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Trudinger-Moser inequalities on a closed Riemannian surface with the action of a finite isometric group. (English) Zbl 1471.30005

Summary: Let \((\sum,g)\) be a closed Riemannian surface, \(W^{1,2}(\Sigma,g)\) be the usual Sobolev space, \(\mathbf{G}\) be a finite isometric group acting on \((\sum,g)\) and \(\mathcal{H}_{\mathbf{G}}\) be the function space including all functions \(u\in W^{1,2}(\Sigma,g)\) with \(\int_\Sigma udv_g=0\) and \(u(\sigma(x))=u(x)\) for all \(\sigma\in\mathbf{G}\) and all \(x\in\Sigma\). Denote the number of distinct points of the set \(\{\sigma(x):\sigma\in\mathbf{G}\}\) by \(I(x)\) and \(\ell=\min_{x\in\Sigma}I(x)\). Let \(\lambda_1^{\mathbf{G}}\) be the first eigenvalue of the Laplace-Beltrami operator on the space \(\mathcal{H}_{\mathbf{G}}\). Using blow-up analysis, we prove that if \(\alpha<\lambda_1^{\mathbf{G}}\) and \(\beta\leq 4\pi\ell\), then there holds \[\sup\limits_{u\in\mathcal{H}_{\mathbf{G}},\int_\Sigma|\nabla_gu|^2dv_g-\alpha\int_\Sigma u^2dv_g\leq 1}\int_\Sigma e^{\beta u^2}dv_g<\infty; \] if \(\alpha<\lambda_1^{\mathbf{G}}\) and \(\beta>4\pi\ell\), or \(\alpha\geq\lambda_1^{\mathbf{G}}\) and \(\beta>0\), then the above supremum is infinity; if \(\alpha<\lambda_1^{\mathbf{G}}\) and \(\beta\leq 4\pi\ell\), then the above supremum can be attained. Moreover, similar inequalities involving higher order eigenvalues are obtained. Our results partially improve original inequalities of J. Moser [Indiana Univ. Math. J. 20, 1077–1092 (1971; Zbl 0213.13001)], L. Fontana [Comment. Math. Helv. 68, No. 3, 415–454 (1993; Zbl 0844.58082)] and W. Chen [Proc. Am. Math. Soc. 108, No. 3, 821–832 (1990; Zbl 0697.53009)].

MSC:

30F10 Compact Riemann surfaces and uniformization
58J05 Elliptic equations on manifolds, general theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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