Chin. Ann. Math. Se. B 39(5), 08, 97 93 DOI: 0.007/s40-08-004-6 Chinese Annals of Mathematics, Seies B c The Editoial Office of CAM and Spinge-Velag Belin Heidelbeg 08 Measue Estimates of Nodal Sets of Polyhamonic Functions Long TIAN Abstact This pape deals with the function u which satisfies k u = 0, whee k is an intege. Such a function u is called a polyhamonic function. The autho gives an uppe bound of the measue of the nodal set of u, and shows some gowth popety of u. Keywods Polyhamonic function, Nodal set, Fequency, Measue estimate, Gowth popety 000 MR Subject Classification 58E0, 53C50 Intoduction The nodal sets ae zeo level sets. We want to study the measue estimates of nodal sets of polyhamonic functions in this pape. In 979, Almgen [] intoduced the fequency concept of hamonic functions. Then in 986 and 987, Gaofalo and Lin[4 5] established the monotonicity fomula of the fequency and the doubling conditions fo solutions of the unifomly second ode elliptic equations, and showed the unique continuation of such solutions by using the doubling conditions. In 000, Han [6] studied the stuctue of the nodal sets of solutions of a class of unifomly high ode elliptic equations. In 003, Han, Hadt and Lin in [7] investigated stuctues and measue estimates of singula sets of solutions of high ode unifomly elliptic equations. In 04, the autho and Yang in [3] gave the measue estimates of nodal sets fo bi-hamonic functions. The classical fequency of a hamonic function is defined as follows. Definition. If u is a hamonic function in B, then fo any, one can define the fequency function of u centeed at the oigin with adius as follows: N() = D() H() = u dx u dσ, (.) whee dσ means the (n )-Hausdoff measue on the sphee. Similaly, one can define the fequency centeed at othe point. Based on this idea, we define the fequency of a polyhamonic function as follows. We fist show some notations in this pape as follows: u = u, u = u,, u k = k u, u k+ = k u = 0. Manuscipt eceived Apil 6, 05. Revised August, 06. Depatment of Applied Mathematics, Nanjing Univesity of Science and Technology, Nanjing 0094, China. E-mail: tianlong98508@63.com This wok was suppoted by the National Natual Science Foundation of China (Nos.40307, 50 9).
98 L. Tian Definition. Suppose that u satisfies that k u = 0, whee k is a positive intege moe than o equal to. Such a function u is called a k-polyhamonic function in the est of this pape. Then we define whee D() = D i (), E() = N() = D()+E(), (.) H() E i (), H() = H i (), D i () = u i dx, E i () = u i u i+ dx, H i () = u i dσ. The function N() is called the fequency of u centeed at the oigin with adius. Similaly, we can define the fequency centeed at othe point. Remak. Noting that fo any j =,,,k, u j is a (k j+)-polyhamonic function, and u k is a hamonic function. Thus one can also define the fequency fo u j as above. We denote such fequency as N j (). It is easy to see that N () = N(), and N k () is just the classical fequency of a hamonic function as in Definition.. Remak. This fequency is in fact the following fom N() =. (.3) u i dσ Hee u iν is u ν and ν is the oute unit nomal on. Now we state the main esults of this pape. Theoem. Let u be a polyhamonic function in B R n, i.e., k u = 0 in B. Then H n ({x B : u(x) = 0}) C N 6 i ()+C, (.4) whee C is a positive constant depending only on n and k. Theoem. Let u be a k-polyhamonic function in the whole space R n. () If the fequency of u centeed at the oigin is bounded in R n, then u is a polynomial. Moeove, if N() < N 0 fo any > 0, then it holds that deg(u) CN 0 +C, (.5) whee deg(u) means the ode of degee of u and C is a positive constant depending only on n and k. In this case, fo any i =,,k, the functions u i ae also polynomials. () If u is a polynomial, then the fequency of u is bounded by the ode of degee of u in the whole space R n. The est of this pape is oganized as follows. In the second section we intoduce some inteesting popeties of the fequency and pove the monotonicity fomula of the fequency. In the thid section, the doubling conditions of the polyhamonic functions ae poved. The foth section gives the measue estimates of nodal sets of polyhamonic functions, i.e., the poof of Theoem.. The last section shows the gowth popety of polyhamonic functions.
Measue Estimates of Nodal Sets of Polyhamonic Functions 99 Monotonicity Fomula of Fequency In this section, we will give some inteesting popeties fo the fequency of polyhamonic functions, and then pove the monotonicity fomula fo this fequency function. Lemma. If u satisfies k u = 0, whee k N and the vanishing ode of u at the oigin is l (k ), then lim N() l (k ). (.) 0 Poof Note that u is k-polyhamonic. So each u i is analytic nea the oigin, thus we may assume that fo each i =,,,k, u i (x) = P i (x) + R i (x), whee P i (x) is a homogeneous polynomial. Assume that the ode of degee of P i (x) is l i, and then R i (x) = o( x li ) as x 0. Because the vanishing ode of u at the oigin is l, it is known that l = l, and fo each i =,3,,k, l i l (i ). Let l 0 = inf{l,l,,l k }. Because each P i (x) is a homogeneous polynomial of degee l i, P i (x) can be witten as P i (x) = li φ i (θ), whee (,θ) is the pola coodinate system. Then u i dx+ k u i u i+ dσ N() = = u i dσ u i dσ = l i li φ i (θ)dσ + k o( li ) li φ i (θ)dσ + k o( li ) l 0 B l0 φ i (θ)dσ +o(l0 ) =, l0 φ i (θ)dσ ) +o(l0 whee dσ = n dω, dω is the (n )-Hausdoff measue on the unit sphee S n. Let 0 in the above fom, one can get lim 0 N() = l 0 l (k ). That is the desied esult. In ode to pove some popeties of the poposed fequency, we need the following two lemmas which can be seen in [9, 3]. Lemma. If u is a hamonic function in, then u dx n u dσ. (.) Lemma.3 Fo any u W, 0 ( ), it holds that u B dx 4 n u dx. (.3) Now we show some popeties of such fequency. Lemma.4 If n,, and u is a k-polyhamonic function as above, then the fequency of u satisfies that N() C, whee C is a positive constant depending only on n.
90 L. Tian Poof Fo any fixed i and, define the function u i and u i as follows: u i = u i+ in, u i = 0 on, u i = 0 in, u i = u i on. So u i = u i +u i. Note that fo any i =,,,k, u i ae hamonic functions, we have u i dσ n u i dσ = n which is pesented in [8, Chapte ]. On the othe hand, the functions u i so fom the Poincaé s inequality, we have B u i dσ 4 n u i dσ. u i dσ, (.4) ae all in W,p 0 ( ), Because and we have u i dσ = u i dσ + u i + u iu i B u i dσ 4 n u i u idσ = 0, u i dσ u i dσ. (.5) We wite the tem u i u i+ dσ as u i u i+ dσ = u i u i+, dσ + u i u i+, dσ + u i u i+, dσ + u i u i+, dσ = I+II+III+IV. Now we will give the estimates of I, II, III and IV sepaately. Fist conside the tem I. By using the fom (.5), we have I ( ) ( ) u i dσ + u i+, dσ 4 n u i dσ + u i+ dσ. Fo IV, by using (.4), we have IV ( ) u n idσ + u i+dσ. Now we focus on II. Also fom the foms (.4) (.5), we have fo any ǫ > 0, it holds that II ǫ u i dσ + ǫ B u i+, dσ ǫ n u i dσ + ǫn u i+ dσ.
Measue Estimates of Nodal Sets of Polyhamonic Functions 9 Similaly, fo III, we have III Cǫ u i+ dσ + ǫn u idσ. So E() ( 4 ǫ n + n ) D()+ ( n + ) H(). ǫn Choose ǫ = 4. Then fom Lemmas..3 and the fact that n,, we have So 4 ǫ n + n = 3 n <. E() CH() +D(). Thus N() = D()+E() H() D() E() C, H() which is the desied esult. Remak. It is obvious that the esult of the above lemmas also hold fo the fequency centeed at othe points. Remak. The fequency of a hamonic function is obviously nonnegative. Fo a polyhamonic function, the fequency may not be nonnegative, but fom Lemma.4, one knows that it also has a lowe bound. Next we will show the monotonicity fomula fo this fequency. Theoem. Let u be a k-polyhamonic function. Then thee exists two positive constants C 0 and C depending only on n and k such that if N() C 0, then it holds that Poof It is easy to check that N () N() C. (.6) N () N() = + D ()+E () D()+E() H () H(). (.7) We calculate D (), E () and H () sepaately. We wite H() as follows: H() = u (x)dσ x = n u (y)dσ y, x= whee dσ x and dσ y ae the (n )-Hausdoff measues on the coesponding sphees. This implies that H i() = (n ) n u i(y)dσ y + n u i (y)u iν (y)dσ y y = y = y =
9 L. Tian = n H i ()+. So H () = Now conside D () and E (). Fist note that whee and Fo D i (), it holds that D i () = + n D () = E () = D i (), E i (), D i() = u i dσ E i() = u i u i+ dσ. H(). (.8) u i x x dx = div( u i x)dx u i u i x l dx x j x j x l u i dx = = n u i dx+ = n D i()+ ( ui )( ui ) x l dx x j x l x j = n D i()+ u i u i x j x l x j x ldσ = n D i ()+ u iν dσ u i x l u i x u i+ dx. x j ( ui x j x l ) dx Fo E i (), we have E i() = u i u i+ dσ = u i u i+ x x dx = div(u i u i+ x)dx = n u i u i+ dx+ u i+ u i xdx+ u i u i+ xdx = n E i()+ u i+ u i xdx+ u i u i+ xdx. Thus D i()+e i() = n (D i ()+E i ())+ u iνdσ + u i+ x u i dx u i x u i+ dx
Measue Estimates of Nodal Sets of Polyhamonic Functions 93 + u i u i+ dx. So D ()+E () D()+E() = n + + u iν dσ ( u i+ x u i dx u i x u i+ dx+ u i u i+ dx) = n + ( u i dx+ u i u i+ dx) u iν dσ + Then we will estimate R, R and R 3 sepaately. R R +R 3 D()+E(). R u i+ x u i dx ( u i+ dx+ ( ) C u i dx+ u i dσ. Fom the simila aguments, we have ) u i dx R u i x u i+ dx ( ) u i dx+ u i+ dx ( ) C u i dx+ u idσ and R 3 u i u i+ dx ( u idx+ ( ) C u i dx+ u i dσ. u i+dx ) Fom the assumption that N() C 0 and the poof of Lemma.4, we have E() CH()+ 3 4 D() C C 0 (D()+ E() )+ 3 4 D(), whee C is the constant in Lemma.4. Choose C 0 lage enough such that ( C C 0 + 3 4 ) C0 C 0 C = 7 8.
94 L. Tian Then D()+E() 8 D(). So R R +R 3 D()+E() CD()+CH() D()+E() C. Thus Fom (.8), we have D ()+E () D()+E() C + n + n u iν dσ. H () H() = n +. u i dσ So we finally get This ends the poof. N () ( N() u iν dσ u i dσ ) C C. Remak.3 The above theoem also holds fo the fequency centeed at othe point, i.e., if u is a polyhamonic function and N(p,) is the fequency of u centeed at the point p with adius, then it holds that dn(p,) d C, (.9) N(p,) if N(p,) C 0, whee C 0 and C ae two positive constants depending only on n and k. Lemma.5 Fo any p B, we have 4 N (p, ) ( p ) C N()+C, (.0) whee C and C ae positive constants depending only on n and k. Poof We only povethe casethat p = 4. Othecasesaesimila. Note that (p) B B3 4 and B (p). Fom Theoem., we have 4 B B 34 (p) u idx 4 CN()+C B (p) u idσ. (.)
Measue Estimates of Nodal Sets of Polyhamonic Functions 95 Now we claim that B5 8 (p) B 58 (p) u idσ 4 CN()+C B (p) B (p) In fact, fom (.3), (.8), Lemma.4 and some diect calculation, we know that d ( d log ) u (p) idσ = n + d (p) d log(h(p,)) Thus and B 34 (p) B (p) u i dx = u idx = C C = n = n + H (p,) H(p, ) + n + u idσ. (.) B u (p) iu iν dσ H(p, ) = N(p,) C. (.3) B 34 (p) B 58 (p) 3 4 5 8 0 u i dσ n u i (p) dσ (p) B5 8 (p) B 58 (p) u i dσ, n u (p) idσ (p) B (p) B (p) u idσ. So the claim (.) holds. Integating (.3) fom to 5 8, we obtain ( log B5 8 (p) B 58 (p) Fom Theoem., we know that 5 8 So Then fom Theoem. again, we get ( u i ) log dσ fo any, and that is the desied esult. B (p) B (p) N(p,) ( d CN p, ) C. ( N p, ) CN()+C. N(p,) CN()+C u i dσ ) = 5 8 N(p,) d.
96 L. Tian 3 Doubling Conditions In this section, we will show the doubling condition of a polyhamonic function u. In fact, fom the poof of Lemma.5, it is easy to see that the following doubling condition holds. Lemma 3. Let u be a k-polyhamonic function, and assume that <. Then it holds that u idσ CN()+C B B u idσ (3.) and u B idx C N()+C u B B idx, (3.) whee C and C ae two positive constants depending only on n and k. Poof We only need to pove the fom (3.). Because one can simply integate (3.) fom 0 to to get (3.). Integating (.3) fom to, we know that and thus ( H() ) ( H() ) log () n log n = H() n H()e N(t) t dt. N(t) dt, t Fom Theoem., we know that N(t) max{cn(),c 0 } CN()+C fo any t (,). Hee C is some positive constant depending only on n and k, and C 0 is the same constant as in Theoem.. So which is the desied esult. H() n H()e CN()+C = CN()+C H(), It is known that the doubling condition fo hamonic functions and bi-hamonic functions as follows. Lemma 3. Let u be a hamonic function and <. Then u dx CN()+C u dx, (3.3) B B whee N() is the fequency of u and C is a positive constant depending only on n. Lemma 3.3 Let u be a bi-hamonic function and <. Then u dx B B 4C(N()+N())+C u dx, (3.4) whee N () is the fequency of u, N () is the fequency of u, and C is a positive constant depending only on n.
Measue Estimates of Nodal Sets of Polyhamonic Functions 97 Lemmas 3. 3.3 can be seen in [9] and [3], espectively. Now we will show the doubling condition fo a polyhamonic function. Theoem 3. Let u be a k-polyhamonic function, i.e., u satisfies that k u = 0 in B R n and assume that <, n. Then it holds that u dx ( ) N i() +C B B CC u dx, (3.5) whee C is a positive constant depending only on n and k. Poof We pove this lemma by the inductions. Assume that we have aleady known that fo any j satisfies k j l, fom (3.5) and the following inequality u j+ dx N i()+c CC i=j+ B u jdx (3.6) holds fo u j. Fom the above two lemmas, we know that fo j = k and j = k, these two inequalities hold. Now we will pove that the inequalities (3.5) and (3.6) hold fo u eplaced by u l and thus the theoem is poved. Noting that u l = u l+, it holds that fo any text function ψ C0 (B ), u l ψdx = u l+ ψdx. (3.7) B B Choose ψ = u l φ, whee φ satisfies and φ = in, φ = 0 outside B, φ < C, φ < C. Put this Ψ into (3.7), we have u l+ u l φ dx = u l (u l φ )dx B B = u lφ dx+4 u l φ u l φdx+ u l u l ( φ +φ φ)dx B B B = u lφ dx 4 u l φ u l φdx u l u l ( φ +φ φ)dx. B B B Thus u lφ dx = u l u l+ φ dx+4 u l φ u l φdx+ u l u l ( φ +φ φ)dx B B B B ( ) ) ( ) ) u l+φ dx u l φ dx +4 u l φ dx u l φ dx B B B B
98 L. Tian ( ) ( ) + u l( φ + φ φ )dx u l ( φ + φ φ )dx. B B Now we conside the estimate of the tem B u l φ dx. u l φ dx B = u l u l+ φ dx u l φ u l φdx B B ( ) ( ) ( ) ( ) u l φ dx u l+ φ dx + u l φ dx u l φ dx B B B B ( ) ( ) u l φ dx u l+ φ dx + u l φ dx+ u l φ dx. B B B B Thus we have ( u l φ dx u lφ dx B B Fom u l+ dx and the doubling condition fo u l, we have u l dx ) ( B N i()+c CC i=l+ B N i()+c CC i=l B ) u l+φ dx +4 u l φ dx. B u l dx u l dx. (3.8) This shows that (3.6) holds fo j = l. Then fom Lemma 3. and the induction assumptions, we have u l dx and thus the desied esult holds by inductions. 4 Measue Estimates of Nodal Sets N i()+c CC i=l B u l dx, (3.9) In this section, we will show the uppe bound of the measue of the nodal set fo a polyhamonic function u, i.e., we will give the poof of Theoem.. To estimate the measue of the nodal set, we need an estimate fo the numbe of zeo points of analytic functions which was fist poved in []. Lemma 4. Suppose that f: B C C is analytic with f(0) = and sup B f N fo some positive constant N. Then fo any (0,), thee holds whee C is a positive constant depending only on. H 0 ({z : f(z) = 0}) CN, (4.)
Measue Estimates of Nodal Sets of Polyhamonic Functions 99 We also need the following pioi estimate. Lemma 4. Let u be a polyhamonic function. Then if <, we have u L () N i()+c k CC whee C is a positive constant depending only on n and k. ( u B R idx ), (4.) Poof Let w i, (x) = Γ(x y)u i+ (y)dy, i =,,,k, whee Γ(x y) = c x y n is the fundamental solution of the Laplace opeato. Then w i, (x) = Γ(x y)u i+ (y)dy C sup u i+ (y). It also holds that So w i, = u i+ in. (u i w i, ) = 0 in. Because u k is a hamonic function, it is known that fo any y, ( u k (y) = u k (z)dz (y) (y) ( C u k B (z)dz B (y) (y) ) CN k()+c ( ) u k(z)dz u k (z)dz ). Thus fo any x, w k, (x) CNk()+C ( ) u k(z)dz. On the othe hand, fom the fact that u k w k, is hamonic in B, we know that fo any x, u k (x) w k, (x) = (u k (z) w k, (z))dz (x) (x) ( ) ( u (x) k (z)dz + (x) (x) ( ) C u k B (z)dz +C sup u k (y) B y B ( k()+n k ())+C CC(N u k (z)dz (x) ) w k, (z) dz ) + CN k()+c ( u k(z)dz )
930 L. Tian (( ) k()+n k ())+C ( CC(N u )) k (z)dz + u k(z)dz. Then fo x, u k (x) u k (x) w k, (x) + w k, (x) (( ) k()+n k ())+C ( CC(N u k (z)dz ) ) + u k (z)dz. That is the desied esult fo u k. Repeat this agument k times, the desied esult can be poved. Now we show the measue estimate of the nodal set {x : u(x) = 0}. Poof of Theoem. Without loss of geneality, we may assume u dx =. B B Then fom Theoem 3. and Lemma.5, it holds that B (p) 6 B 6 (p) u dx 4 C N i() C fo any p B. Then thee exists a point x p B (p) such that 4 6 u(x p ) C N i() C. On the othe hand, fom Lemma 4. and (3.6), one knows that fo any x B 4, u(x) C N i()+c. Choosep j B tobethepointonthej-axisandtakef j(ω;t) = u(x pj +tω)fot ( 5 4 8, ) 5 8, whee ω S n. Then f j is an analytic function with espect to t. Extend it to a complex analytic function f j (ω;z), and keep the uppe bound. Then we have f j (ω;0) C N i() C (4.3) and f j (ω;z) C i()+c N. (4.4) Using Lemma 4., we have That means H 0({ t < 5 }) 8 : u(x p j +tω) = 0 C N i ()+C. H 0 ({t : u(x pj +tω) = 0,x pj +tω B }) C N 6 i ()+C.
Measue Estimates of Nodal Sets of Polyhamonic Functions 93 Then fom the integal geometic fomula, which can be seen in [3, 0], we have and this is the desied esult. H n ({x B : u(x) = 0}) C N 6 i ()+C, (4.5) 5 Gowth Popety of Polyhamonic Functions In this section, we will deive a gowth behavio of the polyhamonic functions in the whole space R n. The esult is witten in Theoem.. Poof of Theoem. Fist assume that N() is bounded, i.e., N() N 0 on R n. Then we need to show that u is a polynomial. Without loss of geneality, assume u B idσ =. (5.) B Fom the mean value fomula and the fact that u k is a hamonic function, we have that ( ) sup u k C u k B dσ B holds fo any >. Fo each i =,,,k, wite u i as u i = u i +u i as in the poof of Lemma.4, i.e., and u i = u i+ in B, u i = 0 on B Then fom the pioi estimate of u i Thus fo u k, it holds that u i = 0 in B, u i = u i on B. sup u i sup u i +sup u i and the mean value popety of u i, we have C sup B u i+ +C ( sup u k C sup u k +C B C (( B 4 Continue these aguments fo k times, we get sup u C k ( B u B idσ B ) ( u k dσ + B 4 B 4 B u i dσ ) ( u B idσ k B k ). B 4 u k dσ ) ) ).. (5.)
93 L. Tian Thus fom Lemma 3. and the assumption (5.), we have that sup u C CN0+C (5.3) holds fo any >. Thus u must be a polynomial and the ode of degee of u is less than o equal to CN 0 +C, whee C is a positive constant depending only on n and k. If a k-polyhamonic function u is a polynomial, then fom the fact that N() = u, i it is easy to check that N() is bounded by the odeof degeeof u. Of couse, fo any i =,,k, the functions u i ae all polynomials. Refeences [] Almgen, F. J., Diichlet s poblem fo muptiple valued functions and the egulaity of Mass minimizing integal cuents, Minimal Submanifolds and Geodesics, Obata, M. (ed.), Noth Holland, Amstedam, 979, 6. [] Donnelly, H. and Feffeman, C., Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93, 988, 6 83. [3] Fedee, H., Geometic Measue Theoy, Spinge-Velag, New Yok, 969. [4] Gaofalo, N. and Lin, F. H., Monotonicity popeties of vaiational integals, A p-weights and unique continuation, Indiana Univ. Math. J., 35, 986, 45 67. [5] Gaofalo, N. and Lin, F. H., Unique continuation fo elliptic opeatos: a geometic-vaiational appoach, Comm. Pue. Appl. Math., 40, 987, 347 366. [6] Han, Q., Schäude estimates fo elliptic opeatos with applications to nodal sets, The Jounal of Geometic Analysis, 0(3), 000, 455 480. [7] Han, Q., Hadt, R. and Lin, F. H., Singula sets of highe ode elliptic equations, Communications in Patial Diffeential Equations, 8, 003, 045 063. [8] Han, Q. and Lin, F. H., Nodal sets of solutions of elliptic diffeential equations, http://nd.edu/qhan/ nodal.pdf. [9] Lin, F. H., Nodal sets of solutions of elliptic and paabolic equations, Comm. Pue. Appl. Math., 44, 99, 87 308. [0] Lin, F. H. and Yang, X. P., Geometic Measue Theoy: An Intoduction, Advanced Mathematics,, Science Pess, Beijing; Intenational Pess, Boston, 00. [] Liu, H. R., Tian, L. and Yang, X. P., The gowth of H-hamonic functions on the Heisenbeg goup, Science China Mathematics, 54(4), 04, 795 806. [] Schoen, R. M. and Yau, S. T., Lectues on Diffeential Geomety, Intenational Pess, Cambidge, MA, 994. [3] Tian, L. and Yang, X. P., Measue estimates of nodal sets of bi-hamonic functions, Jounal of Diffeential Equations, 56(), 04, 558 576.