Minkowski problem Variational characterization for the planar dual Journal of Functional Analysis

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

Variational characterization for the planar dual Minkowski problem

Yong Huang^a, Yongsheng Jiang^b,∗

aInstituteofMathematics,HunanUniversity,Changsha410082,China

bSchoolofStatisticsandMathematics,ZhongnanUniversityofEconomics and Law,Wuhan430073,China

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received21August2018 Accepted6February2019 Availableonline13February2019 CommunicatedbyT.Schlumprecht

MSC:

52A38 35J20 49J40

Keywords:

DualMinkowskiproblem Variationalmethods Functionalinequality

In thispaper, wegive a variational analysis to the planar dualMinkowskiproblemintheSobolevspace.Withthenew variationalcharacterization,wecandealwithexistenceresults for prescribed not necessarily positive data. Meanwhile, functional inequalities and multiple solutions are also obtained.

©2019ElsevierInc.Allrightsreserved.

1. Introduction

In convex geometric analysis, the Brunn–Minkowski theory and its generalization were established after the contribution of Brunn [11], Minkowski [45,46], Hilbert [25], Alekesandrov [2–6], Fenchel & Jessen [20] and so on. The central tasks are to solve Minkowskitypeproblemsandtoestablishrelatedgeometricinequalities,see,forexample

* Correspondingauthor.

E-mailaddresses:[email protected](Y. Huang),[email protected](Y. Jiang).

https://doi.org/10.1016/j.jfa.2019.02.010

0022-1236/©2019ElsevierInc. Allrightsreserved.

[42,51]. Analogy to the classical Brunn–Minkowski theory, the authors in [28] ask the following characterizationproblem.

Dual Minkowski problem.For any q ∈ R, if μ is afinite Borel measure onS^N−1, find necessary and sufficient conditions on μ so that it is the q-th dual curvature measure C˜q(K,·) ofaconvexbodyK inR^N.

WeplantogiveapurevariationalanalysistothedualMinkowskiproblem.Toestab- lish mathematical models for these celebrated problems, the supportfunction u_K and radical functionρK ofaconvexbodyKinR^N withorigininitsinteriorareintroduced by

uK(θ) = max{θ·y:y∈K}, θ∈R^N; ρK(ξ) = max{λ:λξ∈K}, ξ∈R^N\ {0}. Withthese notations,thefirstauthorwithLutwak,Yang andZhangin[28] introduced thenewgeometric measure,theq-thdual curvaturemeasure

C_q(K, η) = 1 n

S^N−1 α^∗_K(η)

ρ^q_K(ξ)dξ,

where α^∗_K denotes reverse radial Gauss image of K insphere S^N−1. Suppose that K hasaC²boundarywitheverywherepositivecurvature,∇denotesthegradientoperator with respect to a frameon S^N−1 andE is thestandard Riemannianmetricmatrix on S^N−1.Thedensityofdual curvaturemeasureC_q(K,·) equals to

(u²_K+|∇u_K|²)^q−N2 u_Kdet(∇²u_K+u_KE),

whichistheintegrandofdualquermassintegral,afundamentalgeometricinvariantinthe dualBrunn–Minkowskitheory[28,41,51].Whenthemeasureμhasadensityfunctiong, thedualMinkowskiproblemisequivalenttothesolvabilityoffollowingMonge–Ampère equation

det(∇²u_K+u_KE) =g(θ)(u²_K+|∇uK|²)^N−q2

u_K , θ∈S^N−1. (1.1)

Analytically, it is nontrivial to give a variational functional for the general nonlinear differential equationwithgradientterms,evenforthestandardmodel

−Δu=c(x) +|∇u|^p. (1.2)

Only whenp= 2,thechangeof variablethatv=e^u−1 isvalidfortransforming (1.2) to alinearequation[31,37] as

−Δv=c(x)(v+ 1),

whichhasavariationalstructure.Besides thefullynonlinearMonge–Ampèreoperator, (1.1) with q = N owns more complicated nonlinear gradient term than the standard model (1.2). The method involving change of variable seems invalid in the process of doingvariationalanalysisto(1.1),evenintwodimensions.Onecouldalsorefertoother applications of variational functional to nonlinear differential equations with gradient terms,suchasthefamousPerelmanfunctionalinSection1.1of[48].

ThefirstauthorwithLutwak,YangandZhang[28] developedaperturbationprocess and geometricvariationalformula after thework ofAleksandrov [4]. LetΩ⊂S^N−1 be aclosed setwhich isnot containedinany closed hemisphere ofS^N−1, ρ0 : Ω→(0,∞) and p: Ω →Rbe two continuous functions. Forδ >0, letρ_t: Ω→ (0,∞) bedefined forξ∈Ω andt∈(−δ,δ) by

logρ_t(ξ) = logρ₀(ξ) +tp(ξ) +o(t, ξ).

Thelogarithmicfamilyofconvexhullsρ_t = conv{ρ_t(ξ)ξ:ξ∈S^N−1},thenitssupport functionisdifferentiablewithrespectto thevariationalvariable,

d dt

t=0

logu_ρt(θ) =p(α^∗_ρ0(θ)). (1.3) It was proved in [28] that (1.1) is the Euler–Lagrange equation with respect to the geometricfunctional

F(f) = ^SN−1g(θ) logf dθ−

S^N−1logu_f(ξ)dξ,

S^N−1g(θ) logf dθ+¹_qlog

S^N−1u⁻_f^q(ξ)dξ, q = 0, f ∈ {f >0 :f ∈C(S^N−1)}, inthesenseoftheAleksandrovsolution[2].ThefunctionalFplaysakeyroleinthestudy ofthesolvabilityof(1.1) [28].Itisalsousefulintheprocessofprovingtheregularitiesof solutionto (1.1) viaaGauss curvatureflowmethodbyLi–Sheng–Wang[39]. Underthe assumption that g is strictly positive,some other important contributions to thedual Minkowskiproblem arealsoobtainedin[7,10,13,24,29,30,36,57,58], andsoon.

The dual Minkowski problem (1.1) with q = N is equivalent to the logarithmic Minkowskiproblem [9,52,59],whichisaspecialcaseoftheLp Minkowskiproblems

det(∇²uK+uKE) =g(θ)u^p−1_K , θ∈S^N−1. (1.4) Important contributions and applications of the Lp Minkowski problem are included in [16,17,19,21,22,26,27,32–36,40,42,47,49,52–55,60,61] and their references. Moreover, problem (1.4) relates to the L_p affine isoperimetric inequalities and Blaschke–Santaló inequality[8,19,38,43,44,54,56].InthespecialcaseN = 2,severalvariationalfunctionals intheSobolevspacewereintroducedtostudy theexistenceof(1.4) in [1,14,21,55] and

functionalinequalitiesweregivenin[15,19,54].However,forthedualMinkowskiproblem (1.1), we know less about the related functional inequalityand variational formula in theSobolevspacethanthecaseofL_p Minkowskiproblem,evenintwodimensions.

In this paper, we develop a new variational formula of the planar dual Minkowski problem in the Sobolev space to give the existence, non-uniqueness, and its related functional inequalitiesofproblem(1.1) in twodimensions,namely,

u+u=g(θ)(u²+|u|²)^2−q2

u , θ∈S¹. (1.5)

Before we go any further, some notations should be introduced. Let k > 1, and H^1,k(S¹) be the usual Sobolev space, which is the completion of the smooth func- tion space C^∞(S¹) with respect to the Sobolev norm. We denote by W^1,k(S¹) = u∈H^1,k(S¹) :u(θ) =u(θ+π), θ∈S¹ .

Themain resultsof thispaperareinthefollowing.

Theorem 1.1. Assume q≥2 be an even number, and g ∈W^1,k(S¹) forsome k > 1. If

S¹g(θ)dθ > 0, then equation (1.5) has a positive solution u ∈ C²(S¹). That is to say there is a solution forthe planar dual Minkowski problem with the prescribed measure gdθ onS¹.

The assumption

S¹g(θ)dθ > 0 implies that g(θ) may equalto zero on a potential interval. Hence,Theorem1.1isquitedifferentfromthepreviousexistenceresultsofthe dual Minkowski problem [10,12,13,28,36,57], in which positive data g is supposed. In fact, theweakerassumption of g hasbeen attemptedby thefirst author withLutwak, Yang and Zhang, it is known as the Lp-Aleksandrov integral curvature problem in all dimensions,see[29].Tosolvethecaseq >2 andgetridoftheconditionthatgisstrictly positive,weapplythevariationalmethodintheSobolevspace,insteadofAleksandrov’s geometric variationalmethodabove-mentioned (see[4,28] fordetails).Indeed,ourvari- ational formula is direct and does not rely on the geometric dual variational formula (1.3).WeproveinSection2belowthat(1.5) withanyevenq≥2 is anEuler–Lagrange equation ofthefunctional

Fq(u) =

⎛

⎝

S¹

u^qdθ−q q/2 i=1

τi

S¹

u^q−2iu²ⁱdθ

⎞

⎠·exp

−q

S¹g(θ) lnudθ

S¹g(θ)dθ

, (1.6)

where

τi= (q/2−1)!

2(2i−1)i!(q/2−i)!, fori= 1,2, . . . , q/2. (1.7)

Forthespecialcaseq= 2,weseethatτ1= 1/2 andthefunctional F₂(u) =

S¹

(u²−u²)dθ·exp

−2

S¹g(θ) lnudθ

S¹g(θ)dθ

isrelatedwiththelogarithmicMinkowskiproblem u+u= g(θ)

u , θ∈S¹. (1.8)

Chenmadeabreakthroughin[14] via firstlyprovinganexistenceresultof(1.8) fornot necessarily positive data g. In [14], a general affine isoperimetric inequality from [15]

playsakeyroleinstudyingthegeometricstructureand themaximumof

Tα,p(u) =

S¹

(αu²−u²)dθ·

⎛

⎝

S¹

g(θ)u^pdθ

⎞

⎠

−²_p

, −2≤p <0,

where αisaparameter; andan approximatingargument is appliedtoget thesolution u_α ofthefollowingapproximateequationof(1.8).

u+αu=g(θ)

u , θ∈S¹.

Then, the solution of (1.8) was obtained in[14] by the limitation of a subsequence of {uα}asα→1.

In the generalcase q > 2, we know less about the related inequalities for studying the geometric structure of F_q, such as the bound of F_q from aboveor below, which is thekey point inthe applicationof criticalpoint theory. So, wehave to establishsome pertinentinequalitiesforestimatingtheboundofF_q.Infact,weshallshowthefollowing interestingfunctionalinequality.

Theorem1.2. Assume q≥2 bean even number and g∈W^1,k(S¹) forsome k >1.Let τi bedefined by(1.7). If

S¹g(θ)dθ >0,thenthere existsconstant Cq≥2π suchthat

⎛

⎝

S¹

u^qdθ−q q/2 i=1

τi

S¹

u^q−2iu²ⁱdθ

⎞

⎠·exp

−q

S¹g(θ) lnudθ

S¹g(θ)dθ

≤Cq, (1.9)

holds for all positive functionu ∈W^1,q(S¹).The equality in (1.9) holds if and only if u=lw,where theconstantl >0,andwisasolution of (1.5).

Toobtain theinequality(1.9), aregularity theory for theweaksolution ofquasilin- ear differential equation is developed; then we prove the following new Poincaré-type inequalitywith itsequality conditions.

S¹

u^qdθ≤q q/2 i=1

τi

S¹

u^q−2iu²ⁱdθ, u∈W^1,q(S¹) withu(θ0) = 0 for someθ0∈S¹.

Basedonthisanalysis,weemployanapproximatingargumenttodeduce(1.9).Byusing the extremal functionsof (1.9), we get the solvability of (1.5). Furthermore, there are other applications of (1.9). For the special casethat q = 2 and min

θ∈S¹g(θ) > 0,(1.9) is equivalenttothelog-Minkowskiinequality[8] fororigin-symmetricconvexbodiesinthe plane. The inequality (1.9) is sharp in the sense that it may be invalid for functions withminimalperiod2πasthefollowingTheorem1.3.Inthissense,theassumptionthat theconvexbodiesareorigin-symmetricis“anecessarycondition”forthelog-Minkowski inequalityin[8].

Theorem 1.3. Let q ≥ 2 be even and τi be defined by (1.7). There exist a series of 2π-periodic positivefunctions{un}⊂H^1,q(S¹) suchthat

n→lim+∞

⎛

⎝

S¹

u^q_ndθ−q q/2 i=1

τ_i

S¹

u^q_n⁻²ⁱu_n²ⁱdθ

⎞

⎠·exp

−q

S¹g(θ) lnu_ndθ

S¹g(θ)dθ

= +∞ (1.10)

holds forany givencontinuouspositivefunctiong.

Inthelastpartofthispaper,wefocusontheuniquenessofsolutionto(1.5).Forthe specialcaseq= 2,(1.5) isthelogarithmicMinkowskiproblemintwodimensions,which hasauniqueπ-periodicsolutionforpositiveg,see, forexample[8,18,21].Ifg≡1,then u= 1 isobviouslyasolutionof(1.5).ViagivinganestimatetothevalueofCqin(1.9), we showthat(1.5) hasthesecondsolutionforanyevennumberq≥6 asfollows.

Theorem 1.4. Assume g ≡ 1. Let q ≥ 6 be even. (1.5) has a non-constant π-periodic solution.

Thepaperisorganizedasfollows.InSection2,wegiveavariationalframeworkinthe Sobolev space for studying theplanar dual Minkowskiproblem (1.5).In Section3, we show the regularity of weaksolutionand anewPoincaré-type inequality.In Section4, we provethemain conclusionsofthis paper.Weuse C,c,Ci,ci for i∈Nto denote the constants whose valuesmaychangefrom lineto line.Weuseo(1),O() todescribe the asymptotic behaviorofvariousquantities.

2. AvariationalframeworkintheSobolevspace

In this section, we establish avariational framework forthe planar dual Minkowski problem withanyevenexponentq≥2.Forr∈Nanda,b∈R,wewillusethebinomial formula

(a+b)^r= r i=0

C_rⁱa^r−ibⁱ, whereC_rⁱ =r!/[i!(r−i)!]. (2.1) Asin[23], wedenote byC⁰(S¹) thecollectionof continuousfunctiononS¹; form∈N andγ∈(0,1],let

C^m(S¹) =

u∈C⁰(S¹) :∂^lu∈C⁰(S¹) for alll= 0,2, . . . , m , C^m,γ(S¹) =

u∈C^m(S¹) : sup

θ1,θ2∈S¹

|∂^lu(θ1)−∂^lu(θ2)|

|θ₁−θ₂|^γ <+∞, for alll= 0,1,2, . . . , m

be theusual differentialfunction spacesand Hölderspaces, respectively.Fork >1,let H^1,k(S¹) andW^1,k(S¹) be twoSobolevspacesintroducedabove.Wedenote by

u=

⎧⎨

⎩

S¹

|u|^k+|u|^kdθ

⎫⎬

⎭

1/k

theusualnormofH^1,k(S¹) andW^1,k(S¹).Foranyevenq≥2,thedualspaceofW^1,q(S¹) isequivalenttoW^1,q/(q−1)(S¹).ThereexistsaconstantC0>0 suchthat

sup

θ∈S¹|u(θ)| ≤C₀u (2.2)

holds for all u ∈ W^1,q(S¹). Moreover, the embedding that W^1,q(S¹) → C^0,γ(S¹) is compactforγ∈[0,(q−1)/q).

Forevenq≥2,wedefineapositiveconeofW^1,q(S¹) as M ={u∈W^1,q(S¹) :u(θ)>0, forθ∈S¹}.

Weaimto getthesolvabilityofplanardualproblem(1.5) viathecriticalpointinM of F_q(u) definedby(1.6).Tostudy theupperboundandcriticalpointsofF_q(u) on M,we introduceaparametertinF_q(u) to getthefollowing approximatefunctional

I_q:=I_q(t, u) =

⎛

⎝

S¹

t^qu^qdθ−q q/2 i=1

τ_i

S¹

t^q−2iu^q−2iu²ⁱdθ

⎞

⎠·exp

−q

S¹g(θ) lnudθ

S¹g(θ)dθ

, (2.3) where thecoefficients {τi}^q/2i=1 aregiven by(1.7). Forthe clearnessofthe following cal- culation,we rewritethesecoefficientsasτ_q/2= 1/[q(q−1)] and

τi= (q/2−1)!

2(2i−1)i!(q/2−i)! = C_q/2ⁱ⁻¹₋₁

2i(2i−1) = C_q/2ⁱ ₋₁

(q−2i)(2i−1) ∀i= 1,2, . . . , q/2−1. (2.4)

When the parameter t = 1, we see that Iq(1,u) equals to Fq(u). The introducing of parametert hereisusefulforapplying approximatingargumentast→1.Fort>0,we see thatthecriticalpointofIq inM isapositivesolutionof

(t²u²+u²)^q−22 (u+t²u) =λg(θ)

u , (2.5)

where λ is amultiplier. Once we obtaina positivesolution w of (2.5) with t = 1 and λ > 0, then λ^−1/qw is obviously the solution of (1.5). If

S¹g(θ)dθ > 0, a standard argument in[50] canbe appliedtoshowthatIqisC¹ onM withadistanceinducedby the normof W^1,q(S¹).Forany t>0,the criticalpoint uof I_q inM isaweaksolution of (2.5) withsomegivenλ∈Rinthesenseof

t^q

S¹

u^q−1ϕdθ−

q/2−1 i=1

τ_it^q−2i

S¹

(q−2i)u^q−2i−1u²ⁱϕ+ 2iu^q−2iu²ⁱ⁻¹ϕdθ

− 1 q−1

S¹

u^q−1ϕdθ=λ

S¹

g(θ)

u ϕdθ, ∀ϕ∈W^1,q(S¹).

(2.6)

Infact,wehavethefollowingtheorem.

Theorem2.1.Assumeq≥2beanevennumber,t>0beaparameterand

S¹g(θ)dθ >0.

Let τ_i be defined by (1.7). If positivefunction u∈W^1,q(S¹) isa critical point of I_q in (2.3),thenuisaweaksolution of (2.5) inthesense of (2.6) with

λ=

⎛

⎝

S¹

t^qu^qdθ−q q/2 i=1

τi

S¹

t^q−2iu^q−2iu²ⁱdθ

⎞

⎠/

S¹

g(θ)dθ. (2.7)

Furthermore, ifu∈W^2,2(S¹),then

t^q

S¹

u^q−1ϕdθ−

q/2−1

i=1

τit^q−2i

S¹

(q−2i)u^q−2i−1u²ⁱϕ+ 2iu^q−2iu²ⁱ⁻¹ϕdθ

− 1 q−1

S¹

u^q−1ϕdθ=

S¹

(t²u²+u²)^q/2−1(u+t²u)ϕdθ, ∀ϕ∈W^1,q(S¹).

(2.8)

Thereforeu(θ)is asolutionof (2.5) foralmosteverywhere θ∈S¹,i.e.

S¹

(t²u²+u²)^q/2−1(u+t²u)ϕdθ=λ

S¹

g(θ)

u ϕdθ, ∀ϕ∈W^1,q(S¹). (2.9)

Proof. Let u ∈ M be a critical point of Iq, then, the Fréchet derivative I_q(t,u) = 0 inW^1,q/(q−1)(S¹).Thisimplies that(2.6) holds withλgiven by(2.7).Inthefollowing, we showthat uisa weaksolutionof (2.5) viaproving that (2.8) and(2.9) holdunder additional assumption u ∈ W^2,2(S¹). For any ϕ ∈ W^1,q(S¹) and i = 1,2,. . . ,q/2, by applyingtheNewton–Leibnizformulaandanapproximatingargument,weobtainthat

S¹

u^q−2iu²ⁱ⁻¹ϕdθ=−(q−2i)

S¹

u^q−2i−1u²ⁱϕdθ−(2i−1)

S¹

u^q−2iu²ⁱ⁻²uϕdθ.

Viaapplyingthisformulaandrewritingtheright-handsideofequation(2.8),wededuce that

the right-hand side of (2.8) =t^q

S¹

u^q−1ϕdθ+

S¹

u^q−2uϕdθ

+

q/2−1 i=1

t^q−2iτ_i

S¹

2i(2i−1)u^q−2iu²ⁱ⁻²uϕ−(q−2i)(1−2i)u^q−2i−1u²ⁱϕdθ,

=t^q

S¹

u^q−1ϕdθ+

q/2−1

i=1

t^q−2iτ_i

S¹

(q−2i)(2i−1)u^q−2i−1u²ⁱϕdθ

(I)

+

S¹

u^q−2uϕdθ+

q/2−1

i=1

t^q−2iτi

S¹

2i(2i−1)u^q−2iu²ⁱ⁻²uϕdθ

(II)

:=(I) + (II).

(2.10) By(2.4), τ_i =C_q/2ⁱ ₋₁/[(q−2i)(2i−1)] for i = 1,2,. . . ,q/2−1. Then wesimplify the formula(I) as

(I) =

S¹

⎛

⎝t^qu^q−1+

q/2−1 i=1

t^q−2iτi(q−2i)(2i−1)u^q−2i−1u²ⁱ

⎞

⎠ϕdθ

=

S¹ q/2−1

i=0

C_q/2ⁱ ₋₁t^q−2iu^q−2i−1u²ⁱϕdθ

=

S¹

(t²u²+u²)^q−22t²uϕdθ.

(2.11)

Fori= 1,2,. . . ,q/2−1,wehaveτi =C_q/2−1ⁱ⁻¹ /[2i(2i−1)] by (2.4),itfollowsthat

(II) =

S¹

u^q−2uϕdθ−q

q/2−1

i=1

t^q−2iτi

S¹

2i(2i−1)u^q−2iu²ⁱ⁻²uϕdθ.

=

S¹

⎛

⎝u^q−2+

q/2−1 i=1

C_q/2ⁱ⁻¹₋₁t^q−2iu^q−2iu²ⁱ⁻²

⎞

⎠uϕdθ.

=

S¹

⎛

⎝u^q−2+

q/2−2

j=0

C_q/2−1^j t^q−2j−2u^q−2j−2u^2j

⎞

⎠uϕdθ.

=

S¹

⎛

⎝^q/2−1

j=0

C_q/2^j ₋₁t^q−2j−2u^q−2j−2u^2j

⎞

⎠uϕdθ.

=

S¹

(t²u²+u²)^q−22 uϕdθ.

(2.12)

Via(2.10)–(2.12) weobtain(2.8).And(2.9) followsfrom (2.6) and(2.8). 2 Forany α < β,wedefinesomeSobolevspacesas

W^1,q(α, β) =

⎧⎨

⎩u∈L¹(α, β) : β α

|u|^q+|u|^qdθ <+∞

⎫⎬

⎭, W₀^1,q(α, β) =

u∈W^1,q(α, β) :u(α) =u(β) = 0 , W^2,2(α, β) =

⎧⎨

⎩u∈L¹(α, β) : β α

|u|²+|u|²dθ <+∞

⎫⎬

⎭.

Inthenextsection,wealsoneedalocalversionof(2.8).

Corollary 2.2.Letq≥2beeven number,τi be definedby (1.7).Assumeu∈W^2,2(α,β) and ϕ∈W₀^1,q(α,β).Then,

t^q β α

u^q−1ϕdθ−

q/2−1 i=1

τit^q−2i β α

(q−2i)u^q−2i−1u²ⁱϕ+ 2iu^q−2iu²ⁱ⁻¹ϕdθ

− 1 q−1

β α

u^q−1ϕdθ= β α

(t²u²+u²)^q/2−1(u+t²u)ϕdθ.

(2.13)

Proof. (2.13) follows by a similar calculation of (2.10)–(2.12). We omit the details here. 2

3. RegularityofweaksolutionandaPoincaré-type inequality

In the rest of this paper, the regularity of weak solutions to a quasilinear elliptic equationwill beusedrepeatedly. However,tothebestoftheauthors’knowledge,there is not a suitable theorem in reference for being applied directly. For the sake of the completeness of our paper, we give the proof of the local differentiability to the weak solutionasfollows.

Lemma 3.1. Let q ≥ 2 be an even number, and t > 0 be a parameter. Assume that f(θ)∈C^0,γ(S¹)forsome γ∈(0,1),andu∈W^1,q(S¹)be aweaksolution of

(t²u²+u²)^q−22 (u+t²u) =f(θ), (3.1) inthesense of

t^q

S¹

u^q−1ϕdθ−

q/2−1 i=1

τ_it^q−2i

S¹

(q−2i)u^q−2i−1u²ⁱϕ+ 2iu^q−2iu²ⁱ⁻¹ϕdθ

− 1 q−1

S¹

u^q−1ϕdθ=

S¹

f(θ)ϕdθ, ∀ϕ∈W^1,q(S¹),

(3.2)

whereτi isdefined by(1.7).If u(θ0)>0forgivenθ0 ∈S¹,then u(θ)iscontinuous at θ=θ₀.

Proof. Forq= 2,(3.1) is alinearequation.Theconclusionis obvious.Inthefollowing part,weassumeq >2.Since 2:=u(θ0)>0 and u∈W^1,q(S¹)⊂C^0,(q−1)/q(S¹),there existsδ >0 such that

|u(θ)−2|< for allθ∈(θ0−δ, θ0+δ). (3.3) Forasmallh∈(0,δ/8),wedenotebyΔ^hv:= Δ^hv(θ)= (v(θ+h)−v(θ))/hthedifference quotient of v. We define a cut-off function ξ ∈ C₀^∞(S¹) such that|ξ(θ)| < 8/δ for all θ∈S¹,ξ(θ)= 1 for|θ−θ0|< δ/4 andξ(θ)= 0 for|θ−θ0|> δ/2.Letϕ= Δ^−h(ξ²Δ^hu) in(3.2),wededucethat

S¹

A(u, u)[Δ^−h(ξ²Δ^hu)]dθ+

S¹

B(u, u)Δ^−h(ξ²Δ^hu)dθ=

S¹

fΔ^−h(ξ²Δ^hu)dθ, (3.4)

whereA(x,y) andB(x,y) arebinary polynomialsdefinedby

A(x, y) =−2 q/2 i=1

iτit^q−2ix^q−2iy²ⁱ⁻¹ and

B(x, y) =t^qx^q−1−

q/2−1

i=1

(q−2i)τit^q−2ix^q−2i−1y²ⁱ. (3.5) By the definition of difference quotient, we have [Δ^−h(ξ²Δ^hu)] = Δ^−h(ξ²Δ^hu) = Δ^−h(ξ²Δ^hu+ 2ξξΔ^hu) and

S¹wΔ^−hvdθ=−

S¹vΔ^hwdθ forany v,w∈L²(S¹). Ap- plying thesepropertiesin(3.4) wededucethat

S¹

Δ^hA(u, u)(ξ²Δ^hu+ 2ξξΔ^hu)dθ+

S¹

ξ²Δ^hB(u, u)Δ^hudθ=−

S¹

fΔ^−h(ξ²Δ^hu)dθ.

(3.6) Andadirectcalculationinduces that

Δ^hA(u(θ), u(θ)) =A1Δ^hu(θ) +A2Δ^hu(θ),

Δ^hB(u(θ), u(θ)) =B1Δ^hu(θ) +B2Δ^hu(θ), (3.7) where

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝

A1:=A1(θ, h) = 1 0

∂

∂xA((1−s)u(θ) +su(θ+h), u(θ+h))ds,

A₂:=A₂(θ, h) = 1 0

∂

∂yA(u(θ),(1−s)u(θ) +su(θ+h))ds,

B₁:=B₁(θ, h) = 1 0

∂

∂xB((1−s)u(θ) +su(θ+h), u(θ+h))ds,

B2:=B2(θ, h) = 1 0

∂

∂yB(u(θ),(1−s)u(θ) +su(θ+h))ds.

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠

(3.8)

Byapplyingformula(3.7) in equation(3.6),weobtainthat

S¹

A2ξ²(Δ^hu)²dθ=−

S¹

A1ξ²Δ^huΔ^hu+ 2A1ξξ(Δ^hu)²+ 2A2ξξΔ^huΔ^hudθ

−

S¹

B1ξ²(Δ^hu)²+B2ξ²Δ^huΔ^hu+fΔ^−h(ξ²Δ^hu)dθ.

(3.9)

From (3.5),weseethat

∂A(x, y)

∂x =−2

q/2−1

i=1

i(q−2i)τit^q−2ix^q−2i−1y²ⁱ⁻¹,

∂A(x, y)

∂y =−2τ1t^q−2x^q−2−2 q/2 i=2

i(2i−1)τit^q−2ix^q−2i−1y²ⁱ⁻²,

∂B(x, y)

∂x = (q−1)t^qx^q−2−

q/2−1 i=1

(q−2i)(q−2i−1)τit^q−2ix^q−2i−2y²ⁱ,

∂B(x, y)

∂y =−2

q/2−1 i=1

i(q−2i)τ_it^q−2ix^q−2i−1y²ⁱ⁻¹.

By applying these formulas and (3.3) in (3.8), we obtain a constant c > 1 depending onlyont,q,δ,,andthatthefollowing estimates

|A₁|+|B₁|+|B₂| ≤c(1 +|u(θ)|+|u(θ+h)|)^q−2, (3.10) 1

c(1 +|u(θ)|+|u(θ+h)|)^q−2≤ −A2≤c(1 +|u(θ)|+|u(θ+h)|)^q−2 (3.11) holdforallθ∈(θ₀−δ/2,θ₀+δ/2).Byapplying(3.10),(3.11) andtheYoung’sinequality in(3.9),wededucethat

1 2c

S¹

ξ²(1 +|u(θ)|+|u(θ+h)|)^q−2(Δ^hu)²dθ

≤C

S¹

(ξ²+|ξξ|)(1 +|u(θ)|+|u(θ+h)|)^q−2(Δ^hu)²dθ+

S¹

fΔ^−h(ξ²Δ^hu)dθ , (3.12) whereC >1 dependsonlyont,q,δand .Leth→0⁺,wehave

S¹

fΔ^−h(ξ²Δ^hu)dθ ≤max

θ∈S¹|f(θ)|

S¹

2|ξξΔ^hu|+|ξ²Δ^hu|dθ+o(1);

itfollowsfromtheYoung’sinequalitythatthereexistsCν >0,dependingonlyonν >0, δandq, suchthat

S¹

fΔ^−h(ξ²Δ^hu)dθ ≤max

θ∈S¹|f(θ)|

⎛

⎝

S¹

2ξ|ξ||Δ^hu|^qdθ+ν

S¹

ξ²|Δ^hu|²dθ+Cν+o(1)

⎞

⎠.

(3.13) Sinceu∈W^1,q(S¹),letν besmallenoughandh→0⁺,from(3.12) and(3.13) wededuce that

θ0+δ/4 θ0−δ/4

(u)²dθ≤C₁

θ0+δ θ0−δ

|u|^qdθ+C₂, (3.14)

whereC1andC2dependonlyont,,δandq.Thatis,u(θ) existsforalmosteverywhere θ∈(θ0−δ/4,θ0+δ/4) and

u∈W^2,2(θ0−δ/4, θ0+δ/4)⊂C^1,1(θ0−δ/4, θ0+δ/4). (3.15) Letζ∈W^1,q(S¹) withcompactsupportin(θ₀−δ/4,θ₀+δ/4),wedefineφ(θ)= (t²u²+ u²)^2−q2 ζ forθ∈(θ0−δ/4,θ0+δ/4) andφ(θ)= 0 forθ∈S¹\(θ0−δ/4,θ0+δ/4).Then itiseasyto checkthatφ∈W^1,q(S¹) bycombining (3.3) and(3.15).Letϕ=φin(3.2), byapplying (2.13) withα=θ₀−δ/4 andβ =θ₀+δ/4 weobtainthat

θ0+δ/4 θ0−δ/4

(u+t²u)ζdθ=

θ0+δ/4 θ0−δ/4

(t²u²+u²)^q/2−1(u+t²u)φdθ

=t^q

θ0+δ/4 θ0−δ/4

u^q−1φdθ− 1 q−1

θ0+δ/4 θ0−δ/4

u^q−1φdθ

−

q/2−1 i=1

τ_it^q−2i

θ0+δ/4 θ0−δ/4

(q−2i)u^q−2i−1u²ⁱφ+ 2iu^q−2iu²ⁱ⁻¹φdθ

=t^q

S¹

u^q−1φdθ− 1 q−1

S¹

u^q−1φdθ

−

q/2−1 i=1

τit^q−2i

S¹

(q−2i)u^q−2i−1u²ⁱφ+ 2iu^q−2iu²ⁱ⁻¹φdθ

=

S¹

f(θ)φdθ=

θ0+δ/4 θ0−δ/4

f(θ)(t²u²+u²)^2−q2 ζdθ.

(3.16)

Itfollowsfrom(3.15) and(3.16) thatu(θ) satisfiesequationu(θ)+t²u(θ)=f(θ)(t²u²+ u²)^2−q2 almosteverywherein(θ₀−δ/4,θ₀+δ/4).By(3.3),(3.15) andtheassumptionf ∈ C^0,γ(S¹),wededucethatf(θ)(t²u²+u²)^2−q2 issmoothenoughover(θ0−δ/4,θ0+δ/4), that astandard regularity theorem canbe applied fordeducing u∈C²(θ0−δ/4,θ0+ δ/4). 2

To estimatethebound offunctionalIq, weneedthefollowinginequalities.

Lemma3.2. Letq >0beanevennumber,u∈W^1,q(S¹)withu(θ0)= 0forsomeθ0∈S¹. Foreach i= 1,2,. . . ,q/2,thereexistsci>0suchthat

S¹

u^q(θ)dθ≤c_i

S¹

u^q−2iu²ⁱdθ. (3.17)

Proof. For each i = 1,2,. . . ,q/2, by using the Newton–Leibniz formula and Hölder inequalitywederive that

S¹

u^q(θ)dθ=

S¹

⎛

⎝ θ θ0

du^2iq(t) dt dt

⎞

⎠

2i

dθ=q 2i

2i

S¹

⎛

⎝ θ θ0

u^2iq−1(t)u(t)dt

⎞

⎠

2i

dθ

≤q 2i

2i

S¹

⎛

⎝(θ−θ0)²ⁱ⁻¹ θ θ0

u^q−2iu²ⁱdt

⎞

⎠dθ

≤q 2i

2i(2π)²ⁱ 2i

S¹

u^q−2iu²ⁱdθ.

Letci= (q/2i)²ⁱ(2π)²ⁱ/2i,weobtain(3.17). 2

The inequality (3.17) is not enough for adirect application to estimate the bound of functional Iq. However, (3.17) is useful to prove the following new Poincaré-type inequality(3.18) withequalityconditions,whichwillplayanessentialroleintheprocess ofstudyingtheboundofI_q.

Lemma3.3. Letθ0∈S¹beagivenpoint,wedenotebyW_θ^1,q₀ =

u∈W^1,q(S¹) :u(θ0) = 0 a subspace of W^1,q(S¹). Assume q > 0 be an even number, τ_i be defined by (1.7) and u∈W_θ^1,q

0 .Then,

S¹

u^qdθ≤q q/2 i=1

τ_i

S¹

u^q−2iu²ⁱdθ. (3.18)

And theequalityin(3.18) holdsif andonly ifu(θ)=l|sin(θ−θ0)| withl∈R.

Proof. Itisclearthat(3.18) holdsforu= 0.Forthecaseu= 0,weconsiderthefollowing eigenvalueproblem

μ:= inf

u∈Wθ^1,q0 ,u =0

q

q/2 i=1

τi

S¹u^q−2iu²ⁱdθ−

S¹u^qdθ

S¹u^qdθ .