Capillary $L_p$-curvature problem

1. Problem Statement

The paper addresses the Capillary $L^p$-Curvature Problem in the half-space $\mathbb{R}^{n+1}_+$. The goal is to establish the existence of a smooth, strictly convex, even capillary hypersurface $\Sigma \subset \mathbb{R}^{n+1}_+$ that satisfies a specific curvature equation with a prescribed contact angle $\theta$ with the boundary plane $\partial \mathbb{R}^{n+1}_+$.

The Equation:
The hypersurface is defined by the condition that its $k$-th elementary symmetric function of the principal curvatures, denoted $S_k$, satisfies:
$$\phi s^{p-1} S_k \circ \tilde{\nu}^{-1} = 1$$
where:

• $\phi \in C^\infty(C_\theta)$ is a given even, positive function defined on the spherical cap $C_\theta$.
• $s$ is the capillary support function.
• $\tilde{\nu}$ is the capillary Gauss map.
• $p$ is a real parameter, and the paper focuses on the range $1 < p < k + 1$.
• $\theta \in (0, \pi/2)$ is the contact angle.

Context:

• The case $p=1$ was solved by Guan and Guan [GG02].
• The case $p \ge k+1$ was solved by various authors using elliptic and parabolic methods.
• The range $1 < p < k + 1$ remained open for the general contact angle case. Previous attempts (e.g., [MWW25e]) established existence but required an additional restriction on the contact angle $\theta$ that depended on $p$.

2. Methodology

The core of the paper is the derivation of a new gradient estimate for the capillary support function $s$. This estimate is crucial for obtaining $C^0$ and $C^2$ bounds, which are necessary to apply degree-theoretic existence arguments.

Key Technical Innovation: The Auxiliary Function

The authors identify that standard gradient estimates used in closed settings or previous capillary works are insufficient for the range $1 < p < k+1$ without imposing angle restrictions.

• Previous Approach: Used the auxiliary function $\Phi = |\nabla u|^2 / u^\gamma$ (where $u = s/\ell$ and $\ell$ is the support function of the model cap). This led to a contradiction only if $\theta$ satisfied a specific $p$-dependent bound.
• New Approach: The authors introduce a weighted auxiliary function:
  $$\Psi = \ell^{2-\gamma} \frac{|\nabla u|^2}{u^\gamma}$$
  where $\ell(\zeta) = \sin^2 \theta - \cos \theta \langle \zeta, e_{n+1} \rangle$ is the support function of the model cap $C_\theta$.
  • Significance: The weight $\ell^{2-\gamma}$ is the critical modification. When $\theta = \pi/2$, $\ell \equiv 1$, and the function reduces to the standard form. However, for general $\theta$, this weight cancels out the terms that previously forced the angle restriction, allowing the gradient estimate to hold for all $\theta \in (0, \pi/2)$.

Proof Strategy (Lemma 3.1)

1. Maximum Principle: Assume $\Psi$ attains a maximum at a point $\xi_0$.
2. Boundary Case ($\xi_0 \in \partial C_\theta$): The authors show that the boundary condition $\nabla_\mu s = \cot \theta s$ leads to a contradiction ($0 < 0$) if the maximum occurs on the boundary, provided $\gamma$ is chosen correctly.
3. Interior Case ($\xi_0 \in \text{int}(C_\theta)$):
   • Compute the first and second derivatives of $\log \Psi$ at the maximum.
   • Utilize the curvature equation $F(\tau^\sharp[s]) = s^{p-1}\phi$ and the Codazzi property of the Hessian.
   • Through careful algebraic manipulation involving the homogeneity of the curvature function $F$ and the specific choice of $\gamma \in (0, \frac{2(p-1)}{k})$, they derive a contradiction for sufficiently large gradient values.
   • This establishes the bound: $|\nabla s|^2 s^{-\gamma} \le C_0 (\max s)^{2-\gamma}$.

3. Key Contributions

1. Removal of Angle Restriction: The primary contribution is proving the existence of solutions for the range $1 < p < k+1$ for all contact angles $\theta \in (0, \pi/2)$. This resolves the limitation of previous works that required $\theta$ to be small or dependent on $p$.
2. Generalized Gradient Estimate: The new gradient estimate relies only on the $k$-homogeneity of the curvature function $F$, not on specific structural properties of $S_k$. This makes the method applicable to a broader class of problems, including:
   • Capillary quotient curvature problems.
   • The Capillary $L^p$-Christoffel–Minkowski problem.
3. Direct $C^0$ and $C^2$ Bounds: The gradient estimate, combined with a geometric lemma (Lemma 3.2), directly implies two-sided bounds for the support function $s$ (i.e., $C^{-1} \le s \le C$) and uniform $C^2$ estimates, bypassing the need for separate, complex arguments for these bounds.

4. Main Results

Theorem 1.1 (Existence):
Let $\theta \in (0, \pi/2)$, $1 < p < k + 1$, and $1 \le k < n$. If $\phi \in C^\infty(C_\theta)$ is an even, positive function, then there exists an even, smooth, strictly convex capillary hypersurface $\Sigma \subset \mathbb{R}^{n+1}_+$ satisfying the curvature equation:
$$\phi s^{p-1} S_k \circ \tilde{\nu}^{-1} = 1$$
The solution is unique up to the degree-theoretic framework referenced.

Theorem 3.3 (A Priori Bounds):
Under the same conditions, any even, smooth, strictly convex solution $s$ satisfies uniform bounds:
$$C_1^{-1} \le s \le C_1$$
where $C_1$ depends only on $\theta, k, p, \gamma$, and $\phi$.

5. Significance and Impact

• Completeness of Theory: This work completes the existence theory for the capillary $L^p$-curvature problem across the full range of $p$ (specifically resolving the difficult $1 < p < k+1$ case) without artificial geometric constraints on the contact angle.
• Methodological Advancement: The introduction of the weighted auxiliary function $\Psi$ provides a robust tool for handling boundary terms in capillary problems. It demonstrates that the "angle obstruction" in gradient estimates is an artifact of the test function choice rather than an intrinsic geometric limitation.
• Broad Applicability: The technique is not limited to $S_k$ curvature but extends to other curvature functions and related problems like the Christoffel–Minkowski problem, offering a unified approach to capillary geometric analysis.
• Regularity: By establishing the necessary $C^0$ and $C^2$ estimates, the paper paves the way for higher-order regularity (via Lieberman's theory) and the application of degree theory to prove existence and uniqueness.

In summary, Hu and Ivaki have solved a long-standing open problem in geometric analysis by developing a novel gradient estimate that eliminates previous restrictions on contact angles, thereby unifying the existence theory for capillary curvature equations in the half-space.