The capillary L_p dual Minkowski problem for p>q in higher dimensions

Abstract

We prove that the capillary L_p dual Minkowski problem in ℝⁿ⁺¹, n≥3, admits a unique positive smooth strictly convex solution for any prescribed positive smooth function f when p>q and the contact angle satisfies θ∈(0,π/2). Earlier results for p>q either required the additional condition q≤1 or were restricted to n=2. We reduce the problem to a Monge--Ampère type equation with a Robin boundary condition on a spherical cap. By constructing a new auxiliary function for the global C² estimate, we overcome these limitations and completely resolve the case of p>q in higher dimensions, i.e., prove existence and uniqueness throughout the case of p>q, without any further restriction on p or q.

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BibTeX RIS

BibTeX

@misc{gao2026capillary,
  title = {{The capillary $L_p$ dual Minkowski problem for $p>q$ in higher dimensions}},
  author = {Ya Gao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.07158},
  eprint = {2609.07158},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07158},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The capillary L_p dual Minkowski problem for p>q in higher dimensions
AU  - Ya Gao
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.07158
UR  - https://arxiv.org/abs/2609.07158
AB  - We prove that the capillary L_p dual Minkowski problem in ℝ^n+1, n≥3, admits a unique positive smooth strictly convex solution for any prescribed positive smooth function f when p>q and the contact angle satisfies θ∈(0,π/2). Earlier results for p>q either required the additional condition q≤1 or were restricted to n=2. We reduce the problem to a Monge--Ampère type equation with a Robin boundary condition on a spherical cap. By constructing a new auxiliary function for the global C^2 estimate, we overcome these limitations and completely resolve the case of p>q in higher dimensions, i.e., prove existence and uniqueness throughout the case of p>q, without any further restriction on p or q.
ER  -

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