On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions

General elliptic boundary value problems with the spectral parameter appearing linearly both in the elliptic equation and in boundary conditions are considered. It is proved that the corresponding matrix operator from the Boutet de Monvel algebra is similar to an almost diagonal operator. This result is applied to prove the completeness and the summability (in the sense of Abel) of the root vectors of this operator.The support of the Rashi Foundation is gratefully acknowledged.The support of the Israel Ministry of Science and Technology is gratefully acknowledged.     

Localized Boundary-Domain Singular Integral Equations Based on Harmonic Parametrix for Divergence-Form Elliptic PDEs with Variable Matrix Coefficients

Employing the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems (BVPs) for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original BVPs is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik–Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.     

Elliptic quasicomplexes in Boutet de Monvel algebra

We consider quasicomplexes of Boutet de Monvel operators in Sobolev spaces on a smooth compact manifold with boundary. To each quasicomplex we associate two complexes of symbols. One complex is defined on the cotangent bundle of the manifold and the other on that of the boundary. The quasicomplex is elliptic if these symbol complexes are exact away from the zero sections. We prove that elliptic quasicomplexes are Fredholm. As a consequence of this result we deduce that a compatibility complex for an overdetermined elliptic boundary problem operator is also Fredholm. Moreover, we introduce the Euler characteristic for elliptic quasicomplexes of Boutet de Monvel operators.     

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

An algebra generated by general pseudodifferential boundary value problems in a cone

The algebra indicated in the title is constructed. By localization outside the vertex of the cone, it reduces to the Boutet de Monvel algebra on a smooth manifold.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 133–161, 1990.     

Elliptic boundary problems on manifolds with polycylindrical ends

We investigate general Shapiro-Lopatinsky elliptic boundary value problems on manifolds with polycylindrical ends. This is accomplished by compactifying such a manifold to a manifold with corners of in general higher codimension, and we then deal with boundary value problems for cusp differential operators. We introduce an adapted Boutet de Monvel's calculus of pseudodifferential boundary value problems, and construct parametrices for elliptic cusp operators within this calculus. Fredholm solvability and elliptic regularity up to the boundary and up to infinity for boundary value problems on manifolds with polycylindrical ends follows.     

Atiyah-Bott-Lefschetz formula for elliptic complexes on manifolds with boundary

Elliptic complexes on a manifold with boundary whose differentials are Boutet de Monvel operators are studied. An Atiyah-Bott-Lefschetz type formula is obtained for such complexes.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 38, pp. 119–183, 1991.     

The spin $^{\bf c}$ Dirac operator on high tensor powers of a line bundle

We study the asymptotic of the spectrum of the Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin–Uribe [13, Theorem 2], which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin [10]. Received: 6 June 2001; in final form: 18 September 2001 / Published online: 28 February 2002     

Fréchet Algebra Techniques for Boundary Value Problems on Noncompact Manifolds: Fredholm Criteria and Functional Calculus via Spectral Invariance

A Boutet de Monvel type calculus is developed for boundary value problems on (possibly) noncompact manifolds. It is based on a class of weighted symbols and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven:

• 1 The algebra G of Green operators of order and type zero is a spectrally invariant Fréchet subalgebra of L(H), H a suitable Hilbert space, i. e.,
• 2 Focusing on the elements of order and type zero is no restriction since there are order reducing operators within the calculus.
• 3 There is a necessary and sufficient criterion for the Fredholm property of boundary value problems, based on the invertibility of symbols modulo lower order symbols, and
• 4 There is a holomorphic functional calculus for the elements of G in several complex variables.

     

Quantization of Compact Symplectic Manifolds

We develop the theory of Berezin–Toeplitz operators on any compact symplectic prequantizable manifold from scratch. Our main inspiration is the Boutet de Monvel–Guillemin theory, which we simplify in several ways to obtain a concise exposition. A comparison with the spin-c Dirac quantization is also included.     

Toeplitz operators and weighted Bergman kernels

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. Finally, we also exhibit a holomorphic continuation of the kernels with respect to the Sobolev parameter to the entire complex plane. Our main tool are the generalized Toeplitz operators of Boutet de Monvel and Guillemin.     

Geometric Arveson–Douglas conjecture

We prove the Arveson–Douglas essential normality conjecture for graded Hilbert submodules that consist of functions vanishing on a given homogeneous subvariety of the ball, smooth away from the origin. Our main tool is the theory of generalized Toeplitz operators of Boutet de Monvel and Guillemin.     

The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C ₀(B, P _1,T ) is the regular value at s = 0 of the zeta function , where B = P ₊ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus)—for example the solution operator of a classical elliptic problem—and P _1,T is a realization of an elliptic differential operator P ₁, having a ray free of eigenvalues. Relative formulas (e.g., for the difference between the constants with two different choices of P _1,T ) have been known for some time and are local. We here determine C ₀(B, P _1,T ) itself (with even-order P ₁), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin, Fedosov–Golse–Leichtnam–Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes a formula shown recently by Paycha and Scott for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving log P _1,T . Since the complex powers of P _1,T lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators can be used.     

Boundary value problems with global projection conditions

Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus.     

The Bergman kernel in a neighborhood of the diagonal for domains that are close to ellipsoids

A class of pseudoconvex domains with weak constraints on the boundary is introduced and for their Bergman kernels one obtains the principal term of the asymptotic expansion near the diagonal. The strictly pseudo-convex domains with C⁴-boundaries belong to the indicated class; for them one has obtained C. Fefferman's principal term for domains with C-boundaries and the error estimate due to L. Boutet de Monvel and J. Sjöstrand.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 163–172, 1991.     

Boundary value problems on manifolds with fibered boundary

We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah–Patodi–Singer problems in subspaces (it contains both as special cases). The boundary conditions in this theory are taken as elements of the C *‐algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems and compute a topological obstruction (similar to Atiyah–Bott obstruction) to the existence of elliptic boundary conditions for a given elliptic operator. Geometric operators with trivial and nontrivial obstruction are given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The modified KdV equation on a finite interval

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann–Hilbert problem in the complex k-plane. This Riemann–Hilbert problem has explicit (x,t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of the initial condition q(x,0)=q₀(x), while the remaining spectral functions are defined in terms of two sets of boundary values. We show that the spectral functions satisfy an algebraic “global relation” that characterize the boundary values in spectral terms. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Decay of solutions of the wave equation with arbitrary localized nonlinear damping

We study the problem of decay rate for the solutions of the initial-boundary value problem to the wave equation, governed by localized nonlinear dissipation and without any assumption on the dynamics (i.e., the control geometric condition is not satisfied). We treat separately the autonomous and the non-autonomous cases. Providing regular initial data, without any assumption on an observation subdomain, we prove that the energy decays at last, as fast as the logarithm of time. Our result is a generalization of Lebeau (in: A. Boutet de Monvel, V. Marchenko (Eds.), Algebraic and Geometric Methods in Mathematical Physics, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1996, pp. 73) result in the autonomous case and Nakao (Adv. Math. Sci. Appl. 7 (1) (1997) 317) work in the non-autonomous case. In order to prove that result we use a new method based on the Fourier-Bross-Iaglintzer (FBI) transform.     

On convergence to steady state of solutions of parabolic equations in unbounded domains

Solutions of the first mixed boundary value problem for a class of parabolic partial differential equations are shown to converge to solutions of associated elliptic boundary value problems as time t → ∞. The method of proof involves the introduction, for the parabolic problem, of a class of generalized solutions which may behave asymptotically, at spatial infinity, like solutions of the associated elliptic equation.