Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 57, pp. 1–22. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOLVING SINGULAR EVOLUTION PROBLEMS IN SUB-RIEMANNIAN GROUPS VIA DETERMINISTIC GAMES PABLO OCHOA, JULIO ALEJO RUIZ Abstract. In this manuscript, we prove the existence of viscosity solutions to singular parabolic equations in Carnot groups. We develop the analysis by constructing appropriate deterministic games adapted to the algebraic and dif- ferential structures of Carnot groups. We point out that the proof of existence does not require a comparison principle and it is based on an Arzela-Ascoli- type theorem. 1. Introduction In the previous decades, there has been a special interest in the study of partial differential equations in non-Euclidean frameworks. In this work, we study the existence of viscosity solutions for singular parabolic equations in Carnot groups, via two-person deterministic games. To motivate the main results, we consider a family of surfaces Mt ⊂ RN , t ≥ 0, given as the zero-level set of a function u: Mt = { p ∈ RN : u(t, p) = 0 } . If we are interested in the movement of Mt by horizontal mean curvature, then the function u is a solution of the PDE ut(t, p) = m1∑ i,j=1 ( δij − XiuXju∑m1 i=1(Xiu)2 ) XiXju. (1.1) This model is known as the horizontal mean curvature flow equation (see [18, 13] for a derivation in Euclidean spaces, and [11, 12] for the corresponding discussion in Carnot groups). The main result of the paper is Theorem 2.1 below, where we establish the existence of viscosity solution to (1.1) by employing two-person deterministic games (see [15] for a stochastic approach to mean curvature flow in sub-Riemannian geometries). We point out that our results apply to a large class of singular equations including (1.1). Existence and comparison results for Carnot groups are less usual than in the Euclidean framework. We highlight that even if one writes equations in Carnot groups in terms of the Euclidean gradient and the Hessian of the unknown and try to apply the Euclidean theory (for instance from [14]), this does not always work. Indeed, the above procedure may introduce degenerate points in the equations or 2010 Mathematics Subject Classification. 35R03, 49L25, 49N70. Key words and phrases. Carnot group; viscosity solutions; differential games. c©2021. This work is licensed under a CC BY 4.0 license. Submitted November 10, 2020. Published June 23, 2021. 1 2 P. OCHOA, J. A. RUIZ EJDE-2021/57 may cause the loss of appropriate continuity. A classical example is the following: consider the sub-elliptic Laplacian operator in the Heisenberg group ∆0 := X2 1 +X2 2 where X1 and X2 are given as in (3.3). In terms of Euclidean derivatives, the sub-elliptic Laplace equation may be written as − tr  1 0 2y 0 1 −2x 2y −2x 4(x2 + y2) ∇2u  = 0. Observe that the matrix  1 0 2y 0 1 −2x 2y −2x 4(x2 + y2)  is not uniformly elliptic for all (x, y, z). The interested reader may consult [27, Remark 1] where we exhibited an example for which the operator in terms of the Euclidean derivatives does not satisfies well-known assumptions on uniform con- tinuity needed and frequently used in the theory of Euclidean viscosity solutions. However, the comparison derived intrinsically in the sub-Rimannian setting in [27] applied to the given example. We also refer the reader to Remark 2.3 in the present article. In sub-Riemannian structures, the singularity of the equation may appear at points where part of the gradient (the horizontal gradient) vanishes. This is not the case of the Euclidean context, where the singularity comes from points where the full gradient vanishes. This facet of sub-Riemannian structures yields difficulties in the study of singular equations. In spite of these facts, there are some interested findings in Carnot groups. For elliptic and uniformly elliptic equations in the Heisenberg group H, comparison results are given in [22], based on a sub-elliptic version of the Euclidean Crandall-Ishii Lemma (see [14] for details). At this point we quote the works [1] and [24] where the authors propose various form of partial nondegeneracy to weaken the uniform ellipticity assumption and apply their results to some sub- elliptic second order equations. Also, in [6], results related to infinite harmonic functions in the Heisenberg group were established. In the general case of Carnot groups, we find the work [5] for the p-Laplacian operator. In the setting of vector fields in RN (with its standard group structure), we refer the reader to the paper [3]. Regarding parabolic equations in Carnot groups, we mention [4], where comparison results for admissible operators in the Heisenberg group where obtained. The word admissible refers to continuous and proper operators F = F(t, p, u, η,X ) which satisfy the following: for each t ∈ [0, T ], there is a modulus of continuity ω : [0,∞]→ [0,∞] so that F(t, q, r, τη,Y)−F(t, p, r, τη,X ) ≤ ω ( dC(p, q) + τ‖η‖2 + ‖X − Y‖ ) , where τ > 0. Hence, the results are not valid for singular equations. In [25], the author provides existence and uniqueness results for the Gauss curvature flow equation of graph in Carnot group in unbounded domains, generalizing the available results in the literature. For singular equations, in [8] it was covered the case of the parabolic p-Laplacian, but the structure of this equation in largely used in the derivation of the uniqueness principle. A remarkable progress was done in [16], where the authors proved existence and a comparison principle (needed EJDE-2021/57 SINGULAR PROBLEMS IN CARNOT GROUPS 3 for existence) for radially symmetric viscosity solutions for the horizontal mean curvature flow equation in the first-order Heisenberg group. The symmetry refers to solutions u = u(t, p1, p2, p3) for which it holds u(t, p1, p2, p3) = u(t, p̃1, p̃2, p3) whenever p21 + p22 = p̃21 + p̃22. We also point out the reference [2] for uniqueness of viscosity solutions of mean curvature flow in two sub-Riemannian structures. In [16], the existence of viscosity solutions is obtained via deterministic games. Here, we extend the findings of [16] to more general singular parabolic equations, by adapting the reasoning of [21] and [19] to the structure of Carnot groups. We point out that, unlike [16], our existence result does not require comparison of solutions and it is based on an Arzela-Ascoli-type theorem from [23]. As a final comment, we mention that the uniqueness of the solutions constructed here constitutes an open problem. The organization of this article is as follows. In Section 2, we introduce the type of equations studied in the paper and the main result concerning existence of solutions. In the end of the section, we provide some applications of our results. In Section 3, we shall introduce the necessary background and notation on Carnot groups as well as the notion of viscosity solutions by means of parabolic jets. In the next Section 4, we provide the proof of the existence result. We end the paper with an Appendix where we prove a technical lemma needed in the analysis of the deterministic games. 2. Main result of the paper and assumptions In this work, we study the existence of solutions for initial-value problems of the form ut + µu+ F(t, p,∇G,0u,∇2,∗ G,0u) = 0, on (0, T )×G, u(0, p) = ψ(p), with p ∈ G. (2.1) Here, T > 0 is fixed and µ ≥ 0 is a parameter. We refer the reader to Section 3 for notation and basic definitions involving Carnot groups. We use the following assumptions: (A1) ψ ∈ BC(G) and for each δ > 0 there are smooth approximations ψ+ δ , ψ − δ with bounded right- and left-invariant horizontal derivatives of first and second order so that ψ − δ ≤ ψ−δ ≤ ψ ≤ ψ + δ ≤ ψ + δ. (A2) F : [0, T ]×G× (Rm1\{0})× Sm1(R)→ R is continuous. (A3) λ0 := supη |F(t, p, η,O)| <∞. (A4) There exists a positive constant λ1 such that F(t, p, η,X )−F(t, p, η, X̂ ) ≤ λ21 2 E+(X̂ − X ), where E+(X ) := max{0, maximum eigenvalue of X}, and there exists a modulus of continuity ω = ω(r) = O(r) as r → 0+ such that: F(t, p, η,X )−F(s, q, η,X ) ≤ ω ( |s− t|+ |p · q−1|G ) , (2.2) for all (t, p), (s, q) ∈ [0, T ]×G, η ∈ Rm1 \ {0}, and X ∈ Sm1(R). (A5) For any r,R > 0, there exists a modulus of continuity ωr,R such that F(t, p, η̂,X )−F(t, p, η,X ) ≤ ωr,R(‖η̂ − η‖), if ‖η̂‖, ‖η‖ ≥ r, ‖X‖ ≤ R. 4 P. OCHOA, J. A. RUIZ EJDE-2021/57 (A6) F∗(t, p, 0,O) = F∗(t, p, 0,O) = 0. The main result of this article read as follows. of the paper is the following existence result. Theorem 2.1. Under assumptions (A1)–(A6), equation (2.1) has a viscosity solu- tion in BUC([0, T ]×G). Remark 2.2. Assumption (A1) on the datum may be satisfied for ψ ∈ BC(G) being constant outside a compact set. That is the setting of [16]. Remark 2.3. In this article, assumptions (A4) and (A5) will be usually applied to: η = ∇G,0φ(p) and X = ∇2,∗ G,0φ(p), for some smooth φ (we do not write the dependence on t). In terms of Euclidean derivatives, there are matrix fields A and M (see [8, Lemma 3.2]) so that ∇G,0φ(p) = A(p)∇φ(p) and ∇2,∗ G,0φ(p) = A(p)∇2φ(p)A(p) +M(p), where ∇φ(p) and ∇2φ(p) denotes the Euclidean gradient and Hessian of φ at p. Hence in order to consider sub-elliptic equations in terms on Euclidean derivatives, it is natural to introduce the operator G(t, p, ηE , XE) := F(t, p, A(p)ηE , A(p)XEA(p) +M(p)), ηE ∈ RN , XE ∈ SN (R) which necessarily depends on t and p (so the existence result from [19] is not applied). Moreover, hypothesis (A5) does not imply, in general, [19, (F4)], since the norm of A(p) is not uniformly bounded in G. 2.1. Some applications. Mean curvature flow equation. If M ⊂ G is a smooth hypersurface, we define Σ(M) as the set of characteristic points of M , that is, the points p ∈ M where the horizontal distribution at p is contained in the tangent space of M at p. The horizontal mean curvature flow is the flow t→Mt in which each point p(t) /∈ Σ(Mt) in the evolving surface moves along the horizontal normal with speed given by the horizontal mean curvature. The equation, outside the characteristic set, may be written as (1.1). For (1.1), the singular operator is given by FMCF (η,X ) = − tr [( I − η ⊗ η ‖η‖2 ) X ] , for η ∈ Rm1 \ {0} and X ∈ Sm1(R). Hence, the existence of solutions follows from Theorem 2.1. Parabolic infinite Laplacian. The infinite Laplacian is connected with the problem of finding minimal Lipschitz extensions, called absolute minimizers. In a Carnot group, we say that a Lipschitz function u in Ω is an absolute minimizer if for every V ⊂ Ω and every Lipschitz function h in V such that u = v on ∂V , it holds ‖∇G,0u‖L∞(V ) ≤ ‖∇G,0v‖L∞(V ). It has been established independently in [2] and [28] that absolutely minimizers are viscosity solutions of the infinite Laplace equation. We consider the parabolic and normalized counterpart ∂tu(t, p) = 1∑m1 i=1(Xiu)2 m1∑ i,j=1 XiuXjXiXju. (2.3) EJDE-2021/57 SINGULAR PROBLEMS IN CARNOT GROUPS 5 Hence, the singular operator is FPIL(η,X ) = − 1 ‖η‖2 〈Xη, η〉, for η ∈ Rm1 \ 〈0〉 and X ∈ Sm1(R). Hence, the existence of solutions is derived from our analysis. 3. Preliminaries 3.1. Carnot groups. Let G be a connected and simply connected Lie group, whose Lie algebra G is real and N-dimensional. We say that G is a Carnot group of step l ≥ 1 if G has a stratification, that is, there exist vector spaces V1, . . . , Vl such that G = V1 ⊕ · · · ⊕ Vl, [V1, Vi] = Vi+1, 1 ≤ i ≤ l − 1, [Vi, Vl] = 0, i = 1, . . . , l. Here, [V1, Vi] stands for the linear subspace generated by the vectors [X,Y ], where X ∈ V1 and Y ∈ Vi. In particular, G is nilpotent. Choose a Riemannian metric with respect to which the Vi are mutually orthogonal. Let mi = dimVi, for i = 1, . . . , l and consider hr = m1 + · · · + mr, 0 ≤ r ≤ l, with h0 = 0. Choose an orthonormal basis of Vi of left-invariant vector fields Xj , j = hi−1 + 1, . . . , hi. Thus, the dimension of G as a manifold is N = hl = m1 + . . .ml. The exponential map exp : G → G is a global diffeomorphism and may be used to define exponential coordinates ϕ in G as follows: any p ∈ G may be written uniquely as p = exp ( p1X1 + · · ·+ pNXN ) , and thus we may put ϕ(p) = (p1, . . . , pN ). In this way, we identify G with (RN , ·), where the group law · is given by the Campbell-Hausdorff formula [9] as ϕi(p · q) = ϕi(p) + ϕi(q) +Ri(p, q), i = 1, . . . , N, (3.1) where Ri depends only on ϕk for k < i. In what follows, we just write pi for ϕi(p). We sometimes use the notation p = (p1,1, . . . , pl,ml). The first layer V1 spanned by the vector fields X1, . . . , Xm1 plays an important role in the theory and it is called the horizontal distribution. Thus, for every p in G: V1,p = span { X1,p, . . . , Xm1,p } . Metric structure on G. If γ : I = [0, 1]→ G is an absolutely continuous curve in G that satisfies γ′(t) ∈ V1,γ(t), for a. e. t ∈ I, we call γ a horizontal path. We define the Carnot-Carathéodory distance on G by dC(p, q) = inf { ‖γ′(t)‖ : γ′(t) ∈ V1,γ(t), ∀t ∈ I, γ(0) = p, γ(1) = q } , where ‖ · ‖ is the norm induced by the Riemann structure on G. Since Carnot groups satisfy the Hörmander’s condition, we get by Chow’s Theorem, that d is well-defined. It is well-known that the topology induced by dC is equivalent to the Euclidean topology. However, dC is not bi-Lipschitz equivalent to the Euclidean distance (see [26]). 6 P. OCHOA, J. A. RUIZ EJDE-2021/57 Calculus on Carnot groups. Consider the Carnot group R×G where we add ∂/∂t to the horizontal frame as X0. For any 1 ≤ k ≤ l, we say that u : R × G → R belongs to Cksub if it is continuous and XIu is continuous for I = (i0, i1, . . . , iN ) so that d(I) = i0d0 + i1d1 + · · ·+ iNdN ≤ k, where d0 = 1 and dm = j if the corresponding vector field belongs to Vj , j = 1, . . . , l. The full (spacial) gradient with respect to the Carnot frame {X1, . . . , XN} will be given by ∇Gu(p) = N∑ i=1 (Xi,pu)Xi,p, p ∈ G. We shall also consider the horizontal and second order horizontal gradients of u, ∇G,0u(p) = h1∑ i=1 (Xi,pu)Xi,p, and ∇G,1u(p) = h2∑ i=h1+1 (Xi,pu)Xi,p. The symmetrized horizontal Hessian matrix, denoted by ∇2,∗ G,0u, has entries ∇2,∗ G,0uij = 1 2 ( XiXju+XjXiu ) , i, j = 1, . . . ,m1, The stratified Taylor expansion of a C2sub-function u at (t, p) ∈ R×G reads as (see [17, Theorem 1.42] or[9, Exercise 6, Chapter 20]): u(s, q) = u(t, p) + ut(t, p)(s− t) + 〈 ∇G,0u(t, p), (p−1 · q)1 〉 + 〈 ∇G,1u(t, p), (p−1 · q)2 〉 + 1 2 〈 ∇2,∗ G,0u(t, p)(p−1 · q)1, (p−1 · q)1 〉 + o(dC(p, q)2 + |s− t|). (3.2) Here, (p−1 · q)1 and (p−1 · q)2 denote the projection of p−1 · q onto V1 and V2, re- spectively. We remark that if instead of choosing a left-invariant frame, we consider a right-invariant basis of the Lie algebra, then we may also define right horizontal derivatives of first and second order (see [17]). Example 3.1. The simplest example of a Carnot group is the Euclidean space with the usual norm (RN , | · |). This is a Carnot group of step 1. Example 3.2. One of the most familiar Carnot groups is the Heisenberg group H, whose background manifold is R2n+1. Given two points p = (p1, . . . , p2n+1) and q = (q1, . . . , q2n+1) in H, we define a group operation by p · q = ( p1 + q1, . . . , p2n + q2n, p2n+1 + q2n+1 + 1 2 2n∑ i=1 (piqi+n − pi+nqi) ) The standard basis of left invariant vector fields of the Heisenberg Lie algebra, denoted by H, is given by Xj = ∂j − pn+j 2 ∂2n+1, j = 1, . . . , n; Yj = ∂j+n + pj 2 ∂2n+1, j = 1, . . . , n; T = ∂2n+1. (3.3) EJDE-2021/57 SINGULAR PROBLEMS IN CARNOT GROUPS 7 Note that [Xj , Yj ] = T for j = 1, . . . , n. Then the vector fieldsX1, . . . , Xn, Y1, . . . , Yn satisfy the Hörmander’s condition. In this case, the stratification of the Lie algebra is given by H = V1 ⊕ V2, where V1 = span{X1, . . . , Xn, Y1, . . . , Yn} and V2 = span{T}. Hence, H is a step- 2 group. For applications of the Heisenberg group to quantum mechanics, the interested reader may consult the monograph [10]. Example 3.3. The Engel group E4 is the Carnot group whose Lie algebra is G = V1 ⊕ V2 ⊕ V3, where V1 = span{X1, X2}, V2 = span{X3} and V3 = span{X4}, with the Carnot frame X1 = ∂1 − p2 2 ∂3 − (p3 2 + p2 12 (p1 + p2) ) ∂4 X2 = ∂2 + p1 2 ∂3 − (p3 2 − p1 12 (p1 + p2) ) ∂4 X3 = ∂3 + 1 2 (p1 + p2)∂4 X4 = ∂4, (3.4) p = (p1, . . . , p4). The Engel group is a Carnot group of step 3. For more details on the Engel group see [20]. 3.2. Jets and viscosity solutions in Carnot groups. We shall recall the defi- nition of parabolic jets in Carnot groups and the notion of viscosity solutions. Parabolic jets. Let u be an upper-semicontinuous function in [0, T ]×G. We define the parabolic superjet of u at the point (t, p) as P2,+u(t, p) = { (a, η, ξ,X ) ∈ R× Rm1 × Rm2 × Sm1(R) such that u(s, q) ≤ u(t, p) + a(s− t) + 〈η, (p−1 · q)1〉+ 〈ξ, (p−1 · q)2〉 + 1 2 〈X (p−1 · q)1, (p−1 · q)1〉+ o ( dc(p, q) 2 + |s− t| ) as (s, q)→ (t, p) } . Similarly, if v is lower semicontinuous in [0, T ]×G, we define the parabolic subjet P2,−v(t, p). It is known that parabolic jets may be seen as appropriate derivatives of test functions touching the given function by above or below. More precisely, if u is upper semicontinuous, we consider K+,2u(t, p) = {( φt(t, p),∇G,0φ(t, p),∇G,1φ(t, p),∇2,∗ G,0φ(t, p) ) so that φ is C2sub and (u− φ)(s, q) ≤ (u− φ)(t, p) for all (s, q) close to (t, p) } and similarly define K−,2v(t, p) for test function touching the lower semicontinuous function v from below. Hence it follows that P2,+u(t, p) = K2,+u(t, p), P2,−v(t, p) = K2,−v(t, p). (3.5) Finally, we shall also consider the theoretic closure of the sets defined above. We define P2,+ u(t, p) as the set of (a, η, ξ,X ) in R × Rm1 × Rm2 × Sm1(R) so that 8 P. OCHOA, J. A. RUIZ EJDE-2021/57 there exists a sequence (tn, pn, an, ηn, ξn,Xn) converging to (t, p, a, η,X ) satisfying (an, ηn,Xn) ∈ P2,+u(tn, pn) for all n. In a similar way, we define P2,− v(t, p). Viscosity solutions. Let Ω ⊂ G be a domain. Observe that in the following defini- tion, the operator F does not depend on ξ ∈ Rm2 . Definition 3.4. An upper semicontinuous function u : [0, T ]× Ω→ R is a subso- lution to the equation ut + F(t, p, u,∇G,0u,∇2,∗ G,0u) = 0 (3.6) in (0, T )×Ω if for every (t, p) ∈ (0, T )×Ω and every subjet (a, η, ξ,X ) ∈ P2,+ u(t, p) it holds a+ F∗(t, p, u(t, p), η,X ) ≤ 0. (3.7) Here, the subscript ∗ stands for the lower semicontinuous envelope of F . Similarly, we say that a lower semicontinuous function v : [0, T ] × Ω → R is a supersolution to (3.6) if for every (t, p) ∈ (0, T ) × Ω and every superjet (a, η, ξ,Y) ∈ P2,− v(t, p) it holds a+ F∗(t, p, u(t, p), η,Y) ≥ 0. Analogously, the superscript ∗ stands for the upper semicontinuous envelope of F . Finally, we say that a continuous function is a viscosity solution to (3.6) if it is a viscosity sub and supersolution. 4. Deterministic games: proof of Theorem 2.1 In this section we employ two-person deterministic games to prove the existence of solutions to ut + µu+ F(t, p,∇G,0u,∇2,∗ G,0u) = 0, on (0, T )×G, u(0, p) = ψ(p), with p ∈ G. under the assumptions listed in Theorem 2.1. Remark 4.1. From (A3) and (A4), we conclude that F has at most linear growth (and at least linear decay). In fact, there exists C = C(λ0, λ1) such that |F(t, p, η,X )| ≤ C(1 + ‖X‖), for (t, p, η,X ) ∈ [0, T ] × G × (Rm1\{0}) × Sm1(R). In addition, from (A4), F is (degenerate) elliptic, since F(·, X̂ ) ≤ F(·,X ) for X̂ ≥ X . We first describe the setting of the game. There are two players, Player I and Player II. Let T > 0 be the maturity of the game. For each ε ∈ (0, 1) let m be the number of steps, m := [ T ε2 ] , where [·] is the integer part function. The players’ choices are the following: • the initial position is p0 = p at t0 = 0; • Player I chooses a pair (η0,X0) ∈ (Rm1\{0})× Sm1(R), with ‖η0‖ ≤ ε−1/4, ‖X0‖ ≤ ε−1/2; EJDE-2021/57 SINGULAR PROBLEMS IN CARNOT GROUPS 9 • for these choices of Player I, Player II chooses a horizontal direction: q0 = (ν0, 0) ∈ G, with ν0 ∈ Rm1 and |q0|G ≤ ε−1/4, where the Carnot gauge is defined as: |p|G := l∑ j=1 mj∑ i=1 |pi,j |1/j ≈ dC(p, 0). (4.1) • Player I moves from p0 to p1 := p0 · δε(q0), where the dilatation δε is given by [δε(p)]i,j = εipi,j ; • the above steps are repeated m times; • at the maturity time T , Player I is at the final position pm and pays to Player II the amount:( 1 1 + µε2 )m ψ(pm) + m−1∑ i=0 ( 1 1 + µε2 )i+1 Rε(T − iε2, pi, qi, ηi,Xi), (4.2) where Rε is the running cost defined in (4.5), and pi, qi, ηi and Xi are the choices of the players at the i-th step. The value uε of the game is obtained by considering that Player I has the objec- tive of minimize (4.2) and Player II wants to maximize it. We extend now the set up of the game to any maturity t and we formalize the definition of uε. Take t ∈ [0, T ] and consider the partition [0, T ] = {0} ∪ ( ∪mk=1 ((k − 1)ε2, kε2] ) . (4.3) Then if t 6= 0, there is a unique kt ∈ {1, 2, . . . ,m} such that t ∈ ((kt − 1)ε2, ktε 2]. If t = 0, we define uε(t, p) := ψ(p). When t 6= 0, we have uε(t, p) := inf η0,X0 sup q0 . . . inf ηkt−1,Xkt−1 sup qkt−1 {( 1 1 + µε2 )kt ψ(pkt) + kt−1∑ i=0 ( 1 1 + µε2 )i+1 Rε(t− iε2, pi, qi, ηi,Xi) } , (4.4) where Rε(t, pj , qj , ηj ,Xj) := −ε〈ηj , νj〉 − ε2 2 〈Xjνj , νj〉 − ε2F (t, pj , ηj ,Xj) . (4.5) We introduce a rigorous definition for the values of the game based on the Dy- namic Programming Principle. Definition 4.2. For ψ ∈ BC(G), we define inductively: uε(0, p) = ψ(p) and for t ∈ ((kt − 1)ε2, ktε 2]: uε(t, p) = 1 1 + µε2 inf η,X sup q { uε(t− ε2, p · δε(q)) +Rε(t, p, q, η,X ) } . When kt = 1, uε(t− ε2, p · δε(q)) is understood as uε(0, p). 10 P. OCHOA, J. A. RUIZ EJDE-2021/57 We start with two technical lemmas regarding some Lipschitz regularity of the value functions for smooth initial datum ψ. Lemma 4.3. Let ψ be smooth and such that the right and left horizontal derivatives of first and second order are bounded in G. Then, there is a constant C = C[ψ] > 0 such that for all p, p̂ ∈ G and t ∈ [0, T ]: |uε(t, p)− uε(t, p̂)| ≤ C|p · p̂−1|G + ktε 2ω ( |p · p̂−1|G ) , where ω is the modulus of continuity from (F3). Proof. We proceed by induction. When t = 0 (kt = 0), we have by the boundedness of the horizontal derivatives and the stratified mean value theorem [17, Theorem 1.41] (modified for right-invariant vector fields, see Remark after [17, Theorem 1.37]) that there is C = C[ψ] > 0 so that |uε(0, p)− uε(0, p̂)| = |ψ(p)− ψ(p̂)| ≤ C|p · p̂−1|G. Assuming for t ∈ ((kt − 1)ε2 and ktε 2], kt ∈ {1, . . . ,m− 1} that |uε(t, p)− uε(t, p̂)| ≤ C|p · p̂−1|G + ktε 2ω ( |p · p̂−1|G ) , for t̃ = t+ ε2 we have uε(t̃− ε2, p · δε(q))− uε(t̃− ε2, p̂ · δε(q)) +Rε(t̃, p, q, η,X )−Rε(t̃, p̂, q, η,X ) ≤ C|p · p̂−1|G + ktε 2ω ( |p · p̂−1|G ) + ε2ω ( |p · p̂−1|G ) ≤ C|p · p̂−1|G + kt̃ε 2ω ( |p · p̂−1|G ) , where kt̃ := kt + 1. Thus uε(t̃− ε2, p · δε(q)) +Rε(t̃, p, q, η,X ) ≤ C|p · p̂−1|G + kt̃ε 2ω ( |p · p̂−1|G ) + uε(t̃− ε2, p̂ · δε(q)) +Rε(t̃, p̂, q, η,X ). Taking supq and then infη,X we derive |uε(t̃, p)− uε(t̃, p̂)| ≤ C|p · p̂−1|G + kt̃ε 2ω ( |p · p̂−1|G ) . � Lemma 4.4. Let ψ be smooth and such that the right and left horizontal derivatives of first and second order are bounded in G. Then, there is C = C(ψ, λ0, λ1) > 0 such that for all p ∈ Ω and t, |uε(t, p)− uε(t− ε2, p)| ≤ C ( 1 1 + µε2 )kt ε2 + (kt − 1)ε2ω(ε2), where ω is the modulus of continuity from (A4). Proof. We proceed again by induction. Since ψ is smooth, and has bounded deriva- tives, we derive from the stratified Taylor formula (3.2) and the fact that δε is horizontal that there is C[ψ] > 0 so that∣∣ψ(p · δε(q))− ψ(p)− 〈∇G,0ψ(p), [δε(q)]1〉 − 1 2 〈∇2,∗ G,0ψ(p)[δε(q)]1, [δε(q)]1〉 ∣∣ ≤ C[ψ]ε3/2 · sup |z|G≤bε3/4,d(I)=2 |XIψ(p · z)−XIψ(p)| for some universal b > 0 ≤ C[ψ]ε2, (4.6) EJDE-2021/57 SINGULAR PROBLEMS IN CARNOT GROUPS 11 where we have used the stratified mean value theorem [17, Theorem 1.41] for the second order derivatives. Hence for t ∈ (0, ε2]: uε(t, p)− ψ(p) ≤ ( 1 1 + µε2 ) inf η,X sup q { ε〈(∇G,0ψ(p)− η), ν〉 + ε2 2 〈 ( ∇2,∗ G,0ψ(p)−X ) ν, ν〉 − ε2F(t, p, η,X ) + C[ψ]ε2 } + µε2 1 + µε2 ψ(p) ≤ ( 1 1 + µε2 ) inf η,X { ε3/4‖∇G,0ψ(p)− η‖ + ε3/2 2 E+ (C0[ψ]I −X ) + Cε2(1 + ‖X‖) + C[ψ]ε2 } + C0[ψ]ε2, where C0[ψ] := max [ ‖ψ‖∞, ‖∇G,0ψ‖∞, ‖∇2,∗ G,0ψ‖∞ ] (4.7) and I ∈ SN denotes the identity matrix. Let ε > 0 be small enough so that C0[ψ] ≤ ε−1/4, then we can choose (η,X ) = (∇G,0ψ(p), C0[ψ]I) (in the case ∇G,0ψ(p) = 0, take an approximating sequence 0 6= ηn → 0) to obtain uε(t, p)− ψ(p) ≤ [C(1 + C0[ψ]) + C[ψ]] ( 1 1 + µε2 ) ε2. (4.8) Next, we show the lower bound. Similarly to the above arguments, for any q with |q|G ≤ ε−1/4 we have (1 + µε2)(uε(t, p)− ψ(p)) ≥ inf η,X { ε〈(∇G,0ψ(p)− η), ν〉+ ε2 2 〈(−C0[ψ]I −X )ν, ν〉 − ε2F(t, p, η,X )− C[ψ]ε2 } − C0[ψ]ε2. Now, we apply Lemma 5.1 from the Appendix with η̂ = ∇G,0ψ(p), X̂ = −C0[ψ]I and R0 := C0[ψ] and K = 1. Then. choosing an appropriate q = q(ε, η, η̂,X , X̂ ), we have in the case ‖∇G,0ψ(p)‖ ≥ 1, that (1 + µε2)(uε(t, p)− ψ(p)) ≥ −ε2F∗(t, p,∇G,0ψ(p),−C0[ψ]I)− h1(ε1/4)ε2 − C[ψ]ε2 − C0[ψ]ε2, and if ‖∇G,0ψ(p)‖ ≤ 1, (1 + µε2)(uε(t, p)− ψ(p)) ≥ −ε2F∗(t, p, 0,−C0[ψ]I)− C[ψ]ε2 − C0[ψ]ε2, for all sufficiently small ε ≤ min[ε1, ε2]. As in (4.8), we have uε(t, p)− ψ(p) ≥ − [ C(1 + C0[ψ]) + h1(ε1/4) + C[ψ] ]( 1 1 + µε2 ) ε2. (4.9) Combining (4.8) and (4.9), we obtain |uε(t, p)− ψ(p)| ≤ C ( 1 1 + µε2 ) ε2. (4.10) Suppose now that |uε(t− ε2, p)− uε(t, p)| ≤ C ( 1 1 + µε2 )kt ε2 + (kt − 1)ε2ω(ε2). 12 P. OCHOA, J. A. RUIZ EJDE-2021/57 Now, taking t̃ = t+ ε, uε(t, p · δε(q))− uε(t− ε2, p · δε(q)) +Rε(t̃, p, η,X )−Rε(t̃− ε2, p, η,X ) ≤ C ( 1 1 + µε2 )kt ε2 + (kt − 1)ε2ω(ε2) + ε2ω(ε2). where we have used (A4) in the latter inequality. Taking supq and infη,X , we derive uε(t̃− ε2, p)− uε(t̃, p) ≤ C ( 1 1 + µε2 )kt+1 ε2 + ktε 2ω(ε2) = C ( 1 1 + µε2 )kt̃ ε2 + (kt̃ − 1)ε2ω(ε2). A similar argument is applied to uε(t̃, p)− uε(t̃− ε2, p). � In the next results we shall appeal to the following constant. For a positive integer K, let us set Cε[ψ,K] := C(1 + C0[ψ]) + hK(ε1/4) + C[ψ], where C is the constant from Remark 4.1, C0 is given by (4.7), C[ψ] by (4.6) and hK is the modulus in Lemma 5.1 from the Appendix. The next proposition establishes the convergence of the value functions. Proposition 4.5. There exist a subsequence {εj}j converging to 0 and a con- tinuous function u so that uεj → u locally uniformly as j → ∞. Moreover, u ∈ BUC([0, T ]×G) and u(0, p) = ψ(p) for all p. Proof. For δ > 0 consider the regularizations ψ±δ ∈ C2 of ψ from assumption (A1), ψ − δ ≤ ψ−δ ≤ ψ ≤ ψ + δ ≤ ψ + δ. (4.11) Lemma 4.3 gives the next estimate for t ∈ (0, ε2], |uε ψ±δ (t, p)− uε ψ±δ (t, p̂)| ≤ Cδ|p · p̂−1|G + ε2ω(|p · p̂−1|G), (4.12) for all p, p̂ and all sufficiently small ε, and where uε ψ±δ denotes the value function with ψ = ψ±δ . Combining (4.12) and (4.11), we conclude that |uε(t, p)− uε(t, p̂)| ≤ Cδ|p · p̂−1|G + ε2ω(|p · p̂−1|G) + δ, for p, p̂ ∈ Ω and all ε ≤ ε′. Here ε′ = ε′(ψ±δ , λ0, λ1) is sufficiently small. Inductively, we derive for t ∈ ((kt − 1)ε2, ktε 2], |uε(t, p)− uε(t, p̂)| ≤ Cδ|p · p̂−1|G + ktε 2ω(|p · p̂−1|G) + δ. (4.13) Now, by Lemma 4.4, the estimate |uε ψ±δ (t, p)− uε ψ±δ (t− ε2, p)| ≤ Cε[ψ±δ ] ( 1 1 + µε2 )kt ε2 + (kt − 1)ε2ω(ε2), holds for each t and all sufficiently small ε where Cε[ψ±δ ] := max[Cε[ψ+ δ , 1], Cε[ψ−δ , 1]]. If 0 ≤ i ≤ j ≤ m, t ∈ ((i− 1)ε2, iε2], and s ∈ ((j − 1)ε2, jε2], then uε ψ±δ (t− ε2, p · δε(q))− uεψ±δ (s− ε2, p · δε(q)) +Rε(t, p, q, η,X )−Rε(s, p, q, η,X ) ≤ uε ψ±δ (t− ε2, p · δε(q))± uεψ±δ (t− ε2 + ε2, p · δε(q))± . . . ± uε ψ±δ ( t− ε2 + (j − i)ε2, p · δε(q) ) − uε ψ±δ (s− ε2, p · δε(q)) + ε2ω(|s− t|) EJDE-2021/57 SINGULAR PROBLEMS IN CARNOT GROUPS 13 ≤ Cε[ψ±δ ](j − i)ε2 + j(j − i)ε2w(ε2) + ε2ω(|s− t|). Hence, taking supq and infη,X , for t ∈ ((i− 1)ε2, iε2], s ∈ ((j − 1)ε2, jε2] we derive that |uε ψ±δ (t, p)− uε ψ±δ (s, p)| ≤ Cε[ψ±δ ](j − i)ε2 + j(j − i)ε2w(ε2) + ε2ω(|s− t|). Therefore, |uε(t, p)− uε(s, p)| ≤ Cε[ψ±δ ](j − i)ε2 +m(j − i)ε2ω(ε2) + ε2ω(|s− t|) + δ. (4.14) Now we prove the proposition. For any t, s ∈ [0, T ] with t ≤ s, there exist i, j such that 0 ≤ i ≤ j ≤ m and jε2 ≤ s < (j + 1)ε2, iε2 ≤ t < (i+ 1)ε2. From (4.13) and (4.14), it follows that |uε(t, p)− uε(s, p̂)| ≤ Cε[ψ±δ ](j − i)ε2 + (j − i)Tω(ε2) + ε2ω(|s− t|) + Cδ|p · p̂−1|G + Tω(|p · p̂−1|G) + 2δ. Set C0[ψ±δ ] := max[C0[ψ+ δ ], C0[ψ−δ ]]. Since iε2 > t− ε2 and jε2 ≤ s, we have |uε(t, p)− uε(s, p̂)| ≤ Cε[ψ±δ ](s− t) +m(s− t)ω(ε2) + ε2ω(|s− t|) + Cδ|p · p̂−1|G + Tω(|p · p̂−1|G) + 2δ. Interchanging s and t, we conclude that |uε(t, p)− uε(s, p̂)| ≤ Cε[ψ±δ ]|s− t|+m|s− t|ω(ε2) + ε2ω(|s− t|) + Cδ|p · p̂−1|G + Tω(|p · p̂−1|G) + 2δ. (4.15) for all s, t ∈ [0, T ]. Using (A4) on ω there is a constant C such that mω(ε2) ≤ C. Also, observe that C[ψ±δ ] := lim ε→0 Cε[ψ±δ ] = C(1 + C0[ψ±δ ]). Next, we take η ∈ (0, 1) and fix δ < η/6. Moreover, we consider ε0 > 0 so that ε0 < η/4 and for all ε < ε0, Cε[ψ±δ ] ≤ C(1 + C0[ψ±δ ]) + δ. Finally, we take r0 > 0 so that |p̂−1 · p|G + |s− t| < r0. Then [ η 4 + C(1 + C0[ψ±δ ])]r0 + ε20ω(r0) + Cδr0 + Tω(r0) < η 2 . Therefore, |p̂−1 · p|G + |s− t| < r0 and ε < ε0 imply |uε(t, p)− uε(s, p̂)| < η. Moreover, the functions uε are uniformly bounded. Indeed, for t ∈ (0, ε2], we have the upper bound (1 + µε2)uε(t, p) ≤ ‖ψ‖∞ + inf η,X sup q Rε(t, p, q, η,X ) ≤ ‖ψ‖∞ + inf η sup q Rε(t, p, q, η,O) = ‖ψ‖∞ + inf η sup q ( −ε〈η, ν〉 − ε2F∗(t, p, η,O) ) . 14 P. OCHOA, J. A. RUIZ EJDE-2021/57 Taking a sequence ηk ↘ 0 with ‖ηk‖ ≤ ε−1/4 and using the lower semicontinuity of F∗ together with (F5), we obtain uε(t, p) ≤ ‖ψ‖∞ for all ε > 0. Now, taking η̂ = 0, X̂ = O and R0 = 1 in Lemma 5.1, there is q (depending on ε, η and X ) so that (1 + µε2)uε(t, p) ≥ −‖ψ‖∞ + inf η,X sup q Rε(t, p, q, η,X ) ≥ −‖ψ‖∞ + inf η,X R∗,ε(t, p, q, 0,O) = −‖ψ‖∞. Therefore, |uε(t, p)| ≤ ‖ψ‖∞. By induction we deduce that for all (t, p), |uε(t, p)| ≤ ‖ψ‖∞. In this way, we may apply [23, Lemma 4.2] to obtain the convergence (up to a subsequence) of uε to some continuous u, locally uniformly in [0, T ]×G. We now prove the final statement. Taking εj → 0 in (4.15), we obtain |u(t, p)− u(s, p̂)| ≤ C(1 + C0[ψ±δ ])|s− t|+ Cδ|p · p̂−1|G + Tω(|p · p̂−1|G) + 2δ. Hence u ∈ BUC([0, T ] × G). Applying (4.15) to s = 0 and p = p̂, and taking the limit εj → 0 we obtain |u(t, p)− ψ(p)| ≤ C(1 + C0[ψ±δ ])|t|+ 2δ. Letting t→ 0 and then δ → 0 it follows that u(0, p) = ψ(p). � Proposition 4.6. The function u is a viscosity subsolution of (2.1). Proof. We argue by contradiction. Then there exist a positive constant θ0 and a smooth function ϕ, such that the following holds in a neighbourhood B0 := [t0− δ, t0 + δ]×BG(p0, r0) of a strict local maximal point P0 = (t0, p0) ∈ (0, T )×G of u− ϕ, ∂tϕ+ µu+ F∗(t, p,∇G,0ϕ,∇2,∗ G,0ϕ) ≥ θ0 > 0, (4.16) where δ and r0 are sufficiently small, with 4δ < T. (4.17) We also assume that maxB0 (u− ϕ) = 0. Let P = (t, p) ∈ B0. Reasoning as in (4.6), we have uε(P ) = ( 1 1 + µε2 ) inf η,X sup q { (uε − ϕ)(t− ε2, p · δε(q)) + ϕ(P )− ε2∂tϕ(P ) + ε〈∇G,0ϕ(P )− η, ν〉+ ε2 2 〈( ∇2,∗ G,0ϕ(P )−X ) ν, ν 〉 − ε2F ( t, p, η,X ) + o(ε2) } . In the sequel, ε is small enough so that ‖∇G,0ϕ‖∞,B0 ≤ ε−1/4, ‖∇2,∗ G,0ϕ‖∞,B0 ≤ ε−1/2 and o(ε2)− ε2θ0 ≤ 0. Using −ϕ ≤ −u in B0 we obtain (uε − ϕ)(P ) ≤ ( 1 1 + µε2 ) inf η,X sup q { (uε − ϕ)(t− ε2, p · δε(q)) EJDE-2021/57 SINGULAR PROBLEMS IN CARNOT GROUPS 15 − ε2[∂tϕ(P ) + µu(P ) + F∗ ( t, p, η,X ) ] + ε〈∇G,0ϕ(P )− η, ν〉+ ε2 2 〈( ∇2,∗ G,0ϕ(P )−X ) ν, ν 〉 + o(ε2) } . Taking the special choices η = ∇G,0ϕ(P ), X = ∇2,∗ G,0ϕ(P ) (if ∇G,0ϕ(P ) = 0, we choose an approximating sequence 0 6= ηk → 0) and appealing to (4.16) we deduce that (uε − ϕ)(P ) ≤ ( 1 1 + µε2 ) sup q { (uε − ϕ)(t− ε2, p · δε(q)) + o(ε2)− ε2θ0 } ≤ ( 1 1 + µε2 ) sup q { (uε − ϕ)(t− ε2, p · δε(q)) } ≤ ( 1 1 + µε2 ) sup q { ((uε)∗ − ϕ)(t− ε2, p · δε(q)) } . Taking a sequence of points Pn = (tn, pn) ∈ B0 converging to P so that (uε)∗(P ) = lim n→∞ uε(Pn) we obtain ((uε)∗ − ϕ)(P ) = lim n→∞ (uε − ϕ)(Pn) ≤ ( 1 1 + µε2 ) lim n→∞ sup q { ((uε)∗ − ϕ)(tn − ε2, pn · δε(q)) } = ( 1 1 + µε2 ) lim n→∞ { ((uε)∗ − ϕ)(tn − ε2, pn · δε(qn)) } ≤ ( 1 1 + µε2 ) ((uε)∗ − ϕ)(t− ε2, p · δε(qε0)) (4.18) for some |qn|G ≤ ε−1/4, where qn → qε0 (up to a subsequence that we do not re label) and where we have used the upper semicontinuity of (uε)∗ − ϕ. Define P ε0 = P0, and, for k ≥ 1, P εk = (tεk, p ε k) as follows: P εk = (tεk−1 − ε2, pεk−1 · δε(qε0(P εk−1))), 1 ≤ k ≤ m. If P ε1 , P ε 2 , . . . , P ε k ∈ B0, from (4.18), we obtain ((uε)∗ − ϕ)(P εk−1) ≤ ( 1 1 + µε2 ) [(uε)∗ − ϕ] (P εk ), and so ((uε)∗ − ϕ)(P ε0 ) ≤ ( 1 1 + µε2 )k [(uε)∗ − ϕ](P εk ). (4.19) Taking n = nε so that nε2 ∈ (δ, 4δ) it follows tεn /∈ [t0 − δ, t0 + δ] (i.e., P εn /∈ B0). In addition, n ≤ m, by the choice (4.17). Hence, there exists a minimal number Kε ≤ m such that P εKε ∈ B0 but P εKε+1 /∈ B0. By compactness, P εKε → P ′ = (t′, p′) ∈ B0\{P0} as ε→ 0 (or equivalently m→∞). Note that 0 < e−µT = e−µmε 2 ≤ ( 1 + µε2 )−m ≤ (1 + µε2 )−Kε ≤ 1. Thus, lim ε→0 (m→∞) ( 1 1 + µε2 )Kε := α ∈ (0, 1]. 16 P. OCHOA, J. A. RUIZ EJDE-2021/57 Consequently, 0 = (u− ϕ)(P0) = lim ε→0 (uε − ϕ)(P ε0 ) ≤ lim ε→0 ( 1 1 + µε2 )Kε ((uε)∗ − ϕ)(P εKε) ≤ α(u− ϕ)(P ′), with P ′ ∈ B0\{P0}. We have a contradiction, since P0 is a strict maximum in B0. � Proposition 4.7. The function u is a viscosity supersolution of (2.1). Proof. Reasoning by contradiction again, there exist θ0 > 0 and a smooth function ϕ, such that the following holds in a neighbourhood B0 of a strict local minimal point P0 = (t0, p0) ∈ (0, T )× Ω of u− ϕ, ∂tϕ+ µu+ F∗(t, p,∇G,0ϕ,∇2,∗ G,0ϕ) ≤ −θ0 < 0. (4.20) We assume that the value of u− ϕ at P0 is 0. Let R0 > 0 be such that ‖∇G,0ϕ‖∞,B0 , ‖∇2,∗ G,0ϕ‖∞,B0 ≤ R0. Suppose first that ∇G,0ϕ(P0) 6= 0. We may assume that there is γ0 > 0 such that ‖∇G,0ϕ‖∞,B0 ≥ γ0 > 0 Hence there exists j0 ∈ N such that γ0 > 1/j0. By Lemma 5.1 and performing a Taylor expansion as in (4.6), we have for P = (t, p) ∈ B0, uε(P ) ≥ 1 1 + µε2 inf η,X { (uε − ϕ)((t− ε2, p · δε(q)) + ϕ(P )− ε2∂tϕ(P ) − ε2F∗ ( t, p,∇G,0ϕ(P ),∇2,∗ G,0ϕ(P ) ) − ε2hj0(ε1/4) + o(ε2) } , for some q ∈ G, with |q|G ≤ ε−1/4. So we obtain (uε − ϕ)(P ) ≥ 1 1 + µε2 inf η,X { (uε − ϕ)(t− ε2, p · δε(q))− ε2 [ ∂tϕ(P ) + µu(P ) + F∗ ( t, p,∇G,0ϕ(P ),∇2,∗ G,0ϕ(P ) ) ] − ε2hj0(ε1/4) + o(ε2) } ≥ 1 1 + µε2 inf η,X {(uε − ϕ)(t− ε2, p · δε(q)) + ε2[θ0 + o(1)− hj0(ε1/4)]} ≥ 1 1 + µε2 inf η,X {(uε − ϕ)(t− ε2, p · δε(q))} ≥ 1 1 + µε2 inf η,X {((uε)∗ − ϕ)(t− ε2, p · δε(q))}, (4.21) where we used −ϕ ≥ −u in B0 and (4.20). By a similar argument as in (4.18), we derive from (4.21) that ((uε)∗ − ϕ)(P ) ≥ 1 1 + µε2 inf η,X {((uε)∗ − ϕ)(t− ε2, p · δε(q))}. (4.22) EJDE-2021/57 SINGULAR PROBLEMS IN CARNOT GROUPS 17 Since for each admissible η and X we have |q|G ≤ ε−1/4, and the function ((uε)∗− ϕ)(t− ε2, p · δε(·)) is lower semicontinuous, the infimum: inf |q|G≤ε−1/4 { ((uε)∗ − ϕ)(t− ε2, p · δε(q)) } is finite. Hence, so is: inf η,X { ((uε)∗ − ϕ)(t− ε2, p · δε(q)) } . Take a sequence (ηn,Xn) so that if qn = qn(ε, ϕ, P, ηn,Xn), then lim n→∞ ((uε)∗ − ϕ)(t− ε2, p · δε(qn)) = inf η,X { ((uε)∗ − ϕ)(t− ε2, p · δε(q)) } . By compactness, there is a point qε0 = qε0(P ), |qε0|G ≤ ε−1/4, so that qε0 = lim n→∞ qn. The lower semicontinuity of ((uε)∗ − ϕ)(t− ε2, p · δε(·)) yields lim n→∞ (uε)∗ − ϕ)(t− ε2, p · δε(qn)) ≥ ((uε)∗ − ϕ)(t− ε2, p · δε(qε0)). Thus, by (4.22), we derive that ((uε)∗ − ϕ)(P ) ≥ ( 1 1 + µε2 ) ((uε)∗ − ϕ)(t− ε2, p · δε(qε0)). (4.23) Next, we consider the case ∇G,0ϕ(P0) = 0. Let F : B0 → R be so that F(·) := ∂tϕ(·) + µu(·) + F∗(·, 0,∇2,∗ G,0ϕ(·)). We can assume F(P ) ≤ −θ0 and that ‖∇G,0ϕ(P )‖ ≤ 1/j for any P ∈ B0 and some positive integer j. Applying Lemma 5.1, for any P ∈ B0 and any η,X admissible, there exists qε0, with |qε0|G ≤ ε−1/4, such that (uε − ϕ)(P ) ≥ 1 1 + µε2 inf η,X { (uε − ϕ)(t− ε2, p · δε(qε0))− ε2 [ ∂tϕ(P ) + µu(P ) + F∗ ( t, p, 0,∇2,∗ G,0ϕ(P ) ) ] + o(ε2) } ≥ 1 1 + µε2 inf |q|G≤ε−1/4 { ((uε)∗ − ϕ)(t− ε2, p · δε(q))− ε2F(P ) + o(ε2) } ≥ 1 1 + µε2 inf |q|G≤ε−1/4 { ((uε)∗ − ϕ)(t− ε2, p · δε(q)) } Thus, there is qε0 = qε0(P ) where the latter infimum is attained. Hence (4.23) holds. We may proceed as in the end of Proposition 4.6 (right after (4.18)) to get a contradiction with the fact that P0 is a strict minimum. � 5. Appendix: A technical lemma for existence We provide the proof of the next lemma which is [19, Lemma 4.6]. We give full details to show that q̄ may be taken horizontal. Lemma 5.1. Let (η̂, X̂ ) ∈ Rm1 × Sm1 (R) and let R0 so that ‖η̂‖, ‖X̂ ‖ ≤ R0. 18 P. OCHOA, J. A. RUIZ EJDE-2021/57 (1) If ‖η̂‖ ≥ K−1 (K ∈ N), then there exists ε1 = ε1(K,R0, λ0, λ1) such that for all (η,X ) ∈ (Rm1\{0})×Sm1 (R), with ‖η‖ ≤ ε−1/4, ‖X‖ ≤ ε−1/2, there exists q = q(ε, η, η̂,X , X̂ ), q = (ν, 0), with |q|G ≤ ε−1/4 such that for all ε ≤ ε1 and all (t, p), Rε(t, p, q, η,X ) ≥ R∗,ε(t, p, q, η̂, X̂ )− ε2hK(ε1/4), (5.1) where R∗,ε is defined as Rε changing F by F∗ and hK(r) := ω(1/2)K,R0 (r), with r ≥ 0. (2) If ‖η̂‖ ≤ K−1 (K ∈ N), then there exists ε2 = ε2(K,R0, λ0, λ1) such that for all (η,X ) ∈ (Rm1\{0})×Sm1 (R), with ‖η‖ ≤ ε−1/4, ‖X‖ ≤ ε−1/2, there exists q = q(ε, η, η̂,X , X̂ ), q = (ν, 0), with |q|G ≤ ε−1/4 such that for each ε ≤ ε2 and all (t, p), Rε(t, p, q, η,X ) ≥ R∗,ε(t, p, q, 0, X̂ ). (5.2) Proof. Assume that η̂ 6= η and X̂ 6= X . Using orthonormal eigenvectors ξ0, ξ1, . . ., ξm1−1 ∈ Rm1 of X − X̂ , we can represent ν with ‖ν‖ ≤ ε−1/4 by ν = m1−1∑ i=0 siξi, where si ∈ R (i = 0, 1, . . . ,m1 − 1) with s20 + . . . + s2m1−1 ≤ ε−1/4. In particular, let ξ0 be the unit eigenvector which gives the maximum eigenvalue of X̂ −X . Thus ε−2[Rε(t, p, q, η,X )−R∗,ε(t, p, q, η̂, X̂ )] is bounded from below by ε−1s0〈η̂ − η, ξ0〉+ ε−1 m1−1∑ i=1 si〈η̂ − η, ξi〉+ 1 2 s20E ( X̂ − X ) + 1 2 m1−1∑ i=1 s2i 〈 ( X̂ − X ) ξi, ξi〉+ [ F ( t, p, η̂, X̂ ) −F ( t, p, η,X )] . (5.3) Case 1: Assume ‖η̂‖ ≥ K−1 for some K ∈ N. (a) If ‖η̂ − η‖ ≤ ε1/4, then ‖η‖ ≥ 1/2K for all sufficiently small ε. In the case E ( X̂ − X ) > 0, we take |s0| = λ1, si = 0 for i = 1, . . . ,m1− 1 in (5.3). Then (5.3) is rewritten as ε−1λ1|〈η̂ − η, ξ0〉|+ λ21 2 E+ ( X̂ − X ) + [ F ( t, p, η̂, X̂ ) −F ( t, p, η̂,X )] + [ F ( t, p, η̂,X ) −F ( t, p, η,X )] , (5.4) where we choose an appropriate sign of s0 so that s0〈η̂ − η, ξ0〉 is non-negative. From (A4), for any η ∈ Rm1\{0}, λ21 2 E+ ( X − X̂ ) + [ F ( t, p, η̂, X̂ ) −F ( t, p, η̂,X )] ≥ 0. (5.5) From (A5) and ‖η‖ ≥ 1/2K it follows F(t, p, η̂,X )−F(t, p, η,X ) ≥ −ω1/2K,R0 (ε1/4) (5.6) where ω1/2K,R0 is a modulus of continuity depending only on K and R0. Thus, from (5.5), (5.6) and (5.4) we obtain ε−2[Rε(t, p, q, η,X )−R∗,ε(t, p, q, η̂, X̂ )] ≥ −hK(ε1/4), (5.7) EJDE-2021/57 SINGULAR PROBLEMS IN CARNOT GROUPS 19 where hK(s) = ω1/2K,R0 (s). Now if E(X̂ −X ) ≤ 0, we take si = 0 for i = 0, 1, . . . ,m1−1 in the formula (5.3). Then F(t, p, η̂, X̂ ) ≥ F(t, p, η̂,X ) for any η ∈ Rm1\{0} holds, since F is degenerate elliptic (see Remark 4.1). From (5.6), the inequality (5.7) is derived (with the same modulus hK). (b) If ‖η̂ − η‖ ≥ ε1/4, then η̂ − η ‖η̂ − η‖ = m1−1∑ i=0 riξi, (5.8) where ri ∈ R with r20 + r21 + . . .+ r2m1−1 = 1. Let us divide this case into two parts. Suppose first that |〈η̂ − η, ξ0〉| ≥ ε1/2 λ1 . Then if E(X̂ − X ) > 0, we choose si so that |s0| = λ1, si = 0 (i = 1, . . . ,m1 − 1) and obtain a formula similar to (5.4): ε−1λ1|〈η̂ − η, ξ0〉|+ λ21 2 E+ ( X̂ − X ) + [ F ( t, p, η, X̂ ) −F ( t, p, η,X )] + [ F ( t, p, η̂, X̂ ) −F ( t, p, η, X̂ )] . Similarly, by (A4) and Remark 4.1, we have for all ε ≤ ε(R0, λ0, λ1), ε−2[Rε(t, p, q, η,X )−R∗,ε(t, p, q, η̂, X̂ )] ≥ ε−1/2 − 2C(1 + ‖X̂ ‖) ≥ ε−1/4 − 2C(1 +R0) ≥ 0. If E(X̂ −X ) ≤ 0, we choose si so that |s0| = ε1/4λ1, si = 0 (i = 1, . . . ,m1− 1), and obtain a similar formula to (5.4). Hence, there is ε1(R0,K, λ0, λ1) so that ε−2[Rε(t, p, q, η,X )−R∗,ε(t, p, q, η̂, X̂ )] ≥ ε−1/2 + λ21 2 ε1/2E(X̂ − X )− 2C(1 + ‖X̂ ‖) ≥ ε−1/4 − λ21 2 ( ε1/2R0 + 1 ) − 2C(1 +R0) ≥ ε−1/4 − λ21 − 2C(1 +R0) ≥ 0, Now we consider the case |〈η̂ − η, ξ0〉| ≤ ε1/2 λ1 . Then from (5.8): |r0| = ∣∣ 〈η̂ − η, ξ0〉 ‖η̂ − η‖ ∣∣ ≤ ε1/4 λ1 =: c0ε 1/4. (5.9) Since r20 + r21 + . . .+ r2m1−1 = 1, we have the inequality 1− c20ε1/2 ≤ 1− r20 = r21 + r22 + . . .+ r2m1−1 ≤ |r1|+ |r2|+ . . .+ |rm1−1|, where we take ε so that c20ε 1/2 < 1/2. This inequality implies that there exists at least one number j0 such that |rj0 | ≥ 1− c20ε1/2 m1 − 1 > 1 2(m1 − 1) . Now we take si so that si = 0 (i 6= 0, j0) in (5.3) to obtain ε−1s0〈η̂ − η, ξ0〉+ ε−1sj0〈η̂ − η, ξj0〉+ s20 2 E ( X̂ − X ) + s2j0 2 〈( X̂ − X ) ξj0 , ξj0 〉 + [ F ( t, p, η, X̂ ) −F ( t, p, η,X )] + [ F ( t, p, η̂, X̂ ) −F ( t, p, η, X̂ )] . (5.10) 20 P. OCHOA, J. A. RUIZ EJDE-2021/57 If E(X̂ − X ) > 0, we chose |s0| = λ1 with s0〈η̂ − η, ξ0〉 ≥ 0. In addition, take |sj0 | = λ1ε 1/4 so that sj0〈η̂− η, ξj0〉 ≥ 0. Then, using also (5.9), (5.10) is rewritten as ε−1λ1|r0|‖η̂ − η‖+ ε−3/4λ1|rj0 |‖η̂ − η‖+ λ21 2 E ( X̂ − X ) + λ21 2 ε1/2〈 ( X̂ − X ) ξj0 , ξj0〉 + [ F ( t, p, η, X̂ ) −F ( t, p, η,X )] + [ F ( t, p, η̂, X̂ ) −F ( t, p, η, X̂ )] . From (A4) and Remark 4.1, there exists ε1(R0,K, λ0, λ1) so that ε−2[Rε(q, η,X )−R∗,ε(q, η̂, X̂ )] ≥ ε−3/4λ1|rj0 |‖η̂ − η‖+ λ21 2 ε1/2〈 ( X̂ − X ) ξj0 , ξj0〉+ [ F ( p, t, η̂, X̂ ) −F ( t, p, η, X̂ )] ≥ λ1ε −1/2 2(m1 − 1) − λ21 2 ε1/2‖X̂ − X‖ − 2C(1 + ‖X̂ ‖) ≥ 0. In the case E(X̂ − X ) ≤ 0, we take s0 = 0 and |sj0 | = λ1ε 1/4 so that sj0〈η̂ − η, ξj0〉 ≥ 0. Then, as in the previous case we have ε−3/4λ1|rj0 |‖η̂ − η‖+ λ21 2 ε1/2〈 ( X̂ − X ) ξj0 , ξj0〉 + [ F ( t, p, η, X̂ ) −F ( t, p, η,X )] + [ F ( t, p, η̂, X̂ ) −F ( t, p, η, X̂ )] ≥ 0. In particular, since ‖η̂‖ ≥ K−1, we see F(t, p, η̂,X ) = F∗(t, p, η̂,X ). Conse- quently if we set ε1 = ε1(K,R0, λ0, λ1), then the formula (5.1) holds with hK(s) = ωR0,1/K(s). Case 2: For ‖η̂‖ ≤ K−1 with K ∈ N, we argue as in case 1 to derive the estimate (5.2). Finally, we consider the case of η = η̂ or X = X̂ for η̂ ∈ Rm1\{0}. We can choose sequences {ηk} ⊂ Rm1\{0} and {Xn} ⊂ Sm1(R) such that ηk 6= η, ηk → η̂, Xk 6= X , Xn → X̂ as k, n→∞, respectively. 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Appendix: A technical lemma for existence Acknowledgements References