Capillary Even Orlicz-Minkowski Problem

• The Capillary Even Orlicz-Minkowski Problem is a geometric boundary problem in the Euclidean half‐space that uses Robin conditions to prescribe capillary Orlicz surface area measures.
• It employs capillary support functions, the capillary Gauss map, and a Monge–Ampère type equation, with horizontal-reflection symmetry replacing full antipodality.
• Existence proofs leverage continuity methods and curvature flows with a priori C0–C2 estimates, establishing smooth symmetric solutions and rigidity under equality conditions.

The capillary even Orlicz-Minkowski problem is a Robin boundary analogue of the classical even Orlicz-Minkowski problem in the Euclidean upper half-space. In its smooth half-space formulation, one seeks a symmetric capillary convex body with prescribed capillary Orlicz surface area measure on a spherical cap, rather than a centrally symmetric convex body with prescribed Orlicz surface area measure on the full sphere. The modern theory is organized around capillary support functions, the capillary Gauss map, a Monge–Ampère equation on the spherical cap with Robin boundary condition, and a symmetry notion adapted to horizontal reflection. Direct volume-normalized smooth existence for even data was established in 2025, and a curvature-flow approach subsequently produced smooth existence without the evenness assumption (Wang et al., 13 Sep 2025, Li et al., 21 Jan 2026).

1. Geometric setting and capillary symmetry

The ambient space is the upper Euclidean half-space

$$\mathbb{R}^{n+1}_+ = \{x\in \mathbb{R}^{n+1}: x_{n+1}>0\},$$

with boundary hyperplane $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$ and

$e=(0,\dots,0,-1).$

A $C^2$ hypersurface $\Sigma\subset \mathbb{R}^{n+1}_+$ with $$\partial \Sigma \subset \partial \mathbb{R}^{n+1}_+$$ is capillary if it meets the boundary hyperplane at a constant contact angle $\theta\in(0,\pi)$, meaning

$$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$

where $\nu$ is the outward unit normal. If $\Sigma$ is smooth, strictly convex, and has positive curvature, the enclosed domain $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$0 together with its flat boundary portion is a capillary convex body; the class of such bodies is denoted $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$1, and $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$2 denotes the subclass whose flat boundary contains the origin in its interior (Wang et al., 13 Sep 2025).

The capillary support-function calculus is built on the spherical cap

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$3

especially the unit cap $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$4. For a smooth strictly convex capillary hypersurface, the capillary Gauss map

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$5

is a diffeomorphism. The support function is therefore defined on $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$6 by

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$7

For the model cap $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$8, its support function is

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$9

The associated capillary support function is

$e=(0,\dots,0,-1).$0

The capillary geometry forces a Robin boundary condition on $e=(0,\dots,0,-1).$1 and an equivalent Neumann condition on $e=(0,\dots,0,-1).$2: $e=(0,\dots,0,-1).$3

$e=(0,\dots,0,-1).$4

where $e=(0,\dots,0,-1).$5 is the outward co-normal of $e=(0,\dots,0,-1).$6 in $e=(0,\dots,0,-1).$7. This boundary condition is the analytic signature of the capillary problem.

The “even” structure is not the antipodal symmetry of the classical theory. For $e=(0,\dots,0,-1).$8, define

$e=(0,\dots,0,-1).$9

A function $C^2$0 on $C^2$1 is even if

$C^2$2

and a capillary convex body is symmetric if its support function is even in this sense. This is the capillary analogue of origin-symmetry in the classical even Minkowski problem, but in half-space geometry the symmetry is horizontal reflection rather than full antipodal symmetry on $C^2$3. The same horizontal-reflection symmetry also appears in the capillary Christoffel-Minkowski problem, where it is imposed as

$C^2$4

for data on $C^2$5 (Hu et al., 12 Apr 2025).

2. Prescribed measure formulation

The capillary even Orlicz-Minkowski problem is formulated by prescribing a capillary Orlicz surface area measure. First, the capillary surface area measure is

$C^2$6

For an admissible Orlicz function $C^2$7, the capillary Orlicz surface area measure is defined by

$C^2$8

for Borel sets $C^2$9. Equivalently,

$\Sigma\subset \mathbb{R}^{n+1}_+$0

The capillary Orlicz-Minkowski problem asks, for a positive smooth function $\Sigma\subset \mathbb{R}^{n+1}_+$1 on $\Sigma\subset \mathbb{R}^{n+1}_+$2, to find $\Sigma\subset \mathbb{R}^{n+1}_+$3 such that

$\Sigma\subset \mathbb{R}^{n+1}_+$4

In support-function variables, this becomes the Monge–Ampère type equation with Robin boundary condition

$\Sigma\subset \mathbb{R}^{n+1}_+$5

Using $\Sigma\subset \mathbb{R}^{n+1}_+$6, the equation can be rewritten as

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

The natural normalization in the direct smooth theory is volume normalization,

$\Sigma\subset \mathbb{R}^{n+1}_+$8

The admissible Orlicz class used in the smooth capillary-even theorem is denoted $\Sigma\subset \mathbb{R}^{n+1}_+$9. It consists of $$\partial \Sigma \subset \partial \mathbb{R}^{n+1}_+$$0 that are $$\partial \Sigma \subset \partial \mathbb{R}^{n+1}_+$$1, strictly increasing, convex, log-concave, and normalized by

$$\partial \Sigma \subset \partial \mathbb{R}^{n+1}_+$$2

together with the conditions

$$\partial \Sigma \subset \partial \mathbb{R}^{n+1}_+$$3

$$\partial \Sigma \subset \partial \mathbb{R}^{n+1}_+$$4

$$\partial \Sigma \subset \partial \mathbb{R}^{n+1}_+$$5

The monotonicity in $$\partial \Sigma \subset \partial \mathbb{R}^{n+1}_+$$6 implies

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

The direct theorem also uses an orthogonality condition. For $$\partial \Sigma \subset \partial \mathbb{R}^{n+1}_+$$8, $$\partial \Sigma \subset \partial \mathbb{R}^{n+1}_+$$9 satisfies the orthogonality condition with respect to $\theta\in(0,\pi)$0 if whenever $\theta\in(0,\pi)$1 satisfies

$\theta\in(0,\pi)$2

then

$\theta\in(0,\pi)$3

In the $\theta\in(0,\pi)$4 case $\theta\in(0,\pi)$5, this condition is trivial (Wang et al., 13 Sep 2025).

A frequent misconception is to treat the capillary problem as a formal restriction of the classical Orlicz-Minkowski problem from $\theta\in(0,\pi)$6 to a subset. The direct capillary theory shows otherwise: the target of the capillary Gauss map is $\theta\in(0,\pi)$7, the support function is defined relative to $\theta\in(0,\pi)$8, and the contact-angle constraint becomes a Robin boundary condition. The geometric weight $\theta\in(0,\pi)$9 is an intrinsic capillary feature rather than a removable normalization artifact.

3. Existence results and historical development

The direct capillary-even Orlicz theorem states that if $$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$0, $$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$1, and $$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$2 is positive and even, satisfying

$$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$3

then there exists a smooth, symmetric capillary convex body

$$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$4

with

$$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$5

such that its support function $$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$6 solves

$$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$7

and $$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$8 satisfies the orthogonality condition with respect to $$\cos(\pi-\theta)=\langle \nu,e\rangle \qquad \text{along } \partial \Sigma,$$9. Moreover, if equality holds in the integral condition and $\nu$0 is strictly convex, then the hypersurface must be the spherical cap

$\nu$1

(Wang et al., 13 Sep 2025).

This result sits in a sequence of capillary Minkowski-type developments. The capillary Christoffel-Minkowski problem for principal radii $\nu$2 established that, for $\nu$3 and $\nu$4, if $\nu$5 is spherically convex on $\nu$6 and the data have horizontal even symmetry, then there exists a smooth strictly convex capillary hypersurface solving

$\nu$7

and the solution is unique within the symmetric class (Hu et al., 12 Apr 2025). The capillary $\nu$8 flow theory then provided a flow approach to the capillary even $\nu$9 Minkowski problem in the Euclidean half-space for all

$\Sigma$0

and to the capillary $\Sigma$1 Minkowski problem for

$\Sigma$2

(Hu et al., 7 Sep 2025). The 2025 Orlicz theorem may therefore be read as the first smooth half-space existence theorem in the genuinely Orlicz setting for even data, while the 2026 capillary Orlicz-Minkowski flow established a new existence result without the evenness assumption and produced a flow approach to smooth solutions (Li et al., 21 Jan 2026).

The $\Sigma$3 specialization is explicitly built into the Orlicz theorem. When

$\Sigma$4

the assumptions $\Sigma$5–$\Sigma$6 hold, and the theorem recovers the capillary even $\Sigma$7-Minkowski problem in volume-normalized form. In particular, for the volume-normalized capillary $\Sigma$8-Minkowski problem, the normalization constant becomes $\Sigma$9 (Wang et al., 13 Sep 2025).

A plausible implication is that the capillary even Orlicz-Minkowski problem occupies the same structural position in half-space geometry that the even Orlicz-Minkowski problem occupies in the classical closed setting: the direct smooth theorem identifies the correct support-function equation, the correct symmetry class, and the correct normalization, while later flow results broaden existence beyond symmetry assumptions.

4. Analytical methods

The direct smooth existence proof is by the continuity method. The interpolating family is

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$00

and the PDE family is

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$01

At $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$02, the obvious solution is

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$03

Let $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$04 be the set of $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$05 for which there exists a positive even solution in the function space $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$06, where $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$07 encodes the Robin boundary condition and the orthogonality condition (Wang et al., 13 Sep 2025).

Closedness is based on capillary-specific a priori estimates. The first is a $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$08 estimate for positive capillary even convex solutions of the normalized equation,

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$09

The upper bound uses evenness to conclude that the Steiner point is at the origin horizontally and then compares with the minimal enclosing spherical cap. The lower bound is obtained in the $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$10 formulation from the PDE at a minimum point and the growth condition $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$11. The gradient estimate is

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$12

proved with the auxiliary function

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$13

where $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$14 is the spherical distance to $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$15. The Neumann condition $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$16 is used to rule out boundary maxima.

The $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$17 theory first reduces interior second derivatives to the boundary double-normal derivative,

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$18

and then proves

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$19

The boundary estimate uses a barrier built from

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$20

with

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$21

and auxiliary functions such as

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$22

The sign conditions required in the second-derivative estimate use the log-concavity of $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$23 and the monotonicity assumption $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$24. Once $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$25 bounds are available, standard theory for fully nonlinear elliptic equations with oblique boundary conditions yields $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$26 estimates and bootstrap to higher regularity (Wang et al., 13 Sep 2025).

Openness is handled by linearization and the implicit function theorem. The nonlinear operator is

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$27

with linearization

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$28

where $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$29. The orthogonality condition is tailored so that the kernel is trivial in the relevant quotient, which gives surjectivity and hence openness.

The flow-based method is different in character. The capillary Orlicz-Minkowski flow in the upper half-space studies an anisotropic capillary Gauss curvature flow, proves long-time existence and asymptotic behavior, and establishes convergence to a stationary solution. Its main contribution is a smooth existence theorem without imposing evenness or symmetry assumptions on the data (Li et al., 21 Jan 2026). In the $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$30 case, the capillary $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$31 Minkowski flows introduced normalized anisotropic capillary Gauss curvature flows, monotone entropy functionals, and $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$32-to-$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$33 estimates for the capillary even $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$34 Minkowski problem, furnishing a direct prototype for a future capillary even Orlicz flow theory (Hu et al., 7 Sep 2025).

A common misconception is that the continuity method and the flow method are interchangeable technical packages. In the current literature they serve different roles. The continuity method provides the direct volume-normalized smooth theorem in the even Orlicz setting, while the flow method first appeared in the capillary $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$35 theory and then reached the capillary Orlicz problem in the non-even setting.

5. Inequalities, rigidity, and uniqueness

The capillary Orlicz theory includes a capillary Orlicz combination. For $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$36, $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$37, $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$38, and $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$39,

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$40

A key point is that this support function again satisfies the Robin condition, so the Orlicz combination remains a capillary convex body. The proof uses the monotonicity consequence of $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$41,

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$42

The capillary Orlicz mixed volume is

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$43

The capillary Orlicz-Minkowski inequality is

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$44

and the capillary Orlicz-Brunn-Minkowski inequality is

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$45

Equality holds for dilates, and if $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$46 is strictly convex then equality holds only for dilates (Wang et al., 13 Sep 2025).

These inequalities yield the rigidity statement in the direct theorem. If $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$47 and $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$48 solves the capillary Orlicz-Minkowski equation, then

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$49

By the capillary Orlicz-Minkowski inequality,

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$50

which explains the admissibility condition

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$51

If equality holds and $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$52 is strictly convex, equality in the inequality forces $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$53 to be a dilate of $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$54, and volume normalization fixes the dilate uniquely as

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$55

General uniqueness for the capillary even Orlicz-Minkowski problem is more limited than the rigidity statement above. The direct theorem gives existence plus rigidity to the spherical cap under the equality condition, but it does not state a blanket uniqueness theorem for arbitrary even data. By contrast, uniqueness mechanisms are well developed in nearby capillary problems. The generalized Hsiung-Minkowski formula and related rigidity theory for anisotropic capillary hypersurfaces yield uniqueness for the anisotropic Orlicz-Christoffel-Minkowski problem and provide a new proof of uniqueness for the $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$56-Minkowski problem with $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$57 in Euclidean capillary convex bodies geometry (Gao et al., 2024). In the capillary Christoffel-Minkowski problem, the symmetric solution is unique within the symmetric class, while in the general case uniqueness holds up to horizontal translation (Hu et al., 12 Apr 2025).

This suggests, but does not prove, that a broader uniqueness theory for capillary Orlicz-Minkowski problems may require either stronger integral identities of Christoffel-Minkowski type or a more refined variational characterization. At present, the conditional rigidity result through the capillary Orlicz-Minkowski inequality is the clearest general uniqueness substitute in the direct even theory.

6. Relation to classical, $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$58, Christoffel-Minkowski, and capacitary theories

The capillary even Orlicz-Minkowski problem belongs to the classical Orlicz-Minkowski lineage, but its half-space geometry fundamentally changes the analytic and geometric objects. In the closed Euclidean setting, Orlicz-Minkowski flows study smooth, strictly convex closed hypersurfaces in $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$59, with support functions on $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$60, and solve regular even Orlicz-Minkowski equations of the form

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$61

Those flows provide long-time existence, smooth strict convexity, and $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$62-subconvergence in the regular even problem, but there is no capillary/contact-angle formulation and no boundary term (Bryan et al., 2020). In the capillary theory, the domain is $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$63, the support function carries a Robin condition, and the symmetry class is horizontal reflection, not antipodal symmetry.

The half-space $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$64 and Christoffel-Minkowski theories supply the immediate precursors of the Orlicz problem. The capillary $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$65-Minkowski problem for $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$66 reduces to

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$67

with full smooth existence and uniqueness for $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$68, existence and uniqueness up to dilation at $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$69, and capillary-even existence for $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$70 when $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$71 is capillary even. That theory also identifies the capillary $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$72-surface area measure

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$73

as the exact power-law template for the Orlicz measure (Mei et al., 12 May 2025). The capillary Christoffel-Minkowski problem, by contrast, prescribes $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$74 of the principal radii on $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$75 with the same Robin boundary condition, and develops the spherical-convexity, centering, and translation-normalization mechanisms later reused by Orlicz-type theories (Hu et al., 12 Apr 2025, Mei et al., 18 Dec 2025).

The capillary problem should also be distinguished from the capacitary Orlicz-Minkowski literature. The $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$76-capacitary Orlicz-Minkowski problem, developed through $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$77-capacitary measures $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$78, Orlicz mixed $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$79-capacities, and equations of the form

$\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$80

belongs to the Brunn-Minkowski theory of $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$81-capacity rather than to half-space capillarity. That theory is structurally relevant because it clarifies first-variation principles and “even” specialization in non-capillary settings, but it concerns capacitary rather than capillary data (Hong et al., 2017, Luo et al., 2018). Likewise, the general dual Orlicz curvature measure and its even-origin-symmetric existence theorems address dual Orlicz-Minkowski problems in $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$82, not capillary free-boundary geometry (Gardner et al., 2018).

A persistent terminological confusion arises precisely here. “Capillary” refers to hypersurfaces in a half-space meeting a boundary at a prescribed contact angle; “capacitary” refers to $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$83-capacity and $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$84-capacitary measures. The capillary even Orlicz-Minkowski problem is part of the former theory, even though both literatures use Orlicz perturbations, Minkowski-type prescription problems, and symmetry assumptions.

From the present body of work, three structural conclusions are clear. First, the capillary even Orlicz-Minkowski problem has a well-defined smooth half-space formulation, with capillary convex bodies, capillary support functions, and prescribed capillary Orlicz surface area measure (Wang et al., 13 Sep 2025). Second, the correct “even” class is horizontal-reflection symmetry on $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$85, not antipodal symmetry on $\partial \mathbb{R}^{n+1}_+=\{x_{n+1}=0\}$86. Third, the direct existence theory is already broader than the classical even-only paradigm, because capillary Orlicz-Minkowski flow subsequently produced smooth existence without the evenness assumption (Li et al., 21 Jan 2026). A plausible implication is that the even problem now functions as the symmetry-normalized core of a larger capillary Orlicz-Minkowski program, in the same way that the classical even theory historically preceded more general nonsymmetric formulations.