Long time behavior of heat kernels of operators with unbounded drift terms

Given a second-order elliptic operator on Rd, with bounded diffusion coefficients and unbounded drift, which is the generator of a strongly continuous semigroup on L²(Rd) represented by an integral, we study the time behavior of the integral kernel and prove estimates on its decay at infinity. If the diffusion coefficients are symmetric, a local lower estimate is also proved.     

Quasi-L p -contractive analytic semigroups generated by elliptic operators with complex unbounded coefficients on arbitrary domains

Let ?? be a domain in ? ^N and consider a second order linear partial differential operator A in divergence form on ?? which is not required to be uniformly elliptic and whose coefficients are allowed to be complex, unbounded and measurable. Under rather general conditions on the growth of the coefficients we construct a quasi-contractive analytic semigroup $(e^{-t A_{V}})_{t\geqslant0}$ on L ²(??,dx), whose generator A _V gives an operator realization of A under general boundary conditions. Under suitable additional conditions on the imaginary parts of the diffusion coefficients, we prove that for a wide class of boundary conditions, the semigroup $(e^{-t A_{V}})_{t\geqslant0}$ is quasi-L ^p -contractive for 1<p<??. Similar results hold for second order nondivergence form operators whose coefficients satisfy conditions similar to those on the coefficients of the operator A, except for some further requirements on the diffusion coefficients. Some examples where our results can be applied are provided.     

Riesz Transforms Associated with Higher-Order Schrödinger Type Operators

Let L = L ₀ + V be a Schrödinger type operator, where L ₀ is a higher order elliptic operator with bounded complex coefficients in divergence form and V is a signed measurable function. Under the strongly subcritical assumption on V, we study the L ^q boundedness of Riesz transform ? ^m L ^?1/2 for q ≤ 2 based on the off-diagonal estimates of semigroup e ^{?t L} . Furthermore, the authors impose extra regularity assumptions on V to obtain the L ^q boundedness of Riesz transform ? ^m L ^?1/2 for some q > 2. In particular, these results are applied to the more interesting Schrödinger operators L = P(D) + V, where P(D) is any homogeneous positive elliptic operator with constant coefficients.     

Decoupling of Modes for the Elastic Wave Equation in Media of Limited Smoothness

We establish a decoupling result for the P and S waves of linear, isotropic elasticity, in the setting of twice-differentiable Lamé parameters. Precisely, we show that the P?S components of the wave propagation operator are regularizing of order one on L ² data, by establishing the diagonalization of the elastic system modulo a L ²-bounded operator. Effecting the diagonalization in the setting of twice-differentiable coefficients depends upon the symbol of the conjugation operator having a particular structure.     

Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients

In this paper we extend recent results on the existence and uniqueness of solutions of ODEs with non-smooth vector fields to the case of martingale solutions, in the Stroock-Varadhan sense, of SDEs with non-smooth coefficients. In the first part we develop a general theory, which roughly speaking allows to deduce existence, uniqueness and stability of martingale solutions for Ld-almost every initial condition x whenever existence and uniqueness is known at the PDE level in the L^∞-setting (and, conversely, if existence and uniqueness of martingale solutions is known for Ld-a.e. initial condition, then existence and uniqueness for the PDE holds). In the second part of the paper we consider situations where, on the one hand, no pointwise uniqueness result for the martingale problem is known and, on the other hand, well-posedness for the Fokker-Planck equation can be proved. Thus, the theory developed in the first part of the paper is applicable. In particular, we will study the Fokker-Planck equation in two somehow extreme situations: in the first one, assuming uniform ellipticity of the diffusion coefficients and Lipschitz regularity in time, we are able to prove existence and uniqueness in the L²-setting; in the second one we consider an additive noise and, assuming the drift b to have BV regularity and allowing the diffusion matrix a to be degenerate (also identically 0), we prove existence and uniqueness in the L^∞-setting. Therefore, in these two situations, our theory yields existence, uniqueness and stability results for martingale solutions.     

A priori error analysis of a discontinuous Galerkin approximation for a kind of compressible miscible displacement problems

A kind of compressible miscible displacement problems which include molecular diffusion and dispersion in porous media are investigated.A symmetric interior penalty discontinuous Galerkin (SIPG) method is applied to the coupled system of flow and transport.Using the induction hypotheses instead of the cut-off operator and the interpolation projection properties,a priori hp error estimates are presented.The error bounds in L2(H1) norm for concentration and in L∞(L2) norm for velocity are optimal in h and suboptimal in p with a loss of power 1/2.     

The structurally stable linear systems on the half-line are those with exponential dichotomies

Let A be a selft-adjoint operator on the Hilbert space L²Ω, ?) = {u ε L_loc²(Ω)|∫_Ω|² ?(x)dx < + ∞} defined by means of a closed, semibounded, sesquilinear form a(·, ·). We obtain a necessary and sufficuents condition for the spectrum of A to be discrete. We apply this result to a Sturm-Liouville problem for an elliptic operator with discontinuous coefficients defined on an unbounded domain and to the study of the spectrum of a Hamiltonian defined by a pseudodifferential operator.     

The Neumann Problem on Unbounded Domains of ℝ d and Stochastic Variational Inequalities

ABSTRACT

An elliptic equation with Neumann boundary conditions and unbounded drift coefficients is studied in a space L ²(? ^d , ν) where ν is an invariant measure. The corresponding semigroup generated by the elliptic operator is identified with the transition semigroup associated with a stochastic variational inequality.     

Systems of p-Laplacian differential inclusions with large diffusion

In this paper we consider coupled systems of p-Laplacian differential inclusions and we prove, under suitable conditions, that a homogenization process occurs when diffusion parameters become arbitrarily large. In fact we obtain that the attractors are continuous at infinity on L²(Ω)×L²(Ω) topology, with respect to the diffusion coefficients, and the limit set is the attractor of an ordinary differential problem.     

Invariant measures and regularity properties of perturbed Ornstein-Uhlenbeck semigroups

This paper deals with perturbations of the Ornstein-Uhlenbeck operator on L²-spaces with respect to a Gaussian measure μ. We perturb the generator of the Ornstein-Uhlenbeck semigroup by a certain unbounded, non-linear drift, and show various properties of the perturbed semigroup such as compactness and positivity. Strong Feller property, existence and uniqueness of an invariant measure are discussed as well.     

The inverse problem for small random perturbations of dynamical systems

What can one get to know about the dynamical system from its small random perturbation? What can one say about solutions of an ordinary differential equation \(\dot x_1 = B(x_1 )\) having some information on its singular perturbation operatorL ^?=?L+(B,?) withL being an elliptic second order operator? These problems are studied in the paper.     

Regularization with differential operators. I. General theory

The method of regularization is used to obtain least squares solutions of the linear equation Kx = y, where K is a bounded linear operator from one Hilbert space into another and the regularizing operator L is a closed densely defined linear operator. Existence, uniqueness, and convergence analyses are developed. An application is given to the special case when K is a first kind integral operator and L is an nth order differential operator in the Hilbert space L²[a, b].     

The L resolvents of elliptic operators with uniformly continuous coefficients

We consider the strongly elliptic operator A of order 2m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous. For 1<p<∞, A is a bounded linear operator from the L^p Sobolev space H^m,p into H^−m,p. We will prove that (A−λ)⁻¹ exists in H^−m,p for some λ and estimate its operator norm.     

Essential self-adjointness of Schrödinger-type operators

The essential self-adjointness of the strongly elliptic operator L = ∑_j,k=1ⁿ (?_j ? ib_j(x)) a_jk(x)(?_k ? ib_k(x)) + q(x) acting on C₀^∞(Rⁿ) is considered, where the matrix (a_jk) is real and symmetric, b_j and q are real, a_jk and b_j are sufficiently smooth, and q?L_loc². It has been shown by Ural'ceva and also Laptev that if q is bounded below and n ? 3 the minimal operator may not be self-adjoint if the principal coefficients rise too rapidly. Thus a theorem of Weyl for ordinary differential operators does not extend to partial differential operators. In this paper it is shown that if q is bounded below and if the principal coefficients are “well behaved” within a sequence of closed shells which go to infinity, then the minimal operator is self-adjoint. It is also shown that a number of results which were known to be true when q is sufficiently smooth may be extended to include the case where q?L_loc². The principal tools used are a distribution inequality due to Tosio Kato and a general maximum principle due to Walter Littman.     

Nonlinear reflecting diffusion process,and the propagation of chaos and fluctuations associated

A nonlinear diffusion satisfying a normal reflecting boundary condition is constructed and a result of propagation of chaos for a system of interacting diffusing particles with normal reflecting boundary conditions is proven. Then a gaussian limit for the fluctuation field which is defined in L₀²(B) of a Wiener type space B is obtained. The covariance of the gaussian limit is computed in terms of a Hilbert-Schmidt operator on L₀²(B).     

Elliptic boundary value problems with λ-dependent boundary conditions

In this paper second order elliptic boundary value problems on bounded domains Ω⊂Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L²(Ω)⊕(L²m(∂Ω)), which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.     

On the A-obstacle problem and the Hausdorff measure of its free boundary

In this paper, we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L ¹-data. We also extend the Lewy?CStampacchia inequalities to the general framework of L ¹-data and show convergence and stability results. We then prove that the free boundary has finite (N ? 1)-Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p?>?2.     

Evolution of a semilinear parabolic system for migration and selection in population genetics

The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape (i.e., a bounded, open domain in R^d). The selection coefficients depend on position and may depend on the gene frequencies; the drift and diffusion coefficients may depend on position. Sufficient conditions are given for the global loss of an allele and for its protection from loss. A sufficient condition for the existence of at least one internal equilibrium is also offered, and the profile of any internal equilibrium in the zero-migration limit is obtained.     

Construction of isospectral deformations of differential operators with periodic coefficients

Isospectral deformations of differential operators with periodic coefficients are constructed by modifying a method due to Burchnall and Chaundy. If the commutant of a differential operator L of order at least two consists of polynomials in L, then L admits holomorphic families of isospectral deformations of every positive dimension. The methods are independent of the order of the operator L.     

Boundary value problem for second-order differential operators with integral conditions

In this article, we study a second-order differential operator with mixed nonlocal boundary conditions combined weighting integral boundary condition with another two-point boundary condition. Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called regular and nonregular cases, we prove that the resolvent decreases with respect to the spectral parameter in L ^p ?(0,?1), but there is no maximal decreasing at infinity for p?>?1. Furthermore, the studied operator generates in L ^p ?(0,?1) an analytic semigroup for p?=?1 in regular case, and an analytic semigroup with singularities for p?>?1 in both cases, and for p?=?1 in the nonregular case only. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with nonregular boundary conditions.