# Research on the convex envelope problem

**The convex envelope problem.** Joint work with R. H. Nochetto (UMD, USA).

Link to our paper: Two-scale methods for convex envelopes.

Given a strictly convex bounded open set $\Omega \subset \mathbb{R}^d$ and a continuous function $f$ defined on $\overline{\Omega}$, and we want to compute its convex envelope $u$ in $\Omega$, i.e. the largest convex function majorized by $f$ in $\overline{\Omega}$. Alternatively, (A. M. Oberman 2007 Proc. Amer. Math. Soc.) showed that the convex envelope $u$ can also be viewed as the viscosity solution of the fully nonlinear, degenerate elliptic PDE

\[\begin{equation} \small T[u;f](x) := \min\left\{ f(x) - u(x), \lambda_1[D^2u](x)\right\} = 0, \; x \in \Omega; \quad u(x) = f(x), \; x \in \partial \Omega, \label{E:pde-int-CE} \end{equation}\]where $\lambda_1[D^2u]$ denotes the smallest eigenvalue of the Hessian $D^2u$.

In Two-scale methods for convex envelopes, we extend the two-scale method of (R. H. Nochetto, D. Ntogkas, W. Zhang 2019 Math. Comp.), originally for the Monge–Ampère equation, to compute the viscosity solution of $\eqref{E:pde-int-CE}$. We introduce a monotone discrete operator with a fine scale $h$ and a coarse scale $\delta$. The convergence is proved using the Barles-Souganidis’ framework (G. Barles, P. E. Souganidis 1991 Asymptotic Anal.), and a pointwise error estimate is also obtained. The main difficulty for the latter comes from the fact that solutions $u$ are generically never better than of class $C^{1,1}(\overline{\Omega})$, which usually induces a large consistency error for the discretization. To overcome this handicap, we utilize the geometric observation that $u$ is flat in at least one direction outside the contact set $\mathcal{C}(f) := { x \in \overline{\Omega}: u(x) = f(x) }$, together with the discrete comparison principle and discrete barrier argument. Our error analysis for solutions $u \in C^{k,\alpha}(\overline{\Omega})$, with $k=0,1$ and $0<\alpha \leq 1$, yields the pointwise error estimates

\[\| u - u_{\delta,h} \|_{L^\infty(\Omega)} \lesssim \Big( |u|_{C^{k,\alpha}(\overline{\Omega})} + |f|_{C^{k,\alpha}(\overline{\Omega})} \Big) \; h^{\frac{(k+\alpha)^2}{2+k+\alpha}},\]and extends to a modified version of the finite difference wide stencil method of (A. M. Oberman 2008 Math. Models Methods Appl. Sci.). The order is linear provided $k=\alpha=1$, which is consistent with our numerical experiments.