## duality argument in PDE

Can anyone please explain the term 'duality argument', or the difference between this term and the weak formulation in PDE analysis? Or give some references?

Occasionally I see this term appears in papers on PDE(existence, uniqueness, continuity proofs of pdes), but I never found a detailed description of it. I hope to get a clear picture on this thing.

Here I am concerned with theoretical analysis in pde, not about numerical schemes (of cos please let me know if they are essentially connected).

Thanks.

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 It would be helpful to give a specific reference that uses this term in a way that you do not understand. – Deane Yang Jun 29 at 16:46 Sorry, I cannot find the exact paper. Just the rough context: For an equation $Au=f$, where $A$ is a differential operator(evolution type or not, Cauchy problem or not), $u,f$ belong to certain topological vector space(test function space $C_0^\infty$,Hilbert space, Banach space, or distribution space). A statement is put into this way: 'By duality argument, we can get the existence of u...' Sorry for the vagueness. Nevertheless I expect 'broad' answers from all aspects. I thank Mr.Anatoly's answer. But I'm not sure his answer is what I want. – pde_bk Jun 29 at 18:52

Such an argument would crucially use properties of the dual problem (e.g., a problem involving the adjoint operator). One can say that the weak formulation is a duality based approach, but typically you would expect a more substantial use of duality in a duality argument.

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Of course the "duality argument" may have different meaning. However a frequent use of the term is as follows. For typical evolution equations, the correct solvability of the Cauchy problem in some space $\Phi$ implies the uniqueness of its solution in the dual space $\Phi'$. This approach gives precise uniqueness classes for equations and systems with constant coefficients. See

Gel'fand, I.M.; Shilov, G.E. Generalized functions. Vol. 3. Theory of differential equations. Academic Press, 1967.

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 Are you referring to Holmgren's uniqueness theorem? – Uday Jun 29 at 15:18 Yes, and to its far-reaching extensions made possible by theory of distributions. – Anatoly Kochubei Jun 29 at 17:37