Theory Of Elliptic Operators Research Articles

Equivariant Brill–Noether theory for elliptic operators and superrigidity of J-holomorphic maps

Abstract The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: Are the intersections nonempty? Are they smooth? What are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of J-holomorphic maps. As an application, we use our framework to give a concise exposition of Wendl’s proof of the superrigidity conjecture.

Generalized Carleson perturbations of elliptic operators and applications

We extend in two directions the notion of perturbations of Carleson type for the Dirichlet problem associated to an elliptic real second-order divergence-form (possibly degenerate, not necessarily symmetric) elliptic operator. First, in addition to the classical perturbations of Carleson type, that we call additive Carleson perturbations, we introduce scalar-multiplicative and antisymmetric Carleson perturbations, which both allow non-trivial differences at the boundary. Second, we consider domains which admit an elliptic PDE in a broad sense: we count as examples the 1-sided NTA (a.k.a. uniform) domains satisfying the capacity density condition, the 1-sided chord-arc domains, the domains with low-dimensional Ahlfors-David regular boundaries, and certain domains with mixed-dimensional boundaries; thus our methods provide a unified perspective on the Carleson perturbation theory of elliptic operators. Our proofs do not introduce sawtooth domains or the extrapolation method. We also present several applications to some Dahlberg-Kenig-Pipher operators, free-boundary problems, and we provide a new characterization of A ∞ A_{\infty } among elliptic measures.

Asymptotic Gluing of Shear-Free Hyperboloidal Initial Data Sets

We present a procedure for asymptotic gluing of hyperboloidal initial data sets for the Einstein field equations that preserves the shear-free condition. Our construction is modeled on the gluing construction in Isenberg et al. (Ann Henri Poincare 11(5):881–927, 2010), but with significant modifications that incorporate the shear-free condition. We rely on the special Holder spaces, and the corresponding theory for elliptic operators on weakly asymptotically hyperbolic manifolds, introduced by the authors in Allen et al. (Commun Anal Geom 26(1):1–61, 2018) and applied to the Einstein constraint equations in Allen et al. (Class Quantum Grav 33(11):115015, 2016).

Liouville type result and long time behavior for Fisher-KPP equation with sign-changing and decaying potentials

This paper concerns the Liouville type result for the general semilinear elliptic equation(S)aij(x)∂iju(x)+Kqi(x)∂iu(x)+f(x,u(x))=0a.e. in RN, where f is of the KPP-monostable nonlinearity, as a continuation of the previous works of the second author [31,32]. The novelty of this work is that we allow fs(x,0) to be sign-changing and to decay fast up to a Hardy potential near infinity. First, we introduce a weighted generalized principal eigenvalue and use it to characterize the Liouville type result for Eq. (S) that was proposed by H. Berestycki. Secondly, if (aij) is the identity matrix and q is a compactly supported divergence-free vector field, we find a condition that Eq. (S) admits no positive solution for K>K⋆, where K⋆ is a certain positive threshold. To achieve this, we derive some new techniques, thanks to the recent results on the principal spectral theory for elliptic operators [14], to overcome some fundamental difficulties arising from the lack of compactness in the domain. This extends a nice result of Berestycki-Hamel-Nadirashvili [5] on the limit of eigenvalues with large drift to the case without periodic condition. Lastly, the well-posedness and long time behavior of the evolution equation corresponding to (S) are further investigated. The main tool of our work is based on the maximum principle for elliptic and parabolic equations however it is far from being obvious to see if the comparison principle for (S) holds or not.

Pseudodifferential calculus on noncommutative tori, II. Main properties

This paper is the second part of a two-paper series whose aim is to give a detailed description of Connes’ pseudodifferential calculus on noncommutative [Formula: see text]-tori, [Formula: see text]. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension [Formula: see text]. This includes the main results mentioned in [2, 5, 11]. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negative order, including a trace-formula for pseudodifferential operators of order [Formula: see text].

Quasi-asymptotically conical Calabi–Yau manifolds

We construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). We do so by first providing a natural compactification of QAC-spaces by manifolds with fibred corners and by giving a definition of QAC-metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on QAC-spaces developed by the second author and Mazzeo, we can in many instances obtain K\"ahler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. This allows us to obtain QAC Calabi-Yau metrics in the K\"ahler classes of these metrics by solving a corresponding complex Monge-Amp\`ere equation.

Shelstad's character identity from the point of view of index theory

Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding L-packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic operators in the framework of K-theory. Our geometric proof of the character identity is evidence that index theory can play a role in the classification of group representations via the Langlands program.

Fredholm theory for elliptic operators on quasi‐asymptotically conical spaces

We consider the mapping properties of generalized Laplace-type operators L = ∇ ∗ ∇ + R on the class of quasi-asymptotically conical (QAC) spaces, which provide a Riemannian generalization of the QALE manifolds considered by Joyce. Our main result gives conditions under which such operators are Fredholm when acting between certain weighted Sobolev or weighted Hölder spaces. These are generalizations of well-known theorems in the asymptotically conical (or asymptotically Euclidean) setting, and also sharpen and extend corresponding theorems by Joyce. The methods here are based on heat kernel estimates originating from old ideas of Moser and Nash, as developed further by Grigor'yan and Saloff-Coste. As demonstrated by Joyce's work, the QAC spaces here contain many examples of gravitational instantons, and this work is motivated by various applications to manifolds with special holonomy.

The commutator of the Kato square root for second order elliptic operators on ℝ n

Let L = −div(A∇) be a second order divergence form elliptic operator, and A be an accretive, n×n matrix with bounded measurable complex coefficients in ℝn. We obtain the Lp bounds for the commutator generated by the Kato square root \(\sqrt L \) and a Lipschitz function, which recovers a previous result of Calderon, by a different method. In this work, we develop a new theory for the commutators associated to elliptic operators with Lipschitz function. The theory of the commutator with Lipschitz function is distinguished from the analogous elliptic operator theory.

Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry

We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable Killing spinor equations on a Courant algebroid. As an application, we propose a unifying framework for metrics with holonomy $\SU(3)$ and solutions of the Strominger system.

Existence of Steady State Solutions with Finite Energy for the Magnetohydrodynamic Equations in the Whole Space R3

We study the steady state Magnetohydrodynamic (MHD) equations in the whole space Following the work of C. Bjorland and M. Schonbek (4) onNavier -Stokes equations in the whole space, we prove the existence of atleast one solution with finite Dirichlet Integral to steady state Magnetohydrodynamic equations in the whole space. Further, we show that these solutions are unique among all solutions with finite energy and finite DirichletIntegral . I. Introduction Magnetohydrodynamics (MHD) is the study of flows of fluids which are electrically conducting and move in a magnetic field. The simplest example of an electrically conducting fluid is a liquid metal like mercury or liquid sodium. MHD treats, in particular, conducting fluids either in liquid form or gaseous form.The equations describing the motion of a viscous incompressible conducting fluid moving in a magnetic field are derived by coupling Navier-Stokes equations with Maxwell's equations together with expression for the Lorentz force. The domain Ω in which the fluid is moving is either a bounded subset of or the whole space . In this paper we restrict our considerations to a domain Ω which is the whole space .During past four or five decades, there have been an extensive study of qualitativeproperties such as existence, uniqueness, regularity and stability of solutions of theMHD equations. This is evident from the work of Duvaut and Lions (1), E. SanchezPalencia (2), Sermange and Temam (3) and other researchers working in the field. Themethods from nonlinear functional analysis such as Galerkin approximation, fixed pointtheorems, monotone and coercive operators, semigroup theory etc have been applied toestablish many a qualitative properties for compressible as well as incompressible MHDflows. The function spaces used are either Holder spaces or Sobolev spaces which are theappropriate function spaces for using these methods and the theory of elliptic operators. In spite of these works, there are very few qualitative results available in the case where the domain is the full space. In the case when domain is a bounded subset of R 3 , it is easy to obtain qualitative results by using Poincare type inequality. But for unbounded domain, one has to use other techniques as were developed by C. Bjorland and M. Schonbek (4). As for MHD flows for incompressible conducting fluids, there are other works where regularity results for MHD flows have been proved ( see references (5-7) ). However, as in the case of Navier-Stokes equations for incompressible fluids, the proof of global regularity remains illusive in this case also. In the present paper, we show that the techniques used in (4) can be extended to prove similar results for steady state Magnetohydrodynamic (MHD) flows. Thus, we consider viscous incompressible Magnetohydrodynamic (MHD) flow governed by the following equations:

Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on a possible location of the transmission eigenvalues. If the index of refraction is real, then we obtain a result on the existence of infinitely many positive ITEs and the Weyl-type lower bound on its counting function. All the results are obtained under the assumption that n(x) − 1 does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued n(x).

Layer Potential Analysis for Pseudodifferential Matrix Operators in Lipschitz Domains on Compact Riemannian Manifolds: Applications to Pseudodifferential Brinkman Operators

R n, n≥ 3, and with L2 boundary data. Shen [47] studied the L p Dirichlet problem for the Stokes system in Lipschitz domains in Rn, n≥ 3, and obtained optimal estimates when n= 3, as well as a weak estimate of solutions for certain range of p when n≥ 4 (see also [48]). Mitrea and Taylor [40–45] developed the potential theory for elliptic operators on Lipschitz domains in Riemannian manifolds and used this theory to treat the

The refined Sobolev scale, interpolation and elliptic problems

The paper gives a detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness. The latter are the isotropic Hörmander spaces $H^{s,\varphi}:=B_{2,\mu}$, with $\mu(\xi)=\langle\xi\rangle^{s}\varphi(\langle\xi\rangle)$ for $\xi\in\mathbb{R}^{n}$. They are parametrized by both the real number $s$ and the positive function $\varphi$ varying slowly at $+\infty$ in the Karamata sense. These spaces form the refined Sobolev scale, which is much finer than the Sobolev scale $\{H^{s}\}\equiv\{H^{s,1}\}$ and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this new scale. Theorems of various type about a solvability of elliptic problems are given. A~local refined smoothness is investigated for solutions to elliptic equations. New sufficient conditions for the solutions to have continuous derivatives are found. Some applications to the spectral theory of elliptic operators are given.

Spectral theory of elliptic operators in exterior domains

Diverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ0⩽|α|,|β|⩽m (−1) α D α a α,β (x)D β , a α,β (·) ∈ C ∞( $$ \bar \Omega $$ ) on smooth (bounded or unbounded) domains Ω in ℝ n with compact boundary ∂Ω are considered. Trace-ideal properties of powers of resolvent differences for these closed realizations of A are proved by using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, and estimates for negative eigenvalues of certain selfadjoint extensions of the nonnegative minimal operator are derived. Our results extend classical theorems due to Vishik, Povzner, Birman, and Grubb.

Spectral asymptotics for Dirichlet elliptic operators with non-smooth coefficients

We consider a $2m$-th-order elliptic operator of divergence form in a domain $\Omega$ of $\mathbb{R}^{n}$, assuming that the coefficients are Holder continuous of exponent $r \in (0,1]$. For the self-adjoint operator associated with the Dirichlet boundary condition we improve the asymptotic formula of the spectral function $e(\tau^{2m},x,y)$ for $x=y$ to obtain the remainder estimate $O(\tau^{n-\theta}+\dist(x,\partial\Omega)^{-1}\tau^{n-1})$ with any $\theta \in (0,r)$, using the $L^{p}$ theory of elliptic operators of divergence form. We also show that the spectral function is in $C^{m-1,1-\varepsilon}$ with respect to $(x,y)$ for any small $\varepsilon > 0$. These results extend those for the whole space $\mathbb{R}^{n}$ obtained by Miyazaki [19] to the case of a domain.

Characters of Fredholm Modules and a Problem of Connes

A large part of research in noncommutative geometry is devoted to various far-reaching generalizations of the Atiyah–Singer index theory of elliptic operators on compact manifolds. Following Kasparov [K1], the notion of an elliptic operator on a manifold is replaced by that of a Fredholm module (or K-cycle) over an algebra of operators on Hilbert space. A number of index theorems for Fredholm modules associated to geometric data have been obtained by Kasparov [K1], Connes (see [C3] and the literature cited therein) and others. In order to handle concrete index problems it is not only necessary to dispose of an index theorem but also to provide an index formula which allows the explicit calculation of indices. The classical index formula of Atiyah and Singer is obtained from the index theorem by applying the Chern character in topological K-homology. This motivates the search for character formulas of Fredholm modules (K-cycles) that define a Chern character on the K-homology groups of an algebra of operators. It was this search for a Chern character in K-homology which led A. Connes to the invention of cyclic cohomology [C2]. He obtained various explicit character formulas [C1,2,3], which depend on the degree of analytic regularity (summability) of the given Fredholm module. Whereas the classical index problems of Atiyah–Singer are all finitely summable in the sense of [C2] there are many examples of Fredholm modules over (noncommutative) algebras which are infinite dimensional (not finitely summable). Characters of Fredholm modules have been calculated in many finitely summable cases, but as far as the author knows the character of a θ-summable, infinitedimensional (unbounded) Fredholm module [C1] has not yet been determined in a single case.

Stochastic homogenization and random lattices

We present some variants of stochastic homogenization theory for scalar elliptic equations of the form − div [ A ( x ε , ω ) ∇ u ( x , ω ) ] = f . These variants basically consist in defining stochastic coefficients A ( x ε , ω ) from stochastic deformations (using random diffeomorphisms) of the periodic setting, as announced in [X. Blanc, C. Le Bris, P.-L. Lions, Une variante de la théorie de l'homogénéisation stochastique des opérateurs elliptiques (A variant of stochastic homogenization theory for elliptic operators), C. R. Acad. Sci. Sér. I 343 (2006) 717–727]. The settings we define are not covered by the existing theories. We also clarify the relation between this type of questions and our construction, performed in [X. Blanc, C. Le Bris, P.-L. Lions, A definition of the ground state energy for systems composed of infinitely many particles, Commun. Partial Differential Equations 28 (1–2) (2003) 439–475; X. Blanc, C. Le Bris, P.-L. Lions, The energy of some microscopic stochastic lattices, Arch. Rat. Mech. Anal. 184 (2) (2007) 303–339], of the energy of, both deterministic and stochastic, microscopic infinite sets of points in interaction.

On m-accretive Schrödinger operators in [formula omitted]-spaces on manifolds of bounded geometry

Let ( M , g ) be a manifold of bounded geometry with metric g. We consider a Schrödinger-type differential expression H = Δ M + V , where Δ M is the scalar Laplacian on M and V is a nonnegative locally integrable function on M. We give a sufficient condition for H to have an m-accretive realization in the space L p ( M ) , where 1 < p < + ∞ . The proof uses Kato's inequality and L p -theory of elliptic operators on Riemannian manifolds.

$L^p$-Theory for Elliptic Operators on $\mathbb R^d$ with Singular Coefficients

We study the generation of an analytic semigroup in L(R) and the determination of the domain for a class of second order elliptic operators with unbounded coefficients in R. We also establish the maximal regularity of type L–L for the corresponding inhomogeneous parabolic equation. In contrast to the previous literature the coefficients of the second derivatives are not required to be strictly elliptic or bounded. Interior singularities of the lower order terms are also discussed.