TITLE: Practical applications of the $L^p$ norm when $p \neq 1,2,\infty$
QUESTION [11 upvotes]: I'm roughly familiar with the concept of $L^p$ norms -- what they represent and how they are computed -- though I am far from educated in functional analysis in general. For reasons that are more or less obvious, in many practical applications, the $L^2$ norm is used (ie, optimization, etc). One also sees the $L^1$ and $L^{\infty}$ norms used from time to time.
However, rarely do I see any mention of, say, the $L^3$ norm, or really any $L^p$ norm where $p$ is something other than 1, 2 or $\infty$ (or, in some interpretations, 0).
Are there any algorithms or applications that have exploited the other norms? Does convergence in an $L^p$ sense mean something different for these values of $p$? Is there any value of $p$ that can lead to a property not obtainable by $p=1,2,\infty$?

REPLY [4 votes]: One example not mentioned in the MO thread: the $L^4$ norm appears in the Ginzburg-Landau formula for the energy of superconductor.
More generally: we want nonlinear variational models to describe phenomena which do not obey linear superposition. Minimization of $L^2$ norm leads to linear PDE. Minimization of $L^1$ and $L^\infty $ norms leads to badly degenerate things that are barely PDE at all. The norms $L^4$ and $L^{\rm dimension}$ emerge as attractive alternatives.