A Trüdinger inequality on surfaces with conical singularities

Abstract

Abstract.In this paper, the author establishes an isoperimetric inequality on surfaces with conical singularities, and by using it, proves a Trüdinger inequality with best constant on such surfaces. The best constants of the Trüdinger inequality are also found for a class of "symmetric" singular matrices. IntroductionRecently Troyanov [1] systematically studied the surfaces with conical singularities, the singularities Picard considered. In order to prescribe Gaussian curvature on such surfaces, he proved the following Trüdinger inequality:(0.1) feb"2dA<cb Js for all u G Hl(S) satisfying ¡s\Vu\2dA < 1 and fsudA = 0 and for all b < b0. Where S is a compact Riemannian surface with conical singularities of divisor ß = J2 ßjXj, dA is the area element of S ,cb isa constant related to b and b0 = 4n min({ 1,1 +ßA . Although he claimed (0.1 ) to be true for b = bQ, it seems that he is unable to guarantee, by his method (Holder inequality), that {cb} is bounded as b -► b0 in the case b0 < 4n. Hence, there are still some problems remaining unsolved:(1) If (0.1) is true for b = b0<4n? (2) Is b0 the best constant? (3) (Posed by Troyanov) If the metric of the surface 5 is invariant under the action of some isometry group G, can the constant b in (0.1) be larger than b0 for the class of t7-equivariant functions? What is the best value of b in this case?

Blow‐up analysis, existence and qualitative properties of solutions for the two‐dimensional Emden–Fowler equation with singular potential

“…We remark that Proposition 3.1 was already known in the case where is substituted by any compact twodimensional Riemannian manifold, see [16,17]. An improvement of Proposition 3.1 has been obtained in [15].…”

“…We will prove the asymptotic symmetry of one-point blow-up solution sequences, that is any unbounded (with respect to n) solution sequence for (16), such that n → 8 . First of all, we note that all blow-up solutions sequences are uniformly bounded in a neighbourhood of the boundary, see Theorem 2.4.…”

“…Finally, we remark that the analysis of (1) and (2) on compact two manifolds is relevant for the prescribed curvature problem with conical singularities, see [15][16][17].…”

“…In case ∈ [0, 8 (1 − )), the existence of a solution is easily obtained by means of a sharp version [16,17,21] of the Moser-Trudinger inequality [22], suitable to be applied to the singular weight (2), see (18). On the other hand, in case 8 (1 − ), the existence problem is more delicate.…”

The inequality of Moser and Trudinger and applications to conformal geometry

“…This is made explicit in the articles [19,22]; in the former it was shown that the constant β 0 is determined by a certain isoperimetric constant associated to twodimensional, piecewise smooth domains and in the latter for two-dimensional orbit folds.…”