238 research outputs found
A remark on energy estimates concerning extremals for Trudinger-Moser inequalities on a disc
In this short note, we generalized an energy estimate due to
Malchiodi-Martinazzi (J. Eur. Math. Soc. 16 (2014) 893-908) and
Mancini-Martinazzi (Calc. Var. (2017) 56:94). As an application, we used it to
reprove existence of extremals for Trudinger-Moser inequalities of
Adimurthi-Druet type on the unit disc. Such existence problems in general cases
had been considered by Yang (Trans. Amer. Math. Soc. 359 (2007) 5761-5776; J.
Differential Equations 258 (2015) 3161-3193) and Lu-Yang (Discrete Contin. Dyn.
Syst. 25 (2009) 963-979) by using another method.Comment: 8 page
A gradient flow for the prescribed Gaussian curvature problem on a closed Riemann surface with conical singularity
In this note, we prove that the abstract gradient flow introduced by
Baird-Fardoun-Regbaoui \cite{BFR}is well-posed on a closed Riemann surface with
conical singularity. Long time existence and convergence of the flow are proved
under certain assumptions. As an application, the prescribed Gaussian curvature
problem is solved when the singular Euler characteristic of the conical surface
is non-positive.Comment: 15 page
Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two
Combining Carleson-Chang's result with blow-up analysis, we prove existence
of extremal functions for certain Trudinger-Moser inequalities in dimension
two. This kind of inequality was originally proposed by Adimurthi and O. Druet,
extended by the author to high dimensional case and Riemannian surface case,
generalized by C. Tintarev to wider cases including singular form and by M. de
Souza and J. M. do \'O to the whole Euclidean space R^2. In addition to the
Euclidean case, we also consider the Riemannian surface case. The results in
the current paper complement that of L. Carleson and A. Chang, M. Struwe, M.
Flucher, K. Lin, and Adimurthi-Druet, our previous ones, and part of C.
Tintarev.Comment: 28 page
Quantization for an elliptic equation with critical exponential growth on compact Riemannian surface without boundary
In this paper, using blow-up analysis, we prove a quantization result for an
elliptic equation with critical exponential growth on compact Riemannian
surface without boundary. Similar results for Euclidean space were obtained by
Adimurthi-Struwe \cite{Adi-Stru}, Druet \cite{Druet}, Lamm-Robert-Struwe
\cite{L-R-S}, Martinazzi \cite{Mart}, Martinazzi-Struwe \cite{Mar-Stru}, and
Struwe \cite{Struwe} respectively.Comment: 43 page
Adams type inequalities and related elliptic partial differential equations in dimension four
Motivated by Ruf-Sani's recent work, we prove an Adams type inequality and a
singular Adams type inequality in the whole four dimensional Euclidean space.
As applications of those inequalities, a class of elliptic partial differential
equations are considered. Existence of nontrivial weak solutions and
multiplicity results are obtained via the mountain-pass theorem and the
Ekeland's variational principle. This is a continuation of our previous work
about singular Trudinger-Moser type inequality.Comment: 27 page
Smoothing metrics on closed Riemannian manifolds through the Ricci flow
Under the assumption of the uniform local Sobolev inequality, it is proved
that Riemannian metrics with an absolute Ricci curvature bound and a small
Riemannian curvature integral bound can be smoothed to having a sectional
curvature bound. This partly extends previous a priori estimates of Ye Li (J.
Geom. Anal. 17 (2007) 495-511; Advances in Mathematics 223 (2010) 1924-1957).Comment: 14 page
A Trudinger-Moser inequality on compact Riemannian surface involving Gaussian curvature
Motivated by a recent work of X. Chen and M. Zhu (Commun. Math. Stat., 1
(2013) 369-385), we establish a Trudinger-Moser inequality on compact
Riemannian surface without boundary. The proof is based on blow-up analysis
together with Carleson-Chang's result (Bull. Sci. Math. 110 (1986) 113-127).
This inequality is different from the classical one, which is due to L. Fontana
(Comment. Math. Helv., 68 (1993) 415-454), since the Gaussian curvature is
involved. As an application, we improve Chen-Zhu's result as follows: A
modified Liouville energy of conformal Riemannian metric has a uniform lower
bound, provided that the Euler characteristic is nonzero and the volume of the
conformal surface has a uniform positive lower bound.Comment: 17 page
Trudinger-Moser inequalities on a closed Riemannian surface with the action of a finite isometric group
Let be a closed Riemannian surface, be the
usual Sobolev space, be a finite isometric group acting on
, and be a function space including all
functions with and
for all and all . Denote
the number of distinct points of the set
by and . Let be the
first eigenvalue of the Laplace-Beltrami operator on the space
. Using blow-up analysis, we prove that if
and , then there holds
if
, or and , then the above supremum is infinity; if
and , then the above supremum
can be attained. Moreover, similar inequalities involving higher order
eigenvalues are obtained. Our results partially improve original inequalities
of J. Moser \cite{Moser}, L. Fontana \cite{Fontana} and W. Chen \cite{Chen-90}.Comment: 24 page
An improved Hardy-Trudinger-Moser inequality
Let be the unit disc in , be the
completion of under the norm
Denote , where stands for
the -norm. Using blow-up analysis, we prove that for any
, ,
and that the above supremum can be
attained by some function with
. This improves an earlier result of
G. Wang and D. Ye [28].Comment: 18 page
Nonexistence of quasi-harmonic sphere with large energy
Nonexistence of quasi-harmonic spheres is necessary for long time existence
and convergence of harmonic map heat flows. Let be a complete
noncompact Riemannian manifolds. Assume the universal covering of
admits a nonnegative strictly convex function with polynomial growth. Then
there is no quasi-harmonic spheres u:\mathbb{R}^n\ra N such that
\lim_{r\ra\infty}r^ne^{-\f{r^2}{4}}\int_{|x|\leq r}e^{-\f{|x|^2}{4}}|\nabla
u|^2dx=0. This generalizes a result of the first named author and X. Zhu
(Calc. Var., 2009). Our method is essentially the Moser iteration and thus very
simple.Comment: 7 page
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