论文标题
对称和正矩阵字段的良好特性具有有界差异
Fine properties of symmetric and positive matrix fields with bounded divergence
论文作者
论文摘要
本文涉及功能\ [\ mathbb {d}（a）= \ int _ {\ mathbb {t}^n} {\ text {det}}^\ frac {1} {1} {n-1} {n-1}（x（x）（A（x））\，dx \]的各种精细属性。此功能是在$ x_p $上定义的，$ x_p $，它是l^p（\ mathbb {t}^n; \ text {sym}^+（n））$的矩阵字段$ a \ in l^p（\ mathbb {t}^n; \ text {div}（div}（div}（a）$）$ A $ A）$。我们首先要纠正[13，推论7]中指出的错误，该错误涉及$ \ mathbb {d}（a）$ $ x_p $中的$ \ mathbb {d}（a）$。我们给出了一个精致的正确语句的证明，我们将使用它来研究$ \ mathbb {d}（a）$当$ a \ in x_ \ frac {n} {n} {n-1} $时的行为，这是$ \ mathbb {d}（d}（a）$的关键可集成性。我们的主要结果之一给出了由$ \ mathbb {d}（a_k）$生成的度量的明确结合，用于此类矩阵字段$ \ {a_k \} _ k $。特别地，它使我们能够在x_ \ frac {n} {n} {n} {n} {n -1} $中的$ \ mathbb {d}（a）$ $ \ mathbb {d}（a）$的上半强度表征$ \ {\ text {div} a_k \} _ k $。我们通过显式示例显示，如果$ p <\ frac {n} {n-1} $，则在$ x_p $中失败。作为我们特征的副产品，我们还恢复并概括了P.-L的结果。狮子[25,26]关于Sobolev嵌入的研究缺乏紧凑。此外，与Monge-ampère理论相比，我们提供了足够的条件，$ \ text {det}^\ frac {1} {n-1} {n-1}（a）$当$ a \ in x__ \ frac {n} {n} {n} {n-1} $时，$ a \ in n $ a \ in S. n} {凸功能$φ$。
This paper is concerned with various fine properties of the functional \[ \mathbb{D}(A) = \int_{\mathbb{T}^n}{\text{det}}^\frac{1}{n-1}(A(x))\,dx \] introduced in [33]. This functional is defined on $X_p$, which is the cone of matrix fields $A \in L^p(\mathbb{T}^n;\text{Sym}^+(n))$ with $\text{div }(A)$ a bounded measure. We start by correcting a mistake we noted in our [13, Corollary 7], which concerns the upper semicontinuity of $\mathbb{D}(A)$ in $X_p$. We give a proof of a refined correct statement, and we will use it to study the behaviour of $\mathbb{D}(A)$ when $A \in X_\frac{n}{n-1}$, which is the critical integrability for $\mathbb{D}(A)$. One of our main results gives an explicit bound of the measure generated by $\mathbb{D}(A_k)$ for a sequence of such matrix fields $\{A_k\}_k$. In particular it allows us to characterize the upper semicontinuity of $\mathbb{D}(A)$ in the case $A \in X_\frac{n}{n - 1}$ in terms of the measure generated by the variation of $\{\text{div } A_k\}_k$. We show by explicit example that this characterization fails in $X_p$ if $p<\frac{n}{n-1}$. As a by-product of our characterization we also recover and generalize a result of P.-L. Lions [25,26] on the lack of compactness in the study of Sobolev embeddings. Furthermore, in analogy with Monge-Ampère theory, we give sufficient conditions under which $\text{det}^\frac{1}{n-1}(A)$ is Hardy when $A \in X_\frac{n}{n - 1}$, generalising the celebrated result of S. Müller [29] when $A=\text{cof } D^2φ$, for a convex function $φ$.
