\pi)$4 pointwise. The boundary condition is inherited from the flow. This establishes both theorems simultaneously.

Relation to prior work and scope of assumptions

Compared with Wang–Zhu's elliptic approach, which requires $\theta \in (0,\pi)$5 even and $\theta \in (0,\pi)$6 convex, increasing, log-concave with $\theta \in (0,\pi)$7, the present result imposes no symmetry on $\theta \in (0,\pi)$8; instead it requires the asymptotic condition linking $\theta \in (0,\pi)$9 to $\theta \in (0,\pi/2]$0. The authors note these two sets of hypotheses are independent, so neither result contains the other. The restriction $\theta \in (0,\pi/2]$1 matches the range in most of the capillary literature but leaves open whether the flow converges for obtuse contact angles $\theta \in (0,\pi/2]$2. Convergence is established only along a subsequence rather than for the full family $\theta \in (0,\pi/2]$3 as $\theta \in (0,\pi/2]$4; uniqueness of the limiting solution and full-time convergence are not addressed. The asymptotic condition also excludes some admissible $\theta \in (0,\pi/2]$5 (e.g., certain growth regimes near $\theta \in (0,\pi/2]$6 or $\theta \in (0,\pi/2]$7), and extending existence beyond it remains open.

Conclusion

This paper extends the flow approach to Minkowski-type problems to the capillary Orlicz setting. By establishing uniform $\theta \in (0,\pi/2]$8, $\theta \in (0,\pi/2]$9, Gauss curvature, and principal curvature estimates for an anisotropic capillary Gauss curvature flow, and by identifying a volume-normalized Lyapunov functional decreasing along the flow, the authors prove smooth long-time existence and subsequential convergence, thereby yielding the first existence theorem for the capillary Orlicz–Minkowski problem without evenness assumptions. The result covers the non-even capillary $L_p$0-Minkowski problem for $L_p$1 as a special case and complements the elliptic, symmetric theory of Wang–Zhu.