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I'm studying a paper and in the introduction appears the following: It is well known that existence of critical points and solvability of Euler-Lagrange equations are related, and there is and extensive literature about critical points which are minimizers, specially for functionals defined on the Sobolev space $W_{0}^{1,p}(\Omega),\; p>1,$ by $$J(u)=\int_{\Omega}\mathcal{F}(x,u,Du) dx,$$ where $\Omega$ is bounded, open subset of $\mathbb{R}^N.$

DOUBT: However, I'm struggling to find this extensive literature, and I also would like to find definition and properties of Euler-Lagrange equations.

Thanks in advance. I appreciate any help.

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  • $\begingroup$ I think any calculus of variations book will have this result, try Dacorogna's Introduction to Calculus of Variations for example. $\endgroup$
    – Chee Han
    Nov 25 at 1:01
  • $\begingroup$ @CheeHan I've studied with this exact book. However, in that book, they assume we have the background I'm asking here. $\endgroup$
    – Silvinha
    Nov 25 at 1:16
  • $\begingroup$ By background, you meant Sobolev spaces? $\endgroup$
    – Chee Han
    Nov 25 at 1:18
  • $\begingroup$ If that's the case, then try Dacorogna's Direct Method in the Calculus of Variations. $\endgroup$
    – Chee Han
    Nov 25 at 1:20
  • $\begingroup$ No, I've read that too. By background, I mean how to build an Euler-Lagrange Equation from a given Functional in Sobolev Spaces. $\endgroup$
    – Silvinha
    Nov 25 at 18:02
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These lecture notes by Piotr Hajłasz might have the introductory level you are looking for:

The lectures will be divided into two almost independent streams. One of them is the theory of Sobolev spaces with numerous aspects which go far beyond the calculus of variations. The second stream is just calculus of variations. The theory of Sobolev spaces is a basic technical tool for the calculus of variations, however it suffices to know only basic results for most of the applications in that context.

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    $\begingroup$ Yes, this will help. I start reading, these lecture notes might help with another good book I found, which is: Semilinear Elliptic Equations for Beginners (Marino Badiale and Enrico Serra). They explain even how $-div$ appears at the beginning of the Euler-Lagrange equations. Thank you very much for your help. $\endgroup$
    – Silvinha
    Nov 25 at 18:10

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