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Let $\Omega$ be a domain in $\mathbb{R}^n$. I am interested in the following critical elliptic partial differential equations (PDEs):

  1. The Yamabe Type Equation (for $n>2$):

\begin{equation} -\Delta u + hu = fu^{2^*-1}, \end{equation}

where $2^*=\frac{2n}{n-2}$ is the critical exponent associated with the Sobolev embedding $W^{1,2} \subset L^{p}$.

  1. The Liouville Equation (for $n=2$):

\begin{equation} -\Delta u + h = f e^{2u}. \end{equation}

In the case of the Liouville equation ($n=2$), it is also considered critical, with $2^* = +\infty$. However, it's worth noting that in this case, the role of the Sobolev embedding is played by the Moser-Trudinger inequality.

These equations naturally arise in conformal geometry, particularly in the context of the evolution of the scalar curvature (or Gaussian curvature for $n=2$) under a conformal change of metric. While these equations have found significant applications in differential geometry, I believe their universal property of conformal invariance suggests potential applications beyond this field.

I am currently preparing a talk for a diverse audience, including individuals from mathematical economics. I would greatly appreciate any references or examples of concrete fields of application for these equations, as their relevance may extend beyond their prominent role in differential geometry and conformal field theory.

Thank you in advance for your input and suggestions.

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    $\begingroup$ There is a slight generalization of the Yamabe equation that appears in general relativity called the Lichnerowicz equation. IT appears when constructing initial data for the Cauchy problem. $\endgroup$ Sep 19, 2023 at 0:12

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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,...

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