Order Elliptic Operators Research Articles

CONTINUITY OF THE DOUBLE LAYER POTENTIAL OF A SECOND ORDER ELLIPTIC DIFFERENTIAL OPERATOR IN SCHAUDER SPACES ON THE BOUNDARY

AbstractWe prove the validity of a regularizing property on the boundary of the double layer potential associated with the fundamental solution of a nonhomogeneous second order elliptic differential operator with constant coefficients in Schauder spaces of exponent greater or equal to two that sharpens classical results of N.M. Günter, S. Mikhlin, V.D. Kupradze, T.G. Gegelia, M.O. Basheleishvili and T.V. Burchuladze, U. Heinemann and extends the work of A. Kirsch who has considered the case of the Helmholtz operator.

A General Dixmier Trace Formula for the Density of States on Open Manifolds

We give an abstract formulation of the Dixmier trace formula for the density of states. This recovers prior versions and allows us to provide a Dixmier trace formula for the density of states of second order elliptic differential operators on manifolds of bounded geometry satisfying a certain geometric condition. This formula gives a new perspective on Roe's index on open manifolds.

Correction: Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part

Correction: Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part

Maximal Function and Riesz Transform Characterizations of Hardy Spaces Associated with Homogeneous Higher Order Elliptic Operators and Ball Quasi-Banach Function Spaces

Maximal Function and Riesz Transform Characterizations of Hardy Spaces Associated with Homogeneous Higher Order Elliptic Operators and Ball Quasi-Banach Function Spaces

Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part

In the present paper we study perturbation theory for the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators L0=div(A0∇)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_0 = \ ext {div}(A_0\ abla )$$\\end{document} and L1=div(A1∇)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1 = \ ext {div}(A_1\ abla )$$\\end{document} such that the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} Dirichlet problem for L0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_0$$\\end{document} is solvable for some p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}; we show that if A0-A1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_0 - A_1$$\\end{document} satisfies certain Carleson condition, then the Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document} Dirichlet problem for L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} is solvable for some q≥p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q \\ge p$$\\end{document}. Moreover if the Carleson norm is small then we may take q=p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=p$$\\end{document}. We use the approach first introduced in Fefferman–Kenig–Pipher ’91 on the unit ball, and build on Milakis–Pipher–Toro ’11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} Dirichlet problem on a bounded Lipschitz domain for an operator L=div(A∇)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L = \ ext {div}(A\ abla )$$\\end{document}, where A satisfies a Carleson condition similar to the one assumed in Kenig–Pipher ’01 and Dindoš–Petermichl–Pipher ’07 but with unbounded antisymmetric part.

Lower bounds at infinity for solutions to second order elliptic equations

We study lower bounds at infinity for solutions to $$ |Pu|\leq M|x|^{-\delta_1}|\nabla u|+M|x|^{-\delta_{0}}|u| $$ where $P$ is a second order elliptic operator. Our results are of quantitative nature and generalize those obtained in [3,6].
 For more information see https://ejde.math.txstate.edu/Volumes/2023/69/abstr.html

Classes of Kernels and Continuity Properties of the Double Layer Potential in Hölder Spaces

We prove the validity of regularizing properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a nonhomogeneous second order elliptic differential operator with constant coefficients in Hölder spaces by exploiting an estimate on the maximal function of the tangential gradient with respect to the first variable of the kernel of the double layer potential and by exploiting specific imbedding and multiplication properties in certain classes of kernels of integral operators and a generalization of a result for integral operators on differentiable manifolds.

Approximations of delocalized eta invariants by their finite analogues

For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose M is a closed smooth spin manifold and \widetilde{M} is a \Gamma -regular covering space of M . Let \langle \alpha \rangle be the conjugacy class of a non-identity element \alpha\in \Gamma . Suppose \{\Gamma_i\} is a sequence of finite-index normal subgroups of \Gamma that distinguishes \langle \alpha \rangle . Let \pi_{\Gamma_i} be the quotient map from \Gamma to \Gamma/\Gamma_i and \langle \pi_{\Gamma_i}(\alpha) \rangle the conjugacy class of \pi_{\Gamma_i}(\alpha) in \Gamma/\Gamma_i . If the scalar curvature on M is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of \widetilde{M} at the conjugacy class \langle \alpha \rangle is equal to the limit of the delocalized eta invariants for the Dirac operators of M_{\Gamma_i} at the conjugacy class \langle \pi_{\Gamma_i}(\alpha) \rangle , where M_{\Gamma_i}= \widetilde{M}/\Gamma_i is the finite-sheeted covering space of M determined by \Gamma_i . In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of M_{\Gamma_i} at the conjugacy class \langle \pi_{\Gamma_i}(\alpha) \rangle converges, under the assumption that the rational maximal Baum–Connes conjecture holds for \Gamma .

A remark on quantitative unique continuation from subsets of the boundary of positive measure

The question of uniqueness of harmonic functions in a domain Ω ⊂ R d \Omega \subset \mathbb {R}^d with boundary ∂ Ω \partial \Omega , satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset of the boundary ω \omega of positive d − 1 d-1 dimensional measure has attracted a lot of attention (see e.g. V. Adolfsson and L. Escauriaza [Comm. Pure Appl. Math. 50 (1997), pp. 935–969]; F.-H. Lin [Comm. Pure Appl. Math. 44 (1991), pp. 287–308]; X. Tolsa [Comm. Pure Appl. Math. 76 (2023), pp. 305–336]). It is essentially a consequence of Carleman estimates when ω \omega contains an open subset of the boundary (uniqueness for second order elliptic operators across an hypersurface). The main open questions (about uniqueness) concern now Lipschitz domains and variable coefficients. Here, using results by Logunov and Malinnikova [Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimensions two and three, Birkhäuser/Springer, Cham, 2018, pp. 333-344; Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018, World Scientific Publishing, Hackensack, NJ, 2018, pp. 2391–2411], we consider the simpler case of W 2 , ∞ W^{2, \infty } domains but prove quantitative uniqueness both for Dirichlet and Neumann boundary conditions. As an application, we deduce quantitative estimates for the Dirichlet and Neumann Laplace eigenfunctions on a W 2 , ∞ W^{2, \infty } domain with boundary.

A primal dual mixed finite element method for inverse identification of the diffusion coefficient and its relation to the Kohn-Vogelius penalty method

We revisit the celebrated Kohn-Vogelius penalty method and discuss how to use it for the unique continuation problem where data is given in the bulk of the domain. We then show that the primal-dual mixed finite element methods for the elliptic Cauchy problem introduced in [1] (E. Burman, M. Larson, L. Oksanen, Primal-dual mixed finite element methods for the elliptic Cauchy problem, SIAM J. Num. Anal., 56(6), 2018) can be interpreted as a Kohn-Vogelius penalty method and modify it to allow for unique continuation using data in the bulk. We prove that the resulting linear system is invertible for all data. Then we show that by introducing a singularly perturbed Robin condition on the discrete level sufficient regularization is obtained so that error estimates can be shown using conditional stability. Finally we show how the method can be used for the identification of the diffusivity coefficient in a second order elliptic operator with partial data. Some numerical examples are presented showing the performance of the method for unique continuation and for impedance computed tomography with partial data.

Regularity and Neumann problems for operators with real coefficients satisfying Carleson conditions

In this paper, we continue the study of a class of second order elliptic operators of the form L=div(A∇⋅) in a domain above a Lipschitz graph in Rn, where the coefficients of the matrix A satisfy a Carleson measure condition, expressed as a condition on the oscillation on Whitney balls. For this class of operators, it is known (since 2001) that the Lq Dirichlet problem is solvable for some 1<q<∞. Moreover, further studies completely resolved the range of Lq solvability of the Dirichlet, Regularity, Neumann problems in Lipschitz domains, when the Carleson measure norm of the oscillation is sufficiently small.We show that there exists preg>1 such that for all 1<p<preg the Lp Regularity problem for the operator L=div(A∇⋅) is solvable. Furthermore 1preg+1q⁎=1 where q⁎>1 is the number such that the Lq Dirichlet problem for the adjoint operator L⁎ is solvable for all q>q⁎.Additionally when n=2, there exists pneum>1 such that for all 1<p<pneum the Lp Neumann problem for the operator L=div(A∇⋅) is solvable. Furthermore 1preg+1q⁎=1 where q⁎>1 is the number such that the Lq Dirichlet problem for the operator L1=div(A1∇⋅) with matrix A1=A/det⁡A is solvable for all q>q⁎.

Edge states for second order elliptic operators in a channel

We present a general framework to study edge states for second order elliptic operators in a half channel.We associate an integer valued index to some bulk materials, and we prove that, for any junction between two such materials, localized states must appear at the boundary whenever the indices differ.

Neumann boundary value problems for elliptic operators with measure-valued coefficients

In this paper, we prove that there exists a unique weak solution to the Neumann boundary value problem for second order elliptic operators whose coefficients are signed measures. We will give a probabilistic representation of the solution. The heat kernel estimates and reflected diffusion processes play important roles.

Realisations of elliptic operators on compact manifolds with boundary

This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of Bär-Ballmann to first order elliptic operators. The space of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calderón projectors which, in the first order case, is equivalent to results of Bär-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven.

On operator estimates in homogenization of nonlocal operators of convolution type

The paper studies a bounded symmetric operator Aε in L2(Rd) with(Aεu)(x)=ε−d−2∫Rda((x−y)/ε)μ(x/ε,y/ε)(u(x)−u(y))dy; here ε is a small positive parameter. It is assumed that a(x) is a non-negative L1(Rd) function such that a(−x)=a(x) and the moments Mk=∫Rd|x|ka(x)dx, k=1,2,3, are finite. It is also assumed that μ(x,y) is Zd-periodic both in x and y function such that μ(x,y)=μ(y,x) and 0<μ−⩽μ(x,y)⩽μ+<∞. Our goal is to study the limit behaviour of the resolvent (Aε+I)−1, as ε→0. We show that, as ε→0, the operator (Aε+I)−1 converges in the operator norm in L2(Rd) to the resolvent (A0+I)−1 of the effective operator A0 being a second order elliptic differential operator with constant coefficients of the form A0=−divg0∇. We then obtain sharp in order estimates of the rate of convergence.

Green function estimates for second order elliptic operators in non-divergence form with Dini continuous coefficients

Two-sided sharp Green function estimates are obtained for second order uniformly elliptic operators in non-divergence form with Dini continuous coefficients in bounded C1,1 domains, which are shown to be comparable to that of the Dirichlet Laplace operator in the domain. The first and second order derivative estimates of the Green functions are also derived. Moreover, boundary Harnack inequality with an explicit boundary decay rate and interior Schauder’s estimates for these differential operators are established, which may be of independent interest.

Generalized principal eigenvalues on $${\mathbb {R}}^{d}$$ of second order elliptic operators with rough nonlocal kernels

We study the generalized eigenvalue problem on the whole space for a class of integro-differential elliptic operators. The nonlocal operator is over a finite measure, but this has no particular structure. Some of our results even hold for singular kernels. The first part of the paper presents results concerning the existence of a principal eigenfunction. Then we present various necessary and/or sufficient conditions for the maximum principle to hold, and use these to characterize the simplicity of the principal eigenvalue.

A new symmetric interior penalty discontinuous Galerkin formulation for the Serre–Green–Naghdi equations

AbstractIn this work, we investigate the construction of a new discontinuous Galerkin discrete formulation to approximate the solution of Serre–Green–Naghdi (SGN) equations in the one‐dimensional horizontal framework. Such equations describe the time evolution of shallow water free surface flows in the fully nonlinear and weakly dispersive asymptotic approximation regime. A new non‐conforming discrete formulation belonging to the family of symmetric interior penalty discontinuous Galerkin methods is introduced to accurately approximate the solutions of the second order elliptic operator occurring in the SGN equations. We show that the corresponding discrete bilinear form enjoys some consistency and coercivity properties, thus ensuring that the corresponding discrete problem is well‐posed. The resulting global discrete formulation is then validated through an extended set of benchmarks, including convergence studies and comparisons with data taken from experiments.

Chernoff approximations of Feller semigroups in Riemannian manifolds

AbstractChernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non‐compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non‐compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.

A path integral formula for certain fourth order elliptic operators

A path integral formula for certain fourth order elliptic operators