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April  2021, 20(4): 1721-1735. doi: 10.3934/cpaa.2021038

Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary

Mengjie Zhang

School of Mathematics, Renmin University of China, Beijing 100872, China

Received  July 2020 Revised  January 2021 Published  March 2021

Fund Project: Supported by the Outstanding Innovative Talents Cultivation Funded Programs 2020 of Renmin University of China

In this paper, we establish several trace Trudinger-Moser inequalities and obtain the corresponding extremals on a compact Riemann surface
$ ( \Sigma,g) $
 with smooth boundary
$ \partial\Sigma $
. To be exact, let
$ \lambda_1(\partial\Sigma) $
 denotes the first eigenvalue of the Laplace-Beltrami operator
$ \Delta _ { g} $
 on
$ \partial \Sigma $
. Moreover, for any
$ 0\leq\alpha<\lambda_1(\partial\Sigma) $
, we set
$ \mathcal { H } = \{ u \in W^{1,2} ( \Sigma, g) : \left(\int _{\Sigma} |\nabla_g u|^2 dv_g -\alpha \int _{\partial\Sigma} {u^2}ds_g \right)^{1/2}\leq 1 \ \, \mathrm{and}\, \int _{\partial\Sigma} {u}\,ds_g = 0 \} $
, where
$ W^{1,2}(\Sigma, g) $
 is the usual Sobolev space. By the method of blow-up analysis, we first prove the supremum
$ \begin{equation*} \sup\limits_{ u \in \mathcal { H } }\int _ { \partial\Sigma } e ^ {\pi u^ 2} ds_g \end{equation*} $
is attained by some function
$ u_\alpha \in \mathcal{H}\cap C^{\infty} \left(\overline{ \Sigma}\right) $
. Further, we extend the result to the case of higher order eigenvalues. The results generalize those of Li-Liu [9] and Yang [19, 20].
Keywords: Trudinger-Moser inequality, Riemann surface, blow-up analysis, extremal function.
Mathematics Subject Classification: Primary: 46E35, 58J05, 58J32.
Citation: Mengjie Zhang. Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1721-1735. doi: 10.3934/cpaa.2021038
References:
[1]

Adimurthi and M. Struwe, Global compactness properties of semilinear elliptic equation with critical exponential growth, J. Funct. Anal., 175 (2000), 125-167.  doi: 10.1006/jfan.2000.3602.  Google Scholar

[2]

T. Aubin, Sur la function exponentielle, C. R. Acad. Sci. Paris Sér. A-B, 270 (1970), A1514-A1516.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and PDEs, Springer, 2011.  Google Scholar

[4]

L. Carleson and S. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127.   Google Scholar

[5]

P. Cherrier, Une inégalité de Sobolev sur les variétés riemanniennes, Bull. Sci. Math., 103 (1979), 353-374.   Google Scholar

[6]

W. Ding, J. Jost, J. Li and G. Wang, The differential equation $\Delta u = 8\pi-8\pi he^u$ on a compact Riemann Surface, Asian J. Math., 1 (1997), 230-248.  doi: 10.4310/AJM.1997.v1.n2.a3.  Google Scholar

[7]

L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68 (1993), 415-454.  doi: 10.1007/BF02565828.  Google Scholar

[8]

Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.   Google Scholar

[9]

Y. Li and P. Liu, Moser-Trudinger inequality on the boundary of compact Riemannian surface, Math. Z., 250 (2005), 363-386.  doi: 10.1007/s00209-004-0756-7.  Google Scholar

[10]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[11]

P. Liu, A Moser-Trudinger Type Inequality and Blow-Up Analysis on Compact Riemannian Surface, Max-Plank Institute, Germany, 2005. Google Scholar

[12]

G. Mancini and L. Martinazzi, Extremals for fractional Moser-Trudinger inequalities in dimension 1 via harmonic extensions and commutator estimates, Adv. Nonlinear Stud., 20 (2020), 599-632.  doi: 10.1515/ans-2020-2089.  Google Scholar

[13]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[14]

B. Osgood, R. Phillips and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal., 80 (1988), 148-211.  doi: 10.1016/0022-1236(88)90070-5.  Google Scholar

[15]

J. Peetre, Espaces d'interpolation et $\mathrm{th\acute{e}or\grave{e}me}$ de Soboleff, Ann. Inst. Fourier, 16 (1966), 279-317.   Google Scholar

[16]

S. Poho$\mathrm{\check{z}}$aev, The Sobolev embedding in the special case $p\ell = n$, Proceedings of the technical scientific conference on advances of scientific reseach 1964-1965, Math. sections, Moscov. Energet. Inst., (1965), 158-170. Google Scholar

[17]

N. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[18]

Y. Yang, Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary, Pacific J. Math., 227 (2006), 177-200.  doi: 10.2140/pjm.2006.227.177.  Google Scholar

[19]

Y. Yang, A sharp form of trace Moser-Trudinger inequality on compact Riemannian surface with boundary, Math. Z., 255 (2007), 373-392.  doi: 10.1007/s00209-006-0035-x.  Google Scholar

[20]

Y. Yang, Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Differ. Equ., 258 (2015), 3161-3193.  doi: 10.1016/j.jde.2015.01.004.  Google Scholar

[21]

Y. Yang and J. Zhou, Blow-up analysis involving isothermal coordinates on the boundary of compact Riemann surface, arXiv: 2009.09626. Google Scholar

[22]

V. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.   Google Scholar

show all references

References:
[1]

Adimurthi and M. Struwe, Global compactness properties of semilinear elliptic equation with critical exponential growth, J. Funct. Anal., 175 (2000), 125-167.  doi: 10.1006/jfan.2000.3602.  Google Scholar

[2]

T. Aubin, Sur la function exponentielle, C. R. Acad. Sci. Paris Sér. A-B, 270 (1970), A1514-A1516.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and PDEs, Springer, 2011.  Google Scholar

[4]

L. Carleson and S. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113-127.   Google Scholar

[5]

P. Cherrier, Une inégalité de Sobolev sur les variétés riemanniennes, Bull. Sci. Math., 103 (1979), 353-374.   Google Scholar

[6]

W. Ding, J. Jost, J. Li and G. Wang, The differential equation $\Delta u = 8\pi-8\pi he^u$ on a compact Riemann Surface, Asian J. Math., 1 (1997), 230-248.  doi: 10.4310/AJM.1997.v1.n2.a3.  Google Scholar

[7]

L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68 (1993), 415-454.  doi: 10.1007/BF02565828.  Google Scholar

[8]

Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.   Google Scholar

[9]

Y. Li and P. Liu, Moser-Trudinger inequality on the boundary of compact Riemannian surface, Math. Z., 250 (2005), 363-386.  doi: 10.1007/s00209-004-0756-7.  Google Scholar

[10]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[11]

P. Liu, A Moser-Trudinger Type Inequality and Blow-Up Analysis on Compact Riemannian Surface, Max-Plank Institute, Germany, 2005. Google Scholar

[12]

G. Mancini and L. Martinazzi, Extremals for fractional Moser-Trudinger inequalities in dimension 1 via harmonic extensions and commutator estimates, Adv. Nonlinear Stud., 20 (2020), 599-632.  doi: 10.1515/ans-2020-2089.  Google Scholar

[13]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[14]

B. Osgood, R. Phillips and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal., 80 (1988), 148-211.  doi: 10.1016/0022-1236(88)90070-5.  Google Scholar

[15]

J. Peetre, Espaces d'interpolation et $\mathrm{th\acute{e}or\grave{e}me}$ de Soboleff, Ann. Inst. Fourier, 16 (1966), 279-317.   Google Scholar

[16]

S. Poho$\mathrm{\check{z}}$aev, The Sobolev embedding in the special case $p\ell = n$, Proceedings of the technical scientific conference on advances of scientific reseach 1964-1965, Math. sections, Moscov. Energet. Inst., (1965), 158-170. Google Scholar

[17]

N. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028.  Google Scholar

[18]

Y. Yang, Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary, Pacific J. Math., 227 (2006), 177-200.  doi: 10.2140/pjm.2006.227.177.  Google Scholar

[19]

Y. Yang, A sharp form of trace Moser-Trudinger inequality on compact Riemannian surface with boundary, Math. Z., 255 (2007), 373-392.  doi: 10.1007/s00209-006-0035-x.  Google Scholar

[20]

Y. Yang, Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Differ. Equ., 258 (2015), 3161-3193.  doi: 10.1016/j.jde.2015.01.004.  Google Scholar

[21]

Y. Yang and J. Zhou, Blow-up analysis involving isothermal coordinates on the boundary of compact Riemann surface, arXiv: 2009.09626. Google Scholar

[22]

V. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.   Google Scholar

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