Bessel Integrals and Fundamental Solutions for 1 0 0 a Generalized Tricomi Operator 2 n a J J. Barros-Neto 8 Rutgers University, Hill Center ] 110 Frelinghuysen Rd, Piscataway, NJ 08854-8019 A C e-mail: jbn@math.rutgers.edu . h Fernando Cardoso t ∗ a m Departmento de Matemática, Universidade Federal de Pernambuco [ 50540-740 Recife, Pe, Brazil 1 e-mail: fernando@dmat.ufpe.br v 5 6 0 1 0 Abstract 1 0 ThemethodofpartialFouriertransformisusedtofindexplicitfor- / h mulas for two remarkable fundamental solutions for a generalized Tri- at comi operator. These fundamental solutions reflect clearly the mixed m type of the Tricomi operator. In proving these results, we establish : explicit formulas for Fourier transforms of some functions involving v i Bessel functions. X r a 1 Introduction Consider the generalized Tricomi operator ∂2 (1.1) P = y∆+ , ∂y2 ∗ Partially supported by CNPq (Brazil) 1 ∂2 where ∆ = n is the Laplace operator. Our aim is to find, via partial j=1 ∂x2 P j Fourier transform with respect to x = (x ,...,x ), fundamental solutions 1 n relative to an arbitrary point (a,0) on the hyperplane y = 0 in Rn+1, that is, distributions that are solutions to Pu = δ(x a,y), − where δ(x a,y) is the Dirac measure concentrated at (a,0). Since P is − invariant under translationsparalleltothathyperplane, itsuffices toconsider the case when the Dirac measure is concentrated at the origin. If n = 1, then (1.1) is the classical Tricomi operator, ∂2 ∂2 (1.2) = y + . T ∂x2 ∂y2 It is known that, for this operator, the equation 9x2 + 4y3 = 0 defines the two characteristic curves that originate at the origin. They divide the plane in two disjoint regions, namely, D = (x,y) R2 : 9x2 +4y3 > 0 , + { ∈ } the region “outside” the characteristics and D = (x,y) R2 : 9x2 +4y3 < 0 , − { ∈ } the region “inside” the characteristics. Note that D is entirely contained in − the hyperbolic region y < 0. In the paper [2] it was shown the existence of the following two fundamental solutions relative to the origin: Γ(1/6) (9x2 +4y3)−1/6 in D  −3 22/3π1/2Γ(2/3) + (1.3) F+(x,y) =  ·   0 elsewhere    and 3Γ(4/3) 9x2 +4y3 −1/6 in D  22/3π1/2Γ(5/6)| | − (1.4) F (x,y) = −    0 elsewhere.    2 The support of F , the closure of D , is, except for the origin, entirely − − contained in the hyperbolic region (y < 0), while the support of F , the + closure of D , consists of the whole elliptic region (y > 0), the parabolic + region (x-axis), and extends to the hyperbolic region up to and included the characteristic curves. Themethodusedin[2]toprovethese resultswas based upon the property that solutions to the equation u = 0 are invariant under the dilation d (x,y) = (t3x,t2y) in R2. T t In [1] we proved formulas (1.3) and (1.4) by using partial Fourier transform with respect to the x variable. In this paper we extend the results of [1] to the generalized Tricomi operator (1.1). Since the dimension of the space variable x is now n > 1, technical difficulties in evaluating Fourier transforms involving Bessel functions do occur. We circumvent them by calculating integrals of the type ∞ I (a,b) = e−ǫtt−λJ (at)J (bt)dt ǫ µ ν Z 0 (Section 3). As a result, we obtain for the operator (1.1) the following fundamental solutions: 3n−2 Γ(n2 − 13) (9 x 2 +4y3)13−n2 in Dn  −22/3πn/2 Γ(2/3) | | + F (x,y) = +    0 elsewhere,    whose supportis theclosure oftheregionDn = (x,y) Rn+1 : 9 x 2+4y3 > + { ∈ | | 0 and } 3nΓ(4) 3 9 x 2 +4y3 13−n2 in Dn  22/3πn/2Γ(4 n) | | | | − F−(x,y) =  3 − 2   0 elsewhere    supportedbytheclosureoftheregionDn = (x,y) Rn+1 : 9 x 2+4y3 < 0 . − { ∈ | | } These fundamental solutions clearly generalize formulas (1.3) and (1.4). In addition, we also show that no matter the parity of the dimension n, F (x,y) does not have support on the boundary of the region D in the − − hyperbolic region and hence, in contrast with what happens for strictly hyperbolic operators – like the wave operator – the Huyghens principle does not hold for the generalized Tricomi operator. 3 It is well known [9] that, for the wave operator, the regularity of the fundamental solutions degenerates as the dimension n increases: a locally constant function, when n = 1; an absolutely continuous measure relative to the Lebesgue measure, when n = 2; a measure carried by the surface of the forward light-cone, when n = 3; and so on. However, for the generalized Tricomi operator they always remain locally integrable. The plan of this paper is as follows. In Section 2 we reduce, via partial Fourier transform, the original problem to an equivalent one of finding fundamental solutions for a second order ordinary differential equation andshow how these can be represented in terms of Airy functions or Bessel functions. In Section 3 we obtain explicit formulas for Fourier transforms of the functions ξ ±νJ ( ξ ), ξ νK ( ξ ), and ξ νN ( ξ ), ξ = (ξ ,...,ξ ), n 1. These ν ν ν 1 n | | | | | | | | | | | | ≥ formulas are used to construct the fundamental solution supported by Dn − (Section 4) and the fundamental solution supported by Dn (Section 5). In + theAppendixthereaderwillfindthedefinitions oftheBessel functionsJ (z), ν I (z), K (z), and N (z), and a list of properties that are needed throughout ν ν ν this work. We also recall the definition of hypergeometric functions and some of their main properties. The method of finding solutions, via partial Fourier transform, to the equation ∂2u (1.5) Pu = y∆u+ = f ∂y2 with, say, f ∞(Rn+1) is a natural one although not particularly new. In ∈ Cc the monograph [5], R. J. P. Groothuizen exhibits a solution u in terms of a Fourier integral operator whose symbol is obtained by the partial Fourier transform analysis employed in this paper. He also obtains fundamental solutions given by Fourier integral operators. Our results are more precise since we give explicit formulas for the fundamental solutions and these are tempered distributions. Consequently, we can obtain more transparent representations for the solution u to the equation (1.5) as a convolution of f with any of the fundamental solutions here described. 2 Preliminaries Consider the more general problem of finding fundamental solutions for the operator (1.1) relative to an arbitrary point (0,b) on the y-axis. That is, one 4 wishes to find distributional solutions to the equation ∂2 (2.1) y∆F + F = δ(x) δ(y b). ∂y2 ⊗ − Partial Fourier transform with respect to x reduces this problem into finding fundamental solutions to the ordinary differential equation (2.2) F˜ y ξ 2F˜ = δ(y b), yy − | | − where F˜ denotes the partial Fourier transform of F, ξ 2 = n ξ2, and | | j=1 j δ(y b) is the Dirac measure concentrated at b. P − One way of solving (2.2) (see [9]) consists in selecting two linearly independent solutions U (ξ,y) and U (ξ,y) to the homogeneous equation 1 2 (2.3) u˜ y ξ 2u˜ = 0 yy − | | so that their Wronskian at y = b (which in the case under consideration is the same as the Wronskian at y = 0) is equal to 1 and in defining − U (ξ,b)U (ξ,y) if y b 2 1 ≥ (2.4) F˜(ξ,y;b) =   U (ξ,b)U (ξ,y) if y b. 1 2 ≤  ˜ It is a matter of verification that F(ξ,y;b) is a solution to (2.2). Equivalently [4], one can obtain a fundamental solution to the equation (2.2) by finding two linearly independent solutions to the homogeneous equation (2.3): u (ξ,y;b), defined for y > b, and u (ξ,y;b), defined for y < b, so 1 2 that (i) the limit of u as y b+ equals the limit of u as y b 1 2 → → − and (ii) the limit of u as y b+ minus the limit of u as y b is equal 1,y 2,y to 1. The function F˜(ξ,y;b)→is now defined by → − u (ξ,y;b) if y b 1 ≥ ˜ (2.5) F(ξ,y;b) =   u (ξ,y;b) if y b. 2 ≤  In what follows we will use at our convenience either one of these two expre- sions for F˜(ξ,y;b). 5 Returning to the original problem (2.1), if we can choose U (ξ,y) and 1 U (ξ,y), in formula (2.4), or u (ξ,y;b) and u (ξ,y;b), in formula (2.5), so 2 1 2 that F˜(ξ,y;b) is a tempered distribution with respect to ξ = (ξ ,...,ξ ), 1 n then its inverse Fourier transform of F˜(ξ,y;b) defines a fundamental solution to the operator (1.1). We now proceed to find linearly independent solutions to the homogeneous equation (2.3). The change of variables z = ξ 2/3y transforms (2.3) | | into the classical Airy’s equation u′′ zu = 0. Two linearly independent − solutions to that equation are Ai(z) and Bi(z) respectively called the Airy functions of the first and second kinds. It is known [6] that these two func- tionscan berepresented interms of Bessel functions oforder 1/3as follows. ± If argz < (2π/3), then | | Ai(z) = z1/2[I (2z3/2) I (2z3/2)] = 1(z)1/2K (2z3/2), 3 −1/3 3 − 1/3 3 π 3 1/3 3 (2.6)   Bi(z) = (z)1/2[I (2z3/2)+I (2z3/2)]. 3 −1/3 3 1/3 3 If argz< (2π/3), then | | Ai( z) = z1/2[J (2z3/2)+J (2z3/2)], − 3 −1/3 3 1/3 3 (2.7)   Bi( z) = (z)1/2[J (2z3/2) J (2z3/2)]. − 3 −1/3 3 − 1/3 3 In the Appendix I, where a brief review of Bessel functions is presented, we show that the following relations hold: 3−2/3 3−4/3 (2.8) Ai(0) = , Ai′(0) = Γ(2/3) −Γ(4/3) and 3−1/6 3−5/6 (2.9) Bi(0) = , Bi′(0) = . Γ(2/3) Γ(4/3) As a consequence, the Wronskian of Ai(z) and Bi(z) evaluated at z = 0 is (2.10) W(Ai(z),Bi(z)) = 1/π. |z=0 Indeed, we have 3−2/3 3−1/6 (cid:12) Γ(2/3) Γ(2/3) (cid:12) (cid:12) (cid:12) 2 3−3/2 1 W(Ai(z),Bi(z)) = (cid:12) (cid:12) = · = , |z=0 (cid:12) (cid:12) Γ(2/3)Γ(4/3) π (cid:12) 3−4/3 3−5/6 (cid:12) (cid:12) (cid:12) (cid:12) −Γ(4/3) Γ(4/3) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 6 because 2π (2.11) Γ(2/3)Γ(4/3) = . 33/2 We now choose the following two linearly independent solutions to the equation (2.3): U (ξ,y) = √π ξ −1/3Ai( ξ 2/3y) and U (ξ,y) = √π ξ −1/3Bi( ξ 2/3y), 1 2 | | | | − | | | | and note that, by virtue of (2.10), the Wronskian of U (ξ,y) and U (ξ,y) is 1 2 equal to 1. Next, according to (2.4), the distribution − π ξ −2/3Bi( ξ 2/3b)Ai( ξ 2/3y) if y b − | | | | | | ≥ ˜ (2.12) F(ξ,y;b) =   π ξ −2/3Ai( ξ 2/3b)Bi( ξ 2/3y) if y b − | | | | | | ≤  is a fundamental solution to the ordinary differential equation (2.2). A fundamental solution F(x,y;b) to the generalized Tricomi operator (1.1) and relative to the point (0,b) would then be the inverse Fourier transform of F˜(ξ,y;b) whenever that Fourier transform exists. We do not know how to obtain an explicit formula (or formulas) for the ˜ inverse Fourier transform of F(ξ,y;b) when b = 0, a problem that merits to 6 be investigated. We conjecture that when b < 0, that is, the point (0,b) is in the hyperbolic region, there exists two fundamental solutions that converge, as b 0, to the fundamental solutions F (x,y) and F (x,y) defined, re- + − → spectively, by the formulas (5.6) and (4.2). The two fundamental solutions described in the monograph [5] do not seem to satisfy these requirements. However, when b = 0,we will show inthe following sections how to obtain from formula (2.12), as well as formulas similar to it, explicit expressions for fundamental solution to (1.1). The case n = 1. For sake of completeness and in order to motivate our work in the forthcoming sections we briefly present the results of the paper [1], for the Tricomi operator (1.2), that is the case when n = 1. If b = 0 and we take into account the values of Ai(0) and Bi(0) as given by (2.8) and (2.9), then formula (2.12) simplifies as follows: π ξ −2/3 | | Ai( ξ 2/3y) if y 0  −31/6Γ(2/3) | | ≥ (2.13) F˜(ξ,y) =     π ξ −2/3 | | Bi( ξ 2/3y) if y 0,  −32/3Γ(2/3) | | ≤     7 where, for simplicity, we wrote F˜(ξ,y) for F˜(ξ,y;0). Now the inverse Fourier transform of both expressions on the right-hand side of (2.13) can be explic- itly calculated. To see this, first consider the representations of Ai(z) and Bi(z) in terms of Bessel functions as given by formulas (2.6) and (2.7): 1 ξ 2/3y 1/2 2 Ai( ξ 2/3y) = | | K ( ξ y3/2) 1/3 | | π (cid:18) 3 (cid:19) 3| | and ξ 2/3( y) 1/2 2 2 Bi( ξ 2/3y) = | | − [J ( ξ ( y)3/2) J ( ξ ( y)3/2)]. −1/3 1/3 | | (cid:18) 3 (cid:19) 3| | − − 3| | − Next by introducing the change of variables s = (2/3)y3/2, whenever y 0, ≥ and t = (2/3)( y)3/2, whenever y 0, rewrite formula (2.13) as − ≤ s α ( )1/3K (s ξ ) if y 0 1/3 · ξ | | ≥  (2.14) F˜(ξ,y) =  | |   t β ( )1/3[J (t ξ ) J (t ξ )] if y 0 −1/3 1/3 · ξ | | − | | ≤  | |  where α and β are constants given by 1 π (2.15) α = and β = . −21/331/3Γ(2/3) −21/335/6Γ(2/3) Theorem 3.1 of [1] proves that F(x,y), the inverse Fourier transform of F˜(ξ,y), is then 3 1 F = F F , + − 2 − 2 a linear combination of the two fundamental solutions F and F respec- + − tively defined by formulas (1.3) and (1.4). To prove this theorem, one re- lies on known formulas (see [4], [7]) for the inverse Fourier transforms of the functions ξ −1/3J ( ξ ), ξ −1/3J ( ξ ),and ξ −1/3K ( ξ ),formulas that 1/3 −1/3 1/3 | | | | | | | | | | | | need be generalized to the case n > 1. 3 Fourier transforms involving Bessel functions As we have indicated in the previous section, we are going to need explicit formulas for the inverse Fourier transforms of the following functions: 8 ξ ±νJ ( ξ ), ξ νK ( ξ ),and ξ νN ( ξ ),where ξ = (ξ ,...,ξ ), n 1. Since, ν ν ν 1 n | | | | | | | | | | | | ≥ when n = 1, explicit formulas for the Fourier transforms are known and can be found, for example, in [4] and [7], we concentrate on the case n > 1. 1. The inverse Fourier transform of ξ νK ( ξ ). ν | | | | Assume that Re ν < 1/2 and define | | 1 (3.1) −1[ ξ νK ( ξ )](x) = eihx,ξi ξ νK ( ξ )dξ. F | | ν | | (2π)n ZR | | ν | | n Theassumptionontherealpartofν securesconvergenceoftheintegralatthe origin. On the other hand, in view of the asymptotic behavior of K ( ξ ) for ν | | largevalues of ξ ,the integral converges absolutely. By introducing spherical | | coordinates, rewrite the integral on the right-hand of (3.1) as ∞ rn+ν−1K (r) eihrx,ωidω dr. ν Z {Z } 0 Sn−1 Since n (2π)2 (3.2) ZSn−1 eihrx,ωidω = |rx|n2−1Jn2−1(r|x|), it follows that x 1−n2 ∞ n (3.3) F−1[|ξ|νKν(|ξ|)](x) = |(2|π)n2 Z0 r2+νJn2−1(r|x|)Kν(r)dr. We now quote the following result found in Watson’s treatise [11]: Lemma 3.1. If Re(µ+1) > Re ν and Re b > Im a then | | | | ∞ (2a)µ(2b)νΓ(µ+ν +1) (3.4) tµ+ν+1J (at)K (bt)dt = . µ ν Z (a2 +b2)µ+ν+1 0 From this lemma, we obtain the following result Theorem 3.1. If Re ν < 1/2 then | | 2ν−1Γ(n +ν) (3.5) −1[ ξ νK ( ξ )](x) = 2 1+ x 2 −n2−ν . F | | ν | | πn2 | | (cid:0) (cid:1) 9 Proof. The integral in (3.3) is the same as the integral in (3.4), where µ = n/2 1, a = x , and b = 1. Clearly the conditions of the lemma are − | | 2 satisfied and Theorem 3.1 follows at once. We remark that, when n = 1, (3.5) becomes 2ν−1Γ(1 +ν) 1 −1[ ξ νK ( ξ )](x) = 2 1+ x 2 −2−ν , F | | ν | | π21 | | (cid:0) (cid:1) a formula found in [4] and [7]. 2. Inverse Fourier transforms of ξ νJ ( ξ ) and ξ −νJ ( ξ ). ν ν | | | | | | | | As before, assume that Re ν < 1/2. Formally, the inverse Fourier trans- | | form of ξ νJ ( ξ ) is ν | | | | 1 −1[ ξ νJ ( ξ )](x) = eihx,ξi ξ νJ ( ξ )dξ. ν ν F | | | | (2π)n ZR | | | | n In general, the integral diverges at and so we introduce a converging ∞ factor e−ǫ|ξ| and take a limit as ǫ 0. Since ξ νJ ( ξ ) is locally integrable, ν it defines a tempered distribution→and the lim| i|t exi|st|s in ′(R ), the space n S of tempered distributions on R . Thus the precise meaning of the inverse n Fourier transform is 1 (3.6) −1[ ξ νJ ( ξ )](x) = lim eihx,ξi−ǫ|ξ| ξ νJ ( ξ )dξ. F | | ν | | (2π)n ǫ→0ZR | | ν | | n By introducing spherical coordinates, one can see that the integral on the right-hand can be written as ∞ e−ǫrrn+ν−1J (r) eihrx,ωidω dr. ν Z {Z } 0 Sn−1 Since n (2π)2 (3.7) ZSn−1 eihrx,ωidω = |rx|n2−1Jn2−1(r|x|), it follows that x 1−n2 ∞ n (3.8) F−1[|ξ|νJν(|ξ|)](x) = |(2|π)n2 lǫi→m0Z0 e−ǫrr2+νJn2−1(r|x|)Jν(r)dr. 10

Bessel Integrals and Fundamental Solutions for a Generalized Tricomi Operator

Similar Bessel Integrals and Fundamental Solutions for a Generalized Tricomi Operator