Up-to-boundary regularity of the convex envelope: minimal hypotheses without uniform convexity or high-order boundary smoothness, and the boundary-regularity versus semiconcavity trade-off for general moduli

The convex envelope $\Gamma_v$ of a continuous function — the largest convex function lying below it — is the central object behind the Aleksandrov–Bakelman–Pucci estimate, the Dirichlet problem for the degenerate Monge–Ampère equation, Hamilton–Jacobi theory, and obstacle formulations. De Philippis and Figalli (Trans. AMS, 2015) proved that $\Gamma_v\in C^{1,\min\{\alpha,\beta\}}_{\mathrm{loc}}(\Omega)$ when $v$ is globally Lipschitz and $(1+\alpha)$-semiconcave and $\partial\Omega$ is of class $C^{1,\beta}$, and that $\Gamma_v\in C^{1,1}(\overline{\Omega})$ when $\Omega$ is uniformly convex with $C^{3,1}$ boundary and $v\in C^{3,1}(\overline{\Omega})$, with both hypothesis sets shown sharp by explicit two-dimensional examples. The unresolved problem, identified by the authors as arising exactly when the contact points supporting the envelope lie close to the boundary, is the matching up-to-boundary theory: characterize the minimal geometric hypotheses on $\Omega$ — weaker than uniform convexity and than $C^{3,1}$ boundary smoothness — and the minimal analytic hypotheses on $v$ that still guarantee $C^{1,\alpha}$ or $C^{1,1}$ regularity of $\Gamma_v$ up to $\partial\Omega$; and classify, for general moduli of continuity, when boundary-induced contacts force a loss of regularity, quantifying how boundary regularity trades against semiconcavity of $v$. Caffarelli–Tang–Wang (2022) settled the $C^{1,\alpha}$ scale on uniformly convex domains under optimal conditions, and Han–Liu–Zhou (2025) proved global $C^{1,\beta/k}$ bounds on planar domains with a finite-order degenerate boundary point; the general-dimensional, non-uniformly-convex, general-moduli regime and the sharp modulus trade-off remain open.

Background

Given an open set $\Omega\subset\mathbb{R}^n$ and a continuous function $v:\overline{\Omega}\to\mathbb{R}$, the convex envelope of $v$ in $\Omega$ is

$$\Gamma_v(x):=\sup\{\ell(x):\ \ell\le v \text{ in }\Omega,\ \ell \text{ affine}\},$$

the largest convex function lying below $v$. Convex functions are automatically locally Lipschitz, but applications need quantitative control of derivatives, so the central question of this literature is when $\nabla\Gamma_v$ is Hölder continuous ($\Gamma_v\in C^{1,\alpha}$ for some $\alpha\in(0,1]$) or Lipschitz ($\Gamma_v\in C^{1,1}$), and up to which sets. The envelope appears throughout analysis and geometry: in the Aleksandrov–Bakelman–Pucci maximum principle for fully nonlinear elliptic equations, in the Dirichlet problem for the degenerate Monge–Ampère equation — where the solution of $\det D^2u=f\ge 0$ with boundary datum $v$ is precisely the envelope of $v$ weighted by $f$ — in Hamilton–Jacobi theory, where viscosity solutions are semiconcave and their singular sets are governed by contact geometry, and as a nonlinear obstacle problem studied numerically by Oberman and Silvestre.

A function $v$ is $(1+\alpha)$-semiconcave in $\overline{\Omega}$, $\alpha\in(0,1]$, if for every $x_0\in\overline{\Omega}$ there is a slope $p_{x_0}$ with $v(x)\le v(x_0)+p_{x_0}\cdot(x-x_0)+C|x-x_0|^{1+\alpha}$ near $x_0$: semiconcave functions are 'differentiable from above' with a controlled defect, which is the natural one-sided regularity of Hamilton–Jacobi solutions. A contact (or touching) point is a point where $v=\Gamma_v$; a supporting slope is the gradient of an affine function touching $v$ from below at such a point. A convex domain $\Omega$ is uniformly convex if for some fixed radius $R$ it is enclosed in balls of radius $R$ tangent to $\partial\Omega$ at each boundary point — a quantitative lower bound on boundary curvature, strictly stronger than smooth strict convexity.

Earlier work treated either the Dirichlet case, where $\Gamma_v$ is determined by boundary data (Caffarelli–Nirenberg–Spruck; Griewank–Rabier; Kirchheim–Kristensen; Oberman–Silvestre), or imposed strong global assumptions on $v$. De Philippis–Figalli (2015) treated the general problem and proved: (a) [Theorem 1.2] if $\Omega$ is bounded convex with $\partial\Omega$ of class $C^{1,\beta}$ and $v$ is globally Lipschitz and $(1+\alpha)$-semiconcave in $\overline{\Omega}$, then $\Gamma_v\in C^{1,\min\{\alpha,\beta\}}_{\mathrm{loc}}(\Omega)$ — an interior result only; (b) [Theorem 1.3] if $\Omega$ is uniformly convex with $\partial\Omega$ of class $C^{3,1}$ and $v\in C^{3,1}(\overline{\Omega})$, then $\Gamma_v\in C^{1,1}(\overline{\Omega})$; (c) both hypothesis sets are sharp, via explicit two-dimensional examples in which the envelope is computed exactly — e.g. a harmonic function equal to $|x|$ on the boundary of the disk has $\Gamma_v=|x|$, which is not $C^1$; on the non-uniformly-convex domain $\{|x|^p+|y|^p\le 1\}$, $1<p<2$, the envelope is no better than $C^{1,p-1}$, so high-order semiconcavity cannot compensate a degenerate boundary; (d) for general moduli of continuity — $v$ being $\omega_1$-semiconcave and $\partial\Omega$ being $C^1$ with modulus $\omega_2$ — the gradient $\nabla\Gamma_v$ has modulus of continuity controlled by $\max\{\omega_1,\omega_2\}$ (up to rescaling) on compact subsets of $\Omega$, again an interior statement; (e) if $v=0$ on $\partial\Omega$ and $\inf_{\Omega}v<0$, no regularity of $\partial\Omega$ whatsoever is needed for interior $C^{1,\alpha}$ regularity [Theorem 4.1], showing the entire difficulty is genuinely a boundary-contact phenomenon.

The proofs expose the structural reason for the gap between (a) and (b): by Carathéodory's theorem, any point of the convex hull of a set in $\mathbb{R}^n$ is a convex combination of at most $n+1$ points of the set, so the regularity of $\Gamma_v$ at a point is controlled by the contact points in its supporting simplex; the authors state that 'the main difficulty when proving the regularity of $\Gamma_v$ arises when [these contact points] are close to the boundary'. Interior contact points transfer the semiconcavity of $v$ to the envelope directly; boundary contact points instead couple the envelope to the fine geometry of $\partial\Omega$, and the only known mechanism to control them — geometric barrier constructions around boundary-flat regions — required uniform convexity and $C^{3,1}$ smoothness. What is missing is the matching theory at the boundary: which geometric assumptions on $\partial\Omega$, weaker than uniform convexity and than high-order smoothness, and which analytic assumptions on $v$, control $\Gamma_v$ up to $\partial\Omega$ — and how the boundary's modulus of continuity trades against the semiconcavity of $v$.

Problem Statement

Determine the up-to-boundary regularity of the convex envelope $\Gamma_v$ in the regime where the supporting contact points of $\Gamma_v$ may lie arbitrarily close to $\partial\Omega$. The single objective is a sharp characterization of this boundary regularity, in the two independent components below: neither component is a prerequisite for the other, and the general-moduli classification (ii) also settles the modulus part of (i).

(i) [Positive direction, minimal hypotheses] Find minimal geometric hypotheses on the bounded convex domain $\Omega$ — strictly weaker in each respect than uniform convexity of $\Omega$ and than $C^{3,1}$-smoothness of $\partial\Omega$, so that neither of these two properties is assumed — and minimal analytic hypotheses on $v$ (in the scale of global Lipschitz continuity and semiconcavity-type moduli) under which $\Gamma_v\in C^{1,\alpha}(\overline{\Omega})$ with its optimal exponent $\alpha\in(0,1)$, or else $\Gamma_v\in C^{1,1}(\overline{\Omega})$, with quantitative estimates on the modulus of continuity of $\nabla\Gamma_v$; minimality is measured in the two scales these hypotheses live in, compared pointwise in each — the analytic hypotheses in the semiconcavity modulus $\omega_1$ of $v$, and the geometric hypotheses in the pair formed by the smoothness modulus $\omega_2$ of $\partial\Omega$ (the $C^1$ boundary modulus of (ii)) and a curvature modulus $\omega_3$ of $\partial\Omega$, a pointwise lower bound on the boundary curvature for which uniform convexity is the case of $\omega_3$ bounded below by a positive constant and strict convexity the case of a finite but possibly degenerating $\omega_3$ — and prove the matching necessity of the hypotheses by explicit examples, as De Philippis–Figalli did for theirs.

(ii) [Classification and trade-off direction] For general moduli of continuity — $v$ being $\omega_1$-semiconcave and $\partial\Omega$ being $C^1$ with modulus $\omega_2$ — classify when boundary-induced contacts force a loss of up-to-boundary regularity, and quantify how much boundary regularity can be traded against improved semiconcavity hypotheses on $v$; equivalently, extend the interior modulus estimate $\max\{\omega_1,\omega_2\}$ of De Philippis–Figalli up to the boundary in a sharp form, identifying the pairs $(\omega_1,\omega_2)$ for which $\nabla\Gamma_v$ has a controlled modulus on $\overline{\Omega}$.

Literature-status note (this describes the state of knowledge, not a restriction of the question): the uniformly convex case in the Hölder scale has been resolved by Caffarelli–Tang–Wang, who proved global $C^{1,\alpha}$ regularity for the envelope under optimal conditions on uniformly convex domains, and Han–Liu–Zhou proved global $C^{1,\beta/k}$ gradient bounds for planar domains whose boundary is strictly convex but only finitely flat at a distinguished point. The problem above is therefore posed in its original generality; its presently open core is the general-dimensional, non-uniformly-convex, general-moduli regime of (i), and the sharp trade-off classification of (ii).

References

1. Guido De Philippis and Alessio Figalli, Optimal regularity of the convex envelope, Transactions of the American Mathematical Society 367 (2015), no. 6, 4407–4422. DOI: 10.1090/S0002-9947-2014-06306-X. https://doi.org/10.1090/S0002-9947-2014-06306-X
2. L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations I. Monge–Ampère equation, Communications on Pure and Applied Mathematics 37 (1984), no. 3, 369–402. DOI: 10.1002/cpa.3160370306. https://doi.org/10.1002/cpa.3160370306
3. A. Griewank and P. J. Rabier, On the smoothness of convex envelopes, Transactions of the American Mathematical Society 322 (1990), no. 2, 691–709. DOI: 10.2307/2001721. https://doi.org/10.2307/2001721
4. Bernd Kirchheim and Jan Kristensen, Differentiability of convex envelopes, Comptes Rendus de l'Académie des Sciences — Série I — Mathématique 333 (2001), no. 8, 725–728. DOI: 10.1016/S0764-4442(01)02117-6. https://doi.org/10.1016/S0764-4442(01)02117-6
5. Adam M. Oberman and Luis Silvestre, The Dirichlet problem for the convex envelope, Transactions of the American Mathematical Society 363 (2011), no. 11, 5871–5886. DOI: 10.1090/S0002-9947-2011-05240-2. https://doi.org/10.1090/S0002-9947-2011-05240-2
6. Qi-Rui Li and Xu-Jia Wang, Regularity of the homogeneous Monge–Ampère equation, Discrete and Continuous Dynamical Systems 35 (2015), no. 12, 6069–6084. DOI: 10.3934/dcds.2015.35.6069. https://doi.org/10.3934/dcds.2015.35.6069
7. Sławomir Dinew, Regularity of the plurisubharmonic envelope in strictly pseudoconvex domains, arXiv preprint arXiv:1703.07727 (2017). https://arxiv.org/abs/1703.07727
8. Jianchun Chu and Bin Zhou, Optimal regularity of plurisubharmonic envelopes on compact Hermitian manifolds, Science China Mathematics 62 (2019), no. 2, 371–380. DOI: 10.1007/s11425-017-9173-0. https://doi.org/10.1007/s11425-017-9173-0
9. J. Ederson M. Braga, Alessio Figalli, and Diego Moreira, Optimal regularity for the convex envelope and semiconvex functions related to supersolutions of fully nonlinear elliptic equations, Communications in Mathematical Physics 367 (2019), no. 1, 1–32. DOI: 10.1007/s00220-019-03370-2. https://doi.org/10.1007/s00220-019-03370-2
10. Wenbo Li and Ricardo H. Nochetto, Two-scale methods for convex envelopes, Mathematics of Computation 91 (2022), no. 333, 111–139. DOI: 10.1090/mcom/3521, arXiv:1812.11519. https://doi.org/10.1090/mcom/3521
11. Luis A. Caffarelli, Lan Tang, and Xu-Jia Wang, Global $C^{1,\alpha}$ regularity for Monge–Ampère equation and convex envelope, Archive for Rational Mechanics and Analysis 244 (2022), no. 1, 127–155. DOI: 10.1007/s00205-022-01757-5. https://doi.org/10.1007/s00205-022-01757-5
12. Qing Han, Jiakun Liu, and Yang Zhou, Global $C^{1,\alpha}$ regularity for Monge–Ampère equations on planar convex domains, Mathematische Annalen 393 (2025), no. 1, 763–805. DOI: 10.1007/s00208-025-03250-2, arXiv:2501.17337. https://doi.org/10.1007/s00208-025-03250-2
13. Egor Dobronravov, Structure of minimal locally concave functions, arXiv preprint arXiv:2608.08783 (2026). https://arxiv.org/abs/2608.08783
14. Guido De Philippis and Alessio Figalli, The Monge–Ampère equation and its link to optimal transportation, Bulletin of the American Mathematical Society 51 (2014), no. 4, 527–580. DOI: 10.1090/S0273-0979-2014-01459-4. https://doi.org/10.1090/S0273-0979-2014-01459-4