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Research article Special Issues

Torque control strategy of electric racing car based on acceleration intention recognition

  • Received: 14 November 2023 Revised: 14 January 2024 Accepted: 18 January 2024 Published: 25 January 2024
  • A torque control strategy based on acceleration intention recognition is proposed to address the issue of insufficient power performance in linear torque control strategies for electric racing cars, aiming to better reflect the acceleration intention of racing drivers. First, the support vector machine optimized by the sparrow search algorithm is used to recognize the acceleration intention, and the running mode of the racing car is divided into two types: Starting mode and driving mode. In driving mode, based on the recognition results of acceleration intention, fuzzy control is used for torque compensation. Based on the results of simulation and hardware in the loop testing, we can conclude that the support vector machine model optimized using the sparrow search algorithm can efficiently identify the acceleration intention of racing drivers. Furthermore, the torque control strategy can compensate for positive and negative torque based on the results of intention recognition, significantly improving the power performance of the racing car.

    Citation: Anlu Yuan, Tieyi Zhang, Lingcong Xiong, Zhipeng Zhang. Torque control strategy of electric racing car based on acceleration intention recognition[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2879-2900. doi: 10.3934/mbe.2024128

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  • A torque control strategy based on acceleration intention recognition is proposed to address the issue of insufficient power performance in linear torque control strategies for electric racing cars, aiming to better reflect the acceleration intention of racing drivers. First, the support vector machine optimized by the sparrow search algorithm is used to recognize the acceleration intention, and the running mode of the racing car is divided into two types: Starting mode and driving mode. In driving mode, based on the recognition results of acceleration intention, fuzzy control is used for torque compensation. Based on the results of simulation and hardware in the loop testing, we can conclude that the support vector machine model optimized using the sparrow search algorithm can efficiently identify the acceleration intention of racing drivers. Furthermore, the torque control strategy can compensate for positive and negative torque based on the results of intention recognition, significantly improving the power performance of the racing car.



    Let (M,g) be a n-dimensional Riemannian manifolds. In the present paper, we focus on the following Monge-Ampère equation on compact Riemannian manifolds (M,g) without boundary:

    S1peφ(ρ2)det(Sij+gijS)detgij=ϕ(z), (1.1)

    for any zM where ρ2=gijSiSj+S2 and Sij+gijS is a (0,2) type Codazzi tensor.

    An equivalent form of (1.1) is

    S1peφ(ρ2)σndetgijdz=ϕ(z)dz, (1.2)

    where σn=λ1λ2λn and {λl}nl=1 is a solution sequence to the following algebraic equation,

    det(Sij+gijSλgij)=0. (1.3)

    Noting that the volume element of Riemannian manifold is dV=detgijdz, we get

    S1peφ(ρ2)σndV=ϕ(z)dz. (1.4)

    The left-side of (1.2) is called the density of p-integral Gaussian curvature measure for the log-convex measure eφ(ρ2)dx for Codazzi tensor in the present paper.

    In particular, if M=S2, p=1, φ0 and S is the support function of hypersurface NR3, the second fundamental form (hij)2×2 of the hypersurface N is given as

    hij=Sij+gijS (1.5)

    for any fixed i,j{1,2} where Sij is the covariant derivation of S of second order. Then the left hand side of (1.4) becomes

    σ2dV (1.6)

    which is associated with the well-known Gauss-Bonnet formula for 2-dimensional Riemannian submanifold without boundary, see pp 358 of Kobayashi and Nomizu [1].

    Lemma A.1(Gauss-Bonnet formula([1])). Let NR3 be an orientable, compact, smooth 2-dimensional submanifold without boundary. Then,

    Nσ2dV=2πχ(N) (1.7)

    where χ(N)Z denotes the Euler characteristic number of N.

    For any fixed n3, one of the geometric interests for the measure σndV is the generalization of classical Gauss-Bonnet formula in higher dimensional space, see Fenchel [2], Allendoerfer [3], Allendoerfer and Weil [4], Chern [5,6], Chern and Lasf [7] and so on.

    Another geometric interest is to the measure σndV in the well-known Steiner-Weyl formula, see Weyl [8], Federer [9], Chern [6], Schneider [10] and so on.

    Since the measure σndV has its geometric origin, and therefore it is natural to get an intrinsic construction of the measure σndV. In polar coordinates, one can formulate the measure σndV as follows:

    ωσndV=νN(rN(ω))dξ (1.8)

    for any Borel set ωSn, where νN and rN are the normal mapping and radial mapping of the hypersurface N,

    ρN(ξ)=max{λ0:λξN},  ξSn, (1.9)

    and

    rN(ξ)=ρN(ξ)ξ,  ξSn, (1.10)

    see Oliker [11,12,13] or Schneider [10]. This observation led Aleksandrov to pose the following classical Aleksandrov problem, see Aleksandrov [14,15], Bakelman [16] or Guan, Li and Li [17].

    Problem A.2 (The classical Aleksandrov problem). For any fixed n1, given a Borel measures μ which is supported on the unit sphere Sn, finds a convex hypersurface NRn+1 such that

    νN(rN(ω))dξ=μ(ω) (1.11)

    for any Borel set ωSn where dξ, νN and rN are the standard n-dimensional spherical Lebesgue measure, normal mapping and radial mapping of the hypersurface N.

    Aleksandrov [14,15] solved Problem A.2. via the mapping argument which is a kind of method of continuity, see also Bakelman [16] or Pogorelov [18]. Later, from the point of view of nonlinear analysis or PDEs theory, Oliker [11,12,13] resolved Problem A.2. Moreover, concerning the regularity or curvature bounds of the hypersurface, Treibergs [19] and Guan and Li [20] also analyzed the Problem A.2. Recently, Huang, Lutwak, Yang and Zhang [21] introduced the so-called p-integral Gaussian curvature measure and posed the Lp Aleksandrov problem which can be stated as follows.

    Problem A.3 (Lp Aleksandrov problem ([21]). For any fixed n1 and pR, given a Borel measures μ which is supported on the unit sphere Sn, find a convex hypersurface NRn+1 such that

    νN(rN(ω))u1pdξ=μ(ω)) (1.12)

    for any Borel set ωSn where dξ, νN, u and rN are the standard spherical Lebesgue measure, normal mapping, support function and radial mapping of the hypersurface N.

    Recently, more and more interesting geometric analysis has been focused on the weighted measure eφ(|x|2)dx, see [22,23,24,25,26,27] and so on [28,29,30,31,32,33,34]. It may be interesting to mention that the convexity of φ can deduce some interesting geometric inequalities for the measure eφ(|x|2)dx, such as Brunn-Minkowski inequality, Prékopa-Leindler inequalities or Blaschke-Santaló inequalities, see [10,35,36,37,38].

    It is interesting to focus on the geometry of weighted measure eφ(|x|2)dx without the assumption of convexity of φ.

    If φ is concave, we call the measure eφ(|x|2)dx a log-convex measure.

    One interest in the geometry of log-convex measure eφ(|x|2)dx is the so-called log-convex density conjecture in geometric measure theory which can be stated as follows.

    Problem A.4 (Log-convex density conjecture). In Rn+1 with a smooth, radial, log-convex density, balls about the origin provide isoperimetric regions of any given volume.

    The Log-convex density conjecture was posed by Brakke and solved by Chambers [39]. More interesting comments on this topic can be referred to [40,41,42,43].

    Motivated by these beautiful results mentioned above, the main focus of the present paper is on Lp Aleksandrov problem for log-convex measure eφ(|x|2)dx in the frame of Riemannian Geometry.

    It is well-known that the main language of Riemannian Geometry is the so-called tensor, see Bishop and Goldberg [44] or Gerretsen [45]. This leads to our consideration on the problem in tensor spaces. By the analysis mentioned above, the core concept is the concept of Gaussian curvature. It is easy to see that the Gaussian curvature of a hypersurface can be calculated by means of the metric and second fundamental form of the hypersurface, see Kobayashi and Nomizu [1]. Therefore, to formulate a natural generalization of the classical Lp Aleksandrov problem in tensor spaces, we need to replace the second fundamental form by some interesting symmetric tensors and pose a natural generalization of Gauss curvature. Since the second fundamental form of any hypersurface in a space of constant curvature satisfies the Codazzi equation, we may say a natural generalization of the second fundamental form of the hypersurface is the so-called Codazzi tensor of Riemannian manifolds in higher dimensional tensor space which is defined as follows.

    For any connected smooth n-dimensional Riemannian manifolds (M,g), we let ST2 be the bundle of smooth symmetric (0,2) type tensor field over M, the covariant differential in the metric g is denoted by X where X is a vector field from the tangle bundle TM. The so-called Codazzi tensor is defined as follows:

    Definition A.5 (Cadazzi tensor [46]). Let A:MST2 be a smooth section. It is called a Codazzi tensor if A satisfies Codazzi equation,

    XA(Y,Z)=YA(X,Z) (1.13)

    for any X,Y,ZTM. The set of Codazzi tensors on M is denoted by Cod(M,g).

    In particular, the second fundamental form of any hypersurface in a space of constant curvature is a Codazzi tensor, see pp. 26 of Kobayashi and Nomizu [1].

    Some basic differential geometric theories about the Codazzi tensor are listed as follows, see [46].

    Let x:MRn+1 be an isometric immersion and assume also that rank A=n at z. Then rank A=n in some neighborhood U of z. Suppose that ξ is the unit normal vector field over x(U).

    Lemma A.6 ([46]).

    (i) Let x:MRn+1 be an isometric immersion and ξ be the unit normal vector field over x(U). Then "support" function f(ξ)=(x,ξ) where (,) is the inner product in Rn+1.

    (ii) the second order covariant differential of f and the coefficient of the second fundamental form (bij)n×n satisfies

    bij=fij+gijf. (1.14)

    Lemma A.7 ([46]). Let (M,g) be a Riemannian manifold of constant sectional curvature Ksec (possibly zero) and ACod(M,g). Then for every point on M, there exists a neighborhood V and a smooth function f:MR such that in V

    (A)ij(f)=fij+Ksecgijf (1.15)

    where (A)ij(f) is the coefficient of A=A(f). In addition, if M is simply connected then such representation is available on the entire M. Conversely, on a manifold of constant curvature Ksec, any smooth function f generates a Codazzi tensor A(f) via Eq (1.15).

    For any fC(M), we let A=(Aij)n×n be the Codazzi tensor generated by f, that is, Aij is given by (1.15). Let

    P0n(f)=det(fij(z)+Ksecgijf)detgij=detAdetg (1.16)

    Remark A.8. It is easy to see that P0n satisfies the following equation

    P0n(f)=λ1λ2λn (1.17)

    where λ1,λ2,,λn are n solutions to the following equation

    det(Aλg)=0. (1.18)

    This means that the geometric meaning of P0n is that P0n is a concept of "Gaussian" curvature for the quadratic form A.

    Oliker and Simon [46] proved the following interesting prescribed Gaussian curvature problem for Codazzi tensor:

    Theorem A.9 ([46]). Let (M,g) be a closed Riemannian manifold with constant sectional curvature Ksec0, and ϕ:M(0,) is a strictly positive C function. If Ksec>0 suppose also that M is not isometrically diffeomorphic to the sphere Sn of Rn+1. Then there exists a (unique) function fC(M) such that is a positive definite Codazzi tensor on M and

    P0n(f)=ϕ. (1.19)

    Suppose the sectional curvature Ksec of M is positive constant, with loss of generality, we may assume that Ksec=1. In (1.5), if we replace the standard Lebesgue dx with eφ(|x|2)dx, the p-integral Gaussian curvature function of Codazzi tensor with log-convex measure eφ(|x|2)dx can be defined as follows:

    f1peφ(ρ2)σndetgij. (1.20)

    Therefore, we let

    Pn,p(f)=P0nf1peφ(ρ2)detgij=f1peφ(ρ2)det(fij(z)+gijf)detgij (1.21)

    and call Pn,p(f) as p-integral Gaussian curvature function for Codazzi tensor with log-convex measure eφ(|x|2)dx.

    In the present paper, we focus on Lp Aleksandrov problem for Codazzi tensor with log-convex measures eφ(|x|2)dx which is stated as follows:

    Problem A.10 (Lp Minkowski problem for Codazzi tensor with log-convex measure). For any fixed n1 and pR, does there exist a Codazzi tensor A whose sectional curvature is 1 and is generated by f such that

    Pn,p(f)=ϕ? (1.22)

    The main result of the present paper can be stated as follows.

    Theorem 1.1. For any fixed n1 and p>n+1, there exist positive constants c,τ and a positive solution SC2,τ(M) to the Eq (1.1) satisfying

    0<c1SC2,τ(M)c< (1.23)

    where τ(0,1), c is independent of S provided the following conditions hold.

    (A.1.) 0<ϕC4(M), φ is a non-negative, radially symmetric, increasing, smooth and concave function in R, 0<ϕC4(M) and

    ϕC4(M)+φC4(0,)<.

    (A.2.)

    limttn+p1eφ(t2)=0,limt0tn+p1eφ(t2)=.

    (A.3.) There exists δ4>0 such that

    mint[a,b]φ(t2)+2φ(t2)t2δ4>0.

    for any compact [a,b](0,).

    (A.4.)

    maxt[a,b]2φ(t2)t2<pn1.

    for any compact [a,b](0,).

    Remark 1.2. The main result of the present paper can be seen as an attempt at some new results on the integral geometry of differential forms which are associated with some interesting invariants arising in geometry and topology. Apart from the beautiful Gauss-Bonnet Theorem, there are many famous theorems which link the analysis, topology and geometry, such as Euler-Poincaré characteristic formula, Riemann-Roch-Hirzebruch-Grothendieck Theorem, Atiyah-Singer index Theorem, Chern-Simons invariants, see Palais [47], Chern-Simons [48], Atiyah [49], Shanahan [50], Mukherjee [51], Moore [52], Gilkey [53], Freed [54] and so on. Following some classical ideas of Steiner, Federer and Chern, it may be interesting to focus on some kinematic formulas for these invariants and consider analogous geometric problems which prescribe differential forms for these invariants. In particular, in the frame of Kähler Manifolds or Symplectic Manifolds, some great results on this topic can be referred to the great works of Patodi [55], Duistermaat [56], Atiyah and Bott [57], Karshon and Tolman [58], Donaldson [59,60,61], Abreu [62], Grossberg and Karshonand [63], Boyer, Calderbank and Tønnesen-Friedman [64] and so on. In our forthcoming study, we will focus on these topics.

    Our proof of Theorem 1.1 is based on the well-known continuous method. We let the set of the positive continuous function on M be C+(M) and

    C={SC2,τ(M):(Sij+gijS)n×n is positive definite}

    The main ingredient is the a priori bounds of solutions to the following auxiliary problem for any SC:

    S1peφ(ρ2)det(Sij(z)+S(z)gij)detgij=tϕ(ξ)+(1t)eφ(1)detgij (1.24)

    for t[0,1].

    Remark 1.3. It is worth mentioning that without the assumption of convexity of φ, some necessary geometric inequalities have not been established which does not guarantee the validity of the classical variational framework for Problem A.10. Therefore, we adopt the well-known continuous method to solve this problem. Moreover, by the a priori bounds (1.23) of S, we get the compactness of the solution set and the curvature estimate of the Codazzi tensor which have their independent interests.

    Remark 1.4. Just like the case of concavity, log-convexity may be defined in the form of Prékopa-Leindler inequality: a function (or functional) f:IR is called log-convex if f satisfies the following condition,

    f(tx+(1t)y)ft(x)f1t(y) (1.25)

    for any t[0,1] and x,yI. The former case can be referred to pp. 369–374 of Schneider [10]. Some inequalities for log-convex functions (or functionals) have been analyzed in [65,66,67,68] and so on. It may be worth mentioning that Klartag [22] and Rotem [28] introduced some geometric notions for log-concave or more general α-concave functions and measures, such as the support function and mean width. Motivated by these interesting results, it may be interesting to introduce similar notions and get more geometric analysis for log-convex measures and this will also be a topic of our future study.

    The remaining part of this paper is arranged as follows: In Section 2, we prove the a priori bounds of S. In Section 3, we prove Theorem 1.1.

    In this section, we consider the a priori bounds of solutions to the following equation on Riemannian manifolds (M,g):

    S1peφ(ρ2)det(Sij+gijS)detgij=ϕ(ξ) (2.1)

    where ρ2=|S|2+S2 and the following conditions hold.

    (A.1.) 0<ϕC4(M), φ is a non-negative, radially symmetric, increasing, smooth and convex function in R, 0<ϕC4(M) and

    ϕC4(M)+φC4(0,)<.

    (A.2.)

    limttn+p1eφ(t2)=0,limt0tn+p1eφ(t2)=.

    (A.3.) There exist δ4>0 such that

    mint[a,b]φ(t2)+2φ(t2)t2δ4>0.

    for any compact [a,b](0,).

    We let the set of the positive continuous function on M be C+(M),

    C={SC2,τ(M):(Sij+gijS)n×n is positive definite}

    and

    ˜C={uC2,τ(M):(uij)n×n is positive definite}

    This main result of this section can be stated as follows,

    Theorem 2.0. For any fixed n1 and p>n+1, we let SCC+(M) be a solution to (2.1). Suppose that (A.1)(A.3) hold. Then there exists a positive constant c, independent of S, such that

    0<c1SC2,τ(M)c<, (2.2)

    where τ(0,1).

    Now, we divide the proof of Theorem 2.0 into following four steps.

    Step (a). For any fixed n1 and p>n+1, we let SCC+(M) be a solution to (2.1). Suppose that (A.1)(A.3) hold. Then there exists a positive constant c such that

    0<c1S(ξ)c< (2.3)

    for any zM.

    Proof of Step (a). We consider the following extremal problem,

    R=maxzMS(z). (2.4)

    It follows from the compactness of M and the continuity of S that there exists z1M such that

    R=S(z1). (2.5)

    It follows from the Eq (2.1) that at the point z=z1,

    0<detgijminzMϕ(z)detgijϕ(z1)Rn+1peφ(R2). (2.6)

    Combining this and condition (A.1), we can see that there exists a positive constant c>0 such that

    Rc<. (2.7)

    We next consider the following extremal problem,

    r=minzMS(z). (2.8)

    Adopting a similar argument mentoned above, we also can see that there exists a positive constant c>0 such that

    rc>0. (2.9)

    (2.7) and (2.9) yield the desired conclusion of Step (a).

    Step (b). For any fixed n1 and p>n+1, we let SCC+(M) be a solution to (2.1). Suppose that (A.1)(A.3) hold. Then there exists a positive constant c such that

    0|S(z)|2c,zM. (2.10)

    Proof of Step (b). The proof is based on Maximum Principle. We let

    v=S2+|S|22=12(S2+ΣijgijSiSj). (2.11)

    Suppose that there exists z0M such that

    v(z0)=maxzMv(z). (2.12)

    Then,

    0=lv=2(SSl+ΣijgijSliSj)=2Σijgji(Sil+Sgil)Sj (2.13)

    for any fixed l{1,2,,n} at the point z0. It follows from Lemma 2.1 that there exists a positive constant c,

    det(Sil+Sgil)=Sp1ϕ(z)eφ(ρ2)detgilSp1ϕ(z)eφ(0)detgilc0detgil (2.14)

    at the point z=z0 where

    c0=eφ(0)minzMSp1(z)ϕ(z)>0. (2.15)

    Noting that g=(gij)n×n is strictly positive, it follows from (2.14) and (2.15) that the matric (Sil+Sgil)n×n is reversible at the point z=z0 and therefore,

    Sl=0 (2.16)

    at the point z=z0 for any fixed l{1,2,,n} due to (2.13) and the positivity of g. From (2.16), we can see that

    |S|2=gijSiSj=0 (2.17)

    at the point z=z0. Therefore, it follows from Lemma 2.1 that there exists a positive constant c, independent of S,

    12|S|2(z)v(z)v(z0)12maxzMS(z)c, (2.18)

    for any zM. This completes the proof of Step (b).

    Before getting the higher order estimates of S, we let

    uij=Sij+gijS,   G(uij)=(detuij)1n (2.19)

    and

    Ψ(z)=(ψ(z)Sp1eφ(ρ2)detgij)1n. (2.20)

    Then, Eq (2.1) becomes

    G(uij)=Ψ. (2.21)

    Step (c). For any fixed n1 and p>n+1, we let SCC+(M) be a solution to (2.1) and u˜CC+(M) be a solution to (2.21). Suppose that (A.1)(A.3) hold. Then there exists a positive constant c, independent of S, such that

    Δuc. (2.22)

    Proof of Step (c). We let H=iuii. Suppose that H achieves it maximum at the point z=z0. Without loss of generality, we may (Hij)n×n is diagonal at the point z=z0. Therefore, at the point z=z0,

    H=0, (2.23)

    and (Hij)n×n is non-positive. We let

    Gij=Guij,Gij,rs=2Guijurs. (2.24)

    for any fixed i,j,s,t{1,2,,n}. Therefore, at the point z=z0,

    0ΣijGijHij=ΣiαGiiHii. (2.25)

    By the commutator identity, we have,

    Hii=Δuiinuii+H. (2.26)

    Putting (2.25) into (2.26), we get

    0ΣiGiiΔuiinΣiGiiuii+HΣiαGii. (2.27)

    Taking the α-th partial derivatives on both sides of (2.21) twice for any fixed α{1,2,,n}, we have

    ΣijGijuijα=Ψα,ΣijstGij,rsuijαursα+ΣijGij(uαα)ij=Ψαα (2.28)

    for any fixed α{1,2,,n}. By the concavity of G, we have

    ΣijstαGij,rsuijαursα0. (2.29)

    This implies that

    ΣiGiiΔuiiΣijGijΔuij+ΣijstαGij,rsuijαursα=ΔΨ. (2.30)

    at the point z=z0. Therefore,

    ΣiGiiΔuiiΔΨ. (2.31)

    at the point z=z0. It follows from Newton-MacLaurin inequality that

    ΣiGii1, (2.32)

    see Guan and Ma [69]. Putting (2.31), (2.32) into (2.27), we have, at the point z=z0,

    0ΔΨnΨ+HΣiGiiΔΨnΨ+HΔΨnΨ. (2.33)

    We let

    r1=minzMρ2(z),R1=maxzMρ2(z). (2.34)

    It follows from Lemmas 2.1 and 2.2 that

    0<r1R1<. (2.35)

    Now, we claim that at the point z=z0,

    ΔΨΨn2δ4nΣijS2ijcnΣijS2ijc (2.36)

    where δ4>0 to be chosen. Indeed, it follows from the definition of Ψ that

    logΨ=logϕ(ξ)n+p1nlogSφ(ρ2)n+1nlogdetgij. (2.37)

    Noting ρ2=|S|2+S2, for any fixed α{1,2,,n}, taking α-th partial derivatives on both sides of (2.37) twice, we have

    ΨαΨ=1n(logϕ)+p1nSα+2nφ(ρ2)(jSjSjα+SSα) (2.38)

    and

    ΔΨΨΣα(ΨααΨΨ2αΨ2)=Σα(1n(logϕ)+p1nSαα+2φ(ρ2)n(ΣjS2jα+SjSjαα+SSαα+S2α)+4n(φ(ρ2)(ΣjSjSjα+SSα)2) (2.39)

    where

    \begin{equation} I_1 = \frac{2\varphi'(\rho^2)}{n}\Sigma_{jl}S^2_{jl} +\frac{4}{n}\varphi''(\rho^2)\Sigma_l(\Sigma_jS_jS_{jl})^2, \end{equation} (2.40)

    and

    \begin{array}{l} I_2 = (\log \phi)''+\frac{2}{n}(\varphi'(\rho^2)+2\varphi''(\rho^2)S^2)|\nabla S|^2 +\frac{1}{n}(2\varphi'(\rho^2)+(p-1))\Delta S \\ +\frac{2\varphi'(\rho^2)}{n}\nabla S\cdot\nabla \Delta S. \end{array} (2.41)

    We now get some estimates of I_2 . Since \phi\in C^4(M) , it follows from Lemmas 2.1 and 2.2 that

    \begin{equation} (\log \phi)''+\frac{2}{n}(\varphi'(\rho^2)+2\varphi''(\rho^2)S^2)|\nabla S|^2\ge -c. \end{equation} (2.42)

    By the definition of H , we have,

    \begin{equation} H = \Delta S+nS. \end{equation} (2.43)

    Therefore, it follows from Lemma 2.1 and Hölder inequality that

    \begin{equation} \begin{split} |\frac{1}{n}(2\varphi'(\rho^2)S+(p-1))\Delta S|& = |\frac{1}{n}(2\varphi'(\rho^2)S+(p-1))(H-nS)|\\ &\le cH+c \le c\sqrt{n}\sqrt{\Sigma_iS^2_{ii}}+c\le c\sqrt{n}\sqrt{\Sigma_{ij}S^2_{ij}}+c \end{split} \end{equation} (2.44)

    for some c which means that

    \begin{equation} \frac{1}{n}(2\varphi'(\rho^2)S+(p-1))\Delta S\ge -c\sqrt{n}\sqrt{\Sigma_{ij}S^2_{ij}}-c. \end{equation} (2.45)

    Moreover, it follows from (2.43), (2.23) and Lemma 2.2 that

    \begin{equation} \begin{split} \frac{2\varphi'(\rho^2)}{n}\nabla S\cdot\nabla \Delta S& = \frac{2\varphi'(\rho^2)}{n}\nabla S\cdot\nabla (H-nS)\\ & = \frac{2\varphi'(\rho^2)}{n}\nabla S\cdot\nabla H-2\varphi'(\rho^2)|\nabla S|^2 = -2\varphi'(\rho^2)|\nabla S|^2\ge -c \end{split} \end{equation} (2.46)

    at the point z = z_0 . Therefore, combining (2.42), (2.45) and (2.46), we have,

    \begin{equation} I_2\ge-c\sqrt{n}\sqrt{\Sigma_{ij}S^2_{ij}}-c \end{equation} (2.47)

    at the point z = z_0 .

    Since \varphi\in C^2 is concave, we have,

    \begin{equation} \varphi''(\rho^2)\le 0 \end{equation} (2.48)

    for any z\in M . Noting that

    \begin{equation} 2\Sigma_l(\Sigma_jS_jS_{jl})^2\le 2\Sigma_l(\Sigma_jS^2_j\Sigma_jS^2_{jl}) = 2|\nabla S|^2\Sigma_{jl}S^2_{jl}\le 2\rho^2\Sigma_{jl}S^2_{jl}. \end{equation} (2.49)

    Therefore,

    \begin{equation} I_1 = \frac{2\varphi'(\rho^2)}{n}\Sigma_{jl}S^2_{jl} +\frac{4}{n}\varphi''(\rho^2)\Sigma_l(\Sigma_jS_jS_{jl})^2 \ge \frac{2}{n}(\varphi'(\rho^2)+2\rho^2\varphi''(\rho^2))\Sigma_{jl}S^2_{jl}. \end{equation} (2.50)

    We let

    \begin{equation} r_1 = \min\limits_{z\in M}\rho(z), R_1 = \max\limits_{z\in M}\rho(z). \end{equation} (2.51)

    It follows from Step (a) and Step (b) that

    \begin{equation} 0 < r_1\le R_1 < \infty. \end{equation} (2.52)

    Therefore, it follows from (A.3) that there exists \delta_4 > 0 such that

    \begin{equation} \min\limits_{z\in M}\varphi'(\rho^2(z))+2\varphi''(\rho^2(z))\rho^2(z)\ge \delta_4 > 0. \end{equation} (2.53)

    Therefore,

    \begin{equation} I_1\ge \frac{2\delta_4}{n}\Sigma_{ij}S^2_{ij}. \end{equation} (2.54)

    Therefore, (2.47) and (2.54) yield

    \begin{equation} I_1+I_2\ge\frac{2\delta_4}{n}\Sigma_{j, \alpha}S^2_{j\alpha}-c\sqrt{n}\sqrt{\Sigma_{ij}S^2_{ij}}-c \end{equation} (2.55)

    at the point z = z_0 . This is the desired inequality (2.36).

    It follows from (2.33) and (2.36) that

    \begin{equation} \Sigma_{ij}S^2_{ij}\le c \end{equation} (2.56)

    at the point z = z_0 . It follows from Hölder inequality that

    \begin{equation} \Delta S = \sqrt{n}\sqrt{\Sigma_iS^2_{ii}}\le \sqrt{n}\sqrt{\Sigma_{ij}S^2_{ij}}\le c \end{equation} (2.57)

    at the point z = z_0 . Combining (2.57), the definition of u and Step (a) , it is easy to get the inequality (2.22) which completes the proof of Step (c) .

    Step (d) . It follows from (2.21) that Eq (2.1) becomes

    \begin{equation} \mathcal{F}(u_{ij}) = 0 \end{equation} (2.58)

    provided \mathcal{F}(u_{ij}) = \mathcal{G}(u_{ij})-\Psi . We let \mathcal{F}_{ij} = \frac{\partial \mathcal{F}}{\partial u_{ij}} . It follows from Step (a) , Step (b) and Step (c) that there exist positive constants \lambda and \Lambda , independent of S , such that

    \begin{equation} 1\le\frac{\Lambda}{\lambda} < \infty , \end{equation} (2.59)

    and

    \begin{equation} 0 < \lambda|\zeta|^2 \le \mathcal{F}_{ij}\zeta_i\zeta_j\le \Lambda|\zeta|^2 , \end{equation} (2.60)

    for any \zeta = (\zeta_1, \zeta_2, \cdots, \zeta_n)\in \mathbb{R}^n . That is,

    (d.ⅰ) (2.58) is elliptic uniformly.

    Now, we claim that

    (d.ⅱ) \mathcal{F} is concave with respect to (u_{ij})_{n\times n} .

    It follows from the definition of \mathcal{F} that it suffices to prove that \mathcal{G} = \det^{\frac{1}{n}} is concave with respect to (u_{ij})_{n\times n} . Indeed, For any u, v\in \mathcal{C} and t\in [0, 1] , we let \{\delta^1_i\}^n_{i = 1} and \{\delta^2_i\}^n_{i = 1} be the eigenvalue sequence of (u_{ij})_{n\times n} and (v_{ij})_{n\times n} respectively. Then \{t\delta^1_i+(1-t)\delta^2_i\}^n_{i = 1} is a eigenvalue sequence of (tu_{ij}+(1-t)v_{ij})_{n\times n} . Moreover, since (u_{ij})_{n\times n} and (v_{ij})_{n\times n} are convex, we have

    \begin{equation} \delta^j_i\ge 0 \end{equation} (2.61)

    for any fixed i\in \{1, 2, \cdots, n\} and j\in \{1, 2\} and therefore,

    \begin{equation} (\prod\limits_{i = 1}^n(t\delta^1_i+(1-t)t\delta^2_i))^{\frac{1}{n}}\ge t(\prod\limits_{i = 1}^n(\delta^1_i)^{\frac{1}{n}} +(1-t)(\prod\limits_{i = 1}^n(\delta^2_i))^{\frac{1}{n}} \end{equation} (2.62)

    for any t\in [0, 1] . Combining the definition of \mathcal{G} and (2.62), we get

    \begin{equation} \mathcal{G}(tu_{ij}+(1-t)v_{ij})\ge t\mathcal{G}(u_{ij})+(1-t)\mathcal{G}(v_{ij}) \end{equation} (2.63)

    which proves that \mathcal{G} = \det^{\frac{1}{n}} is concave with respect to (u_{ij})_{n\times n} and this completes the proof of the claim.

    Then, it follows from (d.ⅰ), (d.ⅱ) and Theorem 17.14 of Gilbarg and Trudinger [70] that there exist \tau_1\in (0, 1) and positive constant c , independent of S , such that

    \begin{equation} \|u\|_{C^{2, \tau_1}(M)}\le c, \end{equation} (2.64)

    (see pp. 457–461 of Gilbarg and Trudinger [70]). Therefore there exist \tau\in (0, 1) and positive constant c , independent of S , such that

    \begin{equation} \|S\|_{C^{2, \tau}(M)}\le c, \end{equation} (2.65)

    This is the desired conclusion of Theorem 2.0.

    This section is devoted to the proof of Theorem 1.1.

    Motivated by [69,71] and so on, we consider the following auxiliary problem with a parameter t\in [0, 1] ,

    \begin{equation} S^{1-p}e^{-\varphi(\rho^2)}\frac{\det(S_{ij}(z)+S(z)g_{ij})}{\sqrt{\det g_{ij}}} = t\phi(z)+(1-t)e^{-\varphi(1)}\sqrt{\det g_{ij}}\triangleq f_t \end{equation} (3.1)

    for any z\in M where \rho^2 = |\nabla S|^2+S^2 = g^{ij}S_iS_j+S^2 , 0 < \phi\in C^4(M) and the following conditions hold.

    (A.1.) 0 < \phi\in C^4(M) , \varphi is a non-negative, radially symmetric, increasing, smooth and convex function in \mathbb{R} , 0 < \phi\in C^4(M) and

    \begin{equation*} \label{300002} \|\phi\|_{C^4(M)}+\|\varphi\|_{C^4(0, \infty)} < \infty. \end{equation*}

    (A.2.)

    \begin{equation*} \label{300003} \lim\limits_{t\to \infty}\frac{t^{n+p-1}}{e^{\varphi(t^2)}} = 0, \lim\limits_{t\to 0}\frac{t^{n+p-1}}{e^{\varphi(t^2)}} = \infty. \end{equation*}

    (A.3.) There exist \delta_4 > 0 such that

    \begin{equation*} \label{300004} \min\limits_{t\in [a, b]}\varphi'(t^2)+2\varphi''(t^2)t^2\ge \delta_4 > 0. \end{equation*}

    for any compact [a, b]\subseteq(0, \infty) .

    (A.4.)

    \begin{equation*} \label{300005} \max\limits_{t\in [a, b]}2\varphi'(t^2)t^2 < p-n-1. \end{equation*}

    for any compact [a, b]\subseteq(0, \infty) .

    We let the set of the positive continuous function on the Riemannian manifolds (M, g) be C_+(M) and

    \begin{equation} \label{3005} \mathcal{C} = \{S\in C^{2, \tau}(M):(S_{ij}+Sg_{ij})_{n\times n} \ \text{is positive definite}\} \\ \;\;\mathcal{I} = \{t\in [0, 1]: S\in \mathcal{C}\cap C_+(M), \ \text{(3.1) is solvable.}\} \end{equation} (3.2)

    Since f_t is a contious function, independent of z , satisfying

    \begin{equation*} \label{3007} 0 < \min\{e^{\varphi(1)}, \min\limits_{z\in M}\phi(z)\}\le f_t(z)\le \max\{e^{\varphi(1)}, \max\limits_{z\in M}\phi(z)\} < \infty, \end{equation*}

    for any t\in [0, 1] and z\in M , adopting some similar arguments in Section 2, we get

    Lemma 3.1. For any fixed n\ge 1 , p > n+1 and t\in [0, 1] , we let S_t\in \mathcal{C}\cap C_+(M) be a solution to (3.1). Suppose that (A.1)\sim(A.3) hold. Then there exists a constant c , independent of t , such that

    \begin{equation*} \label{3101} 0 < c^{-1}\le \|S_t\|_{C^{2, \tau}(M)}\le c, \end{equation*}

    for any t\in [0, 1] and some \tau\in(0, 1) .

    As a corollary of Lemma 3.1, we have,

    Corollary 3.2. For any fixed n\ge 1 , p > n+1 , we let \mathcal{I} is the set defined in (3.2). Suppose that (A.1)\sim(A.3) hold. Then \mathcal{I} is closed.

    Proof. It suffices to show that for any sequence \{t_j\}_{j = 1}^\infty\subseteq\mathcal{I} satisfying

    \begin{equation*} \label{3201} t_j\to t_0, \end{equation*}

    as j\to \infty for some t_0\in [0, 1] , we need to prove t_0\in\mathcal{I} .

    We let S^j be a solutions of problem (3.1) at t = t_j . It follows from the conclusion of Lemma 3.1 that there exists a positive constant c , independent of j such that

    \begin{equation*} \label{3202} \|S^j\|_{C^{2, \tau}(M)}\le c, \end{equation*}

    it follows from Ascoli-Arzela Theorem that up to a subsequence, there exists a S^0\in C^2(M) , such that

    \begin{equation*} \label{3203} \|S^j-S^0\|_{C^2(M)}\to 0 \end{equation*}

    as j\to \infty . It is easy to see that

    \begin{equation} (S^j)^{1-p}\to (S^0)^{1-p}, \rho_j\to \rho_0 \end{equation} (3.3)

    uniformly on M as j\to \infty where (\rho^j)^2 = (S^j)^2+|\nabla S^j|^2 for any j\in \{0, \, \cdots \} . Letting j\to \infty , we can see that (t_0, S^0) is a solution to the following problem:

    \begin{equation} S^{1-p}e^{-\varphi(\rho^2)}\frac{\det(S_{ij}(\xi) +Sg_{ij})}{\sqrt{\det g_{ij}}} = t\phi(z)+(1-t)e^{-\varphi(1)}\sqrt{\det g_{ij}} \end{equation} (3.4)

    for any z\in M . (3.4) implies that t_0\in\mathcal{I} . This is the desired conclusion of Corollary 3.2.

    Lemma 3.3. For any fixed n\ge 1 , p > n+1 , we let \mathcal{I} is the set defined in (3.2) . Suppose that (A.1)\sim(A.4) hold. Then \mathcal{I} is open.

    Proof. Suppose that there exists a \bar{t}\in \mathcal{I} , it suffices to prove t\in \mathcal{I} for any t\in B_\delta(\bar{t})\cap [0, 1] . To achieve this goal, joint with Implicit Function Theorem, we need to analyze the kernel of linearized equation associated to (3.1) . We assume that \bar{S} is a solution to (3.1) at t = \bar{t} . For any \zeta\in M , we let

    \begin{equation} M(S) = S^{1-p}e^{-\varphi(\rho^2)}\rho^{n+1}\frac{\det(S_{ij}(\xi) +Sg_{ij})}{\det g_{ij}}, f_t = t\phi(\xi)+(1-t)e^{-\varphi(1)}\sqrt{\det g_{ij}}, \end{equation} (3.5)
    \begin{equation} G_t(S) = M(S)-f_t, M[\bar{S}](\zeta) = \frac{d}{d\varepsilon}M(\bar{S}+\varepsilon\zeta)|_{\varepsilon = 0}, \end{equation} (3.6)

    and

    \begin{equation} G_t[\bar{S}](\zeta) = \frac{d}{d\varepsilon}G_t(\bar{S} +\varepsilon\zeta)|_{\varepsilon = 0} = \frac{d}{d\varepsilon}M(\bar{S}+\varepsilon\zeta)|_{\varepsilon = 0}. \end{equation} (3.7)

    By the Eq (3.1), we have

    \begin{equation} M(\bar{S}) = f_t. \end{equation} (3.8)

    Taking logarithm on both sides of (3.8), since f_t is independent of \bar{S} , we get,

    \begin{equation} \frac{M'[\bar{S}](\zeta)}{M(\bar{S})} = \frac{1-p}{\bar{S}}\zeta+2\varphi'(\bar{\rho}^2)(\bar{S}\zeta +\nabla\bar{S}\cdot\nabla\zeta)+\bar{P}_{ij}B(\zeta) \end{equation} (3.9)

    where (\bar{P}_{ij})_{n\times n} is the inverse of the matrix (\bar{S}_{ij}+\bar{S}g_{ij})_{n\times n} and

    \begin{equation} B(\zeta) = \zeta_{ij}+\zeta g_{ij}. \end{equation} (3.10)

    We let \zeta = \bar{S}v . Direct Calculation shows that

    \begin{equation} \zeta_i = \bar{S}v_i+\bar{S}_iv \end{equation} (3.11)

    and

    \begin{equation} \zeta_{ij} = \bar{S}v_{ij}+(\bar{S}_iv_j+\bar{S}_jv_i)+\bar{S}_{ij}v. \end{equation} (3.12)

    Therefore, we get

    \begin{equation} \bar{S}\zeta+\nabla \bar{S}\cdot\nabla\zeta = (\bar{S}^2+|\nabla \bar{S}|^2)v+\bar{S}\nabla \bar{S}\cdot\nabla v = \bar{\rho}^2v+\bar{S}\nabla \bar{S}\cdot\nabla v \end{equation} (3.13)

    which implies that

    \begin{equation} \begin{split} \frac{1-p}{\bar{S}}\zeta+(2\varphi'(\bar{\rho}^2))(\bar{S}\zeta+\nabla \bar{S}\cdot\nabla\zeta)& = (1-p+(2\varphi'(\bar{\rho}^2)\bar{\rho}^2)v +2\varphi'(\bar{\rho}^2)\bar{S}\nabla \bar{S}\cdot\nabla v. \end{split} \end{equation} (3.14)

    It follows from (3.12) and (3.10) that

    \begin{equation} \begin{split} B(\zeta)& = \bar{S}v_{ij}+(\bar{S}_iv_j+\bar{S}_jv_i)+(\bar{S}_{ij} +\bar{S}g_{ij})v\\ & = \bar{S}(v_{ij}+g_{ij}v)+(\bar{S}_iv_j+\bar{S}_jv_i) +(\bar{S}_{ij} +\bar{S}g_{ij})v-\bar{S}g_{ij}v \end{split} \end{equation} (3.15)

    and thus,

    \begin{equation} \bar{P}_{ij}B(\zeta) = \bar{S}\bar{P}_{ij}v_{ij}+2\bar{P}_{ij}\bar{S}_iv_j+nv -\bar{S}\Sigma_iP_iv \end{equation} (3.16)

    due to the symmetry of (\bar{P}_{ij})_{n\times n} . Putting (3.14) and (3.16) into (3.9), we have,

    \begin{array}{l} G[\bar{S}](v) = M[\bar{S}](v) = \bar{S}M(\bar{S})\bar{P}_{ij}v_{ij} \\ +2M(\bar{S})\bar{P}_{ij}\bar{S}_jv_i+(2\varphi'(\bar{\rho}^2)+(n+1)\rho^{n-1})M(\bar{S})\bar{S}\nabla \bar{S}\cdot\nabla v\\ \; \; \; +(n+1-p+(2\varphi'(\bar{\rho}^2)\bar{\rho}^2) -\bar{S}\Sigma_iP_{ij}g_{ij})Mv\triangleq a_{ij}v_{ij}+b_iv_i+Nv \end{array} (3.17)

    where

    \begin{equation} a_{ij} = \bar{S}M(\bar{S})\bar{P}_{ij}, b_i = 2M(\bar{S})\bar{P}_{ij}\bar{S}_j-2\varphi'(\bar{\rho}^2)M(\bar{S})\bar{S}\bar{S}_i \end{equation} (3.18)

    and

    \begin{equation} N = (n+1-p+(2\varphi'(\bar{\rho}^2)\bar{\rho}^2)-\bar{S}\Sigma_iP_{ij}g_{ij})M(\bar{S}). \end{equation} (3.19)

    Since \bar{S}, M(\bar{S}) > 0 , (\bar{P}_{ij})_{n\times n} is positive, we see that (a_{ij})_{n\times n} is positive. It follows from Lemma 3.1 that b_i is bounded. By the condition (A.4.) , we have

    \begin{equation} n+1-p+2\varphi'(\bar{\rho}^2)\bar{\rho}^2 < 0. \end{equation} (3.20)

    Since M(\bar{S}) is positive, we have,

    \begin{equation} -\bar{S}\Sigma_{ij}P_{ij}g_{ij})M(\bar{S}) < 0. \end{equation} (3.21)

    Therefore, it follows from (3.20) and (3.21) we get N < 0 . By Strong Maximum Principle for elliptic equations of second order, we see that

    \begin{equation} v\equiv 0 \end{equation} (3.22)

    (see pp. 35 of Gilbarg and Trudinger [70]) and thus,

    \begin{equation} \zeta\equiv 0 \end{equation} (3.23)

    since \bar{S} > 0 . Then by the standard Implicit Function Theorem, for any t\in B_\delta(\bar{t})\cap [0, 1] , there exists a S\in C^{2, \tau}(M) , such that G_t(S) = 0 . This means that t\in \mathcal{I} and completes the proof of Lemma 3.3.

    Final proof of Theorem 1.1. It is easy to see that S\equiv 1 is a solution of (3.1) at t = 0 . This means that \mathcal{I} is not-empty. This, together with Corollary 3.2 and Lemma 3.3, implies that \mathcal{I} = [0, 1] . Taking t = 1 , we get the proof of Theorem 1.1.

    The work was supported by China Postdoctoral Science Foundation (Grant: No.2021M690773). The author would like to thank heartily to the anonymous referees for their invaluable comments which are helpful to improve this paper's quality and to editors' hard work on the publication of the paper and to Professor Daomin Cao and Professor Qiuyi Dai for their useful guidance on nonlinear PDE theory and helpful comments on this work.

    There is no conflict of interest in this work.



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