Smooth counterexamples to strong unique continuation for a Beltrami system in C 2

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1 Smooth couterexamples to strog uique cotiuatio for a Beltrami system i C Adam Coffma Yifei Pa October, 0 Abstract We costruct a example of a smooth map C C which vaishes to ifiite order at the origi, ad such that the ratio of the orm of the z derivative to the orm of the z derivative also vaishes to ifiite order. This gives a couterexample to strog uique cotiuatio for a vector valued aalogue of the Beltrami equatio. Itroductio We will costruct a example of a smooth fuctio u : C C which has a isolated zero of ifiite order at the origi z k uz 0asz 0 for all k 0, ad where the ratio of orms of derivatives u z / u z is small, also vaishig to ifiite order at z = 0. This behavior is obviously differet from that of a map u with u z 0, which would be holomorphic ad could ot have a isolated zero of ifiite order. This vector valued case is also differet from the complex scalar case, where solutios u : C C of the well-kow Beltrami equatio u z = azu z, for small az, also caot vaish to ifiite order at a isolated zero [B, [CH, [AIM, [R. More precisely, we will show i Sectio 4 that i a eighborhood of the origi, uz is a solutio of a Beltrami-type system of differetial equatios, which is liear, elliptic, ad has cotiuous coefficiets very close to those of the Cauchy-Riema system, but does ot have the strog uique cotiuatio property. I Theorem 4., we verify the coefficiets are ot Lipschitz cotiuous. Couterexamples to the weak uique cotiuatio property for elliptic Beltrami systems have bee cosidered by [IVV. The costructio was motivated by a example of Rosay [R ad questios posed by [IS, who were cosiderig the uique cotiuatio problem for systems of equatios from almost complex geometry. MSC 00: 35A0 Primary; 30G0, 3W50, 35J46 Secodary

2 I Sectio, we develop a geeral framework for costructig smooth maps C C vaishig to ifiite order. I Sectio 3, we preset both Rosay s example ad our ew example. I Sectio 5 we state some ope questios. Geeral Setup. Aular cutoff fuctios Start with a real valued fuctio sx whichissmoothor, with s 0o [0, 4, s icreasig o [ 4, 3 4, s =, s =,s = 0, ad s o[3 4,. For r > 0 ad two parameters 0 <r<r ad 0 < Δr <r r, deote the aulus A r,δr = {z = x + iy C : r z r +Δr} cotaied i the disk D r, ad defie a family of fuctios χ r,δr : A r,δr R by the formula χ r,δr z =s z r Δr. At a particular poit z = x, ỹ A r,δr, [ x [χ r,δrx, y x,ỹ = x + y s r x Δr x = s +ỹ r Δr = s z r Δr x z Δr. x,ỹ x x +ỹ The y derivative is similar, ad the z, z derivatives are complex liear combiatios. I particular, = Δr z χ r,δrz = z r z χ r,δrz =s x + iy Δr z Δr z χ r,δrz = z χ r,δrz m 0 Δr for some costat m 0 > 0 ot depedig o r, r or Δr. For higher derivatives of χ r,δr, the followig Lemma is a simplified versio of the Faà di Bruo formula for derivatives of composites. Lemma.. For k 0, there exist polyomials p abc x,x,x 3, q abc x,x,x 3 idexed by a, b, c 0, a + b = k, c k, with costat complex coefficiets ot depedig o r, r, Δr, ors, so that a b z a z b χ r,δrz = k c=0 z r s c pabcz, z, Δr+ z q abc z, z, Δr Δr z k Δr k. Proof. The k = 0 case is trivial ad the k = case is stated above. We record

3 the secod derivatives: z χ r,δrz = z χ r,δrz z r = s Δr z z χ r,δrz = s z z r z 4 z Δr + s Δr 4 z 3 Δr, 3 z r z r Δr 4Δr + s Δr 4 z Δr. The proof for all larger k is by iductio o k; the calculatio is straightforward ad omitted here. It follows as a cosequece of the Lemma that there are positive costats m ab idexed by a, b 0, a + b = k, ad depedig o the choices of s ad r, but ot depedig o r, Δr sothat = max z A r,δr a b z a z b χ r,δrz a b z a z b χ r,δrz m ab z k Δr k m ab r k Δr k. I various cases, i particular k = as i, the r k cabeimprovedwitha smaller expoet, but it is good eough to use later i Lemma.3.. The basic costructio of the examples Let r be a real sequece decreasig with limit = 0. Deote Δr = r r +. Let A deote the closed aulus A = A r+,δr = {z C : r + z r }, so the uio is a disk: D r = A {0}. The aular cutoff fuctios ca be idexed by : χ = χ r+,δr : A R. For N, letp be a icreasig positive iteger sequece; set p0 = 0. Let F be a positive real [ valued sequece; set F 0 =. Defie a fuctio 0 u : D r C by u0 = ad o the aulus A 0, for eve : uz = [ u z u z, u z = F z p, 4 u z = χ zf z p + χ zf +z p+. 5 For odd, switch the formulas for u, u. 3

4 So far, for ay s, r, p, F, the fuctio u is smooth o D r \{0}, ad exteds holomorphically for z r. We also have that u ad u z have o-zero value at every poit of D r \{0}; for eve ad switchig idices if is odd: u z = F p z p. 6 z u.3 Smoothess at the origi A importat property of the examples u : D r C we wat to costruct is that they are smooth at ad ear the origi. It is ot eough to check oly that the compoets vaish to ifiite order. I geeral, as easily costructed examples would show, for fuctios f : R N R M, f ca vaish to ifiite order at the origi: f x lim x 0 x k = 0 for all whole umbers k, but eed ot be smooth. Our approach to provig smoothess of our examples will be to show u, u, ad all their higher partial derivatives approach 0; this implies vaishig to ifiite order, as i the followig Lemma. Lemma.. Give f : R \{ 0} R, suppose f is smooth ad for each j, k =0,,, 3,..., k+j fx, y lim x,y 0 x k y j =0. The extedig f so that f0, 0 = 0 defies a smooth fuctio o R that vaishes to ifiite order at the origi. Proof. f is cotiuous at 0 byhypothesisj = k =0. Toshowf is smooth, we oly eed to show every partial derivative of order l of f exists at 0, ad has value 0; the it follows that for k + j = l, k+j fx,y is cotiuous at 0. x k y j The proof is by iductio o l; suppose for ay o-commutative word x k y j x ka y j b with k + k a +j +...+j b = k +j = l, l fx, y x k y j x ka y j b exists at 0 ad has value 0. The, the x-derivative at the origi is with the y-derivative beig similar: l fx, y x x k y j x ka y j b = lim t 0 = lim t 0 0,0 l f0 + t, 0 x k y j x ka y j b t l x k y j ft, 0 0 t. l f0, 0 x k y j x ka y j b 4

5 Let gt = l x k y ft, 0 for t 0; the lim gt = 0 by hypothesis, ad j t 0 g t = l+ x k+ ft, 0. yj L Hôpital s Rule applies to the above limit: gt lim t 0 t g t l+ = lim = lim t 0 t 0 x k+ ft, 0 = 0. yj The property of vaishig to ifiite order follows from Taylor approximatio at the origi. For our examples u, we wat to choose r, p, adf,sothatu is smooth ad vaishes to ifiite order at 0. The followig criterio for smoothess will be verified for both the Examples i Sectio 3. Lemma.3. If Δr /r Δr +/r + is a bouded sequece ad, for each iteger k 0, lim F +p + k r p+ 4k Δr /r k =0, 7 the u is smooth ad vaishes to ifiite order at the origi. Proof. From Lemma. ad the costructio of u, it is eough to show, for ay o-egative itegers a, b, with a + b = k, that a b a b max u z A z z, max u z A z z both have limit 0 as. The followig estimates for the derivatives assume is eve, ad sufficietly large compared to k. The derivatives of u = F z p 4 are easy: term: z a b u z F pp p k +z p k F p k r p k 8 The derivatives of u z 5 are cosidered oe term at a time. For the first a z z = F z = F b χ zf z p a a +a =a a b χ z z p z a b χ z z z 5 z a z p

6 The sum is over the a terms with may repeated that result from applyig the product rule a times. By Lemma., b χ z z a z m ab z a+b Δr a+b, ad a z p z = p p a + z p a p a z p a. Let m k = max m ab. The a+bk a b χ zf z p z z F a max a a { m ab z a+b Δr a+b } max a a F a m k z k Δr k p a z p a F k m k Δr k p k z p 3k {p a z p a } k F p k r p 3k m k Δr k. 9 Similarly for the secod term of u z, a b χ zf +z p+ z z k F +p + k r p+ 3k m k Δr k. 0 So, the criteria for all the derivatives of u to vaish at the origi are that the expressios 8, 9, ad 0 must all have limit 0 as.thehypothesis 7 is equivalet to 0 0. Comparig 0 to 8 by shiftig the idex i 8 from to +, this scalar multiple of 0 is much larger: F +p + k r p+ 3k Δr k >F +p + k r p+ k +, so if 0 has limit 0, the so does also implies F +r p+ k 0, which is eough to show u vaishes to ifiite order: z k uz 0asz 0. 6

7 Shiftig the idex i 9 from to + gives the followig quatity, which is comparable to 0: k F +p + k r p+ 3k + m k Δr + k < k F +p + k r p+ 4k Δr /r k m k Δr /r k Δr + /r + k, ad uder the additioal hypothesis that also implies Comparig first derivatives Δr /r Δr +/r + is a bouded sequece, 7 We wat to choose F, p, adr so that u z u z is small, as z 0. For z A, eve ad switchig idices if is odd, expadig the derivative ad usig gives: u z = z m 0 F z p + m 0 F + z p+ Δr Δr = m 0 Δr z p [ F z p p+ + F + z p+ p+. Usig 6 ad itroducig a factor g > 0, for z A : [ u z m 0r g F r p p + + F +r p+ p. u z Δr p gf The first fractio i the product is what we wat to make small for large, depedig o p ad Δr. The secod fractio we would like to make bouded, r depedig o F ad r, ad a arbitrary fudge factor g. The role of g is to maagethesizeoff ad simplify the calculatio provig boudedess of the secod factor, possibly at the expese of affectig the rate at which the first factor approaches 0. 3 Examples Example 3.. Rosay s example [R has p =, r = +,ad Δr r =. The becomes: u z u z m 0g [ F + F gf 7

8 The choice, as i [R, F = /, satisfies the recursive formula F = gf, 4 with g =. This simplifies the secod factor of the RHS of 3, so it is easily see to be bouded. The coclusio is u z u C for z A, ad sice log z o A, u z u z C 5 log z for all z D \{0}. To check that u is smooth ad vaishig to ifiite order at the origi, it is eough to verify the coditio of Lemma.3; for each fixed k 0: +/ + k + + 4k lim k =0. The goal of the ext example is to improve upo the order of vaishig of the ratio 5. Example 3.. Cosider r = l+,sor = l.44, r = l3,... This radius shriks much more slowly tha i Example 3.. Sice lim l+ l+ / l + =, there are costats C, c > 0sothatforall: c l + < Δr l + = r l + < C l +. 6 Let p = ; the the iequality becomes: u z u z m 0 l +g c [ [ + [ F + F + l+ l+ +. gf This motivates, i aalogy with the previous Example, this choice of g ad a recursive formula for F as i 4: So F = F 0 =, ad for >, g = l +, F = l + [ F. F = l + [ l + [ l5 4 l4 = [ l + l + l5 l4. 8

9 The the followig sequece of ratios is bouded above because it is coverget as : F l + [ + F + l + [+ gf = F l +[ + l+3 l+ [ F l+ [+ l + l + [ F = + l +3 l + l + [+ C 3. Here we used the elemetary calculus lemma that l+ l is a bouded sequece. The estimate for the ratio of derivatives o A, for eve, becomes: For z A, so u z u z u z u z m 0 l + l + c C 3 = m 0l + C 3 c = C 4l + 7 < C 5l l + z l + l + l + z + exp z +, u z u z C 5 z exp z, for all z D r \{0}. It remais to check that u is smooth, ad vaishes to ifiite order. The hypothesis o Δr of Lemma.3 is satisfied usig 6, so 9

10 for fixed k, cosider the expressio: = < < F +p + k r p+ 4k Δr /r k [ l +3 l + l4 + k + 4k l+ k l+ l+ [ l +3 l + l4 + k k l + + 4k c l+ [ l +3 l + l4 + 3k l + k c k l. ++ 4k The last expressio has limit zero by the Ratio Test: l+4 + l+3 l4 c k l++ 4k + 3k l +3 k l+3 l4 c k l++ 4k +3k l + k = l +4+ l + + 4k + 3k l +3 k l + + 4k + 3k l + k < l +4++k + 3k l + +3+k 0, + 3k agai usig the boudedess of l+. l 4 A Beltrami-type system [ u Ay smooth map u : C C, u = u, satisfies the followig Beltrami-type system of first-order differetial equatios at poits where u z 0: [ [ [ u z u z [u u z = u z z u z u z u z + u z u z [ [ u z u = z u z u z u z u z u 9 z u z u z = Q z [ u z u z. u z u z For u costructedasisectio,otheaulia with eve, u z 0, so the first row of the matrix i 9 is [0 0, ad similarly the secod row is [0 0 for odd. Defie Q0tobethezeromatrix. 0

11 For a matrix Qz defied as i 9 by some fixed fuctio u, the operator L = z Qz is complex liear. If, o some eighborhood of z =0,the z Qz etries are defied ad small eough, the L is elliptic i the sese of [AIM Sectio 7.4. I the followig Theorem, we cosider Qz fortheexampleuz fromexample 3.. If we restrict u ad Q to z i some sufficietly small eighborhood of the origi, u will be a solutio of the elliptic equatio Lu =0. Theorem 4.. For u as i Example 3., let q ij z = etry i the matrix Qz from 9. q ij C D r \{0} C 0 D r ; uī z uj z u z deote the i, j q ij vaishes to ifiite order: z k q ij z 0 as z 0 for ay k 0; The partial derivatives exist at the origi: x q ij0 = y q ij0 = 0; For ay 0 <r<r, q does ot have the Lipschitz property o D r. Proof. The C claim follows from the smoothess of u o D r ad the ovaishig of u z for z 0. The sum q + q + q + q is exactly u z, which vaishes to u z ifiite order as z 0foru as i Example 3.. It follows that each q ij also vaishes to ifiite order, which implies Q ad the etries q ij are cotiuous at the origi, with the previously assiged values q ij 0 = 0. The flatess also implies the existece of all directioal derivatives at the origi of C = R ;for the x directio, x q q ij x, 0 q ij 0, 0 ij = lim =0. x 0 x,y=0,0 x The last claim takes up the rest of the Proof; the pla is to show there is a sequece of poits x C approachig 0 so that u zu z z u z is a ubouded z=x sequece. If q had a Lipschitz property o D r, i.e., q z q z K z z for some K ad all z, z, the its derivatives o D r \{0} would be bouded; the uboudedess of the derivative also directly shows q / C D r. A real variable aalogue of this behavior is the fuctio exp / x cosexp/ x, which vaishes to ifiite order but has ubouded first derivative as x 0. It is eough, ad simpler, to cosider oly which are eve ad sufficietly large. This will ivolve some estimates for derivatives that are more precise tha 9. We choose the sequece x = r + + Δr +0i A ; the by costructio of s ad χ, χ x = χ,adgives z x = χ z x = Δr.From3, χ z z x = χ z x = χ z x = x Δr

12 this is where we use the s = 0 assumptio, to simplify the calculatio. For r as i Example 3., l+ <x < l+,adδr = l+ satisfies 0 <c 6 l + < <C 6 l +. Δr I the followig expressio, u z u z z u z = u z z u z u z + u z u zz u z u z u z z u z u z + u z u z u z 4 l+ = u z z u z u z + u z u zz u z u z + u z u z u z u z u z u zz + u z z u z + u z u zz u z 4, 0 the terms with the largest magitude are the oes ivolvig the secod z- derivatives, u zz ad u zz. Evaluated at poits i the sequece x,theseterms idividually grow at least as fast as some costat multiple of. However, due to some cacellatios, the overall growth rate turs out to be less tha ; the Theorem will be proved by showig oe of the terms is ubouded ad the remaiig terms have a slower rate of growth. The first cacellatio is that the secod ad last terms i the umerator of the secod fractio i 0 are exactly opposites. This leaves two terms with zz-derivatives; usig the power rule o u = F z i the iterior of A gives: u zz = u z. z 0 becomes: u z zu z u zu zu u z + z u zz u z z u z 4 u zu zu z zu z u z 4. We will show that the last of the three terms i is the domiat oe. First, cosider the ratio: u z / u z = F χ z z + χ z + F + χ z z+ + χ + z + F / z. The followig calculatio for z A is similar to that for the estimate 7. [ u χ F z z z + χ z u z F z χ F +[ z z + + χ + z + + F z

13 From Example 3., recallig F =l + F, l+ z l+,ad χ z m0 Δr, it follows that there is some costat C 7 > sothat u z u z C 7 l +. 3 To estimate the deomiators of, u z u u = + z z u z +C 7 l + = u z C 8 l + u z. So we get a lower boud for the third term i, u z u z u z z u z u z 4 ad cosider 4 oe factor at a time. χ z F z χ z u z u = z u z u z = z=x Δr u u z z u z z C8 l +4 u, 4 z 4 F +z+ F z l + x l +3 x + l + c 6 l + l + l +3 l c 4l +. 6 This lower boud 6 is comparable to the upper boud 7, which is used i the ext step to fid a lower boud for u z u z. Note that the expressio has two terms which, usig the equality of χ χ z ad z whe evaluated at x, match two of the terms from u z i 5: u z u z u = z z=x u + z x l + x + l +3 + x + l + u z l + u z x l +3 + l + + l + c 9 l + C 4 C 0 l + c 7 l

14 This lower boud is comparable to the upper boud 3. The remaiig factor from 4 ivolves secod derivatives: = u z z / u z F χ z z z + χ z z χ F + z z z+ + χ z + z + / F z F χ F z z + χ z z z + F + χ F z z z + + χ z + z +. Evaluatig at x agai, for sufficietly large, u z z u z x l + Δr x l +3 x Δr x x x + x Δr Δr l + c6 l + l + C 6l + l + l +3 C 6 l + l l + + c l So, the term from 4 is bouded below by a product icludig factors from 6, 7, ad 8: u z u z u z z c 4 l + c 7 l + c l + 3 u z 4 C8 x l +4 c l

15 From the secod term i, we cosider the followig quatity: u zz u z z χ = F z z + χ z z +χ z +F + χ z z+ χ z + z + + χ + +z + F χ z z + χ z F + χ z z + + χ + z +. The cacellatio of the 4 quatities is the key step. The ratio u zz u z z / u z = F χ z z + χ z 4 +3z χ z χ +F + z χ z+ z +4 +3z + + χ + +z + F / z 5

16 has a upper boud o the x sequece: u zz u z z u z x l + x Δr x l +3 x + + l Δr x x + x Δr + x + + +x Δr C 6 l l + + l + l +3 + C 6 l l l + C 3 l The middle term from, evaluated at poits x, has factors bouded by 7, 7, ad 30: u z u z u z u zz u z /z u 4 x u z u z u zz u z /z = u z u u z x z 4 u z 4 C 4 l + u z 4 x x u zz u z/z u z x c 7 l + 4 C 3l + 3 C 4 l

17 The first term from ivolves the secod z-derivative: u z z χ z F z χ u z = z F +z + F z u z z u z x F F x Δr C 6l + χ z z + + l + F + F x l + l + + l + 3 C 5. The first term from also approaches 0 for large : u z zu z u z z z u u x z χ z z +. + l +3 x + l +3 l + + u x z l + C 5 u z x 3. The coclusio from 0 ad is: u zu z z u z c l + 3 l C 4 l + C +3 5 x > c 6 x 3. 5 Remarks ad Questios Remark 5.. The regularity of the coefficiets is a importat cosideratio i the aalysis of uique cotiuatio properties for some PDEs for example, [LNW for strog UCP, ad [IVV for weak UCP, which is why we preseted the detailed Proof of Theorem 4.. However, we do ot yet uderstad the sharpess of Example 3. ad Theorem 4. for this particular uique cotiuatio problem. The questio that aturally arises is: would improved regularity of Qz i additio to, or istead of, the flatess property imply a strog uique cotiuatio property, or, oppositely, is there some couterexample where Qz is smooth? Remark 5.. [R shows how Example 3. ca be modified so that the origi is ao-isolatedzeroofu; it is a matter of replacig quatities z N i 4, 5 by z N z a for a sequece a ad re-workig the cutoff fuctios χ. Our Example 3. ca be modified i a aalogous way but we have ot worked out all the details. 7

18 Remark 5.3. By a costructio aalogous to 9, the fuctio u from Example 3. also satisfies a real liear, elliptic equatio of the form u z = Q u z. Qz is ot the same as Qz but also has etries vaishig to ifiite order. Remark 5.4. Aother differetial iequality, cosidered by [R, is u z K u α u z, for 0 < α <. Our attempts to use the costructio of Sectio to fid smooth fuctios u satisfyig the iequality ad vaishig to ifiite order at a isolated zero have ot yet met ay success. [R proves a weak uique cotiuatio property for α =, but the strog property remais a ope questio. Refereces [AIM [B [CH [IS [IVV K. Astala, T. Iwaiec, adg. Marti, Elliptic Partial Differetial Equatios ad Quasicoformal Mappigs i the Plae, PMS48, Priceto Uiversity Press, Priceto, 009. MR j:30040, Zbl B. V. Bojarski, Geeralized solutios of a system of differetial equatios of the first order ad elliptic type with discotiuous coefficiets, Uiv. Jyväskylä Dept. of Math. ad Statistics Report Trasl. from Russia, Mat. Sb. N.S MR j:30096, Zbl R. Courat ad D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Differetial Equatios, Wiley, New York - Lodo, 96. MR #46, Zbl S. Ivashkovich ad V. Shevchishi, Local properties of J-complex curves i Lipschitz-cotiuous structures, Math. Z , MR 88746, Zbl T. Iwaiec, G. Verchota, ada. Vogel, The failure of rak-oe coectios, Arch.Ratio.Mech.Aal.63 00, MR j:580, Zbl [LNW C.-L. Li, G. Nakamura, adj.-n. Wag, Optimal three-ball iequalities ad quatitative uiqueess for the Lamésystem, Duke Math. J , MR j:3535, Zbl [R J.-P. Rosay, Uiqueess i rough almost complex structures, ad differetial iequalities. A.Ist.Fourier660 00, MR h:3033, Zbl