Let $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 be a domain and let $$f\in BV_{{\text {loc}}}(\Omega ,{\mathbb {R}}^3)$$ f ∈ B V loc ( Ω , R 3 ) be a homeomorphism such that its distributional adjugate is a finite Radon measure. We show that its inverse has bounded variation $$f^{-1}\in BV_{{\text {loc}}}$$ f - 1 ∈ B V loc . The condition that the distributional adjugate is finite measure is not only sufficient but also necessary for the weak regularity of the inverse.

中文翻译：

---

最小假设下逆的弱正则性

令 $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 是一个域，让 $$f\in BV_{{\text {loc}}}(\Omega ,{\mathbb { R}}^3)$$ f ∈ BV loc ( Ω , R 3 ) 是一个同胚，使得它的分布调节是一个有限氡测度。我们证明它的逆有界变化 $$f^{-1}\in BV_{{\text {loc}}}$$ f - 1 ∈ BV loc 。分布赋形是有限测度的条件对于逆的弱正则性不仅是充分的而且是必要的。