# What is a 'localization argument'?

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.

• 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. Commented Nov 16, 2017 at 23:57
• I added additional information. As you will see, the problem has quite some overhead, which was the reason why I originally left it out. Commented Nov 20, 2017 at 17:51