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

Previous Article
Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems
CPAA Home
This Issue
Next Article
KdV-type equation limit for ion dynamics system

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].

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 and 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.
[2]	T. Aubin, Sur la function exponentielle, C. R. Acad. Sci. Paris Sér. A-B, 270 (1970), A1514-A1516.
[3]	H. Brezis, Functional Analysis, Sobolev Spaces and PDEs, Springer, 2011.
[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.
[5]	P. Cherrier, Une inégalité de Sobolev sur les variétés riemanniennes, Bull. Sci. Math., 103 (1979), 353-374.
[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.
[7]	L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68 (1993), 415-454. doi: 10.1007/BF02565828.
[8]	Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.
[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.
[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.
[11]	P. Liu, A Moser-Trudinger Type Inequality and Blow-Up Analysis on Compact Riemannian Surface, Max-Plank Institute, Germany, 2005.
[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.
[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.
[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.
[15]	J. Peetre, Espaces d'interpolation et $\mathrm{th\acute{e}or\grave{e}me}$ de Soboleff, Ann. Inst. Fourier, 16 (1966), 279-317.
[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.
[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.
[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.
[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.
[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.
[21]	Y. Yang and J. Zhou, Blow-up analysis involving isothermal coordinates on the boundary of compact Riemann surface, arXiv: 2009.09626.
[22]	V. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.

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.
[2]	T. Aubin, Sur la function exponentielle, C. R. Acad. Sci. Paris Sér. A-B, 270 (1970), A1514-A1516.
[3]	H. Brezis, Functional Analysis, Sobolev Spaces and PDEs, Springer, 2011.
[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.
[5]	P. Cherrier, Une inégalité de Sobolev sur les variétés riemanniennes, Bull. Sci. Math., 103 (1979), 353-374.
[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.
[7]	L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68 (1993), 415-454. doi: 10.1007/BF02565828.
[8]	Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ., 14 (2001), 163-192.
[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.
[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.
[11]	P. Liu, A Moser-Trudinger Type Inequality and Blow-Up Analysis on Compact Riemannian Surface, Max-Plank Institute, Germany, 2005.
[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.
[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.
[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.
[15]	J. Peetre, Espaces d'interpolation et $\mathrm{th\acute{e}or\grave{e}me}$ de Soboleff, Ann. Inst. Fourier, 16 (1966), 279-317.
[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.
[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.
[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.
[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.
[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.
[21]	Y. Yang and J. Zhou, Blow-up analysis involving isothermal coordinates on the boundary of compact Riemann surface, arXiv: 2009.09626.
[22]	V. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math. Docl., 2 (1961), 746-749.

[1]	Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963
[2]	Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110
[3]	Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505
[4]	Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure and Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011
[5]	Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378
[6]	Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure and Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006
[7]	Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1747-1756. doi: 10.3934/dcdss.2020452
[8]	Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455
[9]	Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031
[10]	Shengbing Deng, Xingliang Tian. On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022071
[11]	Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941
[12]	Van Hoang Nguyen. The Hardy–Moser–Trudinger inequality via the transplantation of Green functions. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3559-3574. doi: 10.3934/cpaa.2020155
[13]	Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121
[14]	Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212
[15]	Yamin Wang. On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4257-4268. doi: 10.3934/cpaa.2020191
[16]	Sami Aouaoui, Rahma Jlel. Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 781-813. doi: 10.3934/dcds.2021137
[17]	Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391
[18]	Changliang Zhou, Chunqin Zhou. On the anisotropic Moser-Trudinger inequality for unbounded domains in $ \mathbb R^{n} $. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 847-881. doi: 10.3934/dcds.2020064
[19]	C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88
[20]	Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Mengjie Zhang
on Google Scholar
Mengjie Zhang

2021 Impact Factor: 1.273

Tools

Article outline

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Mengjie Zhang
on Google Scholar
Mengjie Zhang

[Back to Top]