A Study on the First Eigenvalue of the P-Laplacian and Critical Sets of Harmonic Functions by Defining Geometric P-Laplacian on Riemannian Manifolds

1. INTRODUCTION

In science, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of second request. It is a nonlinear generalization of the Laplace operator, where p is permitted to range over 1

The variation method, monotonicity, and differentiability for the first eigen value of the p-Laplacian on a n-dimensional shut Riemannian complex whose measurement develops by a summed up geometric flow. It is demonstrated that the first nonzero eigen value is monotonically non-diminishing along the flow under certain geometric conditions and that it is differentiable all over the place. These outcomes provide a brought together approach to the investigation of eigen value variations and applications under numerous geometric flows.

Let (M, g) be an n-dimensional shut Riemannian complex (n >1). Let g(x, t) be a one parameter group of measurements for t ∈ [0, T] and x ∈ M. We state that g(x, t) is a summed up geometric flow in the explodes and h is an overall time-dependent symmetric 2-tensor. Here h is thought to be smooth in the two factors t and x. This is clear since g is likewise smooth in the two factors. The scaling factor 2 in (1) is immaterial while the negative sign might be important in some specific applications to keep the flow either forward or in reverse in time. Two popular examples of geometric flows in this class are: the Ricci flow with h being the Ricci shape tensor, and the mean bend flow with h = H (where H is the mean arch and is the second crucial structure on M). Different examples incorporate Yamabe flow, Ricci-harmonic flow, RicciBourguignon flow. One may impose boundedness condition on tensor h. Truth be told, such boundedness and sign assumptions on h are preserved as long as the flow exists, so it follows that the measurements are consistently same. Precisely, if −K1g ≤ h ≤ K2g, where g(t), t ∈[0, T] is the flow, at that point To see the last limits, we consider the evolution of a vector structure |X|g = g(X, X), X ∈ TxM. From (1) and the boundedness of the tensor h, we have |∂tg(X, X)| ≤ K2g(X, X), which implies (by coordinating from t1 to t2) Taking the exponential of this gauge with t1 =0 and t2 =T yields |g(t)| ≤ eK2T g(0) from which the uniform boundedness of the measurement follows. In this manner, if there holds boundedness assumption The metric g(t) are consistently limited underneath and above forever 0 ≤ t ≤ T under the geometric flow (1). At that point, it doesn't make a difference what metric we use in the contention that follows.

The p-Laplace operator is characterized by For p∈[1, ∞), where div is the difference operator, and the adjoint of inclination (graduate) for the L2 -standard instigated by g on the space of differential structures at the point when p =2, ∆2, g is the standard Laplace-Beltrami operator. The eigen values and the It is notable that the principal image of (2) is nonnegative all over and carefully positive at the neighborhood of the point where ∇ f ≠ 0. We likewise realize that (2) has feeble solutions with only partial routineness of class C1, α,(0 < α < 1) as a rule. Intrigued perusers can locate the traditional papers by Evans and Tolksdorff. Notice that the least eigen value of ∆p,g on shut complex is zero with the corresponding eigen function being a constant. Subsequently, we allude to the infimum of the positive eigen values as the first nonzero eigen value or simply the first eigen value. The first eigen value of ∆p, g is portrayed by the min-max principle Satisfying the following constraint M|f|gp−2fdµg = 0, where dµg is the volume measure on (M,g). Clearly, the infimum doesn't change when one replaces W1, p(M) by C∞(M). The corresponding Eigen function is the energy minimizer of Rayleigh remainder (3) and fulfills the accompanying Euler-Lagrange equation For φ∈C0∞(M) in the cognition of distribution it is notable that p-Laplacian has discrete eigen values yet remains obscure whether it only has discrete eigen values for limited connected domains. Other notable outcomes reveal to us that the first nonzero eigen value is simple and confined. Here the simplicity shows that any nontrivial eigen function corresponding to λp, 1 doesn't change sign and that any two first eigen functions are constant multiple of one another. In contrast to the spectrum of the Laplace-Beltrami operator (the case p=2), the p-Laplacian is nonlinear as a rule. Besides, it isn't known whether λp, 1 or its corresponding eigen function is C1-differentiable (or even locally Lipschitz) along any geometric flow of the structure (1). Notwithstanding, it has been pointed out that the differentiability for the case p=2 is a consequence of eigen value perturbation theory; see, for instance, and the references therein. Consequently, any approach that accepts differentiability of eigen values and eigen functions under the flow must be applied to the case p=2. Presently to keep away from the differentiability assumption on the first eigen value and the corresponding eigen function for the situation p≠2, we will apply strategies introduced in under the Ricci flow to examine the variation and monotonicity of λp,1 (t) =λp,1(t, f(t)), where λp,1(t, f(t)) and f(t) are thought

eigen function only requirements to fulfill certain normalization condition. There are numerous outcomes on the evolution and monotonicity of Eigen values of the Laplace operator on evolving manifolds with or without ebb and flow assumptions. One can find under the Ricci flow, under Ricci-harmonic flow and along unique geometric flow with entropy techniques. The investigation of the properties of eigen values of the p-Laplacian on evolving complex is still youthful. The main point of this paper is to investigate if those known properties of λp, 1 on static measurement and for the case p=2 on evolving metric can be stretched out to different geometric flows. We anyway intend to develop a brought together calculation that can be utilized for this purpose on time-dependent measurements. Many interesting outcomes concerning the conduct of λp, 1 can be found in for static measurements and for evolving measurements along different geometric flows.

We consider solutions to the equation (5) with a consistently elliptic and Lipschitz continuous symmetric lattice, in dimension two. We consider solutions u ∈ W1,ploc (Ω) to the following savage elliptic equation which we will call the anisotropic p-Laplace equation Where Ω is a two-dimensional domain, p fulfills 1

By using the theory of self-adjoint operators, the spectral properties of the linear Laplacian on a domain in a Euclidean space or a manifold have been concentrated widely. Mathematicians generally are interested in the spectrum of the Laplacian on compact manifolds (with or without boundary) or non-compact complete manifolds, since in these two cases the linear Laplacians can be particularly reached out to self-adjoint operators. Nonetheless, the spectrum of the Laplacian on noncompact non-complete manifolds also attracts attention of mathematicians and physicists in the past three decades, since the investigation of the spectral properties of the Dirichlet Laplacian in infinitely extended regions has applications in elasticity, acoustics, electromagnetism, quantum physics, and so forth As of late, the author has proved the presence of discrete spectrum of the linear Laplacian on a class of 4-dimensional rotationally symmetric quantum layers, which is non-compact non-complete manifolds, in under some geometric assumptions therein. A natural generalization of the linear Laplacian is the purported p-Laplacian underneath. Although many outcomes about the linear Laplacian (p=2) have been obtained, many rather basic questions about the spectrum of the nonlinear p-Laplacian remain to be addressed. Let Ω be a limited domain on a n-dimensional Riemannian manifold (M, g). We consider the following nonlinear Dirichlet eigen value problem. Where is the p-Laplacian with 1

Problems involving p-Laplace operator are subject of intensive investigations as they illustrate many of phenomena that happen in nonlinear analysis. Among their applications are singular and non-singular boundary value problems which appear in various branches of mathematical physics. They arise as a model example in the liquid dynamics; glaciology; stellar dynamics; in the theory of electrostatic fields; in the more general context in quantum physics; in the nonlinear elasticity theory as a basic model; and many others. The PDEs involving p-Laplacian are considered in differential geometry in the investigation of critical points for p-harmonic maps between Riemannian manifolds and the eigenvalue problems for p-Laplacian on Riemannian manifolds serve for estimations of the diameter of the manifolds. Eigenvalue problems involving p-Laplacian are applied in functional analysis to infer sharp Poincar'e and Writinger type inequalities, Sobolev embeddings and isoperimetric inequalities. Geometric properties of p-harmonic functions play significant part in the theory of Carnot–Caratheodory groups like Heisenberg group and in the analysis on measurement spaces. One of the problems we experience when investigating p-harmonic equation is that not many explicit solutions are known – affine, quasiradial, radial. Among them radial solutions form the most stretched out nontrivial class in which many properties of p-harmonic world can be recognized? Another motivation to examine radial solutions comes from the seminal paper by Gidas, Ni and Nirenberg who expanded Serrin's moving plane strategy from and proved that at times only radial solutions are admitted. Additionally, it can happen that among the solutions of the PDE are the radial ones regardless of whether the radial solutions are by all account not the only ones. We shall consider radial solutions of the equation We assume that p>1, n>1, R∈(0,∞] (for R=∞ the above equation is defined on Rn), a(.) is nonnegative and belongs to a certain class of functions which will be depicted later, while φ is an arbitrary odd continuous function with the end goal that τφ(τ) is of constant sign for L1 almost all τ 's. In general our equation is given in a non-divergent form. operator, is a quasilinear elliptic partial differential operator of second request. It is a nonlinear generalization of the Laplace operator, where p is allowed to range over 1

1. Valtorta, Daniele. (2012). on the p-Laplace operator on Riemannian manifolds. 2. L. Ma (2006). Eigenvalue monotonicity for the Ricci-Hamilton flow, Ann. Global Anal. Geom. 29, pp. 287-292. 3. A. Abolarinwa, J. Mao (2016). The first eigenvalue of the p-Laplacian on time dependent Riemannian metrics, arxiv.org 1604.05884. 4. J. Wu (2011). First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow, Acta Math. Sinica 27, pp. 1591-1598. 5. X. Cao (2008). First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc. 136, pp. 4075-4078. 6. J-F. Li (2007). Eigenvalues and energy functionals with monotonicity formulae under Ricci flow, Math. Ann. 338, pp. 927– 946. 7. J. Li (2007). Monotonicity formulae under rescaled Ricci flow, arXiv:math/07010.5328v3 [math.DG]. 8. A. Abolarinwa (2019). Eigenvalues of Weighted-Laplacian under the extended Ricci flow, Adv. Geom. 19, pp. 131-143. 9. A. Abolarinwa, O. Adebimpe, E. A. Bakare (2019). Monotonicity formulas for the first

10. Y. Li (2014). Eigenvalues and entropies under the Harmonic-Ricci Flow, Pacific J. Math. 267, pp. 141-184. 11. H. Guo, R. Philipowski, A. Thalmaier (2013). Entropy and lowest eigenvalue on evolving manifolds, Pacific J. Math. 264, pp. 61-82. 12. J. Mao (2014). Eigenvalue inequalities for the p-Laplacian on Riemannian manifold and estimates for the heat kernel, J. Math. Pures Appl. 101, pp. 372-393.

Research Scholar, Mansarovar Global University, Sehore (MP)