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In this paper, combining Nash-Moser iteration and Saloff-Coste type Sobolev inequalities, we establish fundamental estimates for solutions to a class of nonlinear elliptic equations of the form Δ u (x) + a (x) u (x) ln ⁡ u (x) + b (x) u (x) = 0 , which possesses abundant geometric backgrounds. Utilizing these estimates which retrieve more geometric information, we obtain some further properties of such solutions. Especially, we prove a local Liouville type theorem of corresponding constant coefficient equation. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a Liouville theorem of F-harmonic maps between complete Riemannian manifolds with moderate divergent F-energy. We assume that F is a concave function and satisfies a differential inequality. We employ Ara's F-stress-energy tensor and the Hessian comparison theorem to prove the main result. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of T-conformal transformations and T-conformal maps between Riemannian manifolds. Here, T stands for a smooth (1,1)-tensor field defined on the domain of these maps. We start by defining what it means for a map to be T-conformal and also dwell on some basic properties of such type maps. We next specialize our discussion to the situation when the map T satisfies the condition △T = 0. Accordingly, we prove Liouville's theorem for T-conformal maps between space forms Rn(c) as an application under the condition △T = 0. The proof relies upon properties of T-conformal maps proved earlier. Broadly, the paper seeks to provide a general understanding of conformal mappings in the presence of a tensor field T and show how classical results such as Liouville's theorem apply. [ABSTRACT FROM AUTHOR]

We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold (M , g) Δ u + a u (ln u) p + b u ln u = 0 , where a ≠ 0 , b are two constants and p = k 1 2 k 2 + 1 ≥ 2 , here k 1 and k 2 are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we present a unified method for deriving differential Harnack inequalities for positive solutions of the semilinear parabolic equation ∂ t u = Δ V u + H (u) on complete Riemannian manifolds with Bakry-Émery curvature bounded below. This method transforms the problem of deriving differential Harnack inequalities into solving a related ODE system. As its application, on the one hand, we obtain new and improved estimates for logarithmic type equation and Yamabe type equation. On the other hand, we establish some sharp estimates for them under non-negative Bakry-Émery curvature condition. As their natural corollary, we also obtain some sharp Harnack inequalities and Liouville type theorems. [ABSTRACT FROM AUTHOR]

In this paper, we extend the definition of the Jacobi operator of smooth maps, and biharmonic maps via the variation of bienergy between two Riemannian manifolds. We construct an example of (p, f)-biharmonic non (p, f)-harmonic map. We also prove some Liouville type theorems for (p, f)-biharmonic maps. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the gradient estimates for positive weak solutions to the following p-Laplacian equation Δ p u + a u σ = 0 on complete noncompact Riemannian manifold, where a, σ are two nonzero real constants with p ≠ 2. Using the gradient estimate, they can get the corresponding Liouville theorem. On the other hand, by virtue of the Poincaré inequality, they also obtain a Liouville theorem under some integral conditions with respect to positive weak solutions. [ABSTRACT FROM AUTHOR]

We first give a logarithmic gradient estimate for the local positive solutions of Allen–Cahn equation on the complete Riemannian manifolds with Ricci curvature bounded below. As its natural corollary, Harnack inequality and a Liouville theorem for classical positive solutions are obtained. Later, we consider similar estimate under integral curvature condition and generalize previous results to a class nonlinear equations which contain some classical elliptic equations such as Lane–Emden equation, static Whitehead–Newell equation and static Fisher–KPP equation. Last, we briefly generalize them to equation with gradient item under Bakry–Émery curvature condition. [ABSTRACT FROM AUTHOR]

Let (M , F) be a complete foliated Riemannian manifold and all leaves be compact. Let (M ′ , F ′) be a foliated Riemannian manifold of non-positive transversal sectional curvature. Assume that the transversal Ricci curvature Ric Q of M satisfies Ric Q ≥ - λ 0 at all point x ∈ M and Ric Q > - λ 0 at some point x 0 , where λ 0 is the infimum of the spectrum of the basic Laplacian acting on L 2 -basic functions on M. Then every transversally harmonic map ϕ : M → M ′ of finite transversal energy is transversally constant. [ABSTRACT FROM AUTHOR]

In this paper, we discuss two Liouville-type theorems for subelliptic harmonic maps from sub-Riemannian manifolds to Riemannian manifolds. One is the Dirichlet version which states that two subelliptic harmonic maps from a sub-Riemannian manifold with boundary to a regular ball must be same if their restrictions on boundary are same; it is generalized to complete noncompact domains as well. The other is the vanishing-type theorem for finite L p -energy subelliptic harmonic maps on complete noncompact totally geodesic Riemannian foliations which are special sub-Riemannian manifolds. [ABSTRACT FROM AUTHOR]

Let V be a C 1 -vector field on an n -dimensional complete Riemannian manifold (M , g). We prove a Liouville theorem for V -harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative (m , V) -Ricci curvature for m ∈ [ − ∞ , 0 ] ∪ [ n , + ∞ ] into Cartan–Hadamard manifolds, which extends Cheng's Liouville theorem proved in Cheng (1980) for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan–Hadamard manifolds. We also prove a Liouville theorem for V -harmonic maps from complete Riemannian manifolds with non-negative (m , V) -Ricci curvature for m ∈ [ − ∞ , 0 ] ∪ [ n , + ∞ ] into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt et al. (1980) and Choi (1982). Our probabilistic proof of Liouville theorem for several growth V -harmonic maps into Hadamard manifolds enhances an incomplete argument in Stafford (1990). Our results extend the results due to Chen et al. (2012) and Qiu (2017) in the case of m = + ∞ on the Liouville theorem for bounded V -harmonic maps from complete Riemannian manifolds with non-negative (∞ , V) -Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of V -harmonic maps and the recurrence property of Δ V -diffusion processes on manifolds. Our results are new even in the case V = ∇ f for f ∈ C 2 (M). [ABSTRACT FROM AUTHOR]

We prove two Liouville theorems for ancient nonnegative solutions of the heat equation on a complete noncompact Riemannian manifold with Ricci curvature bounded from below by −K, K ≥ 0. If, at any fixed time , such a solution grows subexponentially in space, then it is either constant (when K = 0) or stationary (if K > 0). We also show the optimality of this growth condition through examples. [ABSTRACT FROM AUTHOR]

If f(z) is either a polynomial with no zeroes or a bounded entire function, then a Riemannian metric gf is constructed on the complex plane C. This metric gf is shown to be flat and geodesically complete. Therefore, the Riemannian manifold (C, gf) must be isometric to (C, |dz|²), which implies that f(z) is a constant. [ABSTRACT FROM AUTHOR]

In this paper, we prove a mean value formula for bounded subharmonic Hermitian matrix valued function on a complete Riemannian manifold with nonnegative Ricci curvature. As its application, we obtain a Liouville type theorem for the complex Monge–Ampère equation on product manifolds. [ABSTRACT FROM AUTHOR]

We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term. [ABSTRACT FROM AUTHOR]

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation ut = Δu + au log u in complete noncompact Riemannian manifolds. As its application, we show that if u is a positive solution of the equation ut = Δu and log u is of sublinear growth in both spatial and time directions, then u must be constant. This gradient estimate is sharp since it is well known that u(x, t) = ex+t satisfying ut = Δu [ABSTRACT FROM AUTHOR]

Let M be a C 2 -smooth Riemannian manifold with boundary and N a complete C 2 -smooth Riemannian manifold. We show that each stationary p -harmonic mapping u : M → N , whose image lies in a compact subset of N , is locally C 1 , α for some α ∈ (0 , 1) , provided that N is simply connected and has non-positive sectional curvature. We also prove similar results for minimizing p -harmonic mappings with image being contained in a regular geodesic ball. Moreover, when M has non-negative Ricci curvature and N is simply connected with non-positive sectional curvature, we deduce a gradient estimate for C 1 -smooth weakly p -harmonic mappings from which follows a Liouville-type theorem in the same setting. [ABSTRACT FROM AUTHOR]

We establish a Gaussian upper bound of the heat kernel for the Laplace-Beltrami operator on complete Riemannian manifolds with Bakry-Émery Ricci curvature bounded below. As applications, we first prove an L 1 -Liouville property for non-negative subharmonic functions when the potential function of the Bakry-Émery Ricci curvature tensor is of at most quadratic growth. Then we derive lower bounds of the eigenvalues of the Laplace-Beltrami operator on closed manifolds. An upper bound of the bottom spectrum is also obtained. [ABSTRACT FROM AUTHOR]

Abstract We establish regularity of conformal maps between sub-Riemannian manifolds from regularity of Q -harmonic functions, and in particular we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth in all contact sub-Riemannian manifolds. Together with the recent results in [15] , our work yields a new proof of the smoothness of boundary extensions of biholomorphims between strictly pseudoconvex smooth domains [29]. [ABSTRACT FROM AUTHOR]

In this paper, we consider bounded positive solutions to the Allen- Cahn equation on complete noncompact Riemannian manifolds without boundary. We derive gradient estimates for those solutions. As an application, we get a Liouville type theorem on manifolds with nonnegative Ricci curvature. [ABSTRACT FROM AUTHOR]

By constructing new numerical approximation formulas for the left and right Riemann–Liouville tempered fractional derivatives of order α ∈ ( 0 , 1 ) , a numerical algorithm is proposed to solve the space tempered fractional convection equation. The algorithm is shown to be unconditionally stable and convergent with order O τ 2 + h 2 . Finally, numerical example is provided to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of "good" cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities. [ABSTRACT FROM AUTHOR]

We investigate the Liouville properties of ancient solutions to the V -harmonic map heat flows from complete noncompact Riemannian manifolds with nonnegative Bakry–Emery Ricci curvature. When the target is a simply connected complete Riemannian manifold with nonpositive sectional curvature, then a Liouville theorem of ancient solutions holds under certain growth condition near infinity. When the target is a complete Riemannian manifold with sectional curvature bounded above by a positive constant, we show that if the image of the ancient solution u is contained in a regular ball near infinity, then u is a constant. [ABSTRACT FROM AUTHOR]

In this paper we deal with the uniqueness of nonnegative solutions to systems of quasilinear elliptic differential inequalities on Riemannian manifolds, and we obtain the Liouville type theorems only in terms of the volume growth of geodesic ball. By applying the perturbation and gluing techniques, a counterexample is constructed to show the sharpness of the volume growth condition. [ABSTRACT FROM AUTHOR]

In the first part of this paper we revisit a classical topological theorem by Tischler (1970) and deduce a topological result about compact manifolds admitting a set of independent closed forms proving that the manifold is a fibration over a torus. As an application we reprove the Liouville theorem for integrable systems asserting that the invariant sets or compact connected fibers of a regular integrable system are tori. We give a new proof of this theorem (including the non-commutative version) for symplectic and more generally Poisson manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we study Hamilton–Souplet–Zhang's gradient estimates for positive solutions to the nonlinear parabolic equation u t = Δ u + λ u α on noncompact Riemannian manifolds, where λ , α are two real constants. As an application, we obtain a Liouville-type theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of harmonic maps for Finsler manifolds. Some Liouville theorems on harmonic maps for Finsler manifolds are given. Let M be a complete simply connected Riemannian manifold with non-negative Ricci curvature and M ¯ be a complete Berwald manifold with non-positive flag curvature. The main purpose of this paper is to prove that there exists no non-degenerate harmonic map ϕ from M to M ¯ with ∫ S M e ( ϕ ) d V S M < ∞ , which generalizes the result of Schoen and Yau (1976) from Riemannian manifolds to Berwald manifolds. [ABSTRACT FROM AUTHOR]

We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold without boundary, −Δgui = Hi(u1, · · · , um) on 𝕄, when Δg stands for the Laplace-Beltrami operator, ui : 𝕄 → ℝ and Hi ∈ C1(ℝm) for 1 ≤ i ≤ m. This system is called symmetric if the matrix of partial derivatives of all components of H, that is, ℍ(u) = (∂jHi(u))mi,j=1, is symmetric. We prove a stability inequality and a Poincar´e type inequality for stable solutions using the Bochner-Weitzenb¨ock formula. Then, we apply these inequalities to establish Liouville theorems and flatness of level sets for stable solutions of the above symmetric system under certain assumptions on the manifold and on solutions. [ABSTRACT FROM AUTHOR]

In this article we study a nonlinear elliptic equation by using the maximum principle and cutoff functions, We establish related gradient esti- mates, the Liouville theorem, and the Harnack inequality. [ABSTRACT FROM AUTHOR]

In the present paper we prove Liouville-type theorems: non-existence theorems for some complete Riemannian almost product manifolds and special mappings of complete Riemannian manifolds which generalize similar results for compact manifolds. [ABSTRACT FROM AUTHOR]

In this paper, by the method of J. F. Li and X. J. Xu ( Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456-4491), we shall consider the nonlinear parabolic equation [ABSTRACT FROM AUTHOR]

In this paper, we introduce a horizontal energy functional for maps from a Riemannian manifold to a pseudo-Hermitian manifold. The critical maps of this functional will be called CC-harmonic maps. Under suitable curvature conditions on the domain manifold, some Liouville-type theorems are established for CC-harmonic maps from a complete Riemannian manifold to a pseudo-Hermitian manifold by assuming either growth conditions of the horizontal energy or an asymptotic condition at the infinity for the maps. [ABSTRACT FROM AUTHOR]

In this paper, along the idea of Souplet and Zhang, we deduce a local elliptic-type gradient estimates for positive solutions of the nonlinear parabolic equation: [ABSTRACT FROM AUTHOR]

In this paper, we obtain conditions for the validity of Liouville-type theorems on the triviality of bounded solutions of an elliptic inequality of special form as well as of the stationary Ginzburg-Landau equation for noncompact spherically symmetric Riemannian manifolds. [ABSTRACT FROM AUTHOR]

A Cartan space is a manifold whose cotangent bundle of local coordinates (x, p) is endowed with a smooth regular Hamiltonian K2(x, p) which is positively homogeneous of degree 2 in momenta p. Then the cotangent bundle becomes a semi-Riemannian manifold with a metric G of Sasaki type.We prove that a) for a fixed x = x0 the level hypersurfaces of the function K(x0, p) define a foliation which is totally umbilical and orthogonal with respect to G on the foliation spanned by the Liouville vector field in x0 and b) the level hypersurfaces of the function K(x, p) define also a foliation which is orthogonal on the foliation spanned by the Liouville vector field in (x, p) if and only if the Cartan space is Landsberg. Some other natural foliations on the cotangent bundle are taken into consideration. [ABSTRACT FROM AUTHOR]

Some qualitative properties of positive smooth solutions to a generalized nonlinear parabolic equation involving f -Laplacian (L f) (L f − q (x , t) − ∂ ∂ t) w (x , t) = G (w (x , t)) , are discussed on M f × (− ∞ , + ∞) , where M f is a complete smooth metric measure space (with or without boundary), the potential function q (x , t) is smooth at least C 1 in x and C 0 in t , and G (w (x , t)) is a nonlinear smooth sourcing term. Local and global type space only (elliptic type) gradient estimates are established for this equation under the condition that Bakry-Émery Ricci curvature tensor is bounded from below. As an exploitation of the gradient estimates so derived we obtain a parabolic Harnack inequality and some Liouville type theorems for bounded ancient and eternal solutions. The approach adopted in this paper provides a unified treatment of a large class of nonlinear source terms. To demonstrate this further, cases where G (w) = a w ρ , a ∈ R , ρ ∈ (− ∞ , 0 ] ∪ [ 1 , + ∞) , G (w) = a w | log ⁡ w | γ , a ∈ R , a ≠ 0 , γ > 1 and G (w) = a w (log ⁡ w) γ a ≠ 0 , γ ≥ 1 are considered as specific examples. We further show that all the obtained results also hold on weighted manifolds with compact boundary under some lower boundedness assumptions on mean curvature of the boundary and Bakry-Émery Ricci curvature tensor. [ABSTRACT FROM AUTHOR]

Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we derive a local Hamilton type gradient estimate for a nonlinear heat equation on Riemannian manifolds. As its application, we obtain a Liouville type theorem. [ABSTRACT FROM AUTHOR]

Under appropriate spectral assumptions, we prove two existence results for positive solutions of Lichnerowicz-type equations on complete manifolds. We also give a priori bounds and a comparison result that immediately yields uniqueness for certain classes of solutions. No curvature assumptions are involved in our analysis. [ABSTRACT FROM AUTHOR]

Iln this paper we study existence and uniqueness of solutions for mixed problems consisting non-local Hadamard fractional integrals for coupled systems of Riemann-Liouville fractional differential equations. The existence and uniqueness of solutions is established by using the Banach's contrac-tion principle, while the existence of solutions is derived by applying Leray-Schauder's alternative. Examples illustrating our results are also presented. [ABSTRACT FROM AUTHOR]

Let $(M^{n}, g)$ be an n-dimensional Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions of the porous medium equation posed on $(M^{n}, g)$ with the Ricci curvature bounded from below. Moreover, we also obtain a Liouville type theorem. In particular, the results obtained in this paper generalize those in (Zhu in J. Math. Anal. Appl. 402:201-206, ). [ABSTRACT FROM AUTHOR]

In this paper, we apply a numerical scheme for solving fractional differential equations. Our approach is based on an operational matrix of fractional Riemann-Liouville integration with Legendre basis and zeros of Chebyshev polynomials. In this framework the fractional Burgers equation, modified fractional Korteweg-de Vries equation and fractional wave equation are considered as prototype examples. The results reveal that the present method is very effective and accurate. [ABSTRACT FROM PUBLISHER]

We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. We prove an $$L^{p}$$ Liouville type theorem which is a quantitative integral $$L^{p}$$ estimate of harmonic functions analogous to Karp's theorem for Riemannian manifolds. As corollaries we obtain Yau's $$L^{p}$$ -Liouville type theorem on graphs, identify the domain of the generator of the semigroup on $$L^{p}$$ and get a criterion for recurrence. As a side product, we show an analogue of Yau's $$L^{p}$$ Caccioppoli inequality. Furthermore, we derive various Liouville type results for harmonic functions on graphs and harmonic maps from graphs into Hadamard spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we study a Modica type gradient estimate for smooth solutions to general non-linear Poisson equation where is a complete Riemannian manifold with bounded geometry and non-negative Ricci curvature and f is the derivative of the non-negative smooth function on R. Then we use this gradient estimate to conclude a Liouville theorem. [Copyright &y& Elsevier]

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least $2$ on a closed Riemannian manifold admits an equilibrium measure $\mu _f$, which is balanced and invariant under $f$ and non-atomic, and whose support agrees with the Julia set of $f$. Furthermore, we show that $f$ is strongly mixing with respect to the measure $\mu _f$. We also characterize the measure $\mu _f$ using an approximation property by iterated pullbacks of points under $f$ up to a set of exceptional initial points of Hausdorff dimension at most $n-1$. These dynamical mixing and approximation results are reminiscent of the Mattila–Rickman equidistribution theorem for quasiregular mappings. Our methods are based on the existence of an invariant measurable conformal structure due to Iwaniec and Martin and the ${{\cal {{A}}}}$-harmonic potential theory. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive gradient estimates for Dirac-harmonic maps from complete Riemannian spin manifolds into regular balls in Riemannian manifolds. With these estimates, we can prove Liouville theorems for Dirac-harmonic maps under curvature or energy conditions. [Copyright &y& Elsevier]

Abstract: In this paper we prove a Liouville type theorem for the Witten-Laplacian on a complete Riemannian manifold with nonnegative m-dimensional Bakry–Emery Ricci curvature. [Copyright &y& Elsevier]