We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n -space, where φ ⁢ ( t ) {\varphi(t)} is a non-negative convex function vanishing only at t = 0 {t=0} . We show that this property is always satisfied in dimension n = 2 {n=2} , while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when φ ⁢ ( t ) = c ⁢ t 2 {\varphi(t)=ct^{2}} ) in dimension n ≥ 4 {n\geq 4} . The validity of the quadratic rigidity, which we prove in dimension n = 2 {n=2} , implies the existence of the trace of a divergence-measure vector field ξ on an ℋ 1 {\mathcal{H}^{1}} -rectifiable set S , as soon as its weak normal trace [ ξ ⋅ ν S ] {[\xi\cdot\nu_{S}]} is maximal on S . As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.

中文翻译：

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散度测量向量场的刚性和迹特性

我们考虑欧几里得 n 空间中无散向量场的 φ-刚性性质，其中 φ ⁢ ( t ) {\varphi(t)} 是一个非负凸函数，仅在 t = 0 {t=0 } 。我们证明了这个性质总是在维度 n = 2 {n=2} 中得到满足，而在更高维度上它需要对 φ 进行一些进一步的限制。特别是，我们展示了维度 n ≥ 4 {n\geq 4} 的二次刚度的反例（即当 φ ⁢ ( t ) = c ⁢ t 2 {\varphi(t)=ct^{2}} ）。我们在维度 n = 2 {n=2} 中证明的二次刚度的有效性意味着在 ℋ 1 {\mathcal{H}^{1}} 上存在散度测量向量场 ξ 的迹线-可矫正集 S ，只要它的弱法线迹 [ ξ ⋅ ν S ] {[\xi\cdot\nu_{S}]} 在 S 上最大。作为应用程序，