R E S E A R C H
Open Access
A sharp Trudinger type inequality for
harmonic functions and its applications
Yili Tan
1, Yongli An
2, Hong Wang
1and Jing Liu
3**Correspondence:
[email protected] 3Department of Basic Courses,
Zhejiang New Century College of Economy and Trade, Hangzhou, 311122, China
Full list of author information is available at the end of the article
Abstract
The present paper introduces a sharp Trudinger type inequality for harmonic functions based on the Cauchy-Riesz kernel function, which includes modified Poisson type kernel in a half plane considered by Xu et al. (Bound. Value Probl. 2013:262, 2013). As applications, we not only obtain Morrey representations of continuous linear maps for harmonic functions in the set of all closed bounded convex nonempty subsets of any Banach space, but also deduce the representation for set-valued maps and for scalar-valued maps of Dunford-Schwartz.
Keywords: Trudinger type inequality; Cauchy-Riesz kernel function; modified Poisson type kernel; Morrey representation
1 Introduction
The Trudinger inequality problem (TIP) is generated from the method of mathemati-cal physics and nonlinear programming. It has considerable applications in many fields such as physics, mechanics, engineering, economic decision, control theory and so on. Trudinger inequality is actually a system of partial differential equations. Especially, physi-cists have long been using so-called singular functions such as the Dirac delta functionδ, although these cannot be properly defined within the framework of classical function the-ory. The Dirac delta functionδ(x–ξ) is equal to zero everywhere except at a fixed pointξ. According to the classical definition of a function and an integral, these conditions are in-consistent. In elementary particle physics, one found the need to evaluateδwhen
calcu-lating the transition rates of certain particle interactions []. In [], a definition of product of distributions was given using delta sequences. In [], Bremermann used the Cauchy representations of distributions with compact support to define√δ+and logδ+.
Unfortu-nately, his definition did not carry over to√δand logδ. In , Gel’fand and Shilov [] definedδ(k+)(P) for an infinitely differentiable functionP(x,x, . . . ,xn) such that theP=
hypersurface had no singular points, where
P=P(x,x, . . . ,xp+q) =x+x+· · ·+xp–xp+–· · ·–xp+q, (.)
p+q=nis the dimension of the Euclidean spaceRn, theP= hypersurface was a
hy-percone with a singular point (the vertex) at the origin. Then they also defined the gen-eralized functionsδ(k+)(P) andδ(k+)(P) as in the casesp,q< andp,q= , respectively. By the Sobolev embedding theorem, it was well known that the Sobolev spaceH(G)
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was embedded in all Lebesgue spacesLp(G) for <p<∞but not inL∞(G). Moreover,
δ(k)(P) andδ(k)(P) functions were in the so-called Orlicz space, i.e., their exponential pow-ers were integrable functions. Precisely, Ruf established the Trudinger inequality (see [, Theorem .]). However, the best possible constantβin it was much more interesting and was not exhibited until the paper [] of Li and Ruf. In fact, using the symmetriza-tion argument to reduce to the one-dimensional case, they established a result which is now called the Trudinger inequality. It was refined and extended to many different set-tings. For instance, a singular Trudinger inequality which was an interpolation of Hardy inequality and Trudinger inequality was studied by Su in []. Meanwhile, Su further stud-ied the residue of the generalized functionGλ, whereλwas a nonnegative real number.
Very recently, Yan et al. [] have succeeded to establish the sharp constants and extremal functions of the Trudinger inequality on the Heisenberg group and generalized the dis-tributional product of Dirac’s delta in a hypercone. Furthermore, Li and Vetro [] used a much simpler method of deriving the productf(r– )·δ(k+)(r+ ) for all nonnegative integerkandr= (x
+x+· · ·+xp+q)/. And they found the productPn·δ(k+)(P) as well
as a general productf(P)·δ(k+)(P), wheref was aC∞ -function onR+. The other study of
the products of particular distributions and the development of others’ works can be seen in [, ].
By using augmented Riesz decomposition methods developed by Xie and Viouonu [], the purpose of this paper is to obtain a sharp Trudinger type inequality for harmonic func-tions based on a Cauchy-Riesz kernel function and study the productGl(P)·δ(k+)(P) and then study a more general product off(P)·δ(k+)(P), wheref is aC∞
-function onRand
δ(k+)(G) is the Dirac delta function withk-derivatives. As applications, we not only obtain Morrey representations of continuous linear maps for harmonic functions in the set of all closed bounded convex nonempty subsets of any Banach space, but also deduce the rep-resentation for set-valued maps and for scalar-valued maps of Dunford-Schwartz. Before proceeding to our main results, the following definitions and concepts are required.
2 Preliminaries
Definition . Letx= (x,x, . . . ,xn) be a point inRn, whereRnis then-dimensional
Eu-clidean space. The hypersurfaceG=G(m,x) is defined by
G=G(m,x) = p+
i=
xi m
– p+q
j=p+
xj m
, (.)
wheremis a positive integer.
The hypersurfaceGis due to Kananthai and Nonlaopon []. We observe that putting m= in (.), we obtain
G=G(,x) =
p+
i=
xi –
p+q
j=p+
xj =P(x) =P, (.)
where the quadratic formPis due to Gel’fand and Shilov [] and is given by (.). The hy-persurfaceG= is a generalization of a hyperconeP= with a singular point (the vertex) at the origin.
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Definition . Let gradG= that means there is no singular point onG= . Then we define
δ(k+)(G),φ=
δ(k+)(G)φ(x)dx, (.)
whereδ(k+)is the Dirac delta function with (k+ )-derivatives,φis any real function in
the Schwartz spaceS,x= (x,x, . . . ,xn)∈Rnanddx=dxdxdxn. In a sufficiently small
neighborhoodUof any point (x,x, . . . ,xn) of the hypersurfaceG= , we can introduce a
new coordinate system such thatG= becomes one of the coordinate hypersurfaces. For this purpose, we writeG=uand choose the remaininguicoordinates (i= , , . . . ,n) for
which the Jacobian
D
x u ≤,
where
D
x u =
∂(x,x, . . . ,xp+q)
∂(G,u, . . . ,up+q)
.
Thus (.) can be written as
δ(k+)(G),φ= (–)k+ ∂
k–
∂Gk
φD
u
x G=dudu· · ·dun. (.) The proof of the following lemma is given in [].
Lemma . Given the hypersurface
G= p+
i=
xi m
– p+q
j=p+
xj m
,
where p+q=n and m is a positive integer.If we transform to bipolar coordinates defined by
x=rωp+q, . . . ,xp=rωq+, xq+=sωq–, . . . ,xp+q=sω,
where
p+
i=
ωi =
and
p+q
j=p+
ωj= .
Then the hypersurface G can be written by
G=rm–sm,
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and we obtain
δ(k+)(G),φ= ∞
(m+ )sm
∂ ∂s
k–
sq–mψ(r,s) m
s=r
rp–dr (.) or
δ(k+)(G),φ= (–)k+ ∞
(m+ )sm–
∂ ∂r
k–
ψ(r,s) m
r=s
sq–ds, (.)
where
ψ(r,s) =
s(r)d(p)d(q),
and d(p) and d(q) are the elements of surface area on the unit sphere inRp andRq,
respectively.
Now, we assume thatφvanishes in the neighborhood of the origin, so that these integrals will converge for anyk. Now, for
p+q– m– ≤mk
or
k≥
m+ (p+q– – m),
the integrals in (.) converge for anyφ(x)∈S. Similarly, for
p+q– m– ≤mk–
or
k≥
m– (p+q– m– ),
the integrals in (.) also converge for anyφ(x)∈S. Thus we take (.) and (.) to be the defining equation forδ(k+)(G). On the other hand, if
k≤
m– (p+q– m– ),
we shall defineδ∗(G),φandδ∗(G),φas the regularization of (.) and (.), respec-tively. Forp≤ andq≤, the generalized functionsδ∗(k+)(G) andδ∗(k+)(G) are defined by
δ∗(k+)(G),φ= ∞
(m+ )sm
∂ ∂s
k–ψ
(r,s) m
s=r
rp–dr (.)
for all
k≤
m– (p+q– m– ),
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we have
δ∗(k+)(G),φ= (–)k+ ∞
(m+ )sm–
∂ ∂r
k–ψ
(r,s) m
r=s
ds (.)
for
k≤
m– (p+q– m– ).
In particular, form= , δ∗(k+)(G) is reduced toδ(k+)(G), andδ∗(k+)(G) is reduced to
δ(k+)(G) (see [, p.]).
3 Main results
Assume that bothp≤ andq≤. Let
G(x) =G(x,x, . . . ,xn) =
x+x+· · ·+xp+m–xp++· · ·+xp+qm,
then theG= hypersurface is a hypercone with a singular point (the vertex) at the origin. We start by assuming thatφ(x) vanishes in a neighborhood of the origin. The distribution
δ(k+)(G) is defined by
δ(k+)(G),φ= (–)k+ ∂
k–
∂Gk–
rm–G q
m–φ
G=
rp+qdr d(p)d(q), (.)
which is convergent. Furthermore, if we transform fromGto
s=rm+–Gm+,
then we know that
∂ ∂G= –
(m+ )sm–∂
∂s.
We may write this in the form
δ(k+)(G),φ= (m+ )sm
∂ ∂s
k– φ
m
s=r
rp+qdr d(p)d(q). (.)
Let us now define
ψ(r,s) =
s(r)d(p)d(q).
Hence,
δ(k+)(G),φ= ∞
(m+ )sm
∂ ∂s
k–
sq–mψ(r,s) m
s=r
rp–dr. (.)
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Theorem . The product of Glandδ(k+)(G)exists and
Gl·δ(k+)(G) = ⎧ ⎨ ⎩
(–)l+ (k+)!
k–l+δk–l+(G) if k≥l,
if k<l.
(.)
Proof (.) gives that
Gl·δ(k+)(G),φ
= (–)k+ ∂
k–
∂Gk–
Glrm–G q
m–φ
G=
rp–dr d(p)d(q)
= ∞
(m+ )sm
∂ ∂s
k–
rm–smlψ(r,s) m
s=r
rp+qdr.
Substitutingu=rm–,v=sm+and puttingψ(r,s) =ψ(u,v), we have
Gl·δ(k+)(G),φ
= m
∞ ∂ ∂v k–
(u–v)lvqm++–ψ
(u,v)
u=v
uqm++–du.
It is obvious that
∂k– ∂vk–
(u–v)lvqm++–ψ
(u,v)
u–v
= k i= k i D i
v(u–v)lDkv–i
vqm++–ψ(u,v) u–v
=
i<l
k i D
i
v(u–v)lDkv–i
vqm++–ψ(u,v) u–v
+
k l D
i
v(u–v)lDkv–i
vqm++–ψ
(u,v)
u–v
+
i>l
k i D
i
v(u–v)lDkv–i
vqm++–ψ(u,v) u–v
=I+I+I,
where
Div=∂/∂vi.
It follows that
I=I=
sincei=l. As forI, we obtain
I=
⎧ ⎨ ⎩
(–)l(k+)!
k–l+lD k–l v {v
q+
m+–ψ(u,v)} ifk≥l,
ifk<l.
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SubstitutingIback and using (.), we obtain
Gl·δ(k+)(G) = ⎧ ⎨ ⎩
(–)l(k+)!
k–l δ
k–l+(G) ifk≥l,
ifk<l,
which completes the proof of the theorem.
Example . By letting
m= , n= , p=
in (.),l= andk= in (.), we have
x·δx= –δx.
Obviously, we can extend Theorem . to a more general product as follows.
Theorem . Let f be aC∞-function onR.Then the product of f(G)andδ(k+)(G)exists and
f(G)δ(k+)(G) =
k
i=
k i = (–)
if(i)()δ(k–i)(G).
Proof LetGl=f(G) and use Theorem .. Moreover, note that
∂k–
∂vk–
f(u+v)vqm++–ψ
(u,v)
u–v
=
k+
i=
k
i D
i
vf(u+v)Dkv–i
vqm++–ψ
(u,v)
u+v
=
k+
i=
k
i (–)
if(i)()Dk–i v
vqm++–ψ
(u,v)
u–v.
In particular, we know that
sinG·δ(k+)(G) =
k+
i=
k i (–)
isin(i+ )π
δ
(k–i)(G) (.)
and
eG·δ(k+)(G) =
k+
i=
k
i (–)
iδ(k–i)(G). (.)
Example . By letting
m= , n= , p=
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in (.) andk= in (.), we have
sinx·δx= –δx+δx.
Similarly, by lettingm= ,n= andp= in (.) andk= in (.), we have
ex·δ()x=δ()x– δx+ δ(x) – δx+δx.
4 Numerical simulations
In this section, we give the bifurcation diagrams, phase portraits of model (.) to confirm the above theoretic analysis and show the new interesting complex dynamical behaviors by using numerical simulations. The bifurcation parameters are considered in the following two cases.
In model (.) we chooseμ= .,N= .,β= .,γ= .,h∈[, .] and the initial value (S,I) = (., .). We see that model (.) has only one positive equilibriumE. By
calculation we have
E
S∗,I∗=E(., .),
α= –., α= ., h=
– √,
and
(μ,N,β,h,γ)∈M,
which shows the correctness of Theorem .. From Theorem ., we see that equilibrium E(., .) is stable for
h< – √
,
,
and loses its stability whenh=–
√
, . If
– √,
<h< .,
then there exist the orbits. Moreover, orbits, orbits and period- orbits appear in the rangeh∈[., .). At last, the nperiod orbits disappear and
the dynamical behaviors are from non-period orbits to the chaotic set with the increas-ingh. We also can find that the rangehis decreasing with the doubled increasing of the period orbits, which indicates the Feigenbaum constantδ. The dynamical behavior pro-cesses from period- orbit to chaos sets show the self-similar characteristics. Further, the period-doubling transition leads to the chaos sets.
5 Conclusions
In this paper, we obtained the representation of continuous linear maps in the set of all closed bounded convex nonempty subsets of any Banach space. Meanwhile, we deduced the Riesz integral representation results for set-valued maps, for vector-valued maps of Diestel-Uhl and for scalar-valued maps of Dunford-Schwartz.
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Acknowledgements
We would like to thank the editor, the associate editor and the anonymous referees for their careful reading and constructive comments which have helped us to significantly improve the presentation of the paper. This paper was written during a short stay of the corresponding author at the School of Mathematics of Osaka Kyoyobu University as a visiting professor. He would also like to thank the School of Mathematics and their members for their warm hospitality. This work was supported by the Natural Science Foundation of China (Grant No. 11401160) and the Natural Science Foundation of Hebei Province (No. A2015209040).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YT designed the solution methodology. YT and YA prepared the revised manuscript according to the referee reports. HW participated in the design of the study. JL drafted the manuscript. All authors read and approved the final manuscript.
Author details
1College of Science, North China University of Science and Technology, Tangshan, 063210, China.2College of Information
Engineering, North China University of Science and Technology, Tangshan, 063210, China.3Department of Basic Courses, Zhejiang New Century College of Economy and Trade, Hangzhou, 311122, China.
Publisher’s Note
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Received: 29 May 2017 Accepted: 20 September 2017
References
1. Xu, G, Yang, P, Zhao, T: Dirichlet problems of harmonic functions. Bound. Value Probl.2013, 262 (2013) 2. Gasiorowicz, S: Elementary Particle Physics. Wiley, New York (1966)
3. Antosik, P, Mikusinski, J, Sikorski, R: Theory of Distributions the Sequential Approach. PWN, Warsaw (1973) 4. Bremermann, JH: Distributions, Complex Variables, and Fourier Transforms. Addison-Wesley, Reading (1965) 5. Gelfand, IM, Shilov, GE: Generalized Functions, vol. 1. Academic Press, New York (1964)
6. Ruf, B: A sharp Trudinger-Moser type inequality for unbounded domains inR2. J. Funct. Anal.219(2), 340-367 (2005) 7. Li, Y, Ruf, B: A sharp Trudinger-Moser type inequality for unbounded domains inRn. Indiana Univ. Math. J.57(1),
451-480 (2008)
8. Su, B: Dirichlet problem for the Schrödinger operator in a half space. Abstr. Appl. Anal.2012, Article ID 578197 (2012) 9. Yan, Z, Yan, G, Miyamoto, I: Fixed point theorems and explicit estimates for convergence rates of continuous time
Markov chains. Fixed Point Theory Appl.2015, 197 (2015)
10. Li, Z, Vetro, M: Levin’s type boundary behaviors for functions harmonic and admitting certain lower bounds. Bound. Value Probl.2015, 159 (2015)
11. Pang, S, Ychussie, B: Matsaev type inequalities on smooth cones. J. Inequal. Appl.2015, 108 (2015)
12. Xie, X, Viouonu, CT: Some new results on the boundary behaviors of harmonic functions with integral boundary conditions. Bound. Value Probl.2016, 136 (2016)
