Weak derivative

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space L 1 ( [ a , b ] ) {displaystyle L^{1}()} . See distributions for a more general definition. In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space L 1 ( [ a , b ] ) {displaystyle L^{1}()} . See distributions for a more general definition. Let u {displaystyle u} be a function in the Lebesgue space L 1 ( [ a , b ] ) {displaystyle L^{1}()} . We say that v {displaystyle v} in L 1 ( [ a , b ] ) {displaystyle L^{1}()} is a weak derivative of u {displaystyle u} if, for all infinitely differentiable functions φ {displaystyle varphi } with φ ( a ) = φ ( b ) = 0 {displaystyle varphi (a)=varphi (b)=0} . This definition is motivated by the integration technique of Integration by parts. Generalizing to n {displaystyle n} dimensions, if u {displaystyle u} and v {displaystyle v} are in the space L l o c 1 ( U ) {displaystyle L_{loc}^{1}(U)} of locally integrable functions for some open set U ⊂ R n {displaystyle Usubset mathbb {R} ^{n}} , and if α {displaystyle alpha } is a multi-index, we say that v {displaystyle v} is the α t h {displaystyle alpha ^{th}} -weak derivative of u {displaystyle u} if for all φ ∈ C c ∞ ( U ) {displaystyle varphi in C_{c}^{infty }(U)} , that is, for all infinitely differentiable functions φ {displaystyle varphi } with compact support in U {displaystyle U} . Here D α φ {displaystyle D^{alpha }varphi } is defined as If u {displaystyle u} has a weak derivative, it is often written D α u {displaystyle D^{alpha }u} since weak derivatives are unique (at least, up to a set of measure zero, see below). If two functions are weak derivatives of the same function, they are equal except on a set with Lebesgue measure zero, i.e., they are equal almost everywhere. If we consider equivalence classes of functions such that two functions are equivalent if they are equal almost everywhere, then the weak derivative is unique. Also, if u is differentiable in the conventional sense then its weak derivative is identical (in the sense given above) to its conventional (strong) derivative. Thus the weak derivative is a generalization of the strong one. Furthermore, the classical rules for derivatives of sums and products of functions also hold for the weak derivative. This concept gives rise to the definition of weak solutions in Sobolev spaces, which are useful for problems of differential equations and in functional analysis.