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While reading an article on nonlinear variational problems, the author talked about the possibility to use localization arguments in order to show some coercivity estimates for a given functional. I figured, that what he meant would be something like restricting the functional, which orginally acted on some Sobolev space $W^{1,2}_0(\Omega)$ over a domain $\Omega$, onto one over $W^{1,2}_0(\Omega')$ with $\Omega' \subset\subset \Omega$ being compactly embedded and "small/local enough".

My question: Can you specify what is (generally) meant by localization arguments and give a prototypical example of their use?

Edit (further information): My problem at hand is an intermediate step in the minimization of the socalled Yang-Mills functional. For this, consider some matrix-valued vector field $A \in W^{1,2}(B^n, \mathbb{R}^n\otimes \mathfrak{g})$ with values in a Lie algebra $\mathfrak{g}$. For simplicity, one may think of $\mathfrak{g} = \mathfrak{su}(k)$, the vector space of hermitian trace-less $k\times k$-matrices. We think of $A$ as $n$ matrix-valued functions $(A_i)_{i=1,\ldots,n}$. We then consider the functional $E_0(A) = \int_{B^n} |\mathrm{d}A|^2 + |\mathrm{d}^\ast A|^2 = \int_{B^n} |\mathrm{rot}A|^2 + |\mathrm{div}A|^2$. Here, $\mathrm{d}, \mathrm{d}^\ast$ are the exterior differential, respectively codifferential. This functional is coercive and weakly lower semicontinuous and hence posesses a minimizer. Now, we consider the functional $E(A) = \int_{B^n}|\mathrm{d}A + [A,A]|^2 + |\mathrm{d}^\ast A|^2$, where the first term of the integrand is explicitly given by $\frac14 \sum_{ij} |\partial_i A_j - \partial_j A_i + [A_i,A_j]|^2$. The map $K:W^{1,2}(B^n, \mathbb{R}^n\otimes \mathfrak{g}) \rightarrow L^2(B^n, \mathbb{R}^n\otimes\mathbb{R}^n\otimes \mathfrak{g}), A\mapsto [A,A]$ is (by Sobolev multiplication and embedding theorems) a compact map for dimensions $n\leq 3$. Now the questions is, if the existence of a minimizer for $E_0$ and the compactness of the pertubation $K$ can be somehow used to show existence of a minimizer for $E$. By a private suggestion of a professor, I was told that this is possible by localization arguments. Unfortunately, I was not able to ask more specific about that suggestion.

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  • $\begingroup$ It would be helpful if you could give some details. What type of problem are we dealing with? What type of functional? The term "localization" can be applied to many things. $\endgroup$ – Petr Naryshkin Nov 16 '17 at 23:57
  • $\begingroup$ I added additional information. As you will see, the problem has quite some overhead, which was the reason why I originally left it out. $\endgroup$ – Peter Wildemann Nov 20 '17 at 17:51

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