A note on the boundary behavior for a modiﬁed Green function in the upperhalf space


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1 Zhag ad Pisarev Boudary Value Problems (015) 015:114 DOI /s z RESEARCH Ope Access A ote o the boudary behavior for a modiﬁed Gree fuctio i the upperhalf space Yulia Zhag1 ad Valery Pisarev* * Correspodece: Faculty of Sciece ad Techology, Uiversity of Wollogog, Wollogog, NSW 5, Australia Full list of author iformatio is available at the ed of the article Abstract Motivated by (Xu et al. i Boud. Value Probl. 013:6, 013) ad (Yag ad Re i Proc. Idia Acad. Sci. Math. Sci. 14(): , 014), i this paper we aim to costruct a modiﬁed Gree fuctio i the upperhalf space of the dimesioal Euclidea space, which geeralizes the boudary property of geeral Gree potetial. Keywords: modiﬁed Gree fuctio; capacity; upperhalf space 1 Itroductio ad mai results Let R ( ) deote the dimesioal Euclidea space. The upper halfspace H is the set H = x = (x, x,..., x ) R : x > }, whose boudary ad closure are H ad H respectively. For x R ad r >, let B(x, r) deote the ope ball with ceter at x ad radius r. Set Eα (x) = log x if α = =, if < α <. x α Let Gα be the Gree fuctio of order α for H, that is, Gα (x, y) = Eα (x y) Eα x y, x, y H, x = y, < α, where deotes reﬂectio i the boudary plae H just as y = (y, y,..., y ). I case α = =, we cosider the modiﬁed erel fuctio, which is deﬁed by E,m (x y) = E (x y) E (x y) + (log y m if y <, x = ( y )) if y. I case < α <, we deﬁe Eα,m (x y) = Eα (x y) Eα (x y) m α x = y α+ C if y <, x y ( x y ) if y, 015 Zhag ad Pisarev. This article is distributed uder the terms of the Creative Commos Attributio 4.0 Iteratioal Licese (http://creativecommos.org/liceses/by/4.0/), which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial author(s) ad the source, provide a li to the Creative Commos licese, ad idicate if chages were made.
2 Zhag ad Pisarev Boudary Value Problems (015) 015:114 Page of 7 where m is a oegative iteger, C ω α (t) (ω = ) is the ultraspherical (or Gegebauer) polyomial (see [1]). The expressio arises from the geeratig fuctio for Gegebauer polyomials ( 1 tr + r ) ω = C ω (t)r, (1.1) =0 where r <1, t 1adω > 0. The coefficiet C ω (t) is called the ultraspherical (or Gegebauer) polyomial of degree associated with ω, the fuctio C ω (t) isapolyomial of degree i t. The we defie the modified Gree fuctio G α,m (x, y)by G α,m (x, y)= E,m+1 (x y) E,m+1 (x y ) ifα = =, E α,m+1 (x y) E α,m+1 (x y ) if0<α <, where x, y H ad x y. We remar that this modified Gree fuctio is also used to give uique solutios of the Neuma ad Dirichlet problem i the upperhalf space [ 4]. Write G α,m (x, μ)= G α,m (x, y), H where μ is a oegative measure o H. HereotethatG,0 (x, μ) is othig but the geeral Gree potetial. Let be a oegative Borel measurable fuctio o R R,adset (y, μ)= (y, x) dμ(x) ad (μ, x)= (y, x) E E for a oegative measure μ o a Borel set E R.WedefieacapacityC by C (E)=sup μ ( R ), E H, where the supremum is tae over all oegative measures μ such that S μ (the support of μ) is cotaied i E ad (y, μ) 1foreveryy H. For β 0, δ 0adβ δ, we cosider the erel fuctio α,β,δ (y, x)=x β y δ G α(x, y). Now we prove the followig result. For related results i a smooth coe ad tube, we refer the reader to the papers by Qiao (see [5, 6]) ad LiaoSu (see [7]), respectively. The readers may also fid some related iterestig results with respect to the Schrödiger operator i the papers by Su (see [8]), by Polidoro ad Ragusa (see [9]) ad the refereces therei. Theorem Let + m α + δ + 0. If μ is a oegative measure o H satisfyig H <, (1.) +m α+δ+
3 Zhag ad Pisarev Boudary Value Problems (015) 015:114 Page 3 of 7 the there exists a Borel set E H with properties: x α β+δ+1 (1) lim x 0,x H E (1 + x ) G α,m(x, μ)=0; +m α+δ+ () i( α+β+δ) C α,β,δ (E i )<, i=1 where E i = x E : i x < i+1 }. Remar By usig Lemma 4 below, coditio () i Theorem with α =,β =0,δ =0 meas that E is thi at H i the sese of [10]. Some lemmas Throughout this paper, let M deote various costats idepedet of the variables i questios, which may be differet from lie to lie. Lemma 1 There exists a positive costat M such that G α (x, y) M, where 0< x y α+ α, x =(x 1, x,...,x ) ad y =(y 1, y,...,y ) i H. This ca be proved by a simple calculatio. Lemma Gegebauer polyomials have the followig properties: (1) C ω(t) Cω Ɣ(ω+) (1) = Ɣ(ω)Ɣ(+1), t 1; d () dt Cω (t)=ωcω+1 1 (t), 1; (3) =0 Cω (1)r =(1 r) ω ; (4) C α (t) C α (t ) ( α)c α+ 1 (1) t t, t 1, t 1. Proof (1) ad () ca be derived from [1], p.3. Equality (3) follows from expressio (1.1) by taig t = 1; property (4) is a easy cosequece of the mea value theorem, (1) ad also (). Lemma 3 For x, y R (α = =),we have the followig properties: (1) I m =0 x m 1 x x y +1 =0 ; y + () I =0 x+m+1 m+1 x y x m ; (3) G,m (x, y) G (x, y) M m x y x 1 =1 ; y +1 (4) G,m (x, y) M x y x 1 =m+1. y +1 The followig lemma ca be proved by usig Fuglede (see [11], Théorèm 7.8). Lemma 4 For ay Borel set E i H, we have C α (E)=Ĉ α (E), where Ĉ α (E)=if λ(h), α = α,0,0, the ifimum beig tae over all oegative measures λ o H such that α (λ, x) 1 for every x E. Followig [10], we say that a set E H is αthi at the boudary H if i( α) C α (E i )<, i=1 where E i = x E : i x < i+1 }. x y
4 Zhag ad Pisarev Boudary Value Problems (015) 015:114 Page 4 of 7 3 Proof of Theorem We write [ G α,m (x, μ)= G α (x, y) + G α (x, y) + Gα,m (x, y) G α (x, y) ] G 1 G G 3 + G α,m (x, y) + G α,m (x, y) G 4 G 5 = U 1 (x)+u (x)+u 3 (x)+u 4 (x)+u 5 (x), where G 1 = y H : x y x }, G = y H : y 1, x } < x y 3 x, G 3 = y H : y 1, x y 3 x }, G 4 = y H : y 1, x y >3 x }, G 5 = y H : y <1, x y > x }. We distiguish the followig two cases. Case 1. 0 < α <. By assumptio (1.)wecafidasequecea i } of positive umbers such that lim i a i = ad i=1 a ib i <,where b i = y H: i 1 <y < i+ } Cosider the sets +m α+δ+. E i = x H : i x < i+1 x α β+δ+1 }, (1 + x ) U 1(x) a 1 +m α+δ+ i (i 1)β for i =1,,...Set G = ( B x, x ). x E i The G y H : i 1 < y < i+ }.Letν be a oegative measure o H such that S ν E i,wheres ν is the support of ν.thewehave α,β,δ (y, ν) 1fory H ad H dν a i ( i+1)β x α β+δ+1 H (1 + x ) U 1(x) dν(x) +m α+δ+ Ma i ( i+1)β ( i+1)( α+δ+1) α,β,δ (y, ν) G Ma i ( i+1)β ( i+1)( α+δ+1) i+1 M α+β+δ+ i( α+β+δ) a i b i. y H: i 1 <y < i+ } y δ +m α+δ+ +m α+δ+
5 Zhag ad Pisarev Boudary Value Problems (015) 015:114 Page 5 of 7 So that C α,β,δ (E i ) M i( α+β+δ) a i b i, which yields i( α+β+δ) C α,β,δ (E i )<. i=1 Settig E = i=1 E i, we see that () i Theorem is satisfied ad lim x 0,x H E x α β+δ+1 (1 + x ) +m α+δ+ U 1(x)=0. (3.1) For U (x), by Lemma 1 we have U (x) y Mx G x y α+ Mx α 1 x +m α+δ+ 1 G y δ Mx α 1 x +m α+ G +m α+δ+ +m α+δ+. (3.) Note that C0 ω x y (t) 1. By (3) ad (4) i Lemma,wetaet = x y, t = x y x y i Lemma (4) ad obtai U 3 (x) G 3 m =1 Mx x m x α+ ( α)c y α+ 1 (1) x y x y m =1 α+ y C 1 1 (1) G3 δ y +m α+δ+ +m α+δ+ +m α+δ+ Mx x m. (3.3) Similarly, we have by (3) ad (4) i Lemma U4 (x) G 4 =m+1 Mx x m x y α+ ( α)c α+ 1 (1) x y x y =m+1 α+ y C 1 1 (1) G4 δ+1 1 y +m α+δ+ +m α+δ+ +m α+δ+ Mx x m. (3.4) Fially, by Lemma 1,wehave U 5 (x) Mx α 1 G5. (3.5) +m α+δ+
6 Zhag ad Pisarev Boudary Value Problems (015) 015:114 Page 6 of 7 Combiig (3.1), (3.), (3.3), (3.4) ad(3.5), by Lebesgue s domiated covergece theorem, we prove Case 1. Case. α = =. I this case, U 1 (x), U (x) adu 5 (x) cabeprovedsimilarlyasicase1.hereweomit the details ad state the followig facts: lim x 0,x H E x δ β+1 (1 + x ) m+δ+ U 1(x)=0, (3.6) where E = i=1 E i ad i=1 i(β+δ) C α,β,δ (E i )<, lim x 0,x H x δ β+1 (1 + x ) m+δ+ [ U (x)+u 5 (x) ] =0. (3.7) ByLemma 3(3), we obtai U 3 (x) G 3 m x y x 1 =1 Mx x m y +1 m =1 4 1 y m+δ+ G3 m+δ+ m+δ+ Mx x m. (3.8) ByLemma 3(4), we have U4 (x) G 4 =m+1 Mx x m x y x 1 =m+1 y +1 1 y m+δ+ G4 m+δ+ m+δ+ Mx x m. (3.9) Combiig (3.6), (3.7), (3.8)ad(3.9), we provecase. Hece the proof of the theorem is completed. Competig iterests The authors declare that they have o competig iterests. Authors cotributios All authors cotributed equally to the writig of this paper. All authors read ad approved the fial mauscript. Author details 1 College of Mathematics ad Statistics, Hea Istitute of Educatio, Zhegzhou, , Chia. Faculty of Sciece ad Techology, Uiversity of Wollogog, Wollogog, NSW 5, Australia. Acowledgemets The authors are highly grateful for the referees careful readig ad commets o this paper. This wor was completed while the authors were visitig the Departmet of Mathematical Scieces at the Uiversity of Wollogog, ad they are grateful for the id hospitality of the Departmet. Received: 13 April 015 Accepted: 8 May 015
7 Zhag ad Pisarev Boudary Value Problems (015) 015:114 Page 7 of 7 Refereces 1. Szegö, G: Orthogoal Polyomials. America Mathematical Society Colloquium Publicatios, vol. 3. Am. Math. Soc., Providece (1975). Re, YD, Yag, P: Growth estimates for modified Neuma itegrals i a half space. J. Iequal. Appl. 013, 57 (013) 3. Xu, G, Yag, P, Zhao, T: Dirichlet problems of harmoic fuctios. Boud. Value Probl. 013, 6 (013) 4. Yag, DW, Re, YD: Dirichlet problem o the upper half space. Proc. Idia Acad. Sci. Math. Sci. 14(), (014) 5. Qiao, L: Itegral represetatios for harmoic fuctios of ifiite order i a coe. Results Math. 61, 674 (01) 6. Qiao, L, Pa, GS: Geeralizatio of the PhragméLidelöf theorems for subfuctios. It. J. Math. 4(8), (013) 7. Liao, Y, Su, BY: Solutios of the Dirichlet problem i a tube domai. Acta Math. Si. 57(6), (014) 8. Su, BY: Dirichlet problem for the Schrödiger operator i a half space. Abstr. Appl. Aal. 01, Article ID (01) 9. Polidoro, S, Ragusa, MA: Harac iequality for hypoelliptic ultraparabolic equatios with a sigular lower order term. Rev. Mat. Iberoam. 4(3), (008) 10. Armitage, H: Tagetial behavior of Gree potetials ad cotractive properties of L p potetials. Toyo J. Math. 9, 345 (1986) 11. Fuglede, B: Le théorèm du miimax et la théorie fie du potetiel. A. Ist. Fourier 15, (1965)
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