SOME NONLINEAR DIFFERENTIAL INEQUALITIES AND AN APPLICATION TO HÖLDER CONTINUOUS ALMOST COMPLEX STRUCTURES

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1 SOME NONLINEAR DIFFERENTIAL INEQUALITIES AND AN APPLICATION TO HÖLDER CONTINUOUS ALMOST COMPLEX STRUCTURES ADAM COFFMAN AND YIFEI PAN Abstract. We consider soe second order quasilinear partial differential inequalities for real valued functions on the unit ball and find conditions under which there is a lower bound for the supreu of nonnegative solutions that do not vanish at the origin. As a consequence, for coplex valued functions fz satisfying f/ z = f α,0<α<1, and f0 0, there is also a lower bound for sup f on the unit disk. For each α, we construct a anifold with an α-hölder continuous alost coplex structure where the Kobayashi-Royden pseudonor is not upper seicontinuous. 1. Introduction We begin with an analysis of a second order quasilinear partial differential inequality for real valued functions of n real variables, 1 Δu B u ε 0, where B>0andε [0, 1 are constants. In Section 2, we use a Coparison Principle arguent to show that 1 has no sall solutions, in the sense that there is a unifor lower bound M>0for the supreu of solutions u which are nonnegative on the unit ball and nonzero at the origin. We also consider a generalization of 1: 2 uδu B u 1+ε C u 2 0, and find conditions under which there is a siilar property of no sall solutions, in Theore 2.4. As an application of the results on the inequality 1, we show failure of upper seicontinuity of the Kobayashi-Royden pseudonor for a faily of 4-diensional anifolds with alost coplex structures of regularity C 0,α,0<α<1. This generalizes the α = 1 exaple of [IPR]; it is known 2 Date: February 3, Matheatics Subject Classification. Priary 35R45; Secondary 32F45, 32Q60, 32Q65, 35B05. 1

2 2 ADAM COFFMAN AND YIFEI PAN [IR] that the Kobayashi-Royden pseudonor is upper seicontinuous for alost coplex structures with regularity C 1,α. Our construction of the alost coplex anifolds in Section 4 is very siilar to that of [IPR]; we give the details for the convenience of the reader, and to show how the arguent breaks down as α 1, due to a shrinking radius of the doain. We also take the opportunity in Section 3 to state soe Leas which allow for a ore quantitative description than that of [IPR]. One of the steps in [IPR] is a Maxiu Principle arguent applied to a coplex valued function hz satisfying the equation h/ z = h 1/2,to get the property of no sall solutions. The ain difference between our paper and [IPR] is the use of a Coparison Principle in Section 2 instead of the Maxiu Principle, and we arrive at this result: Theore 1.1. For any α 0, 1, supposehz is a continuous coplex valued function on the closed unit disk, and on the set {z : z < 1,hz 0}, h has continuous partial derivatives and satisfies h 3 z = h α. If h0 0then sup h >S α, where the constant S α > 0 is defined by: 1/1 α 21 α 4 S α =. 2 α 2. Soe differential inequalities Let D R denote the open ball in R n centered at 0 with radius R>0, and let D R denote the closed ball. Lea 2.1. Given constants B>0 and 0 ε<1, let B1 ε ε M = > 0. 22ε + n1 ε Suppose the function u : D 1 R satisfies: u is continuous on D 1, u x 0 for x D 1, on the open set ω = { x D 1 : u x 0}, u C 2 ω, for x ω: 5 If u 0 0,then sup x D 1 u x >M. Δu x Bu x ε 0.

3 NONLINEAR INEQUALITIES AND ALMOST COMPLEX STRUCTURES 3 Proof. Define a coparison function v x =M x 2 1 ε, so v C 2 R n since0 ε<1. By construction of M, itcanbechecked that v is a solution of this nonlinear Poisson equation on the doain R n : Δv x B v x ε 0. Suppose, toward a contradiction, that u x M for all x D 1. For apoint x 0 on the boundary of ω R n,either x 0 =1,inwhichcaseby continuity, u x 0 M = v x 0, or 0 < x 0 < 1andu x 0 =0,sou x 0 v x 0. Since u v on the boundary of ω, the Coparison Principle [GT] Theore 10.1 applies to the subsolution u and the solution v on the doain ω. The relevant hypothesis for the Coparison Principle in this case is that the second ter expression of 5, BX ε, is weakly decreasing, which uses B > 0 and ε 0. To satisfy this technical condition for all X R, we define a function c : R R by cx = BX ε for X 0, and cx =0forX 0. Then c is weakly decreasing in X, v satisfies Δv x+ cv x 0 and u satisfies Δu x+ cu x 0. The conclusion of the Coparison Principle is that u v on ω, however 0 ω and u 0 >v 0, a contradiction. Of course, the constant function u 0 satisfies the inequality 5, and so does the radial coparison function v, so the initial condition u 0 0 is necessary. Exaple 2.2. In the n = 1 case, M = 1 B1 ε 2 1 ε. 21+ε For points c 1, c 2 R, c 1 <c 2, define a function Mx c ε if x c 2 ux = 0 if c 1 x c 2. Mc 1 x 2 1 ε if x c 1 Then u C 2 R, and it is nonnegative and satisfies u = B u ε the case of equality in the n = 1 version of 5. For c 1 < 0 <c 2, this gives an infinite collection of solutions of the ODE u = B u ε which are identically zero in a neighborhood of 0, so the ODE does not have a unique continuation property. For c 1 > 0orc 2 < 0, the function u satisfies u0 0andthe other hypotheses of Lea 2.1, and its supreu on 1, 1 exceeds M even though it can be identically zero on an interval not containing 0.

4 4 ADAM COFFMAN AND YIFEI PAN Exaple 2.3. In the case n =2,B =1,ε = 0, 5 becoes the linear inequality Δu 1 and the nuber M = 1 agrees with Lea 2 of [IPR], 4 which was proved there using a Maxiu Principle arguent. By applying Lea 2.1 to the Laplacian of a power of u, wegetthe following generalization. Theore 2.4. Given constants B>0, C R, andε<1, let B1 ε ε if C ε 22ε C+n1 ε M = 1. 1 ε if C ε B1 ε 2n Suppose the function u : D 1 R satisfies: u is continuous on D 1, u x 0 for x D 1, on the open set ω = { x D 1 : u x 0}, u C 2 ω, for x ω: u xδu x B u x 1+ε + C u x 2. If u 0 0,then sup u x >M. x D 1 Proof. Let μ =in{ε, C}, soμ ε<1, and on the set ω, u xδu x B u x 1+ε + μ u x 2. Consider the function u 1 μ on D 1,sou 1 μ C 0 D 1 C 2 ω, and on the set ω, Δu 1 μ = 1 μu μ 1 uδu μ u 2 1 μu μ 1 Bu 1+ε = 1 μb u 1 μ ε μ/1 μ. Since 1 μb >0, and μ ε<1 = 0 ε μ < 1, Lea 2.1 applies 1 μ to u 1 μ.ifu 0 1 μ 0,then 1 1 μb1 ε μ 1 μ 2 1 ε μ 1 μ sup u 1 μ > = sup u > 22 ε μ 1 μ + n1 ε μ 1 μ B1 ε 2 22ε μ+n1 ε 1 1 ε. Functions satisfying a differential inequality of the for 1 or 2 also satisfy a Strong Maxiu Principle; the only condition is B>0.

5 NONLINEAR INEQUALITIES AND ALMOST COMPLEX STRUCTURES 5 Theore 2.5. Given any open set Ω R n, and any constants B>0, C, ε R, suppose the function u :Ω R satisfies: u is continuous on Ω, on the set ω = { x Ω:u x > 0}, u C 2 ω, on the set ω, u satisfies uδu B u 1+ε C u 2 0. If u x 0 > 0 for soe x 0 Ω, thenu does not attain a axiu value on Ω. Proof. Note that the constant function u 0 is the only locally constant solution of the inequality for B>0. If B = 0 then obviously any constant function would be a solution. Given a function u satisfying the hypotheses, ω is a nonepty open subset of Ω. Suppose, toward a contradiction, that there is soe x 1 Ω with u x u x 1 for all x Ω. In particular, u x 1 u x 0 > 0, so x 1 ω. Letω 1 be the connected coponent of ω containing x 1. For x ω 1, u satisfies the linear, uniforly elliptic inequality Δu x+ Bu x ε 1 u x u x+ C u x u x 0, where the coefficients defined in ters of the given u are locally bounded functions of x, and Bu x ε 1 is negative for all x ω. It follows fro the Strong Maxiu Principle [GT] Theore 3.5 that since u attains a axiu value at x 1,thenu is constant on ω 1. Since the only constant solution is 0, it follows that u x 1 = 0, a contradiction. The next Lea shows how an inequality like 5 with n =2canarise fro a first order PDE for a coplex valued function. Lea 2.6. Consider constants α, γ R with 0 <α<1. Letω C be an open set, and suppose h : ω C satisfies: h C 1 ω, hz 0for all z ω, h z = h α on ω. Then, the following inequality is satisfied on ω: 6 Δ h 1 αγ 41 αγ 2 α 2 h 1 αγ 2. Reark. The special case α = 1, γ = 3 is Lea 1 of [IPR]; its Proof 2 2 there is a long calculation in polar coordinates, which can be generalized to soe other values of α by an analogous arguent. However, using z, z coordinates allows for a shorter calculation.

6 6 ADAM COFFMAN AND YIFEI PAN Proof of Lea 2.6. We first want to show that h is sooth on ω, applying the regularity and bootstrapping technique of PDE to the equation h/ z = h α. We recall the following fact for a ore general stateent, see Theore of [AIM]: for a nonnegative integer l, and0<β<1, if ϕ C l,β loc ω andg : ω C has first derivatives in L2 loc ω and is a solution of g/ z = ϕ, theng C l+1,β loc ω. In our case, ϕ = h α C 1 ω C 0,β loc ω since h C 1 ω and is nonvanishing, and g = h has continuous first derivatives, so we can conclude that g = h C 1,β loc h C 2,β loc ω, etc. ω. Repeating gives that Since the conclusion is a local stateent, it is enough to express ω as a union of open subsets ω k and establish the conclusion on each subset. For each z k ω, there is a sufficiently sall disk ω k containing z k, where real exponentiation of hz is well-defined on ω k, by choosing a single-valued branch of log to define h r =expr logh. The condition h z = h α can be re-written h z = h z = h α = h α/2 hα/2. This leads to h z z = h z z =h α/2 hα/2 z = α h α/2 1 hα/2 h z + h α hα 1 2 = h z z, which is used in a line of the next step. For an arbitrary exponent R, h z z = h /2 h/2 z z 2 h 2 1 h z h 2 + h 2 2 h 2 1 h z = z = 2 z = 2 = 2 h 2 1+ α 2 h 2 + α 2 + h 2 h 2 1 h z [ 2 + α 2 1 h 2 + α 2 2 h z h 2 + α 2 +h 2 + α α h 2 1 h z h 2 1 h z h 2 + α 2 1 h z ] +h h 2 2 h z h z + h 2 h 2 1 h z z. [Re + α 2 h +α 4 h2 h z α h +2α 2 + ] 2 h 2 h z 2.

7 NONLINEAR INEQUALITIES AND ALMOST COMPLEX STRUCTURES 7 With the ai of applying Lea 2.1 to the function h, we consider the expression 8, with real constants B, ε, and 0.Inline9,weassign 7 ε = 1 +2α 2 to be able to cobine like ters, and in line 10, we choose B =4 2 α 2 to coplete the square. 8 Δ h B h ε = 4 h z z B h ε = 2 [Re + α 2 h +α 4 h2 h z α h +2α 2 + ] 2 h 2 h z 2 B h ε 9 = +2α B h +2α 2 +Re 2 + α 2 h +α 4 h2 h z + 2 h 2 h z 2 h α B h 2α 2 + α 2 h α h z + 2 h z 2 10 = h 2 + α 2 h α h z 2 0. Considering the for of 7, it is convenient to choose =1 αγ for soe constant γ 0. The clai of the Lea follows; the γ = 0 case can be checked separately. The paraeter γ can be chosen arbitrarily large; to apply Lea 2.1 to get the no sall solutions result of Theore 1.1, we need the RHS coefficient in 6 to be positive, so γ> 2 α2, and also the RHS exponent 41 α 1 αγ 2 to be nonnegative, so γ 2. In contrast, the α = 1, 2 γ = 3 caseappearinginlea1of[ipr]hasrhsexponent 1. The 2 4 approach of Theore 2 of [IPR] is to use the negative exponent together with the result of Exaple 2.3 to show that assuing h has a sall solution leads to a contradiction. As claied, their ethod can be generalized to apply to other nonpositive exponents, but 2 α2 <γ 2 holds only for 41 α α< Proof of Theore 1.1. Given a continuous h : D 1 C satisfying the hypotheses of Theore 1.1, on the set ω = {z D 1 : hz 0}, h C 1 ω, and the conclusion of Lea 2.6 can be re-written: 11 Δ h 1 αγ 41 αγ 2 α 2 h 1 αγ 1 2 γ. The hypotheses of Lea 2.1 are satisfied with n =2,ux, y = hx + iy 1 αγ,andu 0 0, when the RHS of 11 has a positive coefficient

8 8 ADAM COFFMAN AND YIFEI PAN so γ> 2 α2 and the quantity ε = α γ conclusion of Lea 2.1 is: sup z D 1 hz 1 αγ > M = αγ 2 α2 is in [0, 1 for γ 2. The 2 γ/2 2 γ 1 41 αγ 2 α 2 21 α = sup hz >. z D 1 γ 2 We can optiize this lower bound, using eleentary calculus to show that the axiu value of 41 αγ 2 α2 is achieved at the critical point { } γ 2 γ = 2 α2 > ax 2, 2 α2, and the lower bound for the sup is S 21 α 41 α α as appearing in 4. Note that S α is decreasing for 0 <α<1, with S 1/2 = 4 9, S 2/3 = 1 8,and S α 0asα 1. This Theore is used in the Proof of Theore 4.3. Exaple 2.7. As noted by [IPR], a one-diensional analogue of Equation 3 in Theore 1.1 is the well-known for exaple, [BR] I.9 ODE u x = B ux α for 0 <α<1andb>0, which can be solved explicitly. By an eleentary separation of variables calculation, the solution on an interval where u 0is ux =±1 αbx + C 1 1 α. The general solution on the doain R is, for c 1 <c 2, ux = So u C 1 R, and if u0 0, then 1 α 1 1 α Bx c2 1 1 α if x c 2 0 if c 1 x c 2 1 α 1 1 α Bc1 x 1 1 α if x c 1 sup ux > 1 αb 1 1 α. 1<x<1 3. Leas for holoorphic aps We continue with the D R notation for the open disk in the coplex plane centered at the origin. The following quantitative Leas on inverses of holoorphic functions D R C are used in a step of the Proof of Theore 4.3 where we put a ap D r C 2 into a noral for, 14. Lea 3.1 [G] Exercise I.1.. Suppose f : D 1 D 1 is holoorphic, with f0 = 0, f 0 = δ>0. For any η 0,δ, lets = η; then. δ η 1 ηδ the restricted function f : D η D 1 takes on each value w D s exactly once.

9 NONLINEAR INEQUALITIES AND ALMOST COMPLEX STRUCTURES 9 The hypotheses iply δ 1 by the Schwarz Lea. Lea 3.2. For a holoorphic ap Z 1 : D r D 2 with Z 1 0 = 0, Z 1 0 = 1, ifr>4 2 then there exists a continuous function φ : D 3 1 D r which is holoorphic on D 1 and which satisfies Z 1 φz =z for all z D 1. Reark. It follows fro the Schwarz Lea that r 2, and it follows fro the fact that φ is an inverse of Z 1 that φ0 = 0 and φ 0 = 1. Proof of Lea 3.2. Define a new holoorphic function f : D 1 D 1 by fz = 1 2 Z 1r z, so f0 = 0, f 0 = r, and Lea 3.1 applies with δ = r. If we choose 2 2 η = 3r 3r2,thens =, and the assuption r> 4 2 iplies s> 1. It r follows fro Lea 3.1 that there exists a function ψ : D s D η such that f ψz = z for all z D 1/2 D s ; this inverse function ψ is holoorphic on D 1/2. The claied function φ : D 1 D rη D r is defined by φz =r ψ 1 z, so for z D 2 1, Z 1 φz = Z 1 r ψ 1 2 z = 2 fψ1 2 z = z = z. 4. J-holoorphic disks For S>0, consider the bidisk Ω S = D 2 D S C 2, as an open subset of R 4, with coordinates x =x 1,y 1,x 2,y 2 =z 1,z 2 and the trivial tangent bundle T Ω S T R 4. Consider an alost coplex structure J on Ω S given by a coplex structure operator on T x Ω S of the following for: 12 J x = λ 0 1 λ 0 1 0, where λ :Ω S R is any function. A differentiable ap Z : D r Ω S is a J-holoorphic disk if dz J std = J dz, wherej std is the standard coplex structure on D r C. Let z = x + iy be the coordinate on D r. For J of the for 12, if Zz is defined by coplex valued coponent functions, 13 Z : D r Ω S : Zz =Z 1 z,z 2 z, then the J-holoorphic property iplies that Z 1 : D r D 2 is holoorphic in the standard way.

10 10 ADAM COFFMAN AND YIFEI PAN Exaple 4.1. If the function λz 1,z 2 satisfiesλz 1, 0 = 0 for all z 1 D 2, then the ap Z : D 2 Ω S : Zz =z, 0 is a J-holoorphic disk. Definition 4.2. The Kobayashi-Royden pseudonor on Ω S is a function T Ω S R : x, v x, v K, defined on tangent vectors v T x Ω S to be the nuber { 1 glb r : a J-holoorphic Z : D r Ω S, Z0 = x, dz0 } x = v. Under the assuption that λ C 0,α Ω S, 0 <α<1, it is shown by [IR] and [NW] that there is a nonepty set of J-holoorphic disks through x with tangent vector v as in the Definition, so the pseudonor is a welldefined function. Further, each such disk satisfies Z C 1 D r. At this point we pick α 0, 1 and set λz 1,z 2 = 2 z 2 α. Let S = S α > 0 be the constant defined by forula 4 fro Theore 1.1. Then, Ω S,J is an alost coplex anifold with the following property: Theore 4.3. If 0 b D S then 0,b, 1, 0 K Reark. Since , and 0, 0, 1, 0 2 K 1 by Exaple 4.1, 2 the Theore shows that the Kobayashi-Royden pseudonor is not upper seicontinuous on T Ω S. Proof. Consider a J-holoorphic ap Z : D r Ω S of the for 13, and suppose Z0 = 0,b Ω S and dz0 =1, 0. Then the holoorphic x function Z 1 : D r D 2 satisfies Z 1 0 = 0, Z 1 0 = 1, and Z 2 C 1 D r satisfies Z 2 0 = b. Suppose, toward a contradiction, that there exists such a ap Z with b 0 and r > 4 2. Then Lea 3.2 applies to Z 3 1: there is a reparaetrization φ which puts Z into the following noral for: 14 Z φ :D 1 Ω S z Z 1 φz,z 2 φz = z, fz, where f = Z 2 φ : D 1 D S satisfies f C 0 D 1 C 1 D 1. Fro the fact that Z φ is J-holoorphic on D 1, it follows fro the for 12 of J that if fz =ux, y+ivx, y, then f satisfies this syste of nonlinear Cauchy-Rieann equations on D 1 : 15 du dy = dv dx du dv and + λz, fz = dx dy

11 NONLINEAR INEQUALITIES AND ALMOST COMPLEX STRUCTURES 11 with the initial conditions f0 = b, u x 0 = u y 0 = v x 0 = 0 and v y 0 = λ0,b= 2 b α. The syste of equations iplies 16 f z = 1 2 u + iv+i u + iv x y = 1 2 u x v y + iv x + u y = 1 2 λz, fz = f α. So, Theore 1.1 applies, with f = h. The conclusion is that sup fz z D 1 > S α, but this contradicts fz <S= S α. The previously entioned existence theory for J-holoorphic disks shows there are interesting solutions of the equation 16, and therefore also the inequality 11. Exaple 4.4. For 0 < α < 1, Ω S,J, λz 1,z 2 = 2 z 2 α as above, aapz : D r Ω S of the for Zz =z, fz is J-holoorphic if fx, y =ux, y +ivx, y is a solution of 15. Again generalizing the α = 1 case of [IPR], exaples of such solutions can be constructed for 2 sall r by assuing v 0andudepends only on x, so 15 becoes the ODE u x 2 ux α = 0. This is the equation fro Exaple 2.7; we can conclude that J-holoorphic disks in Ω S do not have a unique continuation property. Acknowledgents. The authors acknowledge the insightful rearks of the referee, whose suggestions led to a shorter proof of Lea 2.6 and an iproveent of Theore 2.4. References [AIM] K. Astala, T. Iwaniec, andg. Martin, Elliptic Partial Differential Equations and Quasiconforal Mappings in the Plane, PMS48, Princeton University Press, Princeton, MR j:30040, Zbl [BR] G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, Ginn & Co., MR #2253, Zbl [G] J. Garnett, Bounded Analytic Functions, rev. firsted., GTM236, Springer, [GT] MR e:30049, Zbl D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer CIM, MR k:35004, Zbl [IPR] S. Ivashkovich, S. Pinchuk, andj.-p. Rosay, Upper sei-continuity of the Kobayashi-Royden pseudo-nor, a counterexaple for Hölderian alost coplex structures, Ark.Mat , MR g:32038, Zbl

12 12 ADAM COFFMAN AND YIFEI PAN [IR] S. Ivashkovich and J.-P. Rosay, Schwarz-type leas for solutions of - inequalities and coplete hyperbolicity of alost coplex anifolds, Ann. Inst. Fourier Grenoble , MR a:32032, Zbl [NW] A. Nijenhuis and W. Woolf, Soe integration probles in alost-coplex and coplex anifolds, Ann.ofMath , MR #6992, Zbl Departent of Matheatical Sciences, Indiana University - Purdue University Fort Wayne, 2101 E. Coliseu Blvd., Fort Wayne, IN, USA E-ail address: CoffanA@ipfw.edu School of Sciences, Nanchang University, Nanchang, P.R.China E-ail address: Pan@ipfw.edu