The classical Minkowski problem for convex bodies has deeply influenced the development of differential geometry. During the past several decades, abundant mathematical theories have been developed for studying the solutions of the Minkowski problem, however, the numerical solution of this problem has been largely left behind, with only few methods available to achieve that goal. In this article, focusing on the two-dimensional Minkowski problem with Dirichlet boundary conditions, we introduce two solution methods, both based on operator-splitting. One of these two methods deals directly with the Dirichlet condition, while the other method uses an approximation of this Dirichlet condition. This relaxation of the Dirichlet condition makes this second method better suited than the first one to treat those situations where the Minkowski and the Dirichlet condition are not compatible. Both methods are generalizations of the solution method for the canonical Monge-Amp\`{e}re equation discussed by Glowinski et al. (Journal of Scientific Computing, 79(1), 1-47, 2019); as such they take advantage of a divergence formulation of the Minkowski problem, well-suited to a mixed finite element approximation, and to the the time-discretization via an operator-splitting scheme, of an associated initial value problem. Our methodology can be easily implemented on convex domains of rather general shape (with curved boundaries, possibly). The numerical experiments we performed validate both methods and show that if one uses continuous piecewise affine finite element approximations of the smooth solution of the Minkowski problem and of its three second order derivatives, these two methods provide nearly second order accuracy for the $L^2$ and $L^{\infty}$ error. One can extend easily the methods discussed in this article, to address the solution of three-dimensional Minkowski problem.

翻译：凸体的经典Minkowski问题深刻影响了微分几何的发展。过去几十年中，研究者为Minkowski问题的解建立了丰富的数学理论，然而该问题的数值求解却长期滞后，仅有少数方法能够实现这一目标。本文针对带Dirichlet边界条件的二维Minkowski问题，引入两种均基于算子分裂的求解方法。其中一种方法直接处理Dirichlet条件，另一种方法则采用对该Dirichlet条件的近似。这种对Dirichlet条件的松弛处理使得第二种方法比第一种更适合处理Minkowski条件与Dirichlet条件不兼容的情形。两种方法均为Glowinski等人（Journal of Scientific Computing, 79(1), 1-47, 2019）提出的标准Monge-Ampère方程求解方法的推广；因此它们利用了Minkowski问题的散度表述形式，该形式非常适合混合有限元逼近，以及通过算子分裂格式对相关初值问题的时间离散化。我们的方法易于在具有较一般形状（可能包含弯曲边界）的凸域上实现。数值实验验证了两种方法的有效性，结果表明：若对Minkowski问题的光滑解及其三个二阶导数采用连续分片仿射有限元逼近，这两种方法在L²和L^∞误差度量下能提供近二阶精度。本文讨论的方法可轻松扩展至三维Minkowski问题的求解。