Strong unique continuation for systems of complex vector fields

In this work we study the property of strong unique continuation, at a given point, for Gevrey solutions to homogeneous systems of PDE defined by complex, real-analytic vector fields in involution. We show that when the system is minimal at the point then the strong unique continuation property holds for Gevrey solutions of order σ∈[1,2]$\sigma \in [1,2]$ and, furthermore, when the minimality property fails to hold then there are non-trivial Gevrey flat solutions of any given order σ>1$\sigma >1$. The case of Gevrey order σ>2$\sigma >2$ is also studied for some particular classes of involutive systems.     

Complex vector fields, unique continuation and the maximum modulus principle

We prove unique continuation and maximum modulus principle for solutions to systems of differential equations and inequalities, involving complex vector fields, under conditions that generalize some weak-pseudoconcavity assumptions for the tangential Cauchy-Riemann complex.     

On unique continuation principles for some elliptic systems

In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully nonlinear operators, such as Pucci's extremal operators, being new even for scalar equations.     

Combinatorial vector fields and dynamical systems

In this paper we introduce the notion of a combinatorial dynamical system on any CW complex. Earlier in [Fo3] and [Fo4], we presented the idea of a combinatorial vector field (see also [Fo1] for the one-dimensional case), and studied the corresponding Morse Theory. Equivalently, we studied the homological properties of gradient vector fields (these terms were defined precisely in [Fo3], see also Sect. 2 of this paper). In this paper we broaden our investigation and consider general combinatorial vector fields. We first study the homological properties of such vector fields, generalizing the Morse Inequalities of [Fo3]. We then introduce various zeta functions which keep track of the closed orbits of the corresponding flow, and prove that these zeta functions, initially defined only on a half plane, can be analytically continued to meromorphic functions on the entire complex plane. Lastly, we review the notion of Reidemeister Torsion of a CW complex (introduced in [Re], [Fr]) and show that the torsion is equal to the value at of one of the zeta functions introduced earlier. Much of this paper can be viewed as a combinatorial analogue of the work on smooth dynamical systems presented in [P-P], [Fra], [Fri1, 2] and elsewhere. Received 2 August 1995; in final form 25 September 1996     

Integrability by quadratures for systems of involutive vector fields

Starting from results and ideas of S. Lie and E. Cartan, we give a systematic and geometric treatment of integrability by quadratures of involutive systems of vector fields, showing how a generalization of the usual multiplier can be constructed with the aid of closed differential forms and enough symmetry vector fields. This leads us to explicit formulas for the independent integrals. These results allow us to identify symmetries with integral invariants in the sense of Poincaré and Cartan. A further (new) result gives the equivalence of integrability by quadratures and the existence of solvable structures, these latter being generalizations of solvable algebras.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1330–1337, October, 1991.     

A unique continuation problem for holomorphic mappings

The authors study analytic discs that are “attached to” a red submanifold having minimal smoothness. They prove a new uniqueness and regularity theorem by using the technique of the Riemann–Hilbert problem. They also present a new method for conatructing families of analytic discs lhat osculate a surface.     

Euler systems in global function fields

In this paper, the construction of Euler systems of cyclotomic units in a general global function fields is explained. As an application, an analogue of Gras’ conjecture in a global function field is proved.     

Invariant submanifolds for systems of vector fields of constant rank

Given a system of vector fields on a smooth manifold that spans a plane field of constant rank, we present a systematic method and an algorithm to find submanifolds that are invariant under the flows of the vector fields. We present examples of partition into invariant submanifolds, which further gives partition into orbits. We use the method of generalized Frobenius theorem by means of exterior differential systems.     

Impulsive control systems with commutative vector fields

We consider variational problems with control laws given by systems of ordinary differential equations whose vector fields depend linearly on the time derivativeu=(u ¹,...,u ^m ) of the controlu=(u ¹,...,u ^m ). The presence of the derivativeu, which is motivated by recent applications in Lagrangian mechanics, causes an impulsive dynamics: at any jump of the control, one expects a jump of the state.The main assumption of this paper is the commutativity of the vector fields that multiply theu . This hypothesis allows us to associate our impulsive systems and the corresponding adjoint systems to suitable nonimpulsive control systems, to which standard techniques can be applied. In particular, we prove a maximum principle, which extends Pontryagin's maximum principle to impulsive commutative systems.     

Weighted gradient inequalities and unique continuation problems

Calculus of Variations and Partial Differential Equations - We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we...     

Nonexistence of global solutions for a class of complex vector fields on two-torus

The goal of this paper is study the global solvability of a class of complex vector fields of the special form L=∂/∂t+(a+ib)(x)∂/∂x, a,b∈C^∞(S¹;R), defined on two-torus T²≅R²/2πZ². The kernel of transpose operator is described and the solvability near the characteristic set is also studied.