Concentration scale - Knowlpedia

A positive function $L(\varepsilon)\to0$ is a concentration scale for a profile family

u_\varepsilon(x)=A(\varepsilon)V\left(\frac{x-x_\varepsilon}{L(\varepsilon)}\right).

For a fixed compactly supported $V$ , the support lies in $x_\varepsilon+L(\varepsilon)\operatorname{supp}V$ . Thus $L$ specifies a spatial length; $A$ specifies amplitude, and $x_\varepsilon$ specifies the center.

Multiple lengths

Different directions may have different lengths $L_i$ . A parameter $q$ with transverse length $q^{1/2}$ is not itself that physical length. Concentration can coexist with a bounded or vanishing integral norm, as the profile norm scaling makes explicit.