Real Analytic Coefficients Research Articles

Determining coefficients for a fractional p-Laplace equation from exterior measurements

We consider an inverse problem of determining the coefficients of a fractional p -Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficients. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.

A local obstruction for elliptic operators with real analytic coefficients on flat germs

A local obstruction for elliptic operators with real analytic coefficients on flat germs

On the microlocal regularity of the analytic vectors for “sums of squares” of vector fields

We prove via FBI-transform a result concerning the microlocal Gevrey regularity of analytic vectors for operators sums of squares of vector fields with real-valued real analytic coefficients of Hörmander type, thus providing a microlocal version, in the analytic category, of a result due to Derridj (Pac J Math 302(2):511–543, 2019) concerning the problem of the local regularity for the Gevrey vectors for sums of squares of vector fields with real-valued real analytic/Gevrey coefficients.

A new microlocal analysis of hyperfunctions

In this work we study microlocal regularity of hyperfunctions defining in this context a class of generalized FBI transforms first introduced for distributions by Berhanu and Hounie. Using a microlocal decomposition of a hyperfunction and the generalized FBI transforms we were able to characterize the wave-front set of hyperfunctions according to several types of regularity. The microlocal decomposition allowed us to recover and generalize both classical and recent results and, in particular, we proved for differential operators with real-analytic coefficients that if the elliptic regularity theorem regarding any reasonable regularity holds for distributions, then it is automatically true for hyperfunctions.

An Inverse Problem for LTV ODE’s

It is well known that any linear combination of products of polynomials and exponentials (a.k.a. Bohl functions) are annihilated by LTI-ODE’s. In this paper, we consider the more general classes of piecewise Bohl functions and of real analytic functions and present the realization of the latter as solutions of linear differential polynomials with real analytic coefficients. In particular, we show that the minimal order depends solely on the multiplicity of the zeros of the given function. We extend the problem further to meromorphic functions without poles on the real line. We briefly comment on piecewise-defined functions and the nonhomogeneous problem.

Multiparameter perturbation theory of matrices and linear operators

We show that a normal matrix A A with coefficients in C [ [ X ] ] \mathbb {C}[[X]] , X = ( X 1 , … , X n ) X=(X_1, \ldots , X_n) , can be diagonalized, provided the discriminant Δ A \Delta _A of its characteristic polynomial is a monomial times a unit. The proof is an adaptation of our proof of the Abhyankar-Jung Theorem. As a corollary we obtain the singular value decomposition for an arbitrary matrix A A with coefficient in C [ [ X ] ] \mathbb {C}[[X]] under a similar assumption on Δ A A ∗ \Delta _{AA^*} and Δ A ∗ A \Delta _{A^*A} . We also show real versions of these results, i.e., for coefficients in R [ [ X ] ] \mathbb {R}[[X]] , and deduce several results on multiparameter perturbation theory for normal matrices with real analytic, quasi-analytic, or Nash coefficients.

On the Gevrey regularity for sums of squares of vector fields, study of some models

The Gevrey hypoellipticity of a class of “sums of squares” with real analytic coefficients is studied in detail.

Analytic hypoellipticity for sums of squares and the Treves conjecture

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson–Treves stratification are symplectic.We produce a model operator, P1, having a single symplectic stratum and prove that it is Gevrey s0 hypoelliptic and not better. See Theorem 2.1 for a definition of s0. We also show that this phenomenon has a microlocal character.We point out explicitly that this is a counterexample to the sufficient part of Treves conjecture and not to the necessary part, which is still an open problem.

Analytic hypoellipticity for sums of squares and the Treves conjecture, II

Abstract We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson–Treves stratification are symplectic. We produce a model operator, P 1 , having a single symplectic stratum and prove that it is Gevrey s 0 hypoelliptic and not better. See Theorem 2.1 for a definition of s 0 . We also show that this phenomenon has a microlocal character. We point out explicitly that this is a counterexample to the sufficient part of Treves conjecture and not to the necessary part, which is still an open problem.

Volume Estimates on the Critical Sets of Solutions to Elliptic PDEs

In this paper we study solutions to elliptic linear equations urn:x-wiley:00103640:media:cpa21708:cpa21708-math-0001 either on or a Riemannian manifold, under the assumption that the coefficient functions aij are Lipschitz bounded. We focus our attention on the critical set and the singular set , and more importantly on effective versions of these. Currently, with just the Lipschitz regularity of the coefficients, the strongest results in the literature say that the singular set is (n–2)–dimensional; however, at this point it has not even been shown that unless the coefficients are smooth. Fundamentally, this is due to the need of an ɛ‐regularity theorem that requires higher smoothness of the coefficients as the frequency increases. We introduce new techniques for estimating the critical and singular set, which avoids the need of any such ɛ‐regularity. Consequently, we prove that if the frequency of u is bounded by Λ, then we have the estimates and , depending on whether the equation is critical or not. More importantly, we prove corresponding estimates for the effective critical and singular sets. Even under the assumption of real analytic coefficients these results are much sharper than those currently in the literature. We also give applications of the technique to give estimates on the volume of the nodal set of solutions and estimates for the corresponding eigenvalue problem.© 2017 Wiley Periodicals, Inc.

Analytic Hypoellipticity and the Treves Conjecture

We are concerned with the problem of the analytic hypoellipticity; precisely, we focus on the real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson-Treves stratification are symplectic. We discuss a model operator, P , (firstly appeared and studied in [3]) having a single symplectic stratum and prove that it is not analytic hypoelliptic. This yields a counterexample to the sufficient part of Treves conjecture; the necessary part is still an open problem.

Estimates for Weierstrass division in ultradifferentiable classes

We study the Weierstrass division theorem for function germs in strongly non-quasianalytic Denjoy–Carleman classes CM. For suitable divisors P(x,t)=xd+a1(t)xd−1+⋯+ad(t) with real-analytic coefficients aj, we show that the quotient and the remainder can be chosen of class CMσ, where Mσ=((Mj)σ)j≥0 and σ is a certain Łojasiewicz exponent related to the geometry of the roots of P and verifying 1≤σ≤d. We provide various examples for which σ is optimal, in particular strictly less than d, which sharpens earlier results of Bronshtein and of Chaumat–Chollet.

Zero Dynamics and Stabilization for Analytic Linear Systems

We combine different system theoretic concepts to solve the feedback stabilization problem for linear time-varying systems with real analytic coefficients. The algebraic concept of the skew polynomial ring with meromorphic coefficients and the geometric concept of (A,B)-invariant time-varying subspaces are invoked. They are exploited for a description of the zero dynamics and to derive the zero dynamics form. The latter is essential for stabilization by state feedback: the subsystem describing the zero dynamics are decoupled from the remaining system which is controllable and observable. The zero dynamics form requires an assumption close to autonomous zero dynamics; this in some sense resembles the Byrnes-Isidori form for systems with strict relative degree. Some aspects of the latter are also proved. Finally, using the zero dynamics form, we show for square systems with autonomous zero dynamics that there exists a linear state feedback such that the Lyapunov exponent of the closed-loop system equals the Lyapunov exponent of the zero dynamics; some boundedness conditions are required, too. If the zero dynamics are exponentially stable this implies that the system can be exponentially stabilized. These results are to some extent also new for time-invariant systems.

Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators

We deal with an inverse obstacle problem for general second order scalar elliptic operators with real principal part and analytic coefficients near the obstacle. We assume that the boundary of the obstacle is a non-analytic hypersurface. We show that, when we put Dirichlet boundary conditions, one measurement is enough to reconstruct the obstacle. In the Neumann case, we have results only for n = 2, 3 in general. More precisely, we show that one measurement is enough for n = 2 and we need 3 linearly independent inputs for n = 3. However, in the case for the Helmholtz equation, we only need n − 1 linearly independent inputs, for any n ≥ 2. Here n is the dimension of the space containing the obstacle. These are justified by investigating the analyticity properties of the zero set of a real analytic function. In addition, we give a reconstruction procedure for each case to recover the shape of obstacle. Although we state the results for the scattering problems, similar results are true for the associated boundary value problems.

Uniqueness properties of solutions to Schrödinger equations

nX j=1 ∂ 2 u ∂x 2 (x) = 0, (harmonic functions) in the unit ball {x ∈ R n : |x| 2, since harmonic functions are still real analytic in {x ∈ R n : |x| < 1}. In fact, it is well-known that if P(x,D) is a linear elliptic differential operator with real analytic coefficients, andP(x,D)u = 0 in a open set ⊂ R n , then u is real analytic in . Hence, the (s.u.c.p.) also holds for such solutions. Through the work of Hadamard [28] on the uniqueness of the Cauchy problem (which is closely related to the strong unique continuation property discussed earlier) it became clear (for applications in nonlinear problems) that it would be desirable to establish the strong unique continuation property for operators whose coefficients are not necessarily real analytic, oreven C ∞ . The first results in this direction were found in the pioneering work of Carleman [9] (when ,

Global solvability of real analytic complex vector fields in two variables

This paper deals with the global solvability of a complex vector field with real analytic coefficients in two real variables. The vector field is assumed to satisfy the Nirenberg–Treves condition ( P ) for local solvability. Normal forms for the vector field near the one-dimensional orbits are obtained and a generalization of the Riemann–Hilbert problem is considered.

On the non-analyticity locus of an arc-analytic function

Let X X be a real analytic manifold. A function f : X → R f:X\to \mathbb {R} is called arc-analytic if it is real analytic on each real analytic arc. In real analytic geometry there are many examples of arc-analytic functions that are not real analytic. They appear while studying the arc-symmetric sets and the blow-analytic equivalence. In this paper we show that the non-analyticity locus of an arc-analytic function is arc-symmetric. We also discuss the behavior of the non-analyticity locus under blowings-up. By a result of Bierstone and Milman, an arc-analytic function f : X → R f:X\to \mathbb {R} that satisfies a polynomial equation with real analytic coefficients, can be made analytic, over any relatively compact subset of X X , by a sequence of blowings-up with smooth centers. We show that these centers can be chosen, at each stage of the resolution, inside the non-analyticity loci.

Gevrey solvability and Gevrey regularity in differential complexes associated to locally integrable structures

In this work we study some properties of the differential complex associated to a locally integrable (involutive) structure acting on forms with Gevrey coefficients. Among other results we prove that, for such complexes, Gevrey solvability follows from smooth solvability under the sole assumption of a regularity condition. As a consequence we obtain the proof of the Gevrey solvability for a first order linear PDE with real-analytic coefficients satisfying the Nirenberg-Treves condition (P).

To the question of the minimal number of inputs for linear differential algebraic systems

We examine a control linear system of ordinary differential equations with an identically degenerate matrix coefficient of the derivative of the unknown vector function. We study the question of the minimal dimension of the control vector when the system could be fully controllable on any segment in the domain of definition. The problem is investigated in the cases of stationary systems and the systems with real analytic and smooth coefficients for which some structural forms can be defined.

Local solvability for a class of evolution equations

Motivated by the celebrated example of Y. Kannai of a linear partial differential operator which is hypoelliptic but not locally solvable, we consider a class of evolution operators with real-analytic coefficients and study their local solvability both in L 2 and in the weak sense. In order to do so we are led to propose a generalization of the Nirenberg–Treves condition ( ψ) which is suitable to our study.