论文标题
广义有限失真和连续性的映射
Mappings of generalized finite distortion and continuity
论文作者
论文摘要
我们研究了sobolev映射$ f \ in w _ {\ mathrm {\ mathrm {loc}}}^{1，n}（ω，\ mathbb {r}^n）$，$ n \ ge 2 $，满足以下一般性的equalition in equality in equality \ [\ [\ [\ lvert df（k） σ（x）\]几乎每个$ x \ in \ mathbb {r}^n $。在这里，$ k \ colonω\ to [1，\ infty）$和$σ\ colonω\ to [0，\ infty）$是可测量的功能。请注意，当$σ\ equiv 0 $时，我们恢复了有限失真的映射类，这些变形始终是连续的。然而，任意解决方案的连续性被证明是一个复杂的问题。在l^\ infty（ω）$中有限的失真$ k \的情况下，我们完全解决了连续性问题，其中连续性的鲜明条件是$σ$在zygmund space $σ\ log^μ（e +σ）\ in L^1 _ _ {\ natrrm {loc}}（loc}}}} $ n $ n $ n中。我们还表明，一个人可以在$ k $上稍微放松对指数类$ \ exp（λk）\ in l^1 _ {\ Mathrm {loc}}（ω）$的界限，并使用$λ> n + 1 $，并且在$σ\ log^g^μ（e + fin）时仍获得连续的解决方案。 l^1 _ {\ mathrm {loc}}（ω）$，$μ>λ$。另一方面，对于所有$ p，q \ in [1，\ infty] $，带有$ p^{ - 1} + q^{ - 1} = 1 $，我们在l^p _ {\ mathrm {loc}}}}（ω）（ω）$ n l^p _ {\ mathrm {loc/k \ n l^p y l^q y y l^p _ {包括一个在l^\ infty _ {\ mathrm {loc}}}（ω）$和l^1 _ {\ Mathrm {loc}}（ω）$的示例。
We study continuity properties of Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (Ω, \mathbb{R}^n)$, $n \ge 2$, that satisfy the following generalized finite distortion inequality \[\lvert Df(x)\rvert^n \leq K(x) J_f(x) + Σ(x)\] for almost every $x \in \mathbb{R}^n$. Here $K \colon Ω\to [1, \infty)$ and $Σ\colon Ω\to [0, \infty)$ are measurable functions. Note that when $Σ\equiv 0$, we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion $K \in L^\infty (Ω)$, where a sharp condition for continuity is that $Σ$ is in the Zygmund space $Σ\log^μ(e + Σ) \in L^1_{\mathrm{loc}}(Ω)$ for some $μ> n-1$. We also show that one can slightly relax the boundedness assumption on $K$ to an exponential class $\exp(λK) \in L^1_{\mathrm{loc}}(Ω)$ with $λ> n+1$, and still obtain continuous solutions when $Σ\log^μ(e + Σ) \in L^1_{\mathrm{loc}}(Ω)$ with $μ> λ$. On the other hand, for all $p, q \in [1, \infty]$ with $p^{-1} + q^{-1} = 1$, we construct a discontinuous solution with $K \in L^p_{\mathrm{loc}}(Ω)$ and $Σ/K \in L^q_{\mathrm{loc}}(Ω)$, including an example with $Σ\in L^\infty_{\mathrm{loc}}(Ω)$ and $K \in L^1_{\mathrm{loc}}(Ω)$.
