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<p>We introduce a capillary <i>k</i>-th area measure for capillary convex bodies in the Euclidean half-space, as a natural boundary analogue of the classical <i>k</i>-th area measure (see, e.g., [39, Chapter 8]). We then formulate the associated capillary Christoffel–Minkowski problem: given a positive density on the spherical cap, find a capillary convex body whose capillary <i>k</i>-th area measure coincides with the prescribed one. Analytically, the problem reduces to a fully nonlinear Hessian equation with a Robin boundary value condition. Under a natural sufficient condition, we prove the existence and uniqueness of smooth solutions (up to horizontal translations).</p>

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The capillary Christoffel–Minkowski problem

  • 期刊论文

  • 发表日期：

    2026年5月16日

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  • Xinqun Mei,
  • Guofang Wang,
  • Liangjun Weng

摘要

We introduce a capillary k-th area measure for capillary convex bodies in the Euclidean half-space, as a natural boundary analogue of the classical k-th area measure (see, e.g., [39, Chapter 8]). We then formulate the associated capillary Christoffel–Minkowski problem: given a positive density on the spherical cap, find a capillary convex body whose capillary k-th area measure coincides with the prescribed one. Analytically, the problem reduces to a fully nonlinear Hessian equation with a Robin boundary value condition. Under a natural sufficient condition, we prove the existence and uniqueness of smooth solutions (up to horizontal translations).