# zbMATH — the first resource for mathematics

Uniqueness of solutions for a mean field equation on torus. (English) Zbl 1105.58005
Summary: We consider on a two-dimensional flat torus $$T$$ the following equation
$\Delta u+\rho \biggl( \frac{e^u}{\int_G e^u}- \frac{1}{|T|}\biggr)= 0.$
When the fundamental domain of the torus is $$(0,a)\times (0,b)$$ $$(a\geq b)$$, we establish that the constants are the unique solutions whenever
$\rho\leq\begin{cases} 8\pi&\text{if }\frac ab\geq \frac \pi4,\\ 32\frac ba &\text{if }\frac ba\leq \frac\pi4, \end{cases}$
and this result is sharp if $$\frac ab\geq \frac \pi4$$. A similar conclusion is obtained for general two-dimensional torus by considering the length of the shortest closed geodesic. These results are derived by comparing the isoperimetric profile of the torus $$T$$ with the one of the two-dimensional canonical sphere which has same volume as $$T$$.

##### MSC:
 5.8e+11 Variational problems in applications to the theory of geodesics (problems in one independent variable)
Full Text:
##### References:
 [1] Bandle, C., Isoperimetric inequalities and applications, (1980), Pitman London · Zbl 0436.35063 [2] Berger, M.; Gauduchon, P.; Mazet, E., Le spectre d’une variété riemannienne, Lecture notes in math., vol. 194, (1971), Springer Berlin/New York · Zbl 0223.53034 [3] Cabré, X.; Lucia, M.; Sanchón, M., A Mean field equation on a torus: one-dimensional symmetry of solutions, Comm. partial differential equations, 30, 1315-1330, (2005) · Zbl 1115.35041 [4] Caglioti, E.; Lions, P.L.; Marchioro, C.; Pulvirenti, M., A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. math. phys., 143, 501-525, (1992) · Zbl 0745.76001 [5] Chang, S.-Y.A.; Chen, C.-C.; Lin, C.-S., Extremal functions for a Mean field equation in two dimension, (), Chapter 4 [6] Chanillo, S.; Kiessling, M., Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Comm. math. phys., 160, 217-238, (1994) · Zbl 0821.35044 [7] Chavel, I., Isoperimetric inequalities. differential geometric and analytic perspectives, Cambridge tracts in math., vol. 145, (2001), Cambridge Univ. Press Cambridge · Zbl 0988.51019 [8] Chen, C.-C.; Lin, C.-S., Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. pure appl. math., 55, 6, 728-771, (2002) · Zbl 1040.53046 [9] Ding, W.; Jost, J.; Li, J.; Wang, G., The differential equation $$\operatorname{\Delta} u = 8 \pi - 8 \pi h e^u$$ on a compact Riemann surface, Asian J. math., 1, 2, 230-248, (1997) [10] Evans, L.C.; Gariepy, R.L., Measure theory and fine properties of functions, Stud. adv. math., (1992), CRC Boca Raton, FL · Zbl 0804.28001 [11] Fontana, L., Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. math. helv., 68, 3, 415-454, (1993) · Zbl 0844.58082 [12] Hong, C.-W., A best constant and the Gaussian curvature, Proc. amer. math. soc., 97, 737-747, (1986) · Zbl 0603.58056 [13] Howards, H.; Hutchings, M.; Morgan, F., The isoperimetric problem on surfaces, Amer. math. monthly, 106, 5, 430-439, (1999) · Zbl 1003.52011 [14] Kiessling, M.K.-H., Statistical mechanics of classical particles with logarithmic interactions, Comm. pure appl. math., 46, 27-56, (1993) · Zbl 0811.76002 [15] Lin, C.-S., Uniqueness of solutions to the Mean field equations for the spherical Onsager vortex, Arch. ration. mech. anal., 153, 2, 153-176, (2000) · Zbl 0968.35045 [16] Lojasiewicz, S., An introduction to the theory of real functions, (1988), Wiley Chichester [17] Lucia, M.; Zhang, L., A priori estimates and uniqueness for some Mean field equations, J. differential equations, 217, 1, 154-178, (2005) · Zbl 1175.35053 [18] Nolasco, M.; Tarantello, G., On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. ration. mech. anal., 145, 2, 161-195, (1998) · Zbl 0980.46022 [19] Onofri, E., On the positivity of the effective action in a theory of random surfaces, Comm. math. phys., 86, 3, 321-326, (1982) · Zbl 0506.47031 [20] Ricciardi, T.; Tarantello, G., On a periodic boundary value problem with exponential nonlinearities, Differential integral equations, 11, 5, 745-753, (1998) · Zbl 1015.34008 [21] Struwe, M.; Tarantello, G., On multivortex solutions in chern – simons gauge theory, Boll. unione mat. ital. sez. B, 1, 8, 109-121, (1998) · Zbl 0912.58046 [22] Suzuki, T., Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. inst. H. Poincaré anal. non linéaire, 9, 367-397, (1992) · Zbl 0785.35045 [23] Tarantello, G., Multiple condensate solutions for the chern – simons – higgs theory, J. math. phys., 37, 3769-3796, (1996) · Zbl 0863.58081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.