Yamabe type equations help in establishing Isoperimetric Inequalities; Isoperimetric Inequalities help in establishing sharp Sobolev Inequalities (cf https://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/S0002-9904-1978-14553-4.pdf).

Sharp Sobolev Inequalities help in improving a priori estimates. Improved a priori estimates help to have a longer observation time (semi-global existence) of a studied objet (gravitational field (https://arxiv.org/pdf/0805.3880.pdf, https://arxiv.org/pdf/1107.0898); velocity field in fluid mechanics,...).

Very often, it's with semi-global existence that one can begin the study of the interesting phenomena: shock formations and development, stability and the dynamic of the system in general.

I picked up here just items in Mathematical General Relativity. Certainly, corresponding items should exist in Economics, Materials Sciences, Robotics,...

Yamabe type equations help in establishing Isoperimetric Inequalities; Isoperimetric Inequalities help in establishing sharp Sobolev Inequalities (cf Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 1182-1238, doi:10.1090/S0002-9904-1978-14553-4).

Sharp Sobolev Inequalities help in improving a priori estimates. Improved a priori estimates help to have a longer observation time (semi-global existence) of a studied object (gravitational field (Demetrios Christodoulou, The Formation of Black Holes in General Relativity, (arXiv:0805.3880), Jonathan Luk, On the Local Existence for the Characteristic Initial Value Problem in General Relativity, (arXiv:1107.0898); velocity field in fluid mechanics,...).

Very often, it's with semi-global existence that one can begin the study of the interesting phenomena: shock formations and development, stability and the dynamic of the system in general.

I picked up here just items in Mathematical General Relativity. Certainly, corresponding items should exist in Economics, Materials Sciences, Robotics,...