A Discrete Leading Symbol and Spectral Asymptotics for Natural Differential Operators

We initiate a systematic study of natural differential operators in Riemannian geometry whose leading symbols are not of Laplace type. In particular, we define a discrete leading symbol for such operators which may be computed pointwise, or from spectral asymptotics. We indicate how this can be applied to the computation of another kind of spectral asymptotics, namely asymptotic expansions of fundamental solutions, and to the computation of conformally covariant operators.     

Heat Kernel Bounds and Homogenization of Elliptic Operators

We obtain upper and lower bounds on the heat kernel of a uniformlyelliptic operator in divergence form with measurable coefficients.From these bounds one can deduce not only the familiar short-timeasymptotics of the heat kernel, but in the case of periodiccoefficients, long-time asymptotics involving homogenizationof the coefficients.     

Regularity Theorems and Heat Kernel for Elliptic Operators

We give a new proof of Aronson's upper gaussian bound on theheat kernel for parabolic equations with time-independent realmeasurable coefficients. This approach also gives gaussian boundsin the case of a complex perturbation of real coefficients andin the case of uniformly continuous and complex-valued coefficients.     

Differential Operators Specifying the Solution of an Elliptic Iterated Equation

Ukrainian Mathematical Journal - We construct differential operators that transform arbitrary holomorphic functions into regular solutions of elliptic equations of the second and higher orders. The...     

Refined Asymptotics for the Infinite Heat Equation with Homogeneous Dirichlet Boundary Conditions

The nonnegative viscosity solutions to the infinite heat equation with homogeneous Dirichlet boundary conditions are shown to converge as t → ∞ to a uniquely determined limit after a suitable time rescaling. The proof relies on the half-relaxed limits technique as well as interior positivity estimates and boundary estimates. The expansion of the support is also studied.     

Heat Kernel Estimates and LP Spectral Independence of Elliptic Operators

Let be an open subset of R^d, and let T_p for p[1, ) be consistentC₀-semigroups given by kernels that satisfy an upper heat kernelestimate. Denoting their generators by A_p, we show that thespectrum (A_p) is independent of p[1, ). We also treat the caseof weighted L^p-spaces for weights that satisfy a subexponentialgrowth condition. An example shows that independence of thespectrum may fail for an exponential weight. 1991 MathematicsSubject Classification 47D06, 47A10, 35P05.     

Elliptic Operators with Singular Coefficients

Mathematical Notes -     

Relaxation of Optimal Control Problem Governed by Semilinear Elliptic Equation with Leading Term Containing Controls

An optimal control problem governed by semilinear elliptic partial differential equation is considered. The equation is in divergence form with the leading term containing controls. A relaxed problem is constructed by homogenization. By studying the G-closure problem, a local representation of admissible set of relaxed control is given. Finally, the maximum principle of relaxed problem is established via homogenization spike variation.     

Perturbations of Pseudodifferential Operators with Negative Definite Symbol

Pseudodifferential operators with negative definite symbols appear as generators of jump-type Markov processes. The purpose of this paper is to treat the large jumps of the process by a perturbation approach for the generator. This is of particular interest since in this way the generators are made accessible to a symbolic calculus of pseudodifferential operators. The main auxiliary result consists of a characterization of tightness of the jump measures in terms of the symbol.     

Perturbations of Pseudodifferential Operators with Negative Definite Symbol

Pseudodifferential operators with negative definite symbols appear as generators of jump-type Markov processes. The purpose of this paper is to treat the large jumps of the process by a perturbation approach for the generator. This is of particular interest since in this way the generators are made accessible to a symbolic calculus of pseudodifferential operators. The main auxiliary result consists of a characterization of tightness of the jump measures in terms of the symbol.     

Parabolic Martin Boundaries and Asymptotic Formulas of Heat Kernels for One-Dimensional Elliptic Operators with Periodic Coefficients

We determine the Martin boundary of R × (???∞?, 0) for operators \(\partial_t+L\), where L is a one-dimensional second-order elliptic operator with periodic coefficients. We also give long-time asymptotic formulas of heat kernels generated by L in the case that the ratio of the spatial variable to the time variable is large or small.     

Recovery of a Rapidly Oscillating Source in the Heat Equation from Solution Asymptotics

Four problems are solved in which a high-frequency source in the one-dimensional heat equation with homogeneous initial–boundary conditions is recovered from partial asymptotics of its solution. It is shown that the source can be completely recovered from an incomplete (two-term) asymptotic representation of the solution. The formulation of each source recovery problem is preceded by constructing and substantiating asymptotics of the solution to the original initial–boundary value problem.     

p-Laplace Schrdinger热方程的椭圆型梯度估计

本文对p-Laplace Schr(o|¨)dinger热方程正连续弱解做了椭圆型梯度估计;作为应用,本文得到了一个关于p-Laplace算子的Liouville型结果;并证明了关于p-LaplaceSchr(o|¨)dinger热方程正连续弱解的Harnack不等式.     

Stable Asymptotics for Elliptic Systems on Plane Domains with Corners

We consider boundary value problems for elliptic systems in the sense of Agmon-Douglis-Nirenberg on plane domains with corners, where the domain, the coefficients of the operators and the right hand sides all depend on a parameter. We construct corner singularities in such a way that the corresponding decomposition of the solution into regular and singular parts is stable, i. e. the regular part and the coefficient of the singular functions depend smoothly on the parameter. The construction of these singular functions continues the paper [3] and generalizes results known for second order scalar boundary value problems — see [4,5] [12].     

On Second-Order Almost-Periodic Elliptic Operators

The paper considers second-order, strongly elliptic, operatorsH with complex almost-periodic coefficients in divergence formon R^d. First, it is proved that the corresponding heat kernelis Hölder continuous and Gaussian bounds are derived withthe correct small and large time asymptotic behaviour on thekernel and its Hölder derivatives. Secondly, it is establishedthat the kernel has a variety of properties of almost-periodicity.Thirdly, it is demonstrated that the kernel of the homogenization of H is the leading term inthe asymptotic expansion of t K_t.     

Cutting and Pasting of Elliptic Operators

If A : C∞E → C∞F is an elliptic operator between sections of vector bundles E, F over a closed smooth n-manifold X, Y a smooth (n – 1)-submanifold of X with trivial normal bundle, and Ψ = (ΨE, ΨF) a pair of automorphisms of E | Y and F | Y inducing a diffeomorphism f of Y and commuting with the principal symbol σ A of A over Y, then an elliptic operator A^Ψ is (uniquely up to operators of lower order) defined between sections of vector bundles E^Ψ, F^Ψ over a closed manifold X^f, all obtained by cutting and pasting the respective objects along Y. The difference index A^Ψ – index A is investigated and the relations with additivity properties of topological invariants and with classical transmission problems are explained.