--- title: Capillary Even Orlicz-Minkowski Problem url: https://www.emergentmind.com/topics/capillary-even-orlicz-minkowski-problem type: topic --- # Capillary Even Orlicz-Minkowski Problem 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 [2509.10859], [2601.14659]. ## 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 \(\widehat{\Sigma}\) together with its flat boundary portion is a capillary convex body; the class of such bodies is denoted \(\mathcal K_\theta\), and \(\mathcal K_\theta^\circ\) denotes the subclass whose flat boundary contains the origin in its interior [2509.10859]. The capillary support-function calculus is built on the spherical cap \[ \mathcal C_{\theta,r}=\{\xi\in \overline{\mathbb R^{n+1}_+}: |\xi-r\cos\theta\, e|=r\}, \] especially the unit cap \(\mathcal C_\theta=\mathcal C_{\theta,1}\). For a smooth strictly convex capillary hypersurface, the capillary Gauss map \[ \widetilde{\nu}:\Sigma\to \mathcal C_\theta,\qquad X\mapsto \nu(X)+\cos\theta\, e \] is a diffeomorphism. The support function is therefore defined on \(\mathcal C_\theta\) by \[ h_{\widehat\Sigma}(\xi) = \big\langle \widetilde{\nu}^{-1}(\xi),\, \xi-\cos\theta\, e\big\rangle. \] For the model cap \(\mathcal C_\theta\), its support function is \[ \ell(\xi)=\langle \xi,\xi-\cos\theta e\rangle =\sin^2\theta+\cos\theta \langle \xi,e\rangle. \] The associated capillary support function is \[ u_{\widehat\Sigma}(\xi)=\frac{h_{\widehat\Sigma}(\xi)}{\ell(\xi)}. \] The capillary geometry forces a Robin boundary condition on \(h\) and an equivalent Neumann condition on \(u\): \[ \nabla_\mu h=\cot\theta\, h \qquad \text{on } \partial \mathcal C_\theta, \] \[ \nabla_\mu u=0 \qquad \text{on } \partial \mathcal C_\theta, \] where \(\mu\) is the outward co-normal of \(\partial \mathcal C_\theta\) in \(\mathcal C_\theta\). This boundary condition is the analytic signature of the capillary problem. The “even” structure is not the antipodal symmetry of the classical theory. For \(\xi=(\xi_1,\dots,\xi_n,\xi_{n+1})\in \mathcal C_\theta\), define \[ \widehat{\xi}=(-\xi_1,\dots,-\xi_n,\xi_{n+1}). \] A function \(f\) on \(\mathcal C_\theta\) is even if \[ f(\xi)=f(\widehat{\xi}), \] 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 \(\mathbb S^n\). The same horizontal-reflection symmetry also appears in the capillary Christoffel-Minkowski problem, where it is imposed as \[ \phi(-\xi_1,\dots,-\xi_n,\xi_{n+1})=\phi(\xi_1,\dots,\xi_n,\xi_{n+1}) \] for data on \(\mathcal C_\theta\) [2504.09320]. ## 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 \[ dS^c(\widehat\Sigma,\xi) = \ell(\xi)\det\big((h_{\widehat\Sigma})_{ij}+h_{\widehat\Sigma}\delta_{ij}\big)\,d\xi. \] For an admissible Orlicz function \(\phi\), the capillary Orlicz surface area measure is defined by \[ S^c_\phi(\widehat\Sigma,\omega) = \int_\omega \phi\!\left(\frac{\ell(\xi)}{h_{\widehat\Sigma}(\xi)}\right) h_{\widehat\Sigma}(\xi)\, \det\!\big((h_{\widehat\Sigma})_{ij}+h_{\widehat\Sigma}\delta_{ij}\big)\, d\xi \] for Borel sets \(\omega\subset \mathcal C_\theta\). Equivalently, \[ \frac{dS^c_\phi(\widehat\Sigma,\xi)}{d\xi} = \phi\!\left(\frac{\ell(\xi)}{h_{\widehat\Sigma}(\xi)}\right) h_{\widehat\Sigma}(\xi)\, \det\!\big((h_{\widehat\Sigma})_{ij}+h_{\widehat\Sigma}\delta_{ij}\big). \] The capillary Orlicz-Minkowski problem asks, for a positive smooth function \(f\) on \(\mathcal C_\theta\), to find \(\widehat\Sigma\in\mathcal K_\theta^\circ\) such that \[ \frac{dS^c_\phi(\widehat\Sigma,\xi)}{d\xi}=f(\xi),\qquad \xi\in \mathcal C_\theta. \] In support-function variables, this becomes the Monge–Ampère type equation with Robin boundary condition \[ \left\{ \begin{aligned} \phi\!\left(\frac{\ell}{h}\right) h\, \det(\nabla^2 h+h\delta_{\mathbb S^n}) &= f && \text{in } \mathcal C_\theta,\\ \nabla_\mu h &= \cot\theta\, h && \text{on } \partial \mathcal C_\theta. \end{aligned} \right. \] Using \(u=h/\ell\), the equation can be rewritten as \[ \left\{ \begin{aligned} \phi\!\left(\frac{1}{u}\right) u\, \det\!\big(\ell\nabla^2u+\cos\theta(\nabla u\otimes e^T+e^T\otimes \nabla u)+u\delta_{\mathbb S^n}\big) &=f &&\text{in } \mathcal C_\theta,\\ \nabla_\mu u&=0 &&\text{on } \partial \mathcal C_\theta. \end{aligned} \right. \] The natural normalization in the direct smooth theory is volume normalization, \[ |\widehat\Sigma|=1. \] The admissible Orlicz class used in the smooth capillary-even theorem is denoted \(\mathcal O\). It consists of \(\phi:[0,\infty)\to[0,\infty)\) that are \(C^2\), strictly increasing, convex, log-concave, and normalized by \[ \phi(0)=0, \] together with the conditions \[ A_1:\quad \lim_{x\to 0^+}\phi'(x)=0, \] \[ A_2:\quad \liminf_{x\to+\infty}\frac{\phi(x)}{x^{n+1}}>0, \] \[ A_3:\quad \frac{d}{dx}\log\frac{\phi(x)}{x}\ge 0,\qquad x>0. \] The monotonicity in \(A_3\) implies \[ \phi'(x)\ge \frac{\phi(x)}{x}>0. \] The direct theorem also uses an orthogonality condition. For \(h,f\in C^2(\mathcal C_\theta)\), \(h\) satisfies the orthogonality condition with respect to \(f\) if whenever \(v\in C^2(\mathcal C_\theta)\) satisfies \[ \int_{\mathcal C_\theta} \left( \frac{\ell\phi'(\ell/h)}{h\phi(\ell/h)}-n-1 \right) \frac{f\,v}{h\phi(\ell/h)}=0, \] then \[ \int_{\mathcal C_\theta} h\,v=0. \] In the \(L_p\) case \(\phi(x)=x^p\), this condition is trivial [2509.10859]. A frequent misconception is to treat the capillary problem as a formal restriction of the classical Orlicz-Minkowski problem from \(\mathbb S^n\) to a subset. The direct capillary theory shows otherwise: the target of the capillary Gauss map is \(\mathcal C_\theta\), the support function is defined relative to \(\xi-\cos\theta e\), and the contact-angle constraint becomes a Robin boundary condition. The geometric weight \(\ell\) 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 \(\phi\in\mathcal O\), \(\theta\in(0,\pi/2)\), and \(f\in C^2(\mathcal C_\theta)\) is positive and even, satisfying \[ \frac{1}{n+1}\int_{\mathcal C_\theta} f \ge \phi\!\left(|\widehat{\mathcal C_\theta}|^{1/(n+1)}\right), \] then there exists a smooth, symmetric capillary convex body \[ \widehat\Sigma\in \mathcal K_\theta^\circ \] with \[ |\widehat\Sigma|=1 \] such that its support function \(h\) solves \[ \left\{ \begin{aligned} \phi\!\left(\frac{\ell}{h}\right)h\det(h_{ij}+h\delta_{ij})&=f &&\text{in }\mathcal C_\theta,\\ \nabla_\mu h&=\cot\theta\,h &&\text{on }\partial\mathcal C_\theta, \end{aligned} \right. \] and \(h\) satisfies the orthogonality condition with respect to \(f\). Moreover, if equality holds in the integral condition and \(\phi\) is strictly convex, then the hypersurface must be the spherical cap \[ |\widehat{\mathcal C_\theta}|^{-1/(n+1)}\,\mathcal C_\theta \] [2509.10859]. This result sits in a sequence of capillary Minkowski-type developments. The capillary Christoffel-Minkowski problem for principal radii \(\sigma_k\) established that, for \(\theta\in(0,\pi/2)\) and \(1\le k n+1 \] [2509.06110]. 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 [2601.14659]. The \(L_p\) specialization is explicitly built into the Orlicz theorem. When \[ \phi(x)=x^p \qquad \text{with } p\ge n+1, \] the assumptions \(A_1\)–\(A_3\) hold, and the theorem recovers the capillary even \(L_p\)-Minkowski problem in volume-normalized form. In particular, for the volume-normalized capillary \(L_{n+1}\)-Minkowski problem, the normalization constant becomes \(1\) [2509.10859]. 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 \[ f_t=(1-t)\phi(1)\ell + t f,\qquad 0\le t\le 1, \] and the PDE family is \[ \left\{ \begin{aligned} \det(h_{ij}+h\delta_{ij})&= \frac{f_t}{h\phi(\ell/h)} &&\text{in }\mathcal C_\theta,\\ \nabla_\mu h&=\cot\theta\, h &&\text{on }\partial\mathcal C_\theta. \end{aligned} \right. \] At \(t=0\), the obvious solution is \[ h=\ell. \] Let \(\mathcal I\subset [0,1]\) be the set of \(t\) for which there exists a positive even solution in the function space \(\mathcal H\), where \(\mathcal H\) encodes the Robin boundary condition and the orthogonality condition [2509.10859]. Closedness is based on capillary-specific a priori estimates. The first is a \(C^0\) estimate for positive capillary even convex solutions of the normalized equation, \[ \frac{1}{C_0}\le h\le C_0. \] 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 \(u=h/\ell\) formulation from the PDE at a minimum point and the growth condition \(A_2\). The gradient estimate is \[ |\nabla h|\le C_1, \] proved with the auxiliary function \[ \Phi=\log\!\left(\frac{|\nabla u|^2}{2}\right)+u+K d, \] where \(d\) is the spherical distance to \(\partial \mathcal C_\theta\). The Neumann condition \(\nabla_\mu u=0\) is used to rule out boundary maxima. The \(C^2\) theory first reduces interior second derivatives to the boundary double-normal derivative, \[ \max_{\mathcal C_\theta} |\nabla^2 h| \le \max_{\partial \mathcal C_\theta} |\nabla^2_{\mu\mu} h| + C_2', \] and then proves \[ \max_{\partial \mathcal C_\theta} |\nabla^2_{\mu\mu} h| \le C_2''. \] The boundary estimate uses a barrier built from \[ \zeta=e^{-d}-1, \] with \[ \zeta|_{\partial\mathcal C_\theta}=0,\qquad \nabla \zeta|_{\partial \mathcal C_\theta}=\mu, \] and auxiliary functions such as \[ Q=\langle \nabla h,\nabla\zeta\rangle-\left(A+\frac12 M\right)\zeta-\cot\theta\, h. \] The sign conditions required in the second-derivative estimate use the log-concavity of \(\phi\) and the monotonicity assumption \(A_3\). Once \(C^2\) bounds are available, standard theory for fully nonlinear elliptic equations with oblique boundary conditions yields \(C^{2,\alpha}\) estimates and bootstrap to higher regularity [2509.10859]. Openness is handled by linearization and the implicit function theorem. The nonlinear operator is \[ \mathcal G(h)=\det(h_{ij}+h\delta_{ij})-\frac{f_t}{h\phi(\ell/h)}, \] with linearization \[ L_h(v)=\sum_{i,j=1}^n c(W)_{ij}(v_{ij}+v\delta_{ij}) +\frac{v}{h^2\phi(\ell/h)} \left( 1-\frac{\ell\phi'(\ell/h)}{h\phi(\ell/h)} \right)f_t, \] where \(W_{ij}=h_{ij}+h\delta_{ij}\). 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 [2601.14659]. In the \(L_p\) case, the capillary \(L_p\) Minkowski flows introduced normalized anisotropic capillary Gauss curvature flows, monotone entropy functionals, and \(C^0\)-to-\(C^\infty\) estimates for the capillary even \(L_p\) Minkowski problem, furnishing a direct prototype for a future capillary even Orlicz flow theory [2509.06110]. 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 \(L_p\) 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 \(\widehat\Sigma_1,\widehat\Sigma_2\in\mathcal K_\theta^\circ\), \(\alpha,\beta\ge0\), \(\alpha^2+\beta^2>0\), and \(\phi\in\mathcal O\), \[ h_{M_\phi(\alpha,\beta;\widehat\Sigma_1,\widehat\Sigma_2)}(\xi) = \inf\left\{ t>0:\, \alpha \phi\!\left(\frac{h_{\widehat\Sigma_1}(\xi)}{t}\right) +\beta \phi\!\left(\frac{h_{\widehat\Sigma_2}(\xi)}{t}\right)\le 1 \right\}. \] 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 \(A_3\), \[ \phi'(x)\ge \frac{\phi(x)}{x}>0. \] The capillary Orlicz mixed volume is \[ V_\phi(\widehat\Sigma_1,\widehat\Sigma_2) = \frac{1}{n+1}\int_{\mathcal C_\theta} \phi\!\left(\frac{h_{\widehat\Sigma_2}}{h_{\widehat\Sigma_1}}\right) h_{\widehat\Sigma_1}\, \det\!\big((h_{\widehat\Sigma_1})_{ij}+h_{\widehat\Sigma_1}\delta_{ij}\big)\,d\xi. \] The capillary Orlicz-Minkowski inequality is \[ V_\phi(\widehat\Sigma_1,\widehat\Sigma_2) \ge V(\widehat\Sigma_1)\, \phi\!\left( \frac{V(\widehat\Sigma_2)^{1/(n+1)}}{V(\widehat\Sigma_1)^{1/(n+1)}} \right), \] and the capillary Orlicz-Brunn-Minkowski inequality is \[ \alpha\, \phi\!\left( \frac{V(\widehat\Sigma_1)^{1/(n+1)}}{V(M_\phi(\alpha,\beta;\widehat\Sigma_1,\widehat\Sigma_2))^{1/(n+1)}} \right) + \beta\, \phi\!\left( \frac{V(\widehat\Sigma_2)^{1/(n+1)}}{V(M_\phi(\alpha,\beta;\widehat\Sigma_1,\widehat\Sigma_2))^{1/(n+1)}} \right) \le 1. \] Equality holds for dilates, and if \(\phi\) is strictly convex then equality holds only for dilates [2509.10859]. These inequalities yield the rigidity statement in the direct theorem. If \(|\widehat\Sigma|=1\) and \(h\) solves the capillary Orlicz-Minkowski equation, then \[ V_\phi(\widehat\Sigma,\widehat{\mathcal C_\theta}) = \frac{1}{n+1}\int_{\mathcal C_\theta} f\,d\xi. \] By the capillary Orlicz-Minkowski inequality, \[ V_\phi(\widehat\Sigma,\widehat{\mathcal C_\theta}) \ge \phi\!\left(|\widehat{\mathcal C_\theta}|^{1/(n+1)}\right), \] which explains the admissibility condition \[ \frac{1}{n+1}\int_{\mathcal C_\theta} f \ge \phi\!\left(|\widehat{\mathcal C_\theta}|^{1/(n+1)}\right). \] If equality holds and \(\phi\) is strictly convex, equality in the inequality forces \(\widehat\Sigma\) to be a dilate of \(\widehat{\mathcal C_\theta}\), and volume normalization fixes the dilate uniquely as \[ \widehat\Sigma = |\widehat{\mathcal C_\theta}|^{-1/(n+1)}\widehat{\mathcal C_\theta}. \] 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 \(L_p\)-Minkowski problem with \(p\ge 1\) in Euclidean capillary convex bodies geometry [2401.12137]. 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 [2504.09320]. 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, \(L_p\), 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 \(\mathbb{R}^{n+1}\), with support functions on \(\mathbb S^n\), and solve regular even Orlicz-Minkowski equations of the form \[ f\,\varphi(h)\,\det(\bar{\nabla}_i\bar{\nabla}_j h+\bar g_{ij}h)=\gamma. \] Those flows provide long-time existence, smooth strict convexity, and \(C^\infty\)-subconvergence in the regular even problem, but there is no capillary/contact-angle formulation and no boundary term [2005.00143]. In the capillary theory, the domain is \(\mathcal C_\theta\), the support function carries a Robin condition, and the symmetry class is horizontal reflection, not antipodal symmetry. The half-space \(L_p\) and Christoffel-Minkowski theories supply the immediate precursors of the Orlicz problem. The capillary \(L_p\)-Minkowski problem for \(p>1\) reduces to \[ \det(\nabla^2 h+h\sigma)=f\, h^{p-1} \quad\text{in } C_\theta,\qquad \nabla_\mu h=\cot\theta\, h \quad\text{on }\partial C_\theta, \] with full smooth existence and uniqueness for \(p>n+1\), existence and uniqueness up to dilation at \(p=n+1\), and capillary-even existence for \(1