Supersolution Verification for the Capillary Problem

Complete verification of 4,821 cases for n=7 to n=100

Overview

This page provides complete verification results for the construction of supersolutions of the one-phase and capillary problems asymptotic to the cone solutions Un,k and Cn,k,θ. The computations here accompany the results of the paper "Firester, B., Tsiamis, R., Wang, Y., Area-minimizing capillary cones".

The construction is possible for all n ≥ 7 and k ≥ 1 in the ranges: 1 ≤ k ≤ n − 2.

All 4,821 cases for n=7 to n=100 and k between 1 and n−2 in the admissible ranges have been rigorously verified using adaptive precision arithmetic and convergence-based sampling.

Mathematical Conditions Verified

For each (n, k, β) triple, five conditions are satisfied with rigorous sign certainty:

  1. r̄ - A < 0 (critical geometric constraint)
  2. P(ξ) > 0 for all ξ ∈ (0, 1]
  3. Q̂(t) < 0 for all t ∈ [0, τ)
  4. K₀(ξ) < 0 for all ξ ∈ (0, 1]
  5. K₁(ξ) < 0 for all ξ ∈ (0, 1]

Error Guarantees:
• Numerical precision error < |value| / 10 (adaptive precision: 8-100 digits)
• Discretization error < |value| / 100 (convergence-based sampling)
• Total error < |value| / 9, ensuring definitive sign certainty

Verified Beta Values

Browse the complete table of verified (n, k, β) values and verification results.

n k β max(r̄ - A) max Q̂ max K₀ max K₁ min P

Download

Complete Verification Results

Machine-readable CSV file with all 4,821 verification results.

Download CSV (916 KB)

Back to Homepage