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On a sharp Sobolev-type inequality on two-dimensional compact manifolds. (English) Zbl 0980.46022
Let \(f\) be a nonnegative bounded function on a compact two-dimensional Riemannian manifold \(M\) and set \[ W= \Biggl\{w\in W^{1,2}(M): \int_M w dV= 0,\;\int_M f\exp(w) dV> 0\Biggr\}. \] The functional \[ I(w)={1\over 2}\|\nabla w\|^2_2- 8\pi\log \int_M f\exp(w) dV \] is defined and bounded from below on \(W\). The authors construct a particularly favourable minimizing sequence for \(I\) with the goal of constructing a minimum. This is accomplished by solving variants of the corresponding Euler-Lagrange equation \[ -\Delta u= 8\pi\Biggl({f\exp(u)\over \int_M f\exp(u) dV}- {1\over\text{vol}(M)}\Biggr). \] For certain functions \(f\) on a flat two-dimensional torus, the minimizing sequence admits a convergent subsequence and thus attains its infimum.

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58J05 Elliptic equations on manifolds, general theory
35J20 Variational methods for second-order elliptic equations
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