The full three-dimensional Newton minimum-resistance problem
Let $D = \{x \in \mathbb{R}^2 : |x| < 1\}$ and, for each height $M > 0$, let $\mathcal U_M$ be the set of concave functions $u\colon D \to [0, M]$, whose gradient $\nabla u$ exists almost everywhere. Newton's resistance functional for a body with circular base and height at most $M$ moving through a rare medium is $$J(u) = \int_D \frac{1}{1 + |\nabla u(x)|^2}\, dx.$$ Newton found the minimizer among radially symmetric functions. Buttazzo, Ferone and Kawohl proved that $J$ attains its minimum on $\mathcal U_M$, and Brock, Ferone and Kawohl that Newton's body of revolution is not a minimizer, so that no minimizer is radially symmetric and, by rotation, minimizers are not unique. Lokutsievskiy, Wachsmuth and Zelikin proved that certain conical parts on the boundary of a body rule out its optimality, and deduced that several bodies proposed in the literature as candidates for optimality are not optimal.
Problem. For each $M > 0$, determine the exact value of $\inf_{u \in \mathcal U_M} J(u)$ and characterize all minimizing functions, up to rotations of $D$. No axial or other symmetry is assumed.
Source: Newton (1687), Principia, among bodies of revolution; among convex bodies without symmetry, Buttazzo and Kawohl (1993).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, concave functions, Newton's problem of minimal resistance, shape optimization
References
@book{newton1687,
author = {Newton, Isaac},
title = {Philosophi{\ae} Naturalis Principia Mathematica},
publisher = {Royal Society},
address = {London},
year = {1687},
}
@article{buttazzo1993,
author = {Buttazzo, Giuseppe and Kawohl, Bernhard},
title = {On {N}ewton's problem of minimal resistance},
journal = {Math. Intelligencer},
volume = {15},
number = {4},
year = {1993},
pages = {7--12},
doi = {10.1007/BF03024318},
}
@article{buttazzo1995,
author = {Buttazzo, Giuseppe and Ferone, Vincenzo and Kawohl, Bernhard},
title = {Minimum problems over sets of concave functions and related questions},
journal = {Math. Nachr.},
volume = {173},
number = {1},
year = {1995},
pages = {71--89},
doi = {10.1002/mana.19951730106},
}
@article{brock1996,
author = {Brock, F. and Ferone, V. and Kawohl, B.},
title = {A symmetry problem in the calculus of variations},
journal = {Calc. Var. Partial Differential Equations},
volume = {4},
number = {6},
year = {1996},
pages = {593--599},
doi = {10.1007/BF01261764},
}
@article{lokutsievskiy2022,
author = {Lokutsievskiy, Lev and Wachsmuth, Gerd and Zelikin, Mikhail},
title = {Non-optimality of conical parts for {N}ewton's problem of minimal resistance in the class of convex bodies and the limiting case of infinite height},
journal = {Calc. Var. Partial Differential Equations},
volume = {61},
number = {1},
year = {2022},
pages = {31},
doi = {10.1007/s00526-021-02118-y},
eprint = {2009.12128},
archivePrefix = {arXiv},
}
@misc{aim-newton-minimum-resistance,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The full three-dimensional Newton minimum-resistance problem}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/245-newton-minimum-resistance.md},
note = {Entry 245 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00538, "The full three-dimensional Newton minimum-resistance problem", clopen.solutions.