Mixed Boundary Value Problems for the Helmholtz Equation in a Quadrant

The main objective is the study of a class of boundary value problems in weak formulation where two boundary conditions are given on the half-lines bordering the first quadrant that contain impedance data and oblique derivatives. The associated operators are reduced by matricial coupling relations to certain boundary pseudodifferential operators which can be analyzed in detail. Results are: Fredholm criteria, explicit construction of generalized inverses in Bessel potential spaces, eventually after normalization, and regularity results. Particular interest is devoted to the impedance problem and to the oblique derivative problem, as well.     

Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces

Generalized Anti-Wick operators are introduced as a class of pseudodifferential operators which depend on a symbol and two different window functions. Using symbols in Sobolev spaces with negative smoothness and windows in so-called modulation spaces, we derive new conditions for the boundedness on L² of such operators and for their membership in the Schatten classes. These results extend and refine results contained in [20], [10], [5], [4], and [14].     

Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators

This paper is devoted to a proof of regularity, near the initial state, for solutions to the Cauchy-Dirichlet and obstacle problem for a class of second order differential operators of Kolmogorov type. The approach used here is general enough to allow us to consider smooth obstacles as well as non-smooth obstacles.     

THE QUASI-GEOSTROPHIC EQUATION AND ITS TWO REGULARIZATIONS

Abstract

A symbolic calculus for the transposes of a class of bilinear pseudodifferential operators is developed. The calculus is used to obtain boundedness results on products of Lebesgue spaces. A larger class of pseudodifferential operators that does not admit a calculus is also considered. Such a class is the bilinear analog of the so-called exotic class of linear pseudodifferential operators and fail to produce bounded operators on products of Lebesgue spaces. Nevertheless, the operators are shown to be bounded on products of Sobolev spaces with positive smoothness, generalizing the Leibniz rule estimates for products of functions.     

Lp-Bounded Pseudodifferential Operators and Regularity for Multi-quasi-elliptic Equations

Using a classical result of Marcinkiewicz and Lizorkin about the L^p-continuity for Fourier multipliers, the authors study the action of a class of pseudodifferential operators with weighted smooth symbol on a family of weighted Sobolev spaces. Results about L^p-regularity for multi-quasi-elliptic pseudodifferential operators are also given.     

Strongly damped wave problems: Bootstrapping and regularity of solutions

The aim of the article is to present a unified approach to the existence, uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in Banach spaces. Our results are applied next to a number of examples appearing in literature, which fall into the class of strongly damped semilinear wave equations. The present work essentially extends the results on the existence and regularity of solutions to such problems. Previously, these problems have been considered mostly within the Hilbert space setting and with the main part operators being selfadjoint. In this article we present a more general approach, involving sectorial operators in reflexive Banach spaces.     

Well-posedness of the martingale problem for some degenerate diffusion processes occurring in dynamics of populations

In this paper we establish the well-posedness in C([0,∞);[0,1]^d), for each starting point x∈[0,1]d, of the martingale problem associated with a class of degenerate elliptic operators which arise from the dynamics of populations as a generalization of the Fleming-Viot operator. In particular, we prove that such degenerate elliptic operators are closable in the space of continuous functions on [0,1]^d and their closure is the generator of a strongly continuous semigroup of contractions.     

On convolution equations with semi-almost periodic symbols on a finite interval

This note concerns a class of Wiener-Hopf operators on a finite interval, acting between Sobolev multi-index spaces. Necessary and sufficient conditions for such an operator to be Fredholm are given, as well as a formula for the index. The argument is based on a reduction procedure of convolution operators on a finite interval to operators of the same type on the half-line.supported by the Netherlands organization for scientific research (NWO)supported in part by NSF Grant 9101143     

An Invariant Harnack Inequality for a Class of Hypoelliptic Ultraparabolic Equations

We show an invariant Harnack inequality for a class of hypoelliptic ultraparabolic operators with underlying homogeneous Lie group structures. As a byproduct we prove a Liouville type theorem for the related stationary operators. We also introduce a notion of link of homogeneous Lie Groups that allows us to show that our results apply to wide classes of operators.     

Symmetry properties for solutions of nonlocal equations involving nonlinear operators

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional p-Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.     

On global hypoellipticity of degenerate elliptic operators

We consider a class of degenerate elliptic operators on a torus and prove that global hypoellipticity is equivalent to an algebraic condition involving Liouville vectors and simultaneous approximability. For another class of operators we show that the zero order term may influence global hypoellipticity. Received August 13, 1997     

Regularity and solvability of linear differential operators in Gevrey spaces

We are interested in the following question: when regularity properties of a linear differential operator imply solvability of its transpose in the sense of Gevrey ultradistributions? This question is studied for a class of abstract operators that contains the usual differential operators with real‐analytic coefficients. We obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity implying global solvability of the transpose), as well as some results in the non‐compact case by means of the so‐called property of non‐confinement of singularities. We provide applications to Hörmander operators, to operators of constant strength and to locally integrable systems of vector fields. We also analyze a conjecture stated in a recent paper of Malaspina and Nicola, which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral.     

Type I Toeplitz algebras

The class of Toeplitz algebras associated to ordered groups is important in the analysis of Toeplitz operators on the generalised Hardy spaces defined by such groups. The conditions under which these Toeplitz algebras are Type I C^*-algebras are investigated.     

Zero measure Cantor spectra for continuum one-dimensional quasicrystals

We study Schrödinger operators on R$\mathbb{R}$ with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral properties of the associated operators. The constant spectrum in the strictly ergodic case coincides with the union of the zeros of the Lyapunov exponent and the set of non-uniformities of the transfer matrices. This result enables us to prove Cantor spectra of zero Lebesgue measure for a large class of operator families, including many operator families generated by aperiodic subshifts.     

A class of elliptic pseudo differential operators generating symmetric Dirichlet forms

It is shown that a special class of symmetric elliptic pseudo differential operators do generate a Feller semigroup and therefore a non-local Dirichlet form.     

Potential analysis for a class of diffusion equations: A Gaussian bounds approach

We axiomatically develop a potential analysis for a general class of hypoelliptic diffusion equations under the following basic assumptions: doubling condition and segment property for an underlying distance and Gaussian bounds of the fundamental solution. Our analysis is principally aimed to obtain regularity criteria and uniform boundary estimates for the Perron-Wiener solution to the Dirichlet problem. As an example of application, we also derive an exterior cone criterion of boundary regularity and scale-invariant Harnack inequality and Hölder estimate for an important class of operators in non-divergence form with Hölder continuous coefficients, modeled on Hörmander vector fields.     

Pointwise estimates for a class of non-homogeneous Kolmogorov equations

We consider a class of ultraparabolic differential equations that satisfy the Hörmander’s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser’s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption. Here we remove that assumption by proving some potential estimates and some ad hoc Sobolev type inequalities for solutions.     

Stefan problems with nonlinear diffusion and convection

We consider a class of Stefan-type problems having a convection term and a pseudomonotone nonlinear diffusion operator. Assuming data in L¹, we prove existence, uniqueness and stability in the framework of renormalized solutions. Existence is established from compactness and monotonicity arguments which yield stability of solutions with respect to L¹ convergence of the data. Uniqueness is proved through a classical L¹-contraction principle, obtained by a refinement of the doubling variable technique which allows us to extend previous results to a more general class of nonlinear possibly degenerate operators.     

Gevrey regularity for the supercritical quasi-geostrophic equation

In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in “critical” Sobolev spaces. Moreover, the Gevrey class that we obtain is “near optimal” and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier–Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the dyadic Littlewood–Paley operators, is established that allow us to exploit the cancellation properties of the equation and circumvent this difficulty.     

Little Hankel operators on the half plane

In this paper we consider a class of weighted integral operators onL ² (0, ) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.