论文标题
关于与椭圆形和取消线性差分运算符相关的方程的局部连续可溶性
On Local Continuous Solvability of Equations Associated to Elliptic and Canceling Linear Differential Operators
论文作者
论文摘要
考虑$ a（x，d）：c^{\ infty}（ω，e）\ rightArrow c^\ infty（ω，f）$ eLLIPTIC且取消订单$ν$的线性差异操作员，具有光滑的复杂系数，$ω\ subset \ subset \ subset \ subbb {r}^nimite vection $ finite vection nimite vection nimite dimemention $ $ f $和$ a^{*}（x，d）$ {is}的伴随。在这项工作中，我们表征了偏微分方程的（本地）连续溶解性$ a^{*}（x，d）v = f $（在分布意义上）对于给定的分布$ f $;更确切地说，我们表明，任何$ x_0 \inΩ$都包含在附近的$ u \ u \ u \ subsetω$中，其中其连续溶解度的特征是以下条件在$ f $上：对于每一个$ε> 0 $，并且任何紧凑型$ k \ subset \ subset \ subset \ subset u $ $ $ a $ complance uncept $ $θ=θ（k，k，，ε$） \ begin {equation} \ nonumber \ left | f（φ）\ right | \ \ leqθ\ | | | _ {w^{ν-1,1}} +ε\ | a（x，x，d）φ\ | _ {l^{1}}，\ end end {equation {equation} $ w^^{ν-1,1} $ for homenogen o s of sybole sobole so $ $ l^$ l^$ l^$ l^1 $ l^1 $ l^1 $ l^1 $ l^1 $ l^1属于$ l^{1}（u）$。该表征意味着并扩展了与向量场椭圆形复合物相关的运算符之前获得的结果（请参阅\ cite {mp}）;我们还为在[4]和[9]中获得的经典差异运算符获得的全局结果提供了局部类似物，用于大量差分运算符。
Consider $A(x,D):C^{\infty}(Ω,E) \rightarrow C^\infty(Ω,F)$ an elliptic and canceling linear differential operator of order $ν$ with smooth complex coefficients in $Ω\subset \mathbb{R}^{N}$ from a finite dimension complex vector space $E$ to a finite dimension complex vector space $F$ and $A^{*}(x,D)$ {its} adjoint. In this work we characterize the (local) continuous solvability of the partial differential equation $A^{*}(x,D)v=f$ (in the distribution sense) for a given distribution $f$; more precisely we show that any $x_0\inΩ$ is contained in a neighborhood $U\subset Ω$ in which its continuous solvability is characterized by the following condition on $f$: for every $ε>0$ and any compact set $K \subset \subset U$, there exists $θ=θ(K,ε)>0$ such that the following holds for all smooth function $φ$ supported in $K$: \begin{equation}\nonumber \left| f(φ) \right| \leq θ\|φ\|_{W^{ν-1,1}} + ε\|A(x,D) φ\|_{L^{1}}, \end{equation} where $W^{ν-1,1}$ stands for the homogenous Sobolev space of all $L^1$ functions whose derivatives of order $ν-1$ belongs to $L^{1}(U)$. This characterization implies and extends results obtained before for operators associated to elliptic complex of vector fields (see \cite{MP}); we also provide local analogues, for a large range of differential operators, to global results obtained for the classical divergence operator in [4] and [9].
