Functional Minimization: When is this heuristic rigorous? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:51:19Z http://mathoverflow.net/feeds/question/81250 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81250/functional-minimization-when-is-this-heuristic-rigorous Functional Minimization: When is this heuristic rigorous? Tom P 2011-11-18T14:27:16Z 2011-12-03T10:22:12Z <p>I'm trying to solve a functional minimization problem of the following form:</p> <p>$$\arg\min_{f:\mathbb{R}\rightarrow [0,1]} h(f)$$ where $h$ is some expression in terms of several integrals over $f$. </p> <p>I have a heuristic calculation that seems indeed to work (to find at least a local minimum): I simply treat $f:\mathbb{R}\rightarrow [0,1]$ as a continuum of variables $f(x)$ for each $x \in \mathbb{R}$, "take the derivative" of $h$ with respect to each variable $f(x)$, set it to zero, and solve. Together, this gives me a functional form for $f$. In my setting, this indeed seems to work, but obviously this technique is not rigorous as stated.</p> <p>My question is, are there any conditions under which functional minimization can be done in this way? When can this technique be made rigorous?</p> http://mathoverflow.net/questions/81250/functional-minimization-when-is-this-heuristic-rigorous/81254#81254 Answer by Antoine Levitt for Functional Minimization: When is this heuristic rigorous? Antoine Levitt 2011-11-18T15:14:15Z 2011-11-18T15:14:15Z <p>This is pretty easy : find out an appropriate Banach space for f (the one that makes your integrals well-defined; usually, some kind of Sobolev space), prove that h is C^1 (with respect to differentiation in Banach spaces), and then standard arguments apply. Keywords you might want to look up are: differentiation in Banach spaces, weak solutions, Sobolev spaces.</p> <p>For instance, if $h(f) = \int f^2 + f'^2$, then $h$ is $C^1$ as a functional in $H^1$, and the solution is a (weak) solution to $f'' + f = 0$</p> http://mathoverflow.net/questions/81250/functional-minimization-when-is-this-heuristic-rigorous/81329#81329 Answer by Igor Khavkine for Functional Minimization: When is this heuristic rigorous? Igor Khavkine 2011-11-19T10:11:48Z 2011-11-19T10:11:48Z <p>A useful and fairly complete reference on this and related questions is Morrey <em>Multiple Integrals in the Calculus of Variations</em>. <a href="http://books.google.com/books?id=-QNKm1PBohsC" rel="nofollow">http://books.google.com/books?id=-QNKm1PBohsC</a></p>