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

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

Received  July 2020 Revised  January 2021 Published  April 2021 Early access  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].
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. DingJ. JostJ. 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. OsgoodR. 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. DingJ. JostJ. 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. OsgoodR. 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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