Weak formulation

Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, an equation is no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain 'test vectors' or 'test functions'. This is equivalent to formulating the problem to require a solution in the sense of a distribution. Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, an equation is no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain 'test vectors' or 'test functions'. This is equivalent to formulating the problem to require a solution in the sense of a distribution. We introduce weak formulations by a few examples and present the main theorem for the solution, the Lax–Milgram theorem. The theorem is named after Peter Lax and Arthur Milgram, who proved it in 1954. Let V {displaystyle V} be a Banach space. We want to find the solution u ∈ V {displaystyle uin V} of the equation where A : V → V ′ {displaystyle Acolon V o V'} and f ∈ V ′ {displaystyle fin V'} , with V ′ {displaystyle V'} being the dual of V {displaystyle V} . This is equivalent to finding u ∈ V {displaystyle uin V} such thatfor all v ∈ V {displaystyle vin V} holds: Here, we call v {displaystyle v} a test vector or test function. We bring this into the generic form of a weak formulation, namely, find u ∈ V {displaystyle uin V} such that by defining the bilinear form