Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 99, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GENERALIZATIONS OF THE DRIFT LAPLACE EQUATION IN THE HEISENBERG GROUP AND GRUSHIN-TYPE SPACES THOMAS BIESKE, KELLER BLACKWELL Abstract. We find fundamental solutions to p-Laplace equations with drift terms in the Heisenberg group and Grushin-type planes. These solutions are natural generalizations of the fundamental solutions discovered by Beals, Gaveau, and Greiner for the Laplace equation with drift term. Our results are independent of the results of Bieske and Childers, in that Bieske and Childers consider a generalization that focuses on the p-Laplace-type equation while we primarily concentrate on a generalization of the drift term. 1. Introduction When studying partial differential equations, one frequent problem under con- sideration concerns establishing a closed-form fundamental solution. While it is often not possible to do so, equations that possess such closed-form solutions spark further study and interest. One of the most well-known examples is the p-Laplace equation in (Euclidean) Rn. In their seminal paper, Capogna, Danielli, and Garo- falo [7] establish the closed-form fundamental solution to the p-Laplace equation in a class of sub-Riemannian spaces called groups of Heisenberg-type. The first author and Gong [6] found a closed-form fundamental solution to the p-Laplace equation in some Grushin-type spaces, which are sub-Riemannian spaces that lack an algebraic group law. Because of this deficiency, the closed-form only holds when the singularity is at certain points. (See Sections 3 and 4 for further discussion concerning the Heisenberg group and Grushin-type planes.) Beals, Gaveau, and Greiner [1] establish a formula for the fundamental solution to the 2-Laplace equation with drift term in a large class of sub-Riemannian spaces. In [5] the first author and Childers expanded these results by invoking a p-Laplace generalization that encompasses the formulas of [1, 7, 6] by generalizing the p- Laplace operator. That paper also included a negative result [5, Theorems 4.1, 4.2]. In this paper, we focus on that negative result and produce a different natural generalization of the p-Laplace equation with drift term by focusing on generalizing the drift term. Our solutions are stable under limits when p → ∞ and when the drift parameter L→ 0 (which is the standard p-Laplace equation). 2010 Mathematics Subject Classification. 53C17, 35H20, 35A08, 22E25, 17B70. Key words and phrases. p-Laplace equation; Heisenberg group; Grushin-type plane; fundamental solution. ©2021. This work is licensed under a CC BY 4.0 license. Submitted December 29, 2020. Published December 20, 2021. 1 2 T. BIESKE, K. BLACKWELL EJDE-2021/99 2. Motivating results 2.1. Heisenberg Group. In the Heisenberg group (See Section 3 for further de- tails and discussion.) the following theorem establishing the fundamental solution of the p-Laplace equation in the Heisenberg group was proved by Capogna, Danielli, and Garofalo [7]. Theorem 2.1 ([7]). Let 1 < p <∞. In the first Heisenberg group H1, let u(x1, x2, x3) = (x2 1 + x2 2)2 + 16x2 3. For p 6= 4, let ηp = 4− p 4(1− p) , and let ζp = { u(x1, x2, x3)ηp p 6= 4 log u(x1, x2, x3) p = 4. Then we have ∆pζp = Cδ0 for some constant C in the sense of distributions. Beals, Gaveau, and Greiner [1] extend this result by finding the fundamental solution to the 2-Laplace equation with a drift term, as shown in the following theorem (cf. [5, Theorem 3.4]). Theorem 2.2 ([1]). Let L ∈ R, |L| 6= 1. Consider the constants η = L− 1 2 and τ = −(L+ 1) 2 together with the functions v(x1, x2, x3) = (x2 1 + x2 2)− 4ix3 and w(x1, x2, x3) = (x1 1 + x2 2) + 4ix3, for defining our main function u2,L(x1, x2, x3) = v(x1, x2, x3)ηw(x1, x2, x3)τ . Then ∆2u2,L + iL[X1, X2]u2,L = Cδ0 for some constant C, in the sense of distri- butions. 2.2. Grushin-type planes. The first author and Gong [6] proved the follow- ing theorem establishing the fundamental solution to the p-Laplace equation in Grushin-type planes Gn. (See Section 4 for further details and discussion.) Theorem 2.3 ([6]). Let 1 < p <∞ and define F (y1, y2) = c2(y1 − a)(2n+2) + (n+ 1)2(y2 − b)2. For p 6= n+ 2, consider τp = n+ 2− p (2n+ 2)(1− p) so that in Gn we have the well-defined function ψp = { F (y1, y2)τp p 6= n+ 2 logF (y1, y2) p = n+ 2. Then ∆pψp = Cδ0 for some constant C, in the sense of distributions. As in the Heisenberg environment, Beals, Gaveau and Greiner [1] extend this result by finding the fundamental solution to the 2-Laplace equation with a drift term, as shown in the following theorem (cf. [5, Theorem 3.2]). EJDE-2021/99 GENERALIZATIONS OF THE DRIFT LAPLACE EQUATION 3 Theorem 2.4 ([1]). Let L ∈ R, |L| 6= 1. Consider the quantities α = −n (2n+ 2) (1 + L) and β = −n (2n+ 2) (1− L). We use these constants with the functions g(y1, y2) = c(y1 − a)n+1 + i(n+ 1)(y2 − b), h(y1, y2) = c(y1 − a)n+1 − i(n+ 1)(y2 − b) for defining our main function f2,L(y1, y2) = g(y1, y2)αh(y1, y2)β . Then ∆2f2,L + iL[Y1, Y2]f2,L = Cδ0 for some constant C, in the sense of distribu- tions. To motivate our study, we make the following key observation. Observation. In the Heisenberg group H1 \ {0}, both the equation and solution of Theorem 2.1 when p = 2 coincides with the equation and solution of Theorem 2.2 when L = 0. In particular, u2,0 = ζ2. Similarly, in Grushin-type planes Gn \ {(a, b)}, both the equation and solution of Theorem 2.3 when p = 2 coincides with the equation and solution of Theorem 2.4 when L = 0. In particular, f2,0 = ψ2. This observation then leads us to state our main question under consideration. Main question. Can we extend the preceding relationship in H \ {0} and in Gn \ {(a, b)} from p = 2 to all p, 1 < p ≤ ∞? Specifically, we have the following goals: • In the case of the Heisenberg group, we wish to find a differential operator Hp,L and a function up,L satisfying: Hp,0 = ∆p and H2,L = ∆2 + iL[X1, X2] with up,0 being the solution of Theorem 2.1 and u2,L being the solution of Theorem 2.2 such that Hp,Lup,L(q) = 0 for q ∈ H1 \ {0}, 1 < p ≤ ∞, and L ∈ R. • In the case of the Grushin-type planes, we wish to find a differential operator Gp,L and a function fp,L satisfying: Gp,0 = ∆p and G2,L = ∆2 + iL[Y1, Y2] with fp,0 being the solution of Theorem 2.3 and f2,L being the solution of Theorem 2.4 such that Gp,Lfp,L(q) = 0 for q ∈ Gn \ {(a, b)}, 1 < p ≤ ∞, and L ∈ R. • Furthermore, we would like fp,L and up,L to be the fundamental solutions to their respective equations. 4 T. BIESKE, K. BLACKWELL EJDE-2021/99 3. Heisenberg group 3.1. Properties. We begin with R3 using the coordinates (x1, x2, x3) and consider the linearly independent vector fields {X1, X2, X3}, defined by: X1 = ∂ ∂x1 − x2 2 ∂ ∂x3 , X2 = ∂ ∂x2 + x1 2 ∂ ∂x3 , X3 = ∂ ∂x3 which obey the relation [X1, X2] = X3. We then have a Lie Algebra denoted h1 that decomposes as a direct sum h1 = V1⊕V2 where V1 = span[X1, X2] and V2 = span[X3]. The Lie algebra is statified; i.e., [V1, V1] = V2 and [V1, V2] = 0. We endow h1 with an inner product 〈·, ·〉H and related norm ‖ · ‖H so that this basis is orthonormal. The corresponding Lie Group is called the general Heisenberg group of dimension 1 and is denoted by H1. With this choice of vector fields the exponential map is the identity map, so that for any p, q in H1, written as p = (x1, x2, x3) and q = (x̂1, x̂2, x̂3) the group multiplication law is given by p · q = ( x1 + x̂1, x2 + x̂2, x3 + x̂3 + 1 2 (x1x̂2 − x2x̂1) ) . The natural metric on H1 is the Carnot-Carathéodory metric given by dC(p, q) = inf Γ ∫ 1 0 ‖γ′(t)‖H dt where the set Γ is the set of all curves γ such that γ(0) = p, γ(1) = q and γ′(t) ∈ V1. By Chow’s theorem (See, for example, [2].) any two points can be connected by such a curve, which makes dC(p, q) a left-invariant metric on H1. Given a smooth function u : H1 → R, we define the horizontal gradient by ∇0u = (X1u,X2u). Additionally, given a vector field F = ∑2 i=1 fiXi + f3X3, we define the Heisenberg divergence of F , denoted divF , by divF = 2∑ i=1 Xifi . A quick calculation shows that when f3 = 0, we have divF = diveucl F where diveucl is the standard Euclidean divergence. The main operator we are concerned with is the horizontal p-Laplacian for 1 < p <∞ defined by ∆pu = div(‖∇0u‖p−2 H ∇0u) = 2∑ i=1 Xi ( ‖∇0u‖p−2 H Xiu ) = p− 2 2 ‖∇0u‖p−4 H 2∑ i=1 Xi‖∇0u‖2HXiu+ ‖∇0u‖p−2 H 2∑ i=1 XiXiu. (3.1) For an extensive treatment of the Heisenberg group, the interested reader is directed to [2, 4, 8, 9, 10, 11, 12, 13] and the references therein. EJDE-2021/99 GENERALIZATIONS OF THE DRIFT LAPLACE EQUATION 5 3.2. Generalization in the Heisenberg group. For the Heisenberg group H1, we consider the parameters η = 4− p+ 2L(1− p) 4(1− p) and τ = 4− p− 2L(1− p) 4(1− p) for L ∈ R with L 6= ± 4− p 2(1− p) . We use these parameters with the functions v(x1, x2, x3) = (x2 1 + x2 2)− 4ix3, w(x1, x2, x3) = (x2 1 + x2 2) + 4ix3 to define our main function up,L(y1, y2) = v(x1, x2, x3)ηw(x1, x2, x3)τ . (3.2) Using this equation, we have the following result. Theorem 3.1. Let 1 < p <∞. On H1, we have Hp,L(up,L) := ∆pup,L + iL[X1, X2] ( ‖∇0up,L‖p−2 H up,L ) = Cδ0 for some constant C, in the sense of distributions. Proof. Suppressing arguments and subscripts, we obtain the following: X1u = 2vη−1wτ−1 ( (ηw + τv)x1 + (ηw − τv)ix2 ) (3.3) X1u = 2vτ−1wη−1 ( (ηv + τw)x1 + (ηv − τw)ix2 ) X2u = 2vη−1wτ−1 ( (ηw + τv)x2 − (ηw − τv)ix1 ) (3.4) X2u = 2vτ−1wη−1 ( (ηv + τw)x2 − (ηv − τw)ix1 ) and so ‖∇0u‖2 = 8(η2 + τ2)vη+τ−1wη+τ−1(x2 1 + x2 2). (3.5) Using the above we have X1(X1u) = 2vη−2wτ−2 ( 2 ( (ηw + τv)x2 1 + (−ηw − τv)ix1x2 )( (η − 1)w + (τ − 1)v ) + 2i ( (ηw + τv)x2 2 + (ηw − τv)ix2 2 )( − (η − 1)w + (τ − 1)v) ) + vw ( 2(x2 1 + x2 2)(τ + η) + (ηw + τv) )) , X2(X2u) = 2vη−2wτ−2 ( 2 ( (ηw + τv)x2 2 + (−ηw + τv)ix1x2 )( (η − 1)w + (τ − 1)v ) + 2i ( (ηw + τv)x1x2 + (−ηw + τv)ix2 1 )( − (η − 1)w + (τ − 1)v) ) + vw ( 2(x2 1 + x2 2)(τ + η) + (ηw + τv) )) . In addition, we have X1‖∇0u‖2 = 16(η2 + τ2)vη+τ−2wη+τ−2 × ( vwx1 + 2(η + τ − 1)(x2 1 + x2 2)2 ( x1 − 4x2x3 )) (3.6) and X2‖∇0u‖2 = 16(η2 + τ2)vη+τ−2wη+τ−2 × ( vwx2 + 2(η + τ − 1)(x2 1 + x2 2)2 ( x2 − 4x1x3 )) (3.7) 6 T. BIESKE, K. BLACKWELL EJDE-2021/99 so that 2∑ j=1 Xj‖∇0u‖2(Xju) = 32(η2 + τ2)v2η+τ−3wη+2τ−3 ( (ηw + τv)vw(x2 1 + x2 2) + 2(η + τ − 1)(x2 1 + x2 2)2 ( (ηw + τv)(x2 1 + x2 2)2 − 4(ηw − τv)ix3 )) and ‖∇0u‖2 ( X1X1u+X2X2u ) = 16(η2 + τ2)v2η+τ−3wη+2τ−3(x2 1 + x2 2) ( 2vw(ηw + τv) + 4vw(η + τ)(x2 1 + x2 2) + 2 ( (η − 1)w + (τ − 1)v ) (ηw + τv)(x2 1 + x2 2) + 2 ( − (η − 1)w + (τ − 1)v ) (ηw − τv)(x2 1 + x2 2) ) . This yields ∆pu = ‖∇0u‖p−4 ( (p− 2) 2 2∑ j=1 Xj‖∇0u‖2(Xju) + ‖∇0u‖2(X1X1f +X2X2u) ) = 2L (4− p)p−2 (1− p)p−2 ( 1 + 4L2(1− p)2 (4− p)2 ) p−2 2 v 1 2 (pη+(p−2)τ−p)w 1 2 ((p−2)η+pτ−p) × (x2 1 + x2 2) p−2 2 (−2L(x2 1 + x2 2) + p4ix3). We then compute iL[X1, X2](‖∇0u‖p−2u) = iL (4− p)p−2 (1− p)p−2 ( 1 + 4L2(1− p)2 (4− p)2 ) p−2 2 (x2 1 + x2 2) p−2 2 × ∂ ∂x3 v 1 2 (p−2)(η+τ−1)+ηw 1 2 (p−2)(η+τ−1)+τ = −2L (4− p)p−2 (1− p)p−2 Big(1 + 4L2(1− p)2 (4− p)2 ) p−2 2 (x2 1 + x2 2) p−2 2 × v 1 2 (pη+(p−2)τ−p)w 1 2 ((p−2)η+pτ−p)(−2L(x2 1 + x2 2) + p4ix3) = −∆pu from which it follows that Hp,Lup,L = 0 on H1 \{0}, away from the singularity. We now consider the normalization vε(x1, x2, x3) := (x2 1 + x2 2) + ε2 − 4ix3, wε(x1, x2, x3) := (x2 1 + x2 2) + ε2 + 4ix3 so that uε(x1, x2, x3) := vε(x1, x2, x3)ηwε(x1, x2, x3)τ . Suppressing arguments and computing similarly as before yields the distribution Hp,Luε = 2 3p−2 2 ε2 (p(4− p) 4(1− p) + L2 ) (η2 + τ2) p−2 2 (x2 1 + x2 2) p−2 2 × v ηp+τ(p−2)−p 2 ε w η(p−2)+τp−p 2 ε . (3.8) EJDE-2021/99 GENERALIZATIONS OF THE DRIFT LAPLACE EQUATION 7 By the argument in [1, Theorem 7.5, (c)], the distribution of (3.8) is determined by the density 2 3p−2 2 (p(4−p) 4(1−p) + L2 ) (η2 + τ2) p−2 2 ( (x1 ε )2 + (x2 ε )2 ) p−2 2 dm (x2 1+x2 2 ε2 ) d(x3 ε2 ) 1 −2i( (x1 ε )2 + (x2 ε )2 + 1− 4ix3 ε2 )− ηp+τ(p−2)−p 2 ( (x1 ε )2 + (x2 ε )2 + 1 + 4ix3 ε2 )− η(p−2)+τp−p 2 (3.9) where dm denotes the Lebesgue measure in the complex plane. Then as ε→ 0 the distribution of (3.9) tends to the δ0 distribution, up to a constant factor. � Observing that L 6= ± 4− p 2(1− p) implies p 6= ∣∣2L+ 4 2L+ 1 ∣∣, ∣∣2L− 4 2L− 1 ∣∣ we have immediately the following corollary. Corollary 3.2. Let p > max{ ∣∣ 2L+4 2L+1 ∣∣, ∣∣ 2L−4 2L−1 ∣∣}. Then the function up,L of (3.2) is a smooth solution to the Dirichlet problem Htp,L(up,L(q)) = 0 q ∈ H1 \ {0} 0 q = 0. 3.3. Limit as p→∞. Recall that the drift p-Laplace equation in the Heisenberg group H1 is given by: Hp,L(u) := ∆pu+ iL[X1, X2] ( ‖∇0u‖p−2 H u ) = 0. A routine expansion of the drift term yields the observation Hp,L(u) = ∆pu+ iL (p− 2 2 ‖∇0u‖p−4 H ( ∂ ∂x3 ‖∇0u‖2H ) u+ ‖∇0u‖p−2 H ∂ ∂x3 u ) = 0. Dividing through by p−2 2 ‖∇0u‖p−4 H and formally taking the limit p→∞, we obtain H∞,L(u) = ∆∞u+ iL[X1, X2](‖∇0u‖2H)u. Considering (3.2) and formally letting p→∞ yields u∞,L(x1, x2, x3) = v(x1, x2, x3) 1+2L 4 w(x1, x2, x3) 1−2L 4 , where we recall the functions v(x1, x2, x3) = (x2 1 + x2 2)− 4ix3, w(x1, x2, x3) = (x2 1 + x2 2) + 4ix3 . Theorem 3.3. The function u∞,L, defined above, is a smooth solution to the Dirichlet problem H∞,Lu∞,L(q) = 0 q ∈ H1 \ {0}, 0 q = 0. Proof. We prove this theorem by letting p → ∞ in (3.3), (3.4), (3.6), and (3.7), and invoking continuity (cf. Corollary 3.2). However, for completeness we compute it formally. We let N = 1 + 2L 4 and T = 1− 2L 4 . 8 T. BIESKE, K. BLACKWELL EJDE-2021/99 Suppressing arguments and subscripts, we compute X1u = 2vN−1wT−1 ( (Nw + Tv)x1 + (Nw − Tv)ix2 ) , X2u = 2vN−1wT−1 ( (Nw + Tv)x2 − (Nw − Tv)ix1 ) , ‖∇0u‖2 = 8(N2 + T 2)vN+T−1wN+T−1(x2 1 + x2 2), X1‖∇0u‖2 = 16(N2 + T 2)vN+T−2wN+T−2 × ( vwx1 + 2(N + T − 1)(x2 1 + x2 2)2 ( x1 − 4x2x3 )) , X2‖∇0u‖2 = 16(N2 + T 2)vN+T−2wN+T−2 × ( vwx2 + 2(N + T − 1)(x2 1 + x2 2)2 ( x2 − 4x1x3 )) , so that ∆∞u = X1‖∇0u‖2X1u+X2‖∇0u‖2X2u = 32(N2 + T 2)v2N+T−3wN+2T−3 ( (Nw + Tv)vw(x2 1 + x2 2) + 2(N + T − 1)(x2 1 + x2 2)2 ( (Nw + Tv)(x2 1 + x2 2)2 − 4(Nw − Tv)ix3 )) = 128iL(N2 + T 2)(x2 1 + x2 2)x3v 2N+T−2wN+2T−2. We also have iL[X1, X2](‖∇0u‖2)u = iLvNwT ∂ ∂x3 ‖∇0f‖2 = −128iL(N2 + T 2)(x2 1 + x2 2)x3v 2N+T−2wN+2T−2. The proof is complete. � We notice that when L = 0, this result was a part of the Ph.D. thesis of the first author [3]. In particular, combined with [3, 4], we have shown the following commutative diagram in H1 \ {0}, Hp,L(up,L) = 0 −−−−→ p→∞ H∞,L(u∞,L) = 0yL→0 yL→0 ∆pup,0 = 0 −−−−→ p→∞ ∆∞u∞,0 = 0 4. Grushin-type planes The Grushin-type planes differ from the Heisenberg group in that Grushin-type planes lack an algebraic group law. We begin with R2, possessing coordinates (y1, y2), a ∈ R, c ∈ R \ {0} and n ∈ N. We use them to construct the vector fields: Y1 = ∂ ∂y1 and Y2 = c(y1 − a)n ∂ ∂y2 . For these vector fields, the only (possibly) nonzero Lie bracket is [Y1, Y2] = cn(y1 − a)n−1 ∂ ∂y2 . Because n ∈ N, it follows that Hörmander’s condition is satisfied by these vector fields. EJDE-2021/99 GENERALIZATIONS OF THE DRIFT LAPLACE EQUATION 9 We will put a (singular) inner product on R2, denoted 〈·, ·〉G, with related norm ‖ · ‖G, so that the collection {Y1, Y2} forms an orthonormal basis. We then have a sub-Riemannian space that we will call gn, which is also the tangent space to a generalized Grushin-type plane Gn. Points in Gn will also be denoted by p = (y1, y2). The Carnot-Carathéodory distance on Gn is defined for points p and q as follows dG(p, q) = inf Γ ∫ ‖γ′(t)‖G dt , with Γ the set of curves γ such that γ(0) = p, γ(1) = q and γ′(t) ∈ span{Y1(γ(t)), Y2(γ(t))}. By Chow’s theorem, this is an honest metric. We shall now discuss calculus on the Grushin-type planes. Given a smooth function f on Gn, we define the horizontal gradient of f as ∇0f(p) = ( Y1f(p), Y2f(p) ) . Using these derivatives, we consider a key operator on C2 G functions, namely the p-Laplacian for 1 < p <∞, given by ∆pf = divG(‖∇0f‖p−2 G ∇0f) = Y1 ( ‖∇0f‖p−2 G Y1f ) + Y2 ( ‖∇0f‖p−2 G Y2f ) = p− 2 2 ‖∇0f‖p−4 G ( Y1‖∇0f‖2GY1f + Y2‖∇0f‖2GY2f ) + ‖∇0f‖p−2 G ( Y1Y1f + Y2Y2f ) . (4.1) 4.1. A Generalization in the Grushin plane. For the Grushin-type planes, we consider the parameters α = n+ 2− p− Ln(1− p) 2(n+ 1)(1− p) and β = n+ 2− p+ Ln(1− p) 2(n+ 1)(1− p) , where L ∈ R with L 6= ±n+ 2− p n(1− p) . We use these constants with the functions g(y1, y2) = c(y1 − a)n+1 + i(n+ 1)(y2 − b), h(y1, y2) = c(y1 − a)n+1 − i(n+ 1)(y2 − b) to define our main function fp,L(y1, y2) = g(y1, y2)αh(y1, y2)β . (4.2) Using this equation, we have the following theorem. Theorem 4.1. Let 1 < p <∞. On Gn, we have Gp,L(fp,L) := ∆pfp,L + iL[Y1, Y2](‖∇0fp,L‖p−2 G fp,L) = Cδ0 for some constant C, in the sense of distributions. 10 T. BIESKE, K. BLACKWELL EJDE-2021/99 Proof. Suppressing arguments and subscripts, we compute the following: Y1f = c(n+ 1)(y1 − a)ngα−1hβ−1(αh+ βg) (4.3) Y1f = c(n+ 1)(y1 − a)ngβ−1hα−1(αg + βh) Y2f = ic(n+ 1)(y1 − a)ngα−1hβ−1(αh− βg) (4.4) Y2f = ic(n+ 1)(y1 − a)ngβ−1hα−1(αg − βh) ‖∇0f‖2 = 2c2(n+ 1)2(y1 − a)2ngα+β−1hα+β−1(α2 + β2). (4.5) Using the above we have Y1(Y1f) = c(n+ 1)(y1 − a)n−1gα−2hβ−2 × ( ngh(αh+ βg) + c(n+ 1)(y1 − a)n+1 × ( (αh+ βg)((α− 1)h+ (β − 1)g) + gh(α+ β) )) , Y2(Y2f) = −c2(n+ 1)2(y1 − a)2ngα−2hβ−2 × ( (αh− βg)((α− 1)h− (β − 1)g)− gh(α+ β) ) , Y1‖∇0f‖2 = 4c2(n+ 1)2(α2 + β2)(y1 − a)2n−1gα+β−2hα+β−2 × ( ngh+ c2(n+ 1)(α+ β − 1)(y1 − a)2n+2 ) , (4.6) Y2‖∇0f‖2 = 4c3(n+ 1)4(α2 + β2)(y1 − a)3n(y2 − b) (α+ β − 1)gα+β−2hα+β−2 (4.7) and 2∑ i=1 Yi‖∇0f‖2(Yif) = 4c3(n+ 1)3(α2 + β2)(y1 − a)3n−1g2α+β−3hα+2β−3 × ( (αh+ βg) ( ngh+ c2(n+ 1)(α+ β − 1)(y1 − a)2n+2 ) + ic(n+ 1)2(y1 − a)n+1(y2 − b)(α+ β − 1)(αh− βg) ) , ‖∇0f‖2(Y1Y1f + Y2Y2f) = 2c3(n+ 1)3(α2 + β2)(y1 − a)3n−1g2α+β−3hα+2β−3 × ( ngh(αh+ βg) + 4c(n+ 1)(y1 − a)n+1gh(αβ) ) , so that ∆pf = ‖∇0f‖p−4 ( (p− 2) 2 2∑ j=1 Yj‖∇0f‖2(Yjf) + ‖∇0f‖2(Y1Y1f + Y2Y2f) ) = −L2 p−2 2 cp−1n2(n+ 1)p−2(y1 − a)n(p−1)−1(α2 + β2) p−2 2 × g 1 2 (αp+β(p−2)−p)h 1 2 (α(p−2)+βp−p)(Lc(y1 − a)n+1 + i(1− p)(n+ 1)(y2 − b)). We then compute iL[Y1, Y2](‖∇0f‖p−2f) = iL2 p−2 2 cp−1n(n+ 1)p−2(y1 − a)n(p−1)−1(α2 + β2) p−2 2 × ∂ ∂y2 ( g 1 2 (αp+β(p−2)−(p−2))h 1 2 (α(p−2)+βp−(p−2)) ) = L2 p−2 2 cp−1n2(n+ 1)p−2(y1 − a)n(p−1)−1(α2 + β2) p−2 2 EJDE-2021/99 GENERALIZATIONS OF THE DRIFT LAPLACE EQUATION 11 × g 1 2 (αp+β(p−2)−p)h 1 2 (α(p−2)+βp−p) × (Lc(y1 − a)n+1 + i(1− p)(n+ 1)(y2 − b)) = −∆pf from which it follows that Gp,Lfp,L = 0 on Gn \ {(a, b)}, away from the singularity. We now consider the normalization gε(y1, y2) := c(y1 − a)n + ε2 + i(n+ 1)(y2 − b), hε(y1, y2) := c(y1 − a)n + ε2 − i(n+ 1)(y2 − b) so that fε(y1, y2) := gε(y1, y2)αhε(y1, y2)β . Suppressing arguments and computing similarly as before yields the distribution Gp,Lfε = −2 p−2 2 ε2((n+ 2− p)− nL2)cp−1n(n+ 1)p−2(α2 + β2) p−2 2 × (y1 − a)n(p−1)−1g αp+β(p−2)−p 2 h α(p−2)+βp−p 2 . (4.8) By the argument in [1, Theorem 7.5, (c)], the distribution of (4.8) is determined by the density − 2 p−2 2 ( (n+ 2− p)− nL2 ) cp−1n(n+ 1)p−2(α2 + β2) p−2 2 × ( y1 − a ε2/(n+1) )n(p−1)−1 dm ( y1 − a ε2/(n+1) ) d (y2 − b ε2 )( 1 −2i ) × ( c ( y1 − a ε2/(n+1) )n+1 + 1 + i(n+ 1) (y2 − b) ε2 )αp+β(p−2)−p 2 × ( c ( y1 − a ε2/(n+1) )n+1 + 1− i(n+ 1) (y2 − b) ε2 )α(p−2)+βp−p 2 (4.9) where dm denotes the Lebesgue measure in the complex plane. Then as ε→ 0 the distribution of (4.9) tends to the δ0 distribution, up to a constant factor. � Observing that L 6= ± n(p− 1) n+ 2− p implies p 6= ∣∣L(n+ 2) + n n+ L ∣∣, ∣∣L(n+ 2)− n n− L ∣∣ we have immediately the following corollary. Corollary 4.2. Let p > max {∣∣L(n+2)+n n+L ∣∣ , ∣∣L(n+2)−n n−L ∣∣}. Then the function fp,L of Equation 4.2 is a smooth solution to the Dirichlet problem Gp,L(fp,L(q)) = 0 q ∈ Gn \ {(a, b)} 0 q = (a, b). 4.2. Limit as p→∞. Recall that the drift p-Laplace equation in the Grushin-type planes Gn is given by Gp,L(f) := ∆pf + iL[Y1, Y2] ( ‖∇0f‖p−2 G f ) = 0. A routine expansion of the drift term yields the observation Gp,L(f) = ∆pf + iLcn(y1 − a)n−1 (p− 2 2 ‖∇0f‖p−4 G ( ∂ ∂y2 ‖∇0f‖2G ) f + ‖∇0f‖p−2 G ∂ ∂y2 f ) = 0. 12 T. BIESKE, K. BLACKWELL EJDE-2021/99 Dividing through by p−2 2 ‖∇0f‖p−4 G and formally taking the limit p→∞, we obtain G∞,L(f) = ∆∞f + iL[Y1, Y2](‖∇0f‖2G)f. Considering (4.2) and formally letting p→∞ yields f∞,L(y1, y2) = g(y1, y2) 1 2(n+1) (1−nL)h(y1, y2) 1 2(n+1) (1+nL) where we recall the functions g(y1, y2) = c(y1 − a)n+1 + i(n+ 1)(y2 − b), h(y1, y2) = c(y1 − a)n+1 − i(n+ 1)(y2 − b). Theorem 4.3. The function f∞,L, defined above, is a smooth solution to the Dirichlet problem G∞,Lf∞,L(q) = 0 q ∈ Gn \ {(a, b)}, 0 q = (a, b). Proof. We prove this theorem by letting p → ∞ in (4.3), (4.4), (4.6), (4.7), and invoking continuity (cf. Corollary 4.2). However, for completeness we compute it formally. We let A = 1 2(n+ 1) (1− nL) and B = 1 2(n+ 1) (1 + nL) and, suppressing arguments and subscripts, compute Y1f = c(n+ 1)(y1 − a)ngA−1hB−1(Ah+Bg), Y2f = ic(n+ 1)(y1 − a)ngA−1hB−1(Ah−Bg), ‖∇0f‖2 = 2c2(n+ 1)2(y1 − a)2ngA+B−1hA+B−1(A2 +B2), Y1‖∇0f‖2 = 4c2(n+ 1)2(A2 +B2)(y1 − a)2n−1gA+B−2hA+B−2 × ( ngh+ c2(n+ 1)(A+B − 1)(y1 − a)2n+2 ) , Y2‖∇0f‖2 = 4c3(n+ 1)4(A2 +B2)(y1 − a)3n(y2 − b) (α+ β − 1)gA+B−2hA+B−2, so that ∆∞f = Y1‖∇0f‖2Y1f + Y2‖∇0f‖2Y2f = 4c3(n+ 1)3(A2 +B2)(y1 − a)3n−1g2A+B−3hA+2B−3 × ( (Ah+Bg) ( ngh+ c2(n+ 1)(A+B − 1)(y1 − a)2n+2 ) + ic(n+ 1)2(y1 − a)n+1(y2 − b)(A+B − 1)(Ah−Bg) ) = 4iLc3(n+ 1)3n2(A2 +B2)(y1 − a)3n−1(y2 − b)g2A+B−2hA+2B−2. We also compute iL[Y1, Y2](‖∇0f‖2)f = iLgAhB(cn(y1 − a)n−1 ∂ ∂y2 ‖∇0f‖2) = −4iLc3(n+ 1)3n2(A2 +B2)g2A+B−2hA+2B−2(y1 − a)3n−1(y2 − b). The proof is complete. � EJDE-2021/99 GENERALIZATIONS OF THE DRIFT LAPLACE EQUATION 13 In particular, combining this with [6], we have shown that the following commu- tative diagram in Gn \ {(a, b)}, Gp,Lfp,L = 0 −−−−→ p→∞ G∞,Lf∞,L = 0yL→0 yL→0 ∆pfp,0 = 0 −−−−→ p→∞ ∆∞f∞,0 = 0 Acknowledgements. This paper is the result of an undergraduate research project by K. Blakwell under the direction of T. Bieske. K. Blakwell would like to thank the University of South Florida Honors College and the Department of Mathematics and Statistics for their support and research opportunities. References [1] Beals, Richard.; Gaveau, Bernard.; Greiner, Peter; On a Geometric Formula for the Funda- mental Solution of Subelliptic Laplacians. Math. Nachr., 181 (1996), 81-163. [2] Belläche, André; The Tangent Space in Sub-Riemannian Geometry. In Sub-Riemannian Ge- ometry; Belläche, André., Risler, Jean-Jacques., Eds.; Progress in Mathematics; Birkhäuser: Basel, Switzerland. 1996, Vol. 144, 1-78. 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B.; Stein, Elias M.; Hardy Spaces on Homogeneous Groups; Princeton University Press: Princeton, NJ. 1982. [10] Gromov, Mikhael; Metric Structures for Riemannian and Non-Riemannian Spaces; Birkhäuser Boston Inc: Boston, 1999. [11] Heinonen, Juha; Calculus on Carnot groups. Fall School in Analysis. Report no. 68, Univ. of Jyväskylä, Jyväskylä, Finland, 1995. [12] Kaplan, Aroldo; Lie Groups of Heisenberg Type. Rend. Sem. Mat. Univ. Politec. Torino 1983 Special Issue, (1984), 117-130. [13] Stein, Elias M.; Harmonic Analysis, Princeton University Press: Princeton, NJ. 1993. Thomas Bieske Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA Email address: tbieske@usf.edu Keller Blackwell School of Engineering Stanford University, Stanford, CA 94305, USA Email address: kellerb@stanford.edu 1. Introduction 2. Motivating results 2.1. Heisenberg Group 2.2. Grushin-type planes Observation Main question 3. Heisenberg group 3.1. Properties 3.2. Generalization in the Heisenberg group 3.3. Limit as p 4. Grushin-type planes 4.1. A Generalization in the Grushin plane 4.2. Limit as p Acknowledgements References