We study the question whether Lipschitz minimizers of \(\int F(\nabla u)\,dx\) in \(\mathbb {R}^n\) are \(C^1\) when F is strictly convex. Building on work of De Silva–Savin, we confirm the \(C^1\) regularity when \(D^2F\) is positive and bounded away from finitely many points that lie in a 2-plane. We then construct a counterexample in \(\mathbb {R}^4\), where F is strictly convex but \(D^2F\) degenerates on the Clifford torus. Finally we highlight a connection between the case \(n = 3\) and a result of Alexandrov in classical differential geometry, and we make a conjecture about this case.

我们研究的问题是，当F是严格凸的时，\（\ mathbb {R} ^ n \）中\（\ int F（\ nabla u）\，dx \）的Lipschitz极小值是否为\（C ^ 1 \）。以De Silva–Savin的工作为基础，当\（D ^ 2F \）为正且与2平面上的有限多个点有界时，我们确认\（C ^ 1 \）正则性。然后，我们在\（\ mathbb {R} ^ 4 \）中构造一个反例，其中F严格是凸的，而\（D ^ 2F \）在Clifford圆环上退化。最后，我们强调案例\（n = 3 \）之间的联系 和亚历山德罗夫（Alexandrov）在经典微分几何中的结果，我们对此情况做出了一个推测。