Liouville's elliptic and hyperbolic equations and their mathematical and physical interpretations

J. Liouville (1853) has solved equation $u_{z\bar z}=e^u$ in closed form. His paper remains to this day one of the most useful works of the nineteenth century, and we relate this fact to the variety of its mathematical and physical interpretations. Liouville’s article simultaneously opened the way to the solution of the elliptic equation (Le) : $\Delta u=K e^{au}$, where u=u(x,y), and of its hyperbolic analogue (Lh) : $u_{xy}= e^u$. G. Darboux (1870) called (Lh) Liouville’s equation and inserted it in a general theory of the solution of second-order PDEs in two variables. E. Picard (1890) gave the same name to (Le); his ideas led to the boundary blow-up problem for (Le). The reception of Liouville’s work in Russia and USSR took a different path. Even though F. Minding had introduced Gauss’ geometry already in the middle of the 19th century, and P.L. Tchebyshev was in close contact with Liouville, equations (Lh), (Le) began to be used in the Soviet Union only in the mid-1930s. Equations (Lh) and (Le) arose again later in numerous unrelated mathematical and physical contexts both in Europe and in USSR. However, it was Liouville’s original solution that opened the way to a new representation of non-integrable problems solutions in any dimension. The present study leads to the following conclusions: first, only a historical perspective gives an adequate view of the mathematical dimensions of Liouville’s equation; second, the evolution of ideas in the works of Russian scientists was different from what we see elsewhere not because of isolation, or language barriers, but because, in some scientific circles in Russia, social conditions made it possible to think independently. See also https://hal.archives-ouvertes.fr/hal-03657588v1
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