Class Of Elliptic Operators Research Articles

The Calderón projector for fibred cusp operators

A Calderón projector for an elliptic operator P on a manifold with boundary X is a projection from general boundary data to the set of boundary data of solutions u of Pu=0. Seeley proved in 1966 that for compact X and for P uniformly elliptic up to the boundary there is a Calderón projector which is a pseudodifferential operator on ∂X. We generalize this result to the setting of fibred cusp operators, a class of elliptic operators on certain non-compact manifolds having a special fibred structure at infinity. This applies, for example, to the Laplacian on certain locally symmetric spaces or on particular singular spaces, such as a domain with cusp singularity or the complement of two touching smooth strictly convex domains in Euclidean space. Our main technical tool is the ϕ-pseudodifferential calculus introduced by Mazzeo and Melrose.In our presentation we provide a setting that may be useful for doing analogous constructions for other types of singularities.

Maximal regularity for gradient systems with boundary degeneracy

We study a class of elliptic operators L that degenerate at the boundary of a bounded open set O ⊂ Rd and possess a symmetrizing invariant measure μ. Such operators are associated with diffusion processes in O which are invariant for time reversal. After showing that the corresponding elliptic equation λφ − Lφ = f has a unique weak solution for any λ > 0 and f ∈ L2(O, μ), we obtain new results for the characterization of the domain of L. Capitolo della scienza: Partial Differentiaol Equations AMS Subject Classification (2000): 35K65,47D10, 60H10

On a class of elliptic operators with unbounded diffusion coefficients

We prove that, for $-\infty <\alpha\leq 2$, $1 < p <\infty$, the operator $L = (1+|x|^2)^\frac{\alpha}{2}\sum_{i,j=1}^N a_{ij}(x)D_{ij}$ generates an analytic semigroup in $L^p(\mathbb{R}^N)$ when the diffusion coefficients $a_{ij}$ admit a limit at infinity.

Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension

The paper is devoted to the exposition of results announced in [1]. We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space ℝn to a similar calculus in the space of sections of a one-dimensional fiber bundle ξ on the 2n-dimensional torus \(\mathbb{T}^{2n} \). This reduction enables us to identify the Schwartz space on ℝn with the space of smooth sections Γ∞(T2n, ξ), compare the Sobolev norms on the corresponding spaces and pseudodifferential operators in them, and describe the class of elliptic operators that reduce to Fredholm operators in Sobolev norms. Thus, for a natural class of elliptic pseudodifferential operators on a noncompact manifold of ℝn, we construct an index formula in accordance with the classical Atya-Singer formula.

A weighted W 2 , p -a priori bound for a class of elliptic operators

We prove a weighted -a priori bound, , for a class of uniformly elliptic second-order differential operators on unbounded domains. We deduce a uniqueness and existence result for the solution of the related Dirichlet problem. MSC: 35J25, 35B45, 35R05.

Uniform Sobolev Inequality along the Sasaki–Ricci Flow

In this paper we prove a uniform Sobolev inequality along the Sasaki–Ricci flow. In the process, we develop the theory of basic Lebesgue and Sobolev function spaces, and prove some general results about the decomposition of the heat kernel for a class of elliptic operators on a Sasaki manifold.

Use of tensor formats in elliptic eigenvalue problems

SUMMARYWe investigate approximations to eigenfunctions of a certain class of elliptic operators in Rd by finite sums of products of functions with separated variables and especially conditions providing an exponential decrease of the error with respect to the number of terms. The consistent use of tensor formats can be regarded as a base for a new class of rank‐truncated iterative eigensolvers. The computational cost is almost linear in the univariate problem size n, while traditional method scale like nd. Tensor methods can be applied to solving large‐scale spectral problems in computational quantum chemistry, for example, the Schrödinger, Hartree–Fock and Kohn–Sham equations in electronic structure calculations. The results of numerical experiments clearly indicate the linear‐logarithmic scaling of the low‐rank tensor method in n. The algorithms work equally well for the computation of both minimal and maximal eigenvalues of the discrete elliptic operators. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Pointwise gradient estimates in exterior domains

We consider a class of elliptic operators \( {\user1{\mathcal{A}}} \) with unbounded coefficients in a smooth exterior domain Ω and we prove that the Cauchy-Neumann problem associated with \( {\user1{\mathcal{A}}} \) admits, for any bounded and continuous initial datum, a unique bounded classical solution. We also provide pointwise gradient estimates for such a solution.

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For L in some class of elliptic operators, we study weighted norm L p inequalities for singular “non-integral” operators arising from L; those are the operators φ ( L ) for bounded holomorphic functions φ, the Riesz transforms ∇ L − 1 / 2 (or ( − Δ ) 1 / 2 L − 1 / 2 ) and its inverse L 1 / 2 ( − Δ ) − 1 / 2 , some quadratic functionals g L and G L of Littlewood–Paley–Stein type and also some vector-valued inequalities such as the ones involved for maximal L p -regularity. For each, we obtain sharp or nearly sharp ranges of p using the general theory for boundedness of Part I and the off-diagonal estimates of Part II. We also obtain commutator results with BMO functions.

Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators

Let M be a manifold and let L be a sufficiently smooth second order elliptic operator in M such that (M, L) is a transient pair. It is first shown that if L is symmetric with respect to some density in M, there exists a positive L-harmonic function in M which dominates L-Green’s function at infinity. Other classes of elliptic operators are investigated and examples are constructed showing that this property may fail if the symmetry assumption is removed. Another part of the paper deals with the existence of critical points for certain L-harmonic functions with periodicity properties. A class of small perturbations of second order elliptic operators is also described.

Phragmen–Lindelöf theorems and the asymptotic behaviour of solutions of quasilinear elliptic equations in slabs

The asymptotic behaviour of solutions of second-order quasilinear elliptic partial differential equations defined on unbounded domains in Rn contained in strips (when n = 2) or slabs (when n &gt; 2) is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data have appropriate asymptotic behaviour at infinity. We prove Phragmèn–Lindelöf theorems for large classes of elliptic operators, including uniformly elliptic operators and operators with well-defined genre, establish exponential decay estimates for uniformly elliptic operators when the Dirichlet boundary data vanish outside a compact set, establish the uniqueness of solutions, and give examples of solutions for non-uniformly elliptic operators which decay but do not decay exponentially. Our principal theorems are proven using special barrier functions; these barriers are constructed by considering an operator associated to our original operator.

Problème de Dirichlet asymptotique et diffusion associés à un opérateur elliptique sur une variété de Cartan-Hadaniard à courbures non minorées

We prove that, in a Cartan-Hadamard manifold, and w.r.t. a class of elliptic operators, the asymptotic Dirichlet problem is solvable if the sectional curvatures satisfy− C e 2 λr ⩽ K ⩽ −1, with λ < 1 4 .

On a class of elliptic operators with nonpolynomial degeneracy on a boundary

On a class of elliptic operators with nonpolynomial degeneracy on a boundary

Essential Spectra of a Class of Elliptic Operators

On determine le spectre essentiel d'une classe d'operateurs elliptiques de type Schrodinger. Pour controler les perturbations on utilise la limitation et la compacite des operateurs pseudo-differentiels relies

Equisummability of eigenfunction expansions under analytic multipliers

In this paper we present some abstract criteria for equisummability of expansions in eienfunctions of certain pairs of elliptic operators on general domains of [w”. The criteria to be developed originate in some cases from general Banach space arguments, and in others from the more specific spatial nature of the differential operators involved. The importance of our analysis of L”-equisummability of two operators is that the question of convergence of the two summability means is reduced essentially to showing that the difference of the modified resolvent operators is uniformly bounded. We apply these criteria to give a simple proof of an equisummability result of Gurarie and Kon [4] for a certain class of elliptic operators whose leading terms are positive and lower order terms have coefficients which are singular on a nowhere dense set. The key technique is to analyze kernels of the resolvents and use ,&‘-radial bounds of the resolvents developed in [3]. The prototypical case for equisummability is found in the study of equiconvergence for differential operators. Classically Haar [ 51, Walsh [lo], Birkhoff [2], and Tamarkin [lo] have shown that the difference between expansions with respect to eigenfunctions of a Sturm-Liouville operator (or Birkhoff series) and the ordinary Fourier series tends to zero uniformly in every finite interval. In this case the multiplier or summator function is the characteristic function. In some one-dimensional cases where such equiconvergence fails, Stone [9], Levitan and Sargsjan [6], and Benzinger [1] have showed equisummability for Riesz typical means of eigenfunction expansions of differential operators, where the summator function is

Some regularity theorems for operators in an enveloping algebra

The regularity properties of linear elliptic differential operators, acting on various function spaces, have been intensively studied in recent years (cf. [2]). In this paper, we consider the corresponding problem for an elliptic operator L in the enveloping algebra of a real Lie algebra r~, acting in a Hilbert space X(V) via the differential &r of a unitary representation r of the Lie group G. This problem was first studied by Nelson and Stinespring [16]. Later Langlands [14] extended their results to include representations on Banach spaces. In both cases the operator h-(L) and its adjoint were studied primarily via the action of the right invariant differential operatorL on the representative functions on G, using the standard elliptic regularity theory. Although this method was sufficient, e.g., to show that h(L) is essentially self-adjoint provided L is elliptic and symmetric (in the case of unitary rr), it did not give the precise domain of the adjoint of i%-(L). This additional information is very useful in the application of elliptic operators, e.g., to constructing analytic vectors for a representation, and in establishing regularity properties for the eigenfunction expansion associated with h(L)* (in case of symmetric L and unitary n). In this paper we establish some domain results for a restricted class of elliptic operators in the enveloping algebra of 0, relative to a unitary representation V. Iiamely, if 1, is of order 2m and is associated with a Hermitian elliptic form (see Section 2 for definition), then the Hilbert-space adjoint of h(L) has as domain the space .%‘zm(~) of 2m-times differentiable vectors for 7r. The proof of this result involves two developments. The first (Section 1) consists of establishing regularity properties for the action of R on the chain