SEMIBOUNDED PERTURBATION OF GREEN FUNCTION IN AN NTA DOMAIN. Hiroaki Aikawa (Received December 10, 1997)

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1 Mem. Fc. Sci. Eng. Shimne Univ. Series B: Mthemticl Science 31 (1998), pp. 1 7 SEMIBOUNDED PERTURBATION OF GREEN FUNCTION IN AN NTA DOMAIN Hiroki Aikw (Received December 1, 1997) Abstrct. Let L = P i,j i( ij j ) be Hölder continuous uniformly elliptic opertor on n NTA domin nd let L V = L + V be generlized Schrödinger opertor for nonnegtive function V. Let G nd G V be the Green function for the Dirichlet problem with respect to L nd L V, respectively. In cse V is function of the distnce to the boundry, necessry nd sufficient condition for G nd G V to hve comprble decy ner the boundry is given. This is bsed on n integrl identity of L -hrmonic functions nd integrl inequlities of L -superhrmonic nd L -subhrmonic functions, which my be of independent interest. 1. Introduction Let L = i,j i( ij j ) be smooth uniformly elliptic opertor on domin D R n, n 2, such tht ij (x) = ji (x) is Hölder continuous on D nd Λ 1 ξ 2 A[ξ] Λ ξ 2 for ll ξ R n, where A[ξ] = i,j ijξ i ξ j nd Λ > 1 is constnt independent of ξ R n nd x D. We lso consider the Schrödinger opertor L V = L + V for nonnegtive function V on D. Let G nd G V be the Green function for the Dirichlet problem in D with respect to L nd L V, respectively. This mens tht for ny y D, G(, y) nd G V (, y) vnish on D nd L G(, y) = L V G V (, y) = δ y, where δ y is the Dirc mesure t y. Since L nd L V re self djoint, it follows tht G nd G V re symmetric. We re interested in the reltionship between G nd G V. It is esy to see tht G V (x, y) G(x, y) for x, y D Mthemtics Subject Clssifiction. 31B5, 31C35, 35B2, 35J1. Key words nd phrses. Semibounded perturbtion, Schrödinger eqution, Green function, NTA domin, Mrtin boundry, Core formul. 1

2 2 HIROAKI AIKAWA Let x D be fixed. We sy tht V is semibounded perturbtion of L V if G(x, y) CG V (x, y) for y D with C > 1 independent of y D (see [13, 3]). The min purpose of this pper is to give necessry nd sufficient condition for V of certin form to be semibounded perturbtion in the cse when D is n NTA domin (see [1]). By the symbol C we denote n bsolute positive constnt whose vlue is unimportnt nd my chnge from line to line. We shll sy tht two positive functions f 1 nd f 2 re comprble, written f 1 f 2, if nd only if there exists constnt C 1 such tht C 1 f 1 f 2 Cf 1. The min result of this pper is the following. Theorem 1. Let D be n NTA domin. Let v(r) be loclly bounded nonnegtive function for < r r = dim(d) such tht v(r) v(2r) nd r 2 v(r) is nondecresing for smll r >. Set V (x) = v(δ(x)) with δ(x) = dist(x, D). Then V is semibounded perturbtion of L V if nd only if (1) rv(r)dr <. It is known tht the Mrtin boundry of n NTA domin with respect to L is the Eucliden boundry nd every boundry point is miniml ([1, Theorem 5.9 nd p.158]). Hence, in view of [14, Theorem 2], we hve the following corollry immeditely. Corollry 1. Let D be n NTA domin nd let v nd V be s bove. If (1) holds, then the Mrtin boundry of D with respect to L V is the Eucliden boundry of D nd every boundry point is miniml. When L = nd D is smooth, Suzuki [14] showed the sufficiency prt of the bove theorem nd we gve n lterntive proof s well s the necessity prt ([4, Theorems 2 nd 3] see lso [3]). Both proofs use the smoothness of D nd do not pply to n NTA domin. Insted, we shll employ the following integrl identity of L -hrmonic functions nd integrl inequlities of L -superhrmonic nd L -subhrmonic functions, which my be of independent interest. These re generliztions of our previous results [1] (see lso [2] nd [5, pp ]), which is bsed on the core formul. Theorem 2. Let D be regulr domin. Let ϕ(t) be nonnegtive function for t >. Let x D nd g(x) = G(x, x). Suppose < b. Then the following sttements hold: (i) If h is L -hrmonic on D, then h(x)ϕ(g(x))a[ g(x)]dx = h(x ) {x D:<g(x)<b} b ϕ(t)dt.

3 SEMIBOUNDED PERTURBATION OF GREEN FUNCTION 3 (ii) If u is L -superhrmonic on D, then b u(x)ϕ(g(x))a[ g(x)]dx u(x ) ϕ(t)dt. {x D:<g(x)<b} (iii) If s is L -subhrmonic on D, then b s(x)ϕ(g(x))a[ g(x)]dx s(x ) ϕ(t)dt. {x D:<g(x)<b} We shll estimte g(x) to obtin some corollries of Theorem 2, which will be needed for the proof of Theorem 1. The upper estimte is esy. The lower estimte is more difficult. If D is n NTA domin, then g(x) cn be estimted by g(x)/δ(x) in certin sense with the help of the boundry Hrnck principle ([1, Theorem 5.1 nd p.138]). For detils see Section 2. For the proof of Theorem 1 we lso need the decy estimte of g(x) ner the boundry. We find < β 1 α < such tht (2) C 1 δ(x) α g(x) Cδ(x) β for x D close to D. The bove inequlities re rther esy. The first inequlity ctully holds for John domin (see e.g. [9, p.185 nd (2.6)]) nd the second does for domin stisfying the cpcity density condition (see e.g. [7]). In generl, β < 1 < α. Only in the cse when D is smooth, we hve α = β = 1, which ws essentil in [4, Theorems 2 nd 3]. Surprisingly, the wek estimte (2) is sufficient for the proof of Theorem 1. Recently, Ancon [8] studied Green functions for elliptic opertors on mnifolds or domins. By using completely different method, he proved better result thn the sufficiency prt of Theorem 1. He proved tht under the condition (1), V becomes bounded purtbtion of L V, i.e., G(x, y) G V (x, y) for ll x, y D. 2. Proof of Theorem 2 nd corollriesi Proof of Theorem 2. The proof is essentilly the sme s in [1]. We shll prove only (i), since the remining cn be proved in the sme fshion. Let h be L -hrmonic on D. Let D t = {x D : g(x) > t} for t >. Observe tht D t is reltively compct subset of D. We see tht the Green function G t for D t with respect to L stisfies tht G t (x, x) = g(x) t. By the Srd theorem (see e.g. [11, Corollry on p.35]), D t is smooth surfce for.e. t >. For such t we see tht the outwrd norml n of D t is given by g/ g. On the other hnd the Poisson integrl formul for D t with respect to L shows tht h(x ) = h(x)( g(x), A(x)n(x))dσ(x), D t

4 4 HIROAKI AIKAWA where A(x)n(x) = ( j ijn j ) is the conorml vector (see [12, (1.4) on p.21]). Hence h(x ) = h(x) A[ g(x)] D t g(x) dσ(x). By the core formul (see e.g. [11, pp.37 39]) b b h(x ) ϕ(t)dt = dt h(x)ϕ(g(x)) A[ g(x)] D t g(x) dσ(x) = h(x)ϕ(g(x))a[ g(x)]dx. Thus the required identity holds. {x D:<g(x)<b} Let us estimte g(x) nd give some corollries of Theorem 2, which will be needed for the proof of Theorem 1. Since g(x) is positive nd L -hrmonic ner D, it follows tht g(x) Cg(x)/δ(x). Since A = ( ij ) is uniformly elliptic, we hve the following corollry from the bove proof. Corollry 2. Let < b. Suppose tht s is positive L -subhrmonic function on {x D : g(x) > }. Then b s(x ) ϕ(t)dt C s(x)ϕ(g(x)) g(x)2 δ(x) 2 dx, where C is independent of s, nd b. {x D:<g(x)<b} The inequlity g(x) Cg(x)/δ(x) does not hold in pointwise sense. However, it holds in certin wek sense for n NTA domin. In [1] this ws observed for L =. The key ingredient ws the boundry Hrnck principle. For generl uniformly elliptic opertor L the boundry Hrnck principle is vilble lso (see [1, Theorem 5.1 nd p.138]) nd hence the sme proof s in [1] works for the present cse. We hve the following corollry. Corollry 3. Let D be n NTA domin. Let ϕ(t) ϕ(2t) for t > nd let < b <. Suppose tht u is positive L -superhrmonic function on D. Then b u(x)ϕ(g(x)) g(x)2 {x D:<g(x)<b} δ(x) 2 dx Cu(x ) ϕ(t)dt, where C is independent of u, nd b. Remrk. Note tht the simplified proofs in [2] nd [5, pp ] use the nlyticity of hrmonic functions ([2, Lemm 1] nd [5, Lemm on p.18]). In generl, L -hrmonic functions need not be nlytic. Insted of the nlyticity, we cn use the unique continution property of those functions for the present sitution.

5 SEMIBOUNDED PERTURBATION OF GREEN FUNCTION 5 3. Proof of Theorem 1 Proof of Theorem 1. Sufficiency: Let us recll the resolvent eqution (3) G(x, y) = G V (x, y) + G V (x, z)v (z)g(z, y)dz for x, y D. D In view of [14, 4] nd [13, Theorem 1.5], it is sufficient to show tht for ny ε > there is compct subset K of D such tht (4) g(z)v (z)g(z, y)dz εg(y) for y D, D\K where we recll g(y) = G(x, y). Suppose (1) holds. Let ϕ(t) = t 2/α 1 v(t 1/α ), in other words, v(r) = r α 2 ϕ(r α ) with α > 1 in (2). By ssumption tϕ(t) is nondecresing. Let b > be sufficiently smll such tht V (z) = v(δ(z)) = δ(z) α 2 ϕ(δ(z) α ) Cg(z)ϕ(g(z))δ(z) 2 for g(z) < b, where (2) nd the monotonicity of tϕ(t) re used. Hence, Corollry 3 with u = G(, y) yields {z D:g(z)<b} By n elementry clcultion g(z)v (z)g(z, y)dz C b Cg(y) {z D:g(z)<b} b ϕ(t)dt. b 1/α ϕ(t)dt = α rv(r)dr <. G(z, y)ϕ(g(z)) g(z)2 δ(z) 2 dz Hence (4) holds with K = {z D : g(z) b} for sufficiently smll b >. Necessity: Suppose V is semibounded perturbtion of L V. Then by the resolvent eqution (3) (5) g(z)v (z)g(z, y)dz Cg(y) for y D. D Let ψ(t) = t 2/β 1 v(t 1/β ), in other words, v(r) = r β 2 ψ(r β ) with < β < 1 in (2). By ssumption tψ(t) is nondecresing. We choose b > sufficiently smll such tht V (z) = v(δ(z)) = δ(z) β 2 ψ(δ(z) β ) Cg(z)ψ(g(z))δ(z) 2 for g(z) < b,

6 6 HIROAKI AIKAWA where (2) nd the monotonicity of tψ(t) re used. Let < < b nd tke y D close to the boundry such tht g(y) <. Then G(, y) is L -hrmonic on {z D : g(z) > }. Hence, Corollry 2 with G(, y) nd ψ(t) in plce of s nd ϕ(t) yields {z D:<g(z)<b} In view of (5) we hve g(z)v (z)g(z, y)dz C b where C is independent of nd hence > b Cg(y) {z D:<g(z)<b} b ψ(t)dt C, ψ(t)dt = β Thus (1) holds. The proof is complete. b 1/β ψ(t)dt. rv(r)dr. G(z, y)ψ(g(z)) g(z)2 δ(z) 2 dz References [1] H. Aikw, Integrbility of superhrmonic functions nd subhrmonic functions, Proc. Amer. Mth. Soc. 12 (1994), [2] H. Aikw, Densities with the men vlue property for hrmonic functions in Lipschitz domin, Proc. Amer. Mth. Soc. 125 (1997), [3] H. Aikw, Norm estimte for the Green opertor with pplictions, Proceedings of Complex Anlysis nd Differentil Equtions, Mrcus Wllenberg Symposium, in honor of Mtts Essén t Uppsl in June 1997 (to pper). [4] H. Aikw, Norm estimte of Green opertor, perturbtion of Green function nd integrbility of superhrmonic functions, preprint. [5] H. Aikw nd M. Essén, Potentil Theory: Selected Topics, Lecture Notes in Mth., vol. 1633, Springer, [6] H. Aikw nd M. Murt, Generlized Crnston-McConnell inequlities nd Mrtin boundries of unbounded domins, J. Anlyse Mth. 69 (1996), [7] A. Ancon, On strong brriers nd n inequlity of Hrdy for domins in R n, J. London Mth. Soc. (2) 34 (1986), [8] A. Ancon, First eigenvlues nd comprison of Green s functions for elliptic opertors on mnifolds or domins, J. Anlyse Mth. (to pper). [9] R. Bñuelos, Intrinsic ultrcontrctivity nd eigenfunction estimtes for Schrödinger opertors, J. Funct. Anl. 1 (1991), [1] D. S. Jerison nd C. E. Kenig, Boundry behvior of hrmonic functions in non-tngentilly ccessible domins, Adv. in Mth. 46 (1982), [11] V. G. Mz y, Sobolev Spces, Springer, [12] C. Mirnd, Prtil Differentil Equtions of Elliptic Type, Springer, 197.

7 SEMIBOUNDED PERTURBATION OF GREEN FUNCTION 7 [13] M. Murt, Semismll perturbtions in the Mrtin theory for elliptic equtions, Isrel J. Mth. (to pper). [14] N. Suzuki, Mrtin boundry for P, Hiroshim Mth. J. 14 (1984), Deprtment of Mthemtics, Shimne University, Mtsue , Jpn E-mil ddress: hikw@riko.shimne-u.c.jp