Usefulness of the $\sigma(L^\infty,L^1)$ topology in the context of differential equations

Question

In Brezis's Functional Analysis book through chapters 3-4, I've seen the $\sigma(L^\infty,L^1)$ topology on $L^\infty$ but did not see (so far) any application of it in differential equations. Is there an example of a differential equation where this topology might be helpful in analyzing the $L^\infty$ norm of the solution for example? Or maybe some other thing I don't know about?

Answer 1

This topology is ubiquitous in modern analysis of PDEs, typically in conjunction with the Banach-Alaoglu theorem. (Bounded sets in $L^\infty$ are relatively compact for the predual $\sigma(L^\infty,L^1)$ / weak-$\ast$ topology.)

Here is one (not so) specific scenario: When trying to prove existence of solutions to non-linear PDEs, one possible and classical approach is to approximate the equation by a sequence of better-behaved ones, for which one can construct a solution $u_n$. If the approximation is well constructed (typically the "structural" properties that one expects formally from the original equation should be reflected in the approximation procedure) then the approximate solutions usually satisfy some "nice" estimates. For example if $\{u_n\}$ satisfies Lipschitz bounds uniformly in $n$ then one can conclude that the derivatives $ \nabla u_n\overset{\ast}{\rightharpoonup} \nabla u$ for the weak-$\ast$ ($\sigma(L^\infty,L^1)$) topology. The Arzelà-Ascoli theorem moreover gives strong uniform convergence $u_n\to u$, which helps passing to the limit in non-linear terms.

For example, if the PDE is quasi-linear elliptic of the form $div(A(u) \nabla u)=f(u)$ then typically one approximates $A_n,f_\approx A,f$ (e.g. by truncation $A_n(u):=A(\min(n,u))$). The above procedure allows to pass to the limit in the weak formulation $$ \int A_n(u_n)\nabla u_n\cdot \nabla\varphi \to \int A(u)\nabla u\cdot \nabla\varphi $$ and $$ \int f_n(u_n)\varphi\to \int f(u)\varphi $$ for any fixed reasonably smooth test-function $\varphi$.

Another such example is the theory of viscosity solutions Due to Crandall-Lions. Here the Hamilton-Jacobi PDE $$ \partial_t u+H(x,u,\nabla u)=0 $$ is usually approximated by a "viscous" Hamilton-Jacobi-Bellman equation $$ \partial_t u_\epsilon+H(x,u_\epsilon,\nabla u_\epsilon)=\epsilon\Delta u_\epsilon, $$ and the solution is constructed as $u=\lim\limits_{\epsilon\to 0}u_\epsilon$.

Answer 2

This is really a comment but will be too long for that. It is also admittedly rather tangential to your question but I hope it can add some context to it and provide useful information. Some of the most important spaces in analysis consist of spaces of bounded functions or operators with the supremum norm. These are often non-separable and non-reflexive Banach spaces. A priori that is not a disadvantage but they do have further negative properties concerning the interaction of their functional analytical properties with those of the underlying spaces—density properties, representability of the dual, tensor-product representations for spaces on products, stability and continuity of certain natural functors. All of these suggest that the norm topology is, in a certain sense, too strong and this has led to the introduction of certain weaker lc topologies which, however, retain the all important property of completeness. At first sight these constructions might appear to be rather ad hoc but in fact they are all examples of so-called mixed topologies (Wiweger) and they have been intensively studied by many mathematicians under a host of names (strict topologies, two-normed spaces, Saks spaces). The general idea is that one replaces the norm by the finest lc topology which agrees on the unit ball with a suitable natural topology on the unbounded elements. Examples of spaces where this has been applied are: bounded continuous functions on a completely regular spaces, bounded holomorphic functions on a complex domain, bounded operators on a Banach spaces (or even between two such), in particular the case of Hilbert spaces and, more generally, von Neumann algebras.

Your case is, of course, that of bounded measurable functions—here one can mix the norm with the weak $\ast$- or the $L^1$-topology (the latter was the original application of this method by Saks—hence the name “Saks space”). The main importance of this to differential equations is that it can happen that the unit ball in such a resulting space is compact, something which can never happen in infinite dimensional Banach spaces, and this is vital in many arguments in ode’s and pde’s. For example, the natural extension of the Peano existence theorem fails for fields with values in a Banach space but there is a version which holds for certain Saks spaces, in particular for the one implicit in your question.