Max an integral with endpoints varying with the extremal function. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T17:48:14Z http://mathoverflow.net/feeds/question/92121 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92121/max-an-integral-with-endpoints-varying-with-the-extremal-function Max an integral with endpoints varying with the extremal function. Pedro Forquesato 2012-03-25T01:30:00Z 2012-03-25T02:21:41Z <p>Hello all,</p> <p>Usually the calculus of variations' take on not variable endpoints means to choose a point in a fixed function, but what I need is for that point to be defined by the function I am maximizing; that is, my problem is to find y(x) to maximize:</p> <p><code>$\int^{k_1}_{k_0}{\int_{y(x)}^{t_1}F(x,y,z)\,dz\,dx}$</code> </p> <p>If it makes it clearer, the geometrical interpretation (if I imagine it correctly) is to choose y(x) which maximizes the volume of the 3-dimensional area (inside a 3-dimensional compact cube) of F(x,y,z) ($F_y$>0), given that the width of the area is in itself given by the choice of y(x).</p> <p>My problem is more complicated (since it involves probability measures), but if someone help me solve this (or point me into the right direction) I believe I can work out the rest.</p> <p>sorry if this question is obvious for some (most), but I'm not a mathematician (economist) and I spent the last week reading several books on calculus of variations and I didn't find anything really helpful to solve my research problem.</p> <p>Thank all, Pedro.</p>