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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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  • $\begingroup$ It would be helpful to give a specific reference that uses this term in a way that you do not understand. $\endgroup$
    – Deane Yang
    Jun 29, 2012 at 16:46
  • $\begingroup$ 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. $\endgroup$
    – pde_bk
    Jun 29, 2012 at 18:52

3 Answers 3

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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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  • $\begingroup$ Are you referring to Holmgren's uniqueness theorem? $\endgroup$
    – Uday
    Jun 29, 2012 at 15:18
  • $\begingroup$ Yes, and to its far-reaching extensions made possible by theory of distributions. $\endgroup$
    – Anatoly Kochubei
    Jun 29, 2012 at 17:37
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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 not for all operator A, there exists even weak solution u. However in somr special case, like A is second order uniformly elliptic,one way to prove the existence is to use Fredholm alternative in Functional analysis,combining with the Gardinger's inequality. For evolution equation,when the spacial part still is second order uniformly elliptic,we can use Galerkin method plus the existence theorem which we stated at the beginging for elliptic operator.

Now, the fredholm alternative is a good tool to consider the relationship for a compactor operator defined on some Hilbert space. For the second order uniformly elliptic operator A, thanks to Gardinger's inequality and Lax-Milmann theorem, we can prove the inverse of A+\lambda I is compact operator on L2 space, for \lambda big enough. And it indeed does give us a kind of complete existence theory for second order uniformly elliptic operators. You can refer to Evans' classical pde book, chapter 6.1, 6.2.

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