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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.

MSC:
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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