Regularity and structure of the representing matrix measure in Lions's identification of the dual cone of convex functions

Lions (C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1385–1390) identified the dual cone of the convex functions in $L^2(\Omega)$ for $\Omega\subset\mathbb{R}^N$ smooth, bounded and strongly convex: a functional $L$ is nonpositive on all convex functions if and only if $L=-\partial_{ij}\mu_{ij}$ in the sense of distributions for some matrix $(\mu_{ij})$ of bounded Radon measures that is symmetric and positive semidefinite as a matrix. This representation yields the Euler–Lagrange equation for variational problems under a convexity constraint, the matrix measure $\mu$ acting as the Lagrange multiplier for the constraint. The open problem is the regularity and structure one can expect for this matrix measure. In dimension one the multiplier is essentially a scalar nonnegative measure and the regularity theory is settled, but for $N\ge 2$ the representing measure is not unique in general, and the sharpness examples in Lions's note exhibit representing measures whose densities are merely $L^2$ (failing $L^p$ for all $p>2$) or that are singular with respect to Lebesgue measure. Whether one can always select a representing measure with improved regularity — absolute continuity, higher integrability, or Sobolev membership — or whether some $L$ force singular or minimally integrable representatives, and how the structure of $\mu$ relates to the geometry of $\Omega$ and to $L$, remains unresolved: literature up to 2024–2025 describes the structure of $\mu$ as still mysterious and develops Abreu-equation approximations precisely to bypass it.

Background

Many variational problems in physics and economics are posed over classes of convex functions: Newton's problem of the minimal resistance of a body and the monopolist's (multidimensional screening) problem in economics both lead to minimizing an integral functional over the set of convex functions on a domain. Writing optimality conditions for such problems is obstructed by the fact that the convexity constraint is an inequality on the second derivatives of the unknown, not a standard pointwise or boundary constraint.

For a closed convex cone $K$ in a Hilbert space, the dual (polar) cone is the set $K^\perp$ of all $L$ with $\langle L,u\rangle\le 0$ for every $u\in K$. When $K$ is the cone of convex functions, elements of $K^\perp$ play the role of generalized Lagrange multipliers for the convexity constraint: the first-order condition of a constrained problem says that the ordinary Euler–Lagrange term of the variation must be balanced by some $L\in K^\perp$ evaluated on admissible perturbations. A usable optimality theory therefore requires an explicit description of $K^\perp$.

It is classical that the second distributional derivatives of a convex function are Radon measures (finite, regular Borel measures) and that the resulting Hessian matrix measure is symmetric positive semidefinite, meaning that $\xi^T(D^2u)\xi\ge 0$ as a measure for every constant vector $\xi$. Lions's 1998 note identifies the dual cone in exactly this language: if $\Omega\subset\mathbb{R}^N$ is smooth, bounded and strongly convex and $K$ is the set of convex functions in $L^2(\Omega)$, then $L\in K^\perp$ if and only if there exist bounded Radon measures $\mu_{ij}$ on $\Omega$, with the matrix $(\mu_{ij})$ symmetric and positive semidefinite as a matrix of measures, such that $L=-\partial_{ij}\mu_{ij}$ in the sense of distributions, the pairing being $\int_\Omega Lv\,dx=-\int\partial_{ij}v\,d\mu_{ij}$ for every $v\in C^2(\overline{\Omega})$ (repeated indices are summed). The note also proves a priori estimates on $\mu$ in terms of $\|L\|_{L^2}$ — weighted integrability against powers of a convex defining function $\Phi$ of $\Omega$ (a smooth function with $\Omega=\{\Phi<0\}$ and $D^2\Phi>0$), a Morrey-type bound (after Morrey's classical estimates controlling a measure's mass in balls) on the total variation of $\mu$, and, when $\mu$ is absolutely continuous, an $L^{N/(N-1)}$ bound on $(\det\mu)^{1/N}$ reminiscent of Monge–Ampère regularity — together with analogous representations for the dual cones in $H^1$ and $H^{-1}$ (the Sobolev space with zero trace and its dual) and for cones combining convexity with positivity and coordinatewise monotonicity. As an application, the minimizer $u$ of a Dirichlet energy with linear forcing terms over convex functions in $H^1_0(\Omega)$ satisfies $-\Delta u=f+\partial_i f_i+\partial_{ij}\mu_{ij}$ together with a complementarity condition: the Lebesgue part of $\mu$ is supported where the convexity of $u$ degenerates ($\operatorname{Tr}(d^2u\cdot\mu^0)=0$ a.e., where $d^2u$ and $\mu^0$ denote the Lebesgue parts of the Hessian of $u$ and of $\mu$). The matrix measure $\mu$ is thus exactly the Lagrange multiplier of the convexity constraint.

What is missing is any real understanding of the regularity of $\mu$. In dimension $N=1$ the multiplier reduces to a scalar nonnegative measure ($L=-\mu''$ with $\mu\ge 0$, determined up to an affine function by positivity), and regularity questions in one dimension — and in the radial case — were settled in subsequent work of Carlier (J. Nonlinear Convex Anal. 3 (2002) 125–143). For $N\ge 2$ the representing measure is not unique in general, and the sharpness examples in Lions's note show that representing measures with limited regularity genuinely occur: densities that belong to $L^2$ but to no $L^p$ with $p>2$, and measures singular with respect to Lebesgue measure. Whether these examples are the last word — that is, whether for every $L\in K^\perp$ one can select a representing $\mu$ with better regularity, and what the structure of the admissible measures is in terms of $\Omega$ and $L$ — is the open problem. Its difficulty is underscored by the later literature: Le (Comm. Pure Appl. Math. 73 (2020) 2248–2283) writes that the structure of $\mu$ is 'still mysterious', and Wang–Zhou (arXiv 2403.04235) note that 'very little is known about the measure $\mu$', which is why minimizer regularity via the Euler–Lagrange equation is 'still inaccessible'; instead, an entire research program approximates the constrained problems by fourth-order Abreu-type equations (equations of Monge–Ampère type involving the logarithm of the Hessian determinant, used as a barrier for the constraint) in order to bypass $\mu$.

Problem Statement

For $\Omega\subset\mathbb{R}^N$ with $N\ge 2$ smooth, bounded and strongly convex, let $K$ denote the set of convex functions in $L^2(\Omega)$, and let $K^\perp$ denote the set of all $L\in L^2(\Omega)$ such that $\int_\Omega Lu\,dx\le 0$ for every $u\in K$. By Lions's theorem, every $L\in K^\perp$ is represented as $L=-\partial_{ij}\mu_{ij}$ in the sense of distributions by a matrix $(\mu_{ij})$ of bounded Radon measures on $\Omega$ that is symmetric and positive semidefinite as a matrix of measures. The problem is to determine the regularity and structure one can expect for this representing matrix measure, that is, to give a rigorous description of the admissible representing measures. The question encompasses the following interconnected aspects, which share this single objective:

(1) Selection and regularity. Since for $N\ge 2$ the representing measure is not uniquely determined by $L$, determine whether one can always select a representing $\mu$ with improved regularity — for instance absolutely continuous with respect to Lebesgue measure, with density in $L^p$ for some $p>2$, or in a Sobolev class — or whether there exist $L\in K^\perp$ for which every representing measure is singular with respect to Lebesgue measure or has minimal integrability. The sharpness examples in Lions's note (representing measures with densities in $L^2$ but in no $L^p$ with $p>2$, and singular representing measures) show that limited regularity occurs for the constructed representatives; whether the full class of representing measures for such $L$ — and for general $L$ — is necessarily of limited regularity, or can be improved by selection, is the question.

(2) Structure. Give a refined description of the class of representing measures for given $L$ (equivalently, of the admissible perturbations $\sigma$ of a representing measure $\mu$, that is, of the symmetric matrices $\sigma=(\sigma_{ij})$ of bounded Radon measures for which $\mu+\sigma$ is again positive semidefinite and represents the same $L$; determining precisely the constraints defining this class — the divergence-type identity $\partial_{ij}\sigma_{ij}=0$ and any further conditions imposed by the pairing in Lions's theorem — is part of the question), of its singular and absolutely continuous parts, of the boundary behavior relative to the defining function of $\Omega$, and of how this structure depends on the geometry of $\Omega$ and on $L$.

(3) Consequences for the multiplier. In the Euler–Lagrange equation $-\Delta u=f+\partial_i f_i+\partial_{ij}\mu_{ij}$ for convexity-constrained variational problems (with the complementarity condition tying the Lebesgue part of $\mu$ to the degeneracy set of $u$'s convexity), determine the corresponding structure and regularity of the multiplier $\mu$.

The one-dimensional case is excluded (there the multiplier is essentially a scalar nonnegative measure and the regularity theory is settled); the problem concerns dimension $N\ge 2$. No restriction on $L$, on $\Omega$ beyond the stated class, or on the form of the answer is intended: the target is a mathematically rigorous characterization — positive selection theorems, negative constructions, or structural descriptions — of the achievable regularity of the representing matrix measure.

References

1. Pierre-Louis Lions, Identification du cône dual des fonctions convexes et applications, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 12, 1385–1390, DOI 10.1016/S0764-4442(98)80397-2, https://doi.org/10.1016/S0764-4442(98)80397-2, Zbl 0924.46059
2. Guillaume Carlier, Thomas Lachand-Robert, Représentation du cône polaire des fonctions convexes et applications, C. R. Math. Acad. Sci. Paris 335 (2002), no. 6, 571–576, DOI 10.1016/S1631-073X(02)02512-8, https://doi.org/10.1016/S1631-073X(02)02512-8
3. Guillaume Carlier, Thomas Lachand-Robert, Representation of the polar cone of convex functions and applications, J. Convex Anal. 15 (2008), no. 3, 535–546, Zbl 1162.46039, https://zbmath.org/?q=an%3A1162.46039
4. G. Carlier, T. Lachand-Robert, Regularity of solutions for some variational problems subject to a convexity constraint, Comm. Pure Appl. Math. 54 (2001), no. 5, 583–594, DOI 10.1002/cpa.3, https://doi.org/10.1002/cpa.3
5. Guillaume Carlier, Calculus of variations with convexity constraint, J. Nonlinear Convex Anal. 3 (2002), no. 2, 125–143, Zbl 1030.49023, https://zbmath.org/?q=an%3A1030.49023
6. Guillaume Carlier, Teresa Radice, Approximation of variational problems with a convexity constraint by PDEs of Abreu type, Calc. Var. Partial Differential Equations 58 (2019), no. 5, Art. 170, DOI 10.1007/s00526-019-1613-1, https://doi.org/10.1007/s00526-019-1613-1
7. Nam Q. Le, Singular Abreu equations and minimizers of convex functionals with a convexity constraint, Comm. Pure Appl. Math. 73 (2020), no. 10, 2248–2283, DOI 10.1002/cpa.21883, arXiv 1811.02355, https://arxiv.org/abs/1811.02355
8. Nam Q. Le, Twisted Harnack inequality and approximation of variational problems with a convexity constraint by singular Abreu equations, Adv. Math. 434 (2023), Art. 109325, DOI 10.1016/j.aim.2023.109325, arXiv 2212.05905, https://arxiv.org/abs/2212.05905
9. Ling Wang, Bin Zhou, $C^{1,\alpha}$ regularity of variational problems with a convexity constraint, arXiv 2403.04235 (2024), https://arxiv.org/abs/2403.04235
10. Young Ho Kim, On the One-Dimensional Singular Abreu Equations, Appl. Math. Optim. 90 (2024), no. 2, Art. 35, DOI 10.1007/s00245-024-10178-7, arXiv 2403.07852, https://arxiv.org/abs/2403.07852
11. Fabio Cavalletti, Michael Westdickenberg, The polar cone of the set of monotone maps, Proc. Amer. Math. Soc. 143 (2015), no. 2, 781–787, DOI 10.1090/S0002-9939-2014-12332-X, arXiv 1305.2692, https://arxiv.org/abs/1305.2692
12. Michael Westdickenberg, Euler-Lagrange equation for a minimization problem over monotone transport maps, Q. Appl. Math. 75 (2017), no. 2, 267–285, DOI 10.1090/qam/1459, https://doi.org/10.1090/qam/1459
13. Krzysztof J. Ciosmak, Matrix Hölder's Inequality and Divergence Formulation of Optimal Transport of Vector Measures, SIAM J. Math. Anal. 53 (2021), no. 6, 6932–6958, DOI 10.1137/20m1367520, arXiv 2109.06588, https://arxiv.org/abs/2109.06588
14. Young Ho Kim, On approximation of convex functionals with a convexity constraint and general Lagrangians, arXiv 2504.07783 (2025), https://arxiv.org/abs/2504.07783, to appear in Proc. R. Soc. Edinb. Sect. A (Math.)
15. Chong Gu, Nam Q. Le, A Green's function approach to linearized Monge-Ampère equations in divergence form and application to singular Abreu type equations, arXiv 2511.15621 (2025), https://arxiv.org/abs/2511.15621, to appear in J. Lond. Math. Soc.
16. Yann Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375–417, DOI 10.1002/cpa.3160440402, https://doi.org/10.1002/cpa.3160440402