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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 |
$ ( \Sigma,g) $ |
$ \partial\Sigma $ |
$ \lambda_1(\partial\Sigma) $ |
$ \Delta _ { g} $ |
$ \partial \Sigma $ |
$ 0\leq\alpha<\lambda_1(\partial\Sigma) $ |
$ \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 \} $ |
$ W^{1,2}(\Sigma, g) $ |
$ \begin{equation*} \sup\limits_{ u \in \mathcal { H } }\int _ { \partial\Sigma } e ^ {\pi u^ 2} ds_g \end{equation*} $ |
$ u_\alpha \in \mathcal{H}\cap C^{\infty} \left(\overline{ \Sigma}\right) $ |
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.
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