On Aronson's Upper Bounds for Heat Kernels

Let L be a uniformly elliptic operator in divergence form onR^d, and let p(t,x,y) be the fundamental solution to the heatequation for L. A new proof is given of Aronson's upper bound: 2000 Mathematics Subject Classification35J15, 60J60.     

Gradient estimates and Harnack inequalities for diffusion equation on Riemannian manifolds

We derive the gradient estimates and Harnack inequalities for positive solutions of the diffusion equation u _t = Δu ^m on Riemannian manifolds. Then, we prove a Liouville type theorem.     

Circle Actions and Morse Theory on Quaternion-Kahler Manifolds

In this paper we study quaternion-Kähler manifolds endowedwith an isometric S¹-action. We consider the corresponding momentmap µ and prove that the only compact quaternion-Kählermanifold with positive scalar curvature which admits an isometriccircle action free on µ^–1(0) is the quaternionicprojective space HPⁿ.     

Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds

In this paper, we consider the generalized solutions of the inequality $$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$ on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality ? Δu ≥ u ^q, q > 1, on spherically symmetric noncompact Riemannian manifolds.     

Perturbation Solutions of the Whittaker-Hill Equation

When the Helmholtz equation ²V+k²V = 0 is separated in the generalparaboloidal co-ordinate system, the three ordinary differentialequations obtained each take, after a suitable change of variable,the form of the Whittaker-Hill equation. For the case k²<0,a considerable amount is known about the periodic solutionsof this equation. For k²>0, however, very little is so farknown. In this paper solutions of the Whittaker-Hill equationfor small positive k² are derived. These are the first explicitsolutions to be obtained for the case k²>0, and they couldbe employed to solve the Dirichlet or Neumann problem for ageneral paraboloid when k² is small. Three limiting cases arenoted, involving the reduction of the solutions to Mathieu functionsand the reduction of the co-ordinate system to the rotation-paraboloidalsystem.     

Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

This paper is devoted to the study of an error estimate of thefinite volume, approximation to the solution u L(R_N x R) ofthe equation u_t + div(Vf(u)) = 0, where v is a vector functiondepending on time and space. A 'h' error estimate for an initialvalue in BV(R^N) is shown for a large variety of finite volumemonotonous flux schemes, with an explicit or implicit time discretization.For this purpose, the error estimate is given for the generalsetting of approximate entropy solutions, where the error isexpressed in terms of measures in R^N and R^N x R. The study ofthe implicit schemes involves the study of the existence anduniqueness of the approximate solution. The cases where an 'h'error estimate can be achieved are also discussed.     

Global gradient estimate on graph and its applications

Continuing our previous work(ar Xiv:1509.07981v1),we derive another global gradient estimate for positive functions,particularly for positive solutions to the heat equation on finite or locally finite graphs.In general,the gradient estimate in the present paper is independent of our previous one.As applications,it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs.These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al.[J.Differential Geom.,99,359–409(2015)].In many topics,such as eigenvalue estimate and heat kernel estimate(not including the Liouville type theorems),replacing the Li–Yau inequality by the global gradient estimate,we can get similar results.     

The Conormal Derivative Problem for Elliptic Equations with BMO Coefficients on Reifenberg Flat Domains

We study the inhomogeneous conormal derivative problem for thedivergence form elliptic equation, assuming that the principalcoefficients belong to the BMO space with small BMO semi-normsand that the boundary is -Reifenberg flat. These conditionsfor the W^{1, p}-theory not only weaken the requirements on thecoefficients but also lead to a more general geometric conditionon the domain. In fact, the Reifenberg flatness is the minimalregularity condition for the W^{1, p}-theory. 2000 MathematicsSubject Classification 35R05 (primary), 35J15 (secondary).     

Riemann流形上改进的p-Laplace方程的梯度估计

本文研究Riemann流形上的改进的p-Laplace方程,运用截断函数的估计、Hessian比较定理和Laplace比较定理,得到该方程正解的梯度估计.并应用该结论,得到在Riemann流形上关于改进的p-Laplace方程正解的Harnack不等式和Liouville型定理.     

Backlund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magnetostatic atmosphere

The equations of magnetostatic equilibria for a plasma in agravitational field are investigated analytically. For equilibriawith an ignorable spatial coordinate, the equations reduce toa single nonlinear elliptic equation for the magnetic potentialu known as the Grad-Shafranov equation. By specifying the arbitraryfunctions in this equation, a Liouville equation is obtained.Bäcklund transformations are described and applied to obtainexact solutions for the Liouville equation modelling an isothermalmagnetostatic atmosphere, in which the current density J isproportional to the exponential of the magnetic potential andmoveover falls off exponentially with distance vertical to thebase with an e-folding distance equal to the gravitational scaleheight.     

Gradient estimates on the weighted p-Laplace heat equation

In this paper, by a regularization process we derive new gradient estimates for positive solutions to the weighted p-Laplace heat equation when the m-Bakry–Émery curvature is bounded from below by ?K for some constant $K\ge 0$. When the potential function is constant, which reduce to the gradient estimate established by Ni and Kotschwar for positive solutions to the p-Laplace heat equation on closed manifolds with nonnegative Ricci curvature if $K\searrow 0$, and reduce to the Davies, Hamilton and Li–Xu's gradient estimates for positive solutions to the heat equation on closed manifolds with Ricci curvature bounded from below if $p=2$.     

Existence Theorems for the Linear, Space-inhomogeneous Transport Equation

This paper studies existence problems in L¹ for the linear,space-inhomogeneous Boltzmann equation with periodic or (perfectly)absorbing boundary conditions under realistic assumptions onthe cross-sections. By an iteration technique, solutions arefirst constructed to an integral equation variant of the transportequation in the case of bounded impact parameters and an L¹type of cross-sections. They are then used to study the existenceof solutions of a measure form of the transport equation inthe case of unbounded impact parameters. These solutions conservemass. Estimates of their higher moments are also given. In particularthe results hold for inverse kth-power forces with 3 < k 5.     

Stationary Critical Points of the Heat Flow in Spaces of Constant Curvature

The paper considers stationary critical points of the heat flowin sphere S^N and in hyperbolic space H^N, and proves severalresults corresponding to those in Euclidean space R^N which havebeen proved by Magnanini and Sakaguchi. To be precise, it isshown that a solution u of the heat equation has a stationarycritical point, if and only if u satisfies some balance lawwith respect to the point for any time. In Cauchy problems forthe heat equation, it is shown that the solution u has a stationarycritical point if and only if the initial data satisfies thebalance law with respect to the point. Furthermore, one point,say x₀, is fixed and initial-boundary value problems are consideredfor the heat equation on bounded domains containing x₀. It isshown that for any initial data satisfying the balance law withrespect to x₀ (or being centrosymmetric with respect to x₀)the corresponding solution always has x₀ as a stationary criticalpoint, if and only if the domain is a geodesic ball centredat x₀ (or is centrosymmetric with respect to x₀, respectively).     

Integral stable manifolds in Banach spaces

We establish the existence of smooth integral stable manifoldsfor sufficiently small perturbations of nonuniform exponentialdichotomies in Banach spaces. We also consider the case of anonautonomous dynamics given by a sequence of C¹ maps. The optimalsmoothness of the manifolds is obtained at the same time astheir existence, using a convenient lemma of Henry. Furthermore,we obtain not only the exponential decay of the dynamics alongthe stable manifolds, but also of its derivative. In addition,we give a characterization of the stable manifolds in termsof the maximal exponential growth rate that is allowed, we discusshow the manifolds vary with the perturbations, and we discusstheir equivariance with respect to a sequence of linear operators.     

Increased Accuracy Cubic Spline Solutions to Two-Point Boundary Value Problems

The relation between finite difference approximation and cubicspline solutions of a two-point boundary value problem for thedifferential equation y' +f(x)y^'+g(x)y = r(x) has been consideredin a previous paper. The present paper extends the analysisto the integral equation formulation of the problem. It is shownthat an improvement in accuracy (local truncation error O(h⁶)rather than O(h⁴)) now results from a cubic spline approximationand that for the particular case f(x) 0 the resulting recurrencerelations have a form and accuracy similar to the well-knownNumerov formula. For this case also a formula with local truncationerror O(h⁸) is derived.     

Cocompact Spherical-Euclidean Spaceform Groups of Infinite VCD

We exhibit closed manifolds M covered by S^2n–1 x R^k forall n 2 and for sufficiently large k, with fundamental groupsof infinite virtual cohomological dimension. These examplesare based on results of Raghunathan on lattices in covers ofspin and symplectic groups, and address a problem first raisedby Wall.     

On steady and large time solutions of the semi-discrete Moving Finite Element equations for one-dimensional diffusion problems

This paper considers how the Moving Finite Element (MFE) methodapproxim ates the steady and large time solutions of a familyof linear diffusion equations in one space dimension. In particular,it is shown that any steady solution to the Moving Finite Elementequations must satisfy the stationary equations for a best approximationto the steady solution of the PDE from the manifold of free-knotlinear splines, in some problem dependent norm. For the special case of the inhomogeneous linear heat equationit is also shown that, under certain conditions, the only steadyMFE solution is the unique global best fit to the true steadysolution, in the H¹ semi-norm. It is also demonstrated numericallythat these steady solutions are stable attractors. Finally,a numerical study of the large time solutions of the homogeneouslinear heat equation is undertaken and it is demonstrated thatthe MFE solutions appear to possess a rather novel temporalaccuracy property.     

A Critical Phenomenon for Sublinear Elliptic Equations in Cone-Like Domains

The authors of this paper study positive supersolutions to theelliptic equation -u = c|x|^–su^p in Cone-like domains ofR^N (N 2), where p, s R and c > 0. They prove that in thesublinear case p < 1 there exists a critical exponent p_*> 1 such that the equation has a positive supersolution ifand only if – < p < p_*. The value of p_* is determinedexplicitly by s and the geometry of the cone. 2000 MathematicsSubject Classification 35J60 (primary), 35B05, 35R45 (secondary).     

The Existence of Periodic and Subharmonic Solutions of Subquadratic Second Order Difference Equations

By critical point theory, a new approach is provided to studythe existence of periodic and subharmonic solutions of the secondorder difference equation where f C(R x R^m, R^m), f(t+M,z)+f(t,z) for any (t, z)R x R^mand M is a positive integer. This is probably the first timecritical point theory has been applied to deal with the existenceof periodic solutions of difference systems.     

The norm of the Riemann-Liouville operator on Lp[0,1]: a probabilistic approach

We obtain explicit lower and upper bounds for the norm of theRiemann–Liouville operator V^s on L^p[0, 1] which are asymptoticallysharp, thus completing previous results by Eveson. Similar statementsare shown with respect to the norms ||V^s f||_p, whenever f satisfiescertain smoothness properties. It turns out that the correctrate of convergence of ||V^s f||_p as s depends both on theinfimum of the support of f and on the degree of smoothnessof f. We use a probabilistic approach which allows us to giveunified proofs.