Weak Solution - General Case

General Case

The general idea which follows from this example is that, when solving a differential equation in u, one can rewrite it using a so-called test function, such that whatever derivatives in u show up in the equation, they are "transferred" via integration by parts to . In this way one obtains solutions to the original equation which are not necessarily differentiable.

The approach illustrated above works for equations more general than the wave equation. Indeed, consider a linear differential operator in an open set W in Rn

where the multi-index (α₁, α₂, ..., α_n) varies over some finite set in Nn and the coefficients are smooth enough functions of x.

The differential equation P(x, ∂)u(x) = 0 can, after being multiplied by a smooth test function with compact support in W and integrated by parts, be written as

where the differential operator Q(x, ∂) is given by the formula

The number

shows up because one needs α₁ + α₂ + ... + α_n integrations by parts to transfer all the partial derivatives from u to in each term of the differential equation, and each integration by parts entails a multiplication by −1.

The differential operator Q(x, ∂) is the formal adjoint of P(x, ∂) (see also adjoint of an operator for the concept of adjoint).

In summary, if the original (strong) problem was to find a |α|-times differentiable function u defined on the open set W such that

(a so-called strong solution), then an integrable function u would be said to be a weak solution if

for every smooth function with compact support in W.