Elliptic Partial Differential Operators Research Articles

Spectral shift functions and Dirichlet-to-Neumann maps

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh–Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.

Spherical Harmonics Expansion of Fundamental Solutions and Their Derivatives for Homogeneous Elliptic Operators

In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.

Quantum heat traces

We study new invariants of elliptic partial differential operators acting on sections of a vector bundle over a closed Riemannian manifold that we call the relativistic heat trace and the quantum heat traces. We obtain some reduction formulas expressing these new invariants in terms of some integral transforms of the usual classical heat trace and compute the asymptotics of these invariants. The coefficients of these asymptotic expansion are determined by the usual heat trace coefficients (which are locally computable) as well as by some new global invariants.

Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh M-functions, and a generalized index of unbounded meromorphic operator-valued functions

We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schrödinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl–Titchmarsh M-functions and Donoghue-type M-functions corresponding to closed extensions of symmetric operators belong to it.The principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of Robin realizations of non-self-adjoint Schrödinger operators, and more abstract pairs of closed operators in Hilbert spaces with the generalized index of the corresponding energy dependent Dirichlet-to-Neumann maps and abstract Weyl–Titchmarsh M-functions, respectively.

Heat determinant on manifolds

We introduce and study new invariants associated with Laplace type elliptic partial differential operators on manifolds. These invariants are constructed by using the off-diagonal heat kernel; they are not pure spectral invariants, that is, they depend not only on the eigenvalues but also on the corresponding eigenfunctions in a non-trivial way. We compute the first three low-order invariants explicitly.

Some properties of a class of elliptic partial differential operators

We prove Schwarz–Pick type estimates and coefficient estimates for a class of functions induced by the elliptic partial differential operators. Then we apply these results to obtain a Landau type theorem.

Existence and spectral theory for weak solutions of Neumann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients

In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated with second order linear degenerate elliptic partial differential operators X with rough coefficients, of the form X=−div(P∇)+HR+S′G+F, where the n×n matrix function P=P(x) is nonnegative definite and allowed to degenerate, R, S are families of subunit vector fields, G, H are vector valued functions and F is a scalar function. We operate in a geometric homogeneous space setting and we assume the validity of certain Sobolev and Poincaré inequalities related to a symmetric nonnegative definite matrix of weights Q=Q(x) that is comparable to P; we do not assume that the underlying measure is doubling. We give a maximum principle for weak solutions of Xu≤0, and we follow this with a result describing a relationship between compact projection of the degenerate Sobolev space QH1,p, related to the matrix of weights Q, into Lq and a Poincaré inequality with gain adapted to Q.

The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains ofRm(m≥3) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

Stability of square root domains associated with elliptic systems of PDEs on nonsmooth domains

We discuss stability of square root domains for uniformly elliptic partial differential operators La,Ω,Γ=−∇⋅a∇ in L2(Ω), with mixed boundary conditions on ∂Ω, with respect to additive perturbations. We consider open, bounded, and connected sets Ω∈Rn, n∈N\\{1}, that satisfy the interior corkscrew condition and prove stability of square root domains of the operator La,Ω,Γ with respect to additive potential perturbations V∈Lp(Ω)+L∞(Ω), p>n/2.Special emphasis is put on the case of uniformly elliptic systems with mixed boundary conditions.

Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions II

The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for self-adjoint $L^2(\Om; d^n x)$-realizations, of divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$.

Stochastic representation of solutions to degenerate elliptic and parabolic boundary value and obstacle problems with Dirichlet boundary conditions

We prove existence and uniqueness of stochastic representations for solutions to elliptic and parabolic boundary value and obstacle problems associated with a degenerate Markov diffusion process. In particular, our article focuses on the Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate, elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to terminal/boundary value or obstacle problems for the parabolic Heston operator correspond to value functions for American-style options on the underlying asset.

Holomorphic extension of fundamental solutions of elliptic linear partial differential operators of higher order with analytic coefficients

We prove that every fundamental solution of an elliptic linear partial differential operator with analytic coefficients in Ω open subset of Rn and simple complex characteristics can be holomorphically continued at least locally as a multi-valued analytic function x⟶E(x,y) in Cn up to the complex bicharacteristic conoid. This holomorphic extension is ramified around the bicharacteristic conoid and belongs to the Nilsson class.

Computation of eigenvalues by numerical upscaling

We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we consider benchmark multi-scale eigenvalue problems in reservoir modeling and material science. We compute a low-dimensional generalized (possibly mesh free) finite element space that preserves the lowermost eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of $$\text {H}^1_0(\Omega )$$H01(Ω) by means of a certain Clement-type quasi-interpolation operator.

Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions

One of the principal topics of this paper concerns the realization of self-adjoint operators L Θ,Ω in L 2(Ω; d n x) m , m, n ∈ ℕ, associated with divergence form elliptic partial differential expressions L with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains Ω ⊂ ℝ n . In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions L which act as $$Lu = - \left( {\sum\limits_{j,k = 1}^n {\partial _j } \left( {\sum\limits_{\beta = 1}^m {a_{j,k}^{\alpha ,\beta } \partial _k u_\beta } } \right)} \right)_{1 \leqslant \alpha \leqslant m} , u = \left( {u_1 , \ldots ,u_m } \right).$$ The (nonlocal) Robin-type boundary conditions are then of the form $$v \cdot ADu + \Theta [u|_{\partial \Omega } ] = 0{\text{ on }}\partial \Omega ,$$ where Θ represents an appropriate operator acting on Sobolev spaces associated with the boundary ∂Ω of Ω, ν denotes the outward pointing normal unit vector on ∂Ω, and $$Du: = \left( {\partial _j u_\alpha } \right)_{_{1 \leqslant j \leqslant n}^{1 \leqslant \alpha \leqslant m} } .$$ Assuming Θ ≥ 0 in the scalar case m = 1, we prove Gaussian heat kernel bounds for L Θ,Ω, by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on ∂Ω. We also discuss additional zero-order potential coefficients V and hence operators corresponding to the form sum L Θ,Ω + V.

Solvability and stationarity for the optimal control of variational inequalities with point evaluations in the objective functional

AbstractMotivated by applications in economics and engineering, we consider the optimal control of a variational inequality with point evaluations of the state variable in the objective. This problem class constitutes a specific mathematical program with complementarity constraints (MPCC). In our context, the problem is posed in an adequate function space and the variational inequality involves second order linear elliptic partial differential operators. The necessary functional analytic framework complicates the derivation of stationarity conditions whereas the non‐convex and non‐differentiable nature of the problem challenges the design of an efficient solution algorithm. In this paper, we present a penalization and smoothing technique to derive first order type conditions related to C‐stationarity in the associated Sobolev space setting. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

We prove weak and strong maximum principles, including a Hopf lemma, for C 2 subsolutions to equations defined by linear, second-order, linear, elliptic partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the C 2 subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol's vanishing locus. We obtain uniqueness and a priori maximum principle estimates for C 2 solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators with partial Dirichlet or Neumann boundary conditions. We also prove weak maximum principles and uniqueness for W 1, 2 solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions.

Liouville comparison principles for solutions of semilinear elliptic second-order partial differential inequalities

The purpose of this work is to obtain new Liouville comparison principles for entire weak solutions of semilinear elliptic second-order partial differential inequalities of the forms and on ℝ n , where n ≥ 2, q > 0 and ℒ is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergence form, in terms of the behaviour of the coefficients of the operator ℒ at infinity in a function space directly associated with ℒ.

A Family of Fundamental Solutions of Elliptic Partial Differential Operators with Real Constant Coefficients

We present a construction of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. The aim is to write detailed expressions for such functions. Such expressions are then exploited to prove regularity properties in the frame of Schauder spaces and jump properties of the corresponding single layer potentials.

Bounds on the non-real spectrum of differential operators with indefinite weights

Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and $$\infty $$ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm–Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.

Stability of Exponential Operator Splitting Methods for Noncontractive Semigroups

Operator splitting methods constitute an attractive class of time discretization schemes for evolution equations. Due to their computational advantages, they are widely used in scientific computation. Here we are interested in linear parabolic problems where the evolution is given by an analytic semigroup. Apart from the simple situation where the single operators generate semigroups of contractions, the stability of operator splitting methods in Banach spaces can be a delicate problem. In this paper we derive a Banach space framework that allows us to prove the stability of the Lie--Trotter splitting for noncontractive semigroups under reasonable assumptions on the involved operators. It is noteworthy that the same assumptions also imply stability of the Strang--Marchuk splitting. The stability result is verified with the help of a particular representation of the local error combined with an appropriate commutator bound. Our abstract stability result can be applied, in particular, to the dimension splitting of strongly elliptic partial differential operators with sufficiently smooth coefficients. A considerable part of the paper is devoted to the elaboration of this example and the verification of our assumptions.