Necessary and sufficient conditions for the chain rule in W 1 , 1 loc ( R N ; R d ) and BV loc ( R N ; R d )
نویسندگان
چکیده
In this paper we prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space W 1,1 loc ( R ;R ) and in the space of functions of bounded variation BVloc ( R ;R ) .
منابع مشابه
Necessary and sufficient conditions for the chain rule in W 1 , 1 loc ( R N ; R d ) and BV
In this paper we prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space W 1,1 loc R ;R and in the space of functions of bounded variation BVloc R ;R .
متن کاملConstruction of a surface pencil with a common special surface curve
In this study, we introduce a new type of surface curves called $D$-type curve. This curve is defined by the property that the unit Darboux vector $vec{W}_{0} $ of a surface curve $vec{r}(s)$ and unit surface normal $vec{n} $ along the curve $vec{r}(s)$ satisfy the condition $leftlangle vec{n} ,vec{W}_{0} rightrangle =text{constant}$. We point out that a $D$-type curve is a geodesic curve or an...
متن کاملAbsolute continuity on tracks and mappings of Sobolev spaces
The present paper is concerned with the circumstances under which a function g(x,t ,...,t ) provides, via composition, a ^ 1 m mapping between Sobolev spaces. That is, we examine conditions which ensure that for every system of functions u_,...,u e W _ (Q) 1 m 1, q (where W, (Q) is the class of L functions with L summable i,q q q strong first derivatives on the domain Q, c: R )S the composite f...
متن کاملSome results on the symmetric doubly stochastic inverse eigenvalue problem
The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an $n$-tuple $sigma=(1,lambda_{2},lambda_{3},ldots,lambda_{n})in mathbb{R}^{n}$ with $|lambda_{i}|leq 1,~i=1,2,ldots,n$, to be the spectrum of an $ntimes n$ symmetric doubly stochastic matrix $A$. If there exists an $ntimes n$ symmetric doubly stochastic ...
متن کاملThe (R,S)-symmetric and (R,S)-skew symmetric solutions of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2
Let $Rin textbf{C}^{mtimes m}$ and $Sin textbf{C}^{ntimes n}$ be nontrivial involution matrices; i.e., $R=R^{-1}neq pm~I$ and $S=S^{-1}neq pm~I$. An $mtimes n$ complex matrix $A$ is said to be an $(R, S)$-symmetric ($(R, S)$-skew symmetric) matrix if $RAS =A$ ($ RAS =-A$). The $(R, S)$-symmetric and $(R, S)$-skew symmetric matrices have a number of special properties and widely used in eng...
متن کامل