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Let $F: \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be the Lagrangian. We say that $f \in X$ is a local minimizer of the variational integral if for all compact sets $C \subset \mathbb{R}^n$ we have $$\int_C F(x,f(x),\nabla f(x))dx = \inf\bigg\{\int_C F(x,u(x),\nabla u(x))dx \colon u \in X \bigg\} $$ where $X$ is some function space ($C^2$ or a Sobolev space, etc.)

Unlike the classical variational integral that takes place on a bounded set, here I am interested in functions defined in all of the space, since the integral of $\int_{\mathbb{R}^n} F(x,u(x),\nabla u(x))dx$ might be infinite, I think it's appropiate to try to find a local minimizer.

Some questions I have in mind for this problem are when there exists local minimizers, when they are unique and regularity but I haven't been able to find any references for this kind of problem or related problems. I appreciate any reference about this or related problems.

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    $\begingroup$ To get useful answers to this you would need to be more specific. Regularity in this context is not particularly interesting, the theory is the same as in the compact case. The main difficulty is existence vs. nonexistence and behavior of minimizing sequences. A reasonable starting point might be the series of papers by P.L. Lions, "The concentration-compactness principle in the calculus of variations," which has many examples. $\endgroup$
    – user378654
    Commented Feb 22, 2023 at 23:51
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    $\begingroup$ To add to this, a similar definition is quite common in the theory of minimal surfaces, which often deals with non-compact surfaces (e.g. Simon's cone). Also the name "local minimizer" is normally used to talk about functions $f$ that minimize the full integral in some neighborhood of $f$ in $X$. $\endgroup$
    – mlk
    Commented Feb 23, 2023 at 10:56

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