sup inf inequality on manifold of dimension 3

Transcription

1 Mathematca Aeterna, Vol., 0, no. 0, 3-6 sup nf nequalty on manfold of dmenson 3 Samy Skander Bahoura Department of Mathematcs, Patras Unversty, 6500 Patras, Greece samybahoura@yahoo.fr Abstract We gve an estmate of type sup nf on remannan manfold of dmenson 3 for the prescrbed curvature equaton. Mathematcs Subject Classfcaton: 53C, 35J60 35B45 35B50 Keywords: sup nf, remannan manfold, dmenson 3, prescrbed curvature. Introducton and Man Results In dmenson 3, the scalar curvature equaton s: 8 u + R g u = V u 5, u > 0. Where R g s the scalar curvature and V s a functon (called the prescrbed scalar curvature). We consder three postve real number a, b, A and we suppose V lpschtzan: (E) 0 < a V (x) b < + and V L (M) A. (C) The equaton (E) was studed a lot when M = Ω R n or M = S n see for example [], [6], [9]. In these cases we have some nequaltes of type sup nf. The correspondng equaton n dmenson, on open set Ω of R, s: u = V e u, (E ) The equaton (E ) was studed a lot and we can fnd many mportant results about a pror estmates n [3], [4], [7], [0], and [3]. In the case V and M compact, the equaton (E) s Yamabe equaton. T.Aubn and R.Schoen have proved the exstence of soluton n ths case, see for example [] and [8].

2 4 Samy Skander Bahoura When M s a compact remannan manfold, there s some compactness results for the equaton (E) see [-]. L and Zhu, see [], proved that the energy s bounded, and, f we assume M not dffeormorfc to the three sphere, the solutons are unformly bounded. They use the postve mass theorem. Now, f we suppose M a remannan manfold (not necessarly compact) and V, L and Zhang [] proved that the product sup nf s bounded. Here, we gve an equalty of type sup nf for the equaton (E) wth general condtons (C). We have: Theorem. For all compact set K of M and all postve numbers a,b,a, there s a postve constant c, whch depends only on, a,b,a,k,m,g such that: (sup u) /3 nf u c, K M for all u soluton of (E) wth condtons (C). As a consequence of the prevous theorem, we have an estmate of the maxmum f we control the mnmum of the solutons: Corollary. For all compact set K of M and all postve numbers a,b,a, m, there s a postve constant c, whch depends only on, a,b,a, m, K,M,g such that: sup u c, f nf u m > 0, K M for all u soluton of (E) wth condtons (C). Note that n our work, we have not assumpton on energy or boundary condton f we assume the manfold M wth boundary. Next, n the proof of the prevous theorem, we can replace the scalar curvature by any smooth functon f, but here we proof the result wth R g the scalar curvature. Proof of the Theorem Part I: The metrc and the laplacan n polar coordnates. Let (M, g) a Remannan manfold. We note g x,j the local expresson of the metrc g n the exponental map centered n x. We are concernng by the polar coordnates expresson of the metrc. By usng Gauss lemma, we can wrte:

3 sup nf nequalty 5 g = ds = dt + g k j(r, θ)dθ dθ j = dt + r g k j(r, θ)dθ dθ j = g x,j dx dx j, n a polar chart wth orgn x", ]0, ɛ 0 [ U k, wth (U k, ψ) a chart of S n. We can wrte the element volume: then, dv g = r n È g k drdθ... dθ n = È [det(g x,j )]dx... dx n, dv g = r n È [det(g x,j )][exp x (rθ)]α k (θ)drdθ... dθ n, where, α k s such that, dσ Sn = α k (θ)dθ... dθ n. (Remannan volume element of the la sphere n the chart (U k, ψ) ). Then, È gk = α k (θ) È [det(g x,j )], Clearly, we have the followng proposton: Proposton. Let x 0 M, there exst ɛ > 0 and f we reduce U k, we have: r g k j(x, r, θ) + r θ m g k j(x, r, θ) Cr, x B(x 0, ɛ ) r [0, ɛ ], θ U k. and, r g k (x, r, θ) + r θ m g k (x, r, θ) Cr, x B(x 0, ɛ ) r [0, ɛ ], θ U k. Remark: r [log È g k ] s a local functon of θ, and the restrcton of the global functon on the sphere S n, r [log È det(g x,j )]. We wll note, J(x, r, θ) = È det(g x,j ). Let s wrte the laplacan n [0, ɛ ] U k, = rr + n r + r [log È g r k ] r + r È g k È θ ( gθ θj gk θ j). We have,

4 6 Samy Skander Bahoura = rr + n r + r log J(x, r, θ) r + r r È g k È θ ( gθ θj gk θ j). We wrte the laplacan ( radal and angular decomposton), = rr + n r + r [log J(x, r, θ)] r Sr(x), r where Sr(x) s the laplacan on the sphere S r (x). We set L θ (x, r)(...) = r Sr(x)(...)[exp x (rθ)], clearly, ths operator s a laplacan on S n for partcular metrc. We wrte, and, L θ (x, r) = gx,r,sn, = rr + n r + r [J(x, r, θ)] r r r L θ(x, r). If, u s functon on M, then, ū(r, θ) = u[exp x (rθ)] s the correspondng functon n polar coordnates centered n x. We have, u = rr ū + n r ū + r [J(x, r, θ)] r ū r r L θ(x, r)ū. Part II: "Blow-up" and "Movng-plane" methods The "blow-up" technque Let, (u ) a sequence of functons on M such that, 8 u + R g u = V u 5, u > 0, (E) We argue by contradcton and we suppose that sup /3 nf s not bounded. We assume that: c, R > 0 u c,r soluton of (E) such that: R[ sup u c,r ] /3 nf u c,r c, B(x 0,R) M (H)

5 sup nf nequalty 7 Proposton. There exst a sequence of ponts (y ), y x 0 and two sequences of postve real number (l ), (L ), l 0, L +, such that f we consder v (y) = u [exp y (y)], we have: u (y ) 0 < v (y) β /, β. Œ / v (y), unformly on every compact set of R 3. + y l [u (y )] /3 nf M u + Proof: We use the hypothess (H), we can take two sequences R > 0, R 0 and c +, such that, R [ sup u ] /3 nf u c +, B(x 0,R ) M Let, x B(x 0, R ), such that sup B(x0,R ) u = u (x ) and s (x) = [R d(x, x )] / u (x), x B(x, R ). Then, x x 0. We have, Set : max s (x) = s (y ) s (x ) = R / u (x ) c +. B(x,R ) l = R d(y, x ), ū (y) = u [exp y (y)], v (z) = u [exp y (z/[u (y )] )]. u (y ) Clearly, y x 0. We obtan: L = l (c ) / [u (y )] = [s (y )] c / c c / = c / +. If z L, then y = exp y [z/[u (y )] ] B(y, δ l ) wth δ = (c ) d(y, y ) < R d(y, x ), thus, d(y, x ) < R and, s (y) s (y ), we can wrte, u (y)[r d(y, y )] / u (y )(l ) /. / and

6 8 Samy Skander Bahoura But, d(y, y ) δ l, R > l and R d(y, y ) R δ l > l δ l = l ( δ ), we obtan, 0 < v (z) = u (y) u (y ) l / /. l ( δ ) / We set, β =, clearly β. δ The functon v s soluton of: g jk [exp y (y)] jk v k h g jk È g [exp y (y)] j v + R g[exp y (y)] [u (y )] 4 v = V v 5, By elleptc estmates and Ascol, Ladyzenskaya theorems, (v ) converge unformely on each compact to the functon v soluton on R 3 of, 8 v = V (x 0 )v 5, v(0) =, 0 v /, Wthout loss of generalty, we can suppose V (x 0 ) = 4. By usng maxmum prncple, we have v > 0 on R 3, the result of Caffarell- Œ / Gdas-Spruck ( see [5]) gve, v(y) =. We have the same propretes + y for v n the prevous paper []. Polar coordnates and "movng-plane" method Let, w (t, θ) = e / ū (e t, θ) = e t/ u [exp y (e t θ)], et a(y, t, θ) = log J(y, e t, θ). Lemma.3 The functon w s soluton of: tt w t a t w L θ (y, e t ) + cw = V w 5, wth, c = c(y, t, θ) = + ta λe t, Proof: We wrte:

7 sup nf nequalty 9 t w = e 3t/ r ū + w, tt w = e 5t/ rr ū + e rū t + w. t a = e t r log J(y, e t, θ), t a t w = e 5t/ [ r log J r ū ] + taw. the lemma s proved. Now we have, t a = tb b, b (y, t, θ) = J(y, e t, θ) > 0, We can wrte, b tt ( È b w ) L θ (y, e t )w + [c(t) + b / b (t, θ)]w = V w N, where, b (t, θ) = tt ( b ) = tt b b 4(b ) ( tb 3/ ). Let, w = È b w. Lemma.4 The functon w s soluton of: tt w + gy,e t, S ( w ) + θ ( w ). θ log( È b ) + (c + b / b c ) w = = V w 5, b where, c = [ gy ( b,e b t, ) + θ log( b ) ]. Sn Proof: We have: But, tt w È b gy,e t, S w + (c + b ) w = V b w 5,

8 0 Samy Skander Bahoura gy,e t, S ( È b w ) = È b gy,e t, S w θ w. θ È b + w gy,e t, S ( È b ), and, we deduce, θ ( È b w ) = w θ È b + È b θ w, È b gy,et,s w = gy ( w,e t, ) + θ ( w ). θ log( È b ) c w, S wth c = [ gy ( b,e b t, ) + θ log( b ) ]. The lemma s proved. S The "movng-plane" method: Let ξ a real number, and suppose ξ t, we set t ξ w (t ξ, θ). = ξ t and w ξ (t, θ) = We have, tt w ξ + gy,e tξ S ( w )+ θ ( w ξ ). θ log( È b ) w ξ +[c(t ξ )+b / (t ξ,.)b (t ξ ) c ξ ] w ξ = = V ξ b ξ! ( w ξ ) 5. By usng the same arguments than n [], we have: Proposton.5 ) w (λ, θ) w (λ + 4, θ) k > 0, θ S. For all β > 0, there exst c β > 0 such than: ) c β e t/ w (λ + t, θ) c β e t/, t β, θ S. We set, Z = tt (...) + gy,e t, S (...) + θ (...). θ log( È b ) + (c + b / b c )(...)

9 sup nf nequalty Remark: In the opertor Z, by usng the proposton 3, the coefcent c + b / b c satsfy: c + b / b c k > 0, for t << 0, t s fundamental f we want to apply Hopf maxmum prncple. Goal: Lke n [], we have ellptc second order operator, here t s Z, the goal s to use the "movng-plane" method to have a contradcton. For ths, we must have: We wrte: Z ( w ξ w ) 0, f w ξ w 0. Z ( w ξ w ) = ( gy,e tξ,s gy,e t, S )( w ξ )+ q q +( θ,e t ξ θ,e t)(w ξ ). θ,e t ξ log( b ξ )+ θ,e t( w ξ ). θ,e t ξ [log( b ξ ) log È b ]+ + θ,e tw ξ.( θ,e t ξ θ,e t) log È b [(c+b / b c ) ξ (c+b / b c )] w ξ + +V ξ Clearly, we have: b ξ! ( w ξ ) 5 V b w 5. ( ) Lemma.6 b (y, t, θ) = 3 Rcc y (θ, θ)e t +..., R g (e t θ) = R g (y )+ < R g (y ) θ > e t Accordng to proposton. and lemma.6, we have Proposton.7 Z ( w ξ w ) V b ( ) [( w ξ ) 5 w 5 ] + ( w ξ ) 5 V ξ V + +C e t e tξ θ w ξ + θ( w ξ ) + Rcc y [ w ξ + ( w ξ ) 5 ] + R g (y ) w ξ +C e 3tξ e 3t.

10 Samy Skander Bahoura Proof: We use proposton., we have: a(y, t, θ) = log J(y, e t, θ) = log b, t b (t) + tt b (t) + tt a(t) Ce t, and, then, θj b + θj,θ k b + t,θj b + t,θj,θ k b Ce t, t b (t ξ ) t b (t) C e t e tξ, on ], log ɛ ] S, x B(x 0, ɛ ) Locally, gy,e t, S = L θ (y, e t ) = È gk (e t, θ) θ j θ l[ gθl (e t, θ) È g k (e t, θ) θ j]. Thus, n [0, ɛ ] U k, we have, 6 A = 44È gk È 3 3 ξ θ j θ l( gθl g k θ j) 5 È gk È θ j θ l( gθl g k 7 θ j) 5 ( w ξ ) then, A = B + D wth, and, B = h g θl θ j (e tξ, θ) g θl θ j (e t, θ) θ l θ j wξ (t, θ), D = 4È gk (e tξ, θ) θ j θ l[ gθl (e tξ, θ) È g k (e tξ, θ)] È gk (e t, θ) θ j θ l[ gθl (e t, θ) È 3 g k (e t, θ)] 5 θ j w ξ (t, we deduce, A C k e t e tξ θ w ξ + θ( w ξ ), We take C = max{c, q} and f we use ( ), we obtan proposton.7. We have,

11 sup nf nequalty 3 Then, c(y, t, θ) = + 4 ta + R g e t, (α ) b (t, θ) = tt ( È b ) = tt b b 4(b ) ( tb 3/ ), (α ) c = [ gy ( È b,e b t, ) + θ log( È b ) ], (α 3 ) Sn t c(y, t, θ) = tta + e t R g (e t θ) + e 3t < R g (e t θ) θ >, by proposton, t c + t b + t b + t c K e t, Now, we consder the functon, w (t, θ) = w (t, θ) [u (y )] /3 mn M u e t, and λ > > 0. For t t = (/3) log u (y ), we have: w (t, θ) = e t " b (t, θ)e t/ u o exp y (e t θ) [u (y )] /3 # mn M u e t [u (y )] /3 mn M u > 0, We set, µ = [u (y )] /3 mn M u. We use proposton.5 and the same arguments than n [], we obtan: Lemma.8 There exsts ν < 0 such that for µ ν : w µ (t, θ) w (t, θ) 0, (t, θ) [µ, t ] S, We set, λ = log u (y ), then, Lemma.9 w (λ, θ) w (λ + 4, θ) > 0.

12 4 Samy Skander Bahoura Proof of lemma.9: Clearly: w (λ, θ) w (λ + 4, θ) = w (λ, θ) w (λ + 4, θ) + µ e λ (e 4 ), we deduce lemma.9 from proposton.5. Let, ξ = sup{µ λ +, w ξ (t, θ) w (t, θ) 0, (t, θ) [ξ, t ] S }. The real ξ exsts (see []), f we use ( ), we have: w ξ (t, θ) + θ w ξ (t, θ) + θ w ξ (t, θ) C(R), (t, θ) ], log R] S, We can wrte: Z ( w ξ w ) = Z ( w ξ w ) µ Z (e tξ e t ), Z (e tξ e t ) = [ 4 3 ta R g e t + b / b c ](e tξ e t ) c (e tξ e t ), wth c > 0, because t a + t b + tt b + t,θj b + t,θj,θ k b C e t <, for t very small. We use proposton.7, to obtan on, [ξ, t ] S, Z ( w ξ w ) c V [( w ξ wth c > 0. ) 5 w 5 ]+ V ξ V (w ξ ) 5 +[µ c C (R)](e tξ e t ) 0, Lke n [], after usng Hopf maxmum prncple, we have, We have: We deduce, sup θ S [ w ξ (t, θ) w (t, θ)] = 0. w ξ (t, θ ) w (t, θ ) = 0,. [u ( y )] /3 nf M u c,.

13 sup nf nequalty 5 t s n contradcton wth proposton. ACKNOWLEDGEMENTS. Ths work was done when the author was n Greece at Patras. The author s grateful to Professor Athanase Cotsols, the Department of Mathematcs of Patras Unversty and the IKY Foundaton for hosptalty and the excellent condtons of work. References [] T. Aubn. Some Nonlnear Problems n Remannan Geometry. Sprnger- Verlag 998 [] S.S Bahoura. Majoratons du type sup u nf u c pour l équaton de la courbure scalare sur un ouvert de R n, n 3. J. Math. Pures. Appl.(9) no, 9, [3] H. Brezs, YY. L Y-Y, I. Shafrr. A sup+nf nequalty for some nonlnear ellptc equatons nvolvng exponental nonlneartes. J.Funct.Anal.5 (993) [4] H.Brezs and F.Merle, Unform estmates and blow-up bhavor for solutons of u = V e u n two dmensons, Commun Partal Dfferental Equatons 6 (99), [5] L. Caffarell, B. Gdas, J. Spruck. Asymptotc symmetry and local behavor of semlnear ellptc equatons wth crtcal Sobolev growth. Comm. Pure Appl. Math. 37 (984) [6] C-C.Chen, C-S. Ln. Estmates of the conformal scalar curvature equaton va the method of movng planes. Comm. Pure Appl. Math. L(997) [7] C-C.Chen, C-S. Ln. A sharp sup+nf nequalty for a nonlnear ellptc equaton n R. Commun. Anal. Geom. 6, No., -9 (998). [8] J.M. Lee, T.H. Parker. The Yamabe problem. Bull.Amer.Math.Soc (N.S) 7 (987), no., [9] YY. L. Prescrbng scalar curvature on S n and related Problems. C.R. Acad. Sc. Pars 37 (993) Part I: J. Dffer. Equatons 0 (995) Part II: Exstence and compactness. Comm. Pure Appl.Math.49 (996) [0] YY. L. Harnack Type Inequalty: the Method of Movng Planes. Commun. Math. Phys. 00,4-444 (999).

14 6 Samy Skander Bahoura [] YY. L, L. Zhang. A Harnack type nequalty for the Yamabe equaton n low dmensons. Calc. Var. Partal Dfferental Equatons 0 (004), no., [] YY.L, M. Zhu. Yamabe Type Equatons On Three Dmensonal Remannan Manfolds. Commun.Contem.Mathematcs, vol. No. (999) -50. [3] I. Shafrr. A sup+nf nequalty for the equaton u = V e u. C. R. Acad.Sc. Pars Sér. I Math. 35 (99), no.,