n"/> n"/> n"/> Advances in Pure Mathematics, 2013, 3, 153-158 http://dx.doi.org/10.4236/apm.2013.31A021 Published Online January 2013 (http://www.scirp.org/journal/apm) Regularity of Solutions to an Integral Equation on a Half-Space nR* Linfen Cao1, Zhaohui Dai2 1College of Mathematics and Information Science, Henan Normal University, Xinxiang, China 2Department of Comp u t e r Science, Henan Normal University, Xinxiang, China Email: caolf2010@yahoo.com, dzhsoft@sina.com Received September 18, 2012; revised October 21, 2012; accepted October 29, 2012 ABSTRACT nRIn this paper, we discuss the integral equation on a half space    11 d,n>0, .pnxx RnnRuxuy yuxy xy0n (0.1) where 11,, ,nn, xxxnR is the reflection of th e point x about the x. We study the regularity for the positive solutions of (0.1). A regularity lifting method by contracting operators is used in proving the boundedness of solutions, and the Lipschitz continuity is d erived by combinations of contracting and shrinking operators introduced by Ma-Che n-L i ( [ 1 ]). Keywords: Regularity Lifting; HLS Inequality; Contracting Operators; Shrinking Operators 1. Introduction Let be the upper half Euclidean space nR12,,,| 0.nnnnRxxx xRx nR In this paper we consider the regularity of positive solution of the following integral equation in   11 d, .npnnnRuxuyyx Rxy xy 1pnR(1.1) where . It relates closely to the higher-order PDEs with Navier boundary conditions in :  2in;0, on.pnnuu Ruuu R   12,0u (1.2) D. Li and R. Zhuo proved the following result: Proposition 1.1. () Let  be an even number and npnux2. If is the smooth solution of the inte- uxgral Equation (1.1), then satisfies the PDEs (1.2). In particular, when  and 22npn, Chen and Li () showed the equivalence between the integral Equation (1.1) and partial differential Equation (1.2). For more results concerning integral equations, see [4-6]. Firstly, in this paper we have the boundedness for the positive solutions of (1.1) by using the contracting op-erators. Theorem 1.1. Let u be a solution of (1.1). If npn, and 1np nuL R, then u is in rn nLR LR1r for any . Remark 1. In , the authors prov ed that Theorem 1.1 is true for the critical case npn. While our result also covers subcritical case nnpnn and super critical case npn. Then we employ the brand new method which is the combinations of contracting and shrinking operators in-troduced by Ma-Chen-Li () to derive the Lipschitz continuity of solutions. *This work is supported by grant No.11001076 and 11171091 of NSFC, NSF of Henan Provincial Education Committee (No. 2011A110008) and Foundation for University Key T ea c h e r o f H en a n Prov i n c e. Copyright © 2013 SciRes. APM L. F. CAO, Z. H. DAI 154 Theorem 1.2. Under the same conditions of Theorem 1.1, u is Lipschitz continuous in . nRLL2. Estimate by Contracting Operators In this section, we obtain estimate for positive solu-tions to the equation (0.1) by using the contracting op-erators. To prove the Theorem 1.1, we need the follow-ing equivalent form of Hardy-Littlewood-Sobolev ine-quality. Lemma 2.1. Let nr nnrgLR for nrn. Define  d.ng yy1nRTg xxy (2.1) Then .nr nnrL Rgrn 1r,,rnLRTgC nr Proof of Theorem 1.1: The proof is divided into two steps. Step 1. We first show that , uxL Rn, xR .ax ux. Define 1p Then  dayuy y11nnnRux xy xy For a positive number A, defin e  ,if0, elsewherAax axax ,ore.AxA .BAax ax Let ax Obviously, B, and vanishes outside the ball ax ABax0AB. Define  d.yv yy11nAAnnRTv xaxy xy  d.11nABnnRFxaxy xyyuyy .AAxF x1rn The Equation (0.1) can be rewritten as ux Tu We will show that, for any , 1) TA is a contracting map from to rLRrnLR for A large, and rnLR2) AFx is in . rnvLR1) Assume , then 11 d.nAAnnRTvayv yyxy xy For any nrn, we apply Hardy-Littlewood-Sobo- lev inequality and Hölder inequality to obtain  .nr nrnnn rnnrAA ALRLRL RLRTv CavCav Since nnaxLR, by the definition of Axa , one can choose a large number A, such that 1.2nnALRCa and hence arrives at  1.2rn rnALR LRTv v :rn rnAR LRTLThat is  is a contracting op- erator. 2) Consider 11 d.nABnnRxayuyyxy xyF For any nrn, we apply Hardy-Littlewood-Sobo- lev inequality and Hölder inequality to obtain  .nrrnnsn tnnrAB BLRL RLRLRFCau Cau We require 11,, 1.nr stnrst By the bounded-ness of Ba, we see that s can be arbitrary. Since 1np nuL R, we take 1npt, and hence  11,as ,122np nprsnp pps we see 12nppAFL0, for any small . Obviously, 112np npp1p2p2p since . If , we are done. If , repeat the above Copyright © 2013 SciRes. APM L. F. CAO, Z. H. DAI 155process and after a few steps, we arrive at ,.nrnrnux L R Step 2. In this step we will show that .nL Rnux For any point xR, we divide the integral into two parts 1111\\1211111d1dnnnnBxnnRBxpnBxpnRBxux xy xyxy xyuyyxyuyyxyIIddppuyyuyy Consider 2I. Since 1nxynR, and by the result in Step 1, , for rux Lnrn, we have 21IC. For 1I, we apply Hölder inequality   11111qnqBx BxIyxy11ddqqqpquyy Choose appropriate q, so that nqn, and hence  11Bx xy12dqnqyC Since ,nrnrnux L R, 1qpqBxuy113dqqy C We conclude that .nL RuxnLR0,1 nux CRux 3. Lipschitz Continuity by Combinations of Contracting Operators and Shrinking Operators In the previous section we showed that the solution of (0.1) is in . In this section, we will use the regularity lifting by combinations of contracting and shrinking operators to prove , the space of Lipschitz continuous functions with norm  0,1 supnnCRLR xyvx vyvv xy (3.1) To prove the Theorem 1.2, we need introduce the fol-lowing definition, property and a more general Regular-ity Lifting Theorem on the combined use of contracting and shrinking operators. Let V be a Hausdorff topological vector space. Sup-pose there are two extented norms (i.e. the norm of an element in V might be infinity) defined on V, ::and:: .XYXvVvYvVv  Definition. (“XY-pair”) Suppose X, Y are two normed subspaces described above, X and Y are called “XY-pair”, if whenever the sequence nuXnuu with in X and nYuC will imply . uYRemark 2. The “XY-pair” are quite common, here we choose rn0,1 nRYCXLRr for 1, and XY:TXX with the norm defined in (3.1). Theorem 3.1. (Regularity Lifting Theorem) Suppose Banach spaces X, Y are an “XY-pair”, and let and be closed subsets of X and Y respectively. Suppose is a contraction: ,,forsome0 1;XXTfTgfgf g X:TYY and is shrinking: ,,forsome01.YYTggg  Yfor some.Sf Tf FFDefine XY:.SXY XYinuTuF Moreover, assume that Then there exists a solution u of equation X.uYand more importantly, The proof and some applications of Theorem 3.1 can be found in [1,7,8]. nProof of Theorem 1.2: For any xR, by elemen-tary calculus one can verify that 110dddd11 d.ntnppnnBxR xypnRttuyy uyyttuyynxy   It follows that the solution of (0.1) only differs by a constant multiple from the solution of the following equation Copyright © 2013 SciRes. APM L. F. CAO, Z. H. DAI APM 156 Copyright © 2013 SciRes. 1dd.pnty ytand 0dttpBx Bxux uy yu  (3.2) Hence, for convenience of argument, we prove that every positive solution u of (3.2) is Lipschitz continuous. Let 2LLv unvXLRX and 0,1 nvY CRY2.LLv u0 For every , define 1ddd.ntyy t0ttppBx BxTv xvy yv  1ddd.ttppnBx BxtFxuy yuy yt .vTvF Then obviously , u is a solution of the equation .Sv Tv FWrite  TXTYF We will show that for suffi-ciently small, 1) is a contracting operator from to X. 2) is a shrinking operator from to Y. XY:SXYXY,fg and . 3) 1) For any Xn and for any xR   , we have 1dddttpppp nBx BxtTfx Tgxfyg yyfyg yyt   0Thus     1011121110111210dddddddd.ttttppppnBx BxppnBx BxpttnLLppLLLLtTfxTg xfygyyfygyyttpyfygy yyfygy yttCuf gBxBxttCu fgCufgt   0 and consequently Here we applied the Mean Value Theorem with both 1y2yfy and valued between and gy, and tB denotes the volume of the ball tB. Choose sufficiently small such that 12,4pLCu1 1.4LLTf TgfgTXv Therefore  is a contracting operator from to X for such a small . Y,n, then for any x2) Assume zR ,  1010dddd ddddttt tttppp pnBx BzBxBzpppp nBx BxtTvxTvzvyy vyyvyyvyyttvyvyzxyvyvyz xyt    the last equality above is from the fact dddtt tpp pBz BxBxvyy vyzxyvyzxy   Therefore   0,1 0,111001 11 21 1001 1110ddddddddd.t tt tppnnBx Bxp pn nBx Bxp pttnLC LCttTvxTvzzxyvyvyz xyttttpvvy vyzxypvvyvyzxytttCuvzxB xBxCuvzxt   ppvyvy        L. F. CAO, Z. H. DAI 157 Again choosing  sufficiently small, we derive by0,11.4Czv Combining this with the estimate in 1), we arrive at supxzTv xTvzx0,1 0,11,.2CCTvvvY Hence T is a shrinking rator from Y to Y. 3) To show F is Lipschitz continuous, we split it into two parts: opettIx Ix111112ddddddttppnBx BxppnBx BxtFxuy yuy yttuyyuyyt   For the first part, we have  111 dttppBx BzIx Izu yyu  111111ddd dd1.t tp pnBx BzppnnLLtyyuyyu yy ttCutxzxzCuxzt    useHere wed the fact that 1the volume of\\.ntt ttBx BzBz BxCtxz It follows that 11 .sup IxIzC (3.3) xz xzFor the second part, we use a different approach. Write xz, then  11133dddd d,,ttt tppp pnBx BzBzBxtIzuyyu yyu yyu yy tIxz Izx   1Ix    1111 11111dddd dd1dd111d1tttttpnB zBzn npppnn nBzBz Bzpnyztu uyytt tuyyuyy uyytt tuyyyz       12d.npRCuyy C (3.4) Similarly, 1n13111d,dd dttpp pnBx BztIxzuyyuyyyytt   131111dd,dd ddtt ttpp ppnnBzBxB xBxttIzxuyy uyyuyy uyytt   1111111ddd1 dddd11 dttt tnppnnBx BxnppnnBx Bxttuyy uyyttttuyy uyytt         12311nC (3.5) Note that here we applied the Mean Value Theorem with both 1fore, and 2 valued between 0 and . Combining (3.4) and (3.5), we have  223 3,,IxIz IxzIzxC  The same inequality holds for  22IzIx. There- 22 .supxzIx IzCxz Also from the definition of  (3.6) Fx, we immediately have Copyright © 2013 SciRes. APM L. F. CAO, Z. H. DAI 158 .Fu (3.7) y (LLObviousl (3.3), 3.6) and (3.7) imply that Fx is Li ith (3 ipschitz continuous, and this together w.7)mply .FXY Finally, to see that S maps XY tself, we only need to verify that  into iif2, thenLL .Lvu Tvu  (3.8) In fact, L10ttnBx BxpvtvBxBx10.ttnLpLtCudddpptTvxvy yy ydt  Choosing  sufficiently smallut independent of v),w 1), 2) and 3), by the The3.1 and Remark 2, we conclude that the solutiois Lipschitz continuous. This completes the proTh. Usually, contracting operators are used to lift regties. For a linear operator, if it is “shrinking”, then it is contracting. While for nonlinear problems, as were seen in Section 3, sometimes it is very difficult or even im-possible to prove that it is contracting in a given functiospace. However, one can show that it is “shrinking”, acan still lift the regularity of solutions in many cases. The ge la4. Acknowledgements enxiong Chen for his discussions. Society, Vol. 138, 2010, pp. 2779-2791. doi:10.1090/S (b e can guarantee (3.8). So far we have verifiedorem n u of (0.1) of of the eorem 1.2ulari- n nd neral Regurity Lifting Theorem is applied for inte- gral equations and system of integral equations associ- ated with Bessel potentials and Wolff potentials (see  and ), and therefore arrive at higher regularity as Lipschitz continuity of solutions. Most of this work was completed when the first author was visiting the Department of Mathematics, Yeshiva University, and she would like to thank the hospitality of the Department. Besides, the authors would like to ex- press their gratitude to Professor Whospitality and many valuableREFERENCES  C. Ma, W. Chen and C. 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