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Why Sobolev spaces are so important in study of partial differential equations? What could have light up the mind of researchers to use these spaces to analyze PDEs?

IgotiT
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    Because they have very nice and convenient properties and this allows, e.g., for very general solution results. The Poisson problem with right-hand side in $L^2(\Omega)$ always has a solution in $H_0^1(\Omega)$. – gerw May 03 '16 at 06:48
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    You can check a very nice answer here : https://math.stackexchange.com/questions/1327750/significance-of-sobolev-spaces-for-numerical-analysis-pdes – Ahmed May 10 '17 at 02:37

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