Duality in Differential Equations

  1. The vector space DN (Ω) of all distributions in Ω is the dual space of the test function space D(Ω).

  2. Distributions and test functions are symmetric in Green's identity [Joh, p.94, (1.2b); p.97, (1.18)].

    1. (Existence of solutions-Conjugates). y is the solution of the Newman problem iff j is the solution of the Dirichlet problem, where y+ij is analytic.
    2. (Uniqueness of solutions-Twins). u and du/dn are twin factors in [Joh, p.95, (1.4)].

  4. Locally compact commutative groups [Po3, p.236, l.-5-p.237, l.5].

  5. Different outlooks.
    1. Mechanics. (Particle, Position) (Wave, Momentum).
    2. Topology. Compact discrete [Po3, p.251, Theorem 39].

  6. [Cou, p.165, Example (a)]
    1. (base line, perimeter) ( base line, area).
    2. The greatest area the least perimeter.

  7. Suppose we have two statements A and B. They have not been proved to be true yet. However, the proof of A using B and the proof of B using A have the same pattern, then we say that A and B are formally dual. For example, Statement 1 and Statement 2 in are formally dual before Statement 1 is proved without using unproven statements. If Statement A is proved without using unproven statements, and if we dualize each statement of the proof, we obtain a proof of Statement B, then we say that A and B are actually dual. For Example, the above two Statements 1 and 2 are actually dual. [Cou, p.258, Exercise 2] provides another example because the two extremal problems result in the same Euler equation.

  8. For the geometric meaning of [Fomi, p.73, (20)], read [p.2, \S 1.1 Legendre and Young--Fenchel transforms, Geometric interpretation] of the following website: dual.pdf. The replacement of the independent variables (x,y,y') with (x,y,p) in one statement leading to an equivalent statement illustrates the relation of duality between points and lines.

  9. [Fomi, p.75, l.-8-p.76, l.-11] proves that the y(x) giving the extremum of [Fomi, p.75, (24)] and the y(x) giving the extremum of [Fomi, p.75, (27)] are the same. In [Fomi, pp.75--77, sec.18.2], for the process from the extremum problem of a functional to the Euler equations, the description with independent variables (x,y, y') is one viewpoint; the description with independent variables (x,y, p) is an equivalent dual viewpoint. [Fomi, pp.75-77, sec.18.2] illustrates that there are two statements in the process whose two descriptions are equivalent; in fact, the two descriptions of any statement in the entire process are equivalent.