A nonlinear Korn inequality estimates the distance between two immersions from an open subset of ${\mathbb{R}}^n$ into the Euclidean space ${\mathbb{R}}^k$, $k⩾n⩾1$, in terms of the distance between specific tensor fields that determine the two immersions up to a rigid motion in ${\mathbb{R}}^k$. We establish new inequalities of this type in two cases: when k = n, in which case the tensor fields are the square roots of the metric tensor fields induced by the two immersions, and when k = 3 and n = 2, in which case the tensor fields are defined in terms of the fundamental forms induced by the immersions. These inequalities have the property that their constants depend only on the open subset over which the immersions are defined and on three scalar parameters defining the regularity of the immersions, instead of constants depending on one of the immersions, considered as fixed, as up to now.

非线性Korn不等式估计了一个开放子集的两次浸入之间的距离 ${\mathbb{[R}}^{ñ}$ 进入欧几里得空间 ${\mathbb{[R}}^{ķ}$， $ķ⩾ñ⩾\mathrm{1个}$，根据特定张量场之间的距离来确定两次沉入，直到在 ${\mathbb{[R}}^{ķ}$。我们在两种情况下建立了这种类型的新不等式：当k = n时，张量场是两次浸没引起的度量张量场的平方根，当k = 3且n = 2时，在这种情况下张量场是根据浸没引起的基本形式定义的。这些不等式的性质是，它们的常数仅取决于定义浸入的开放子集和定义浸入规则性的三个标量参数，而不是取决于至今为止被视为固定的浸入之一的常数。 。