flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

a pocket guide to nonlinear differential equations in musielak orlicz spaces abstract the musielak orlicz setting unifies variable exponent orlicz weighted sobolev and double phase spaces they inherit technical difficulties resulting from general growth and inhomogeneity in this survey we present an overview of developments of the theory of existence of pdes in the setting including reflexive and non reflexive cases as well as isotropic and anisotropic ones particular attention is paid to problems with data below natural duality in absence of lavrentiev s phenomenon

liouville type theorem for quasi linear elliptic inequality δpu uσ 0 on riemannian manifolds abstract we studied the uniqueness of a nonnegative solution of the quasi linear elliptic inequality p u u 0 on a connected geodesically complete riemannian manifold x where is a parameter and p u div u p 2 u we proved that if p 1 p 1 and for some x 0 x lim inf t 0 t p 1 1 b x 0 r r p 1 p 1 p 1 p t 2 p 1 d r then inequality has no nontrivial nonnegative weak solutions we also studied the weighed case and the sharpness of the main result

null controllability of linear systems with constrained controls the paper considers the problem of steering the state of a linear time varying system to the origin when the control is subject to magnitude constraints necessary and sufficient conditions are given for global constrained controllability as well as a necessary and sufficient condition for the existence of a control satisfying the constraints which steers the system to the origin from a specified initial epoch x 0 t 0 the global result does not require zero to be an interior point of the control set omega and the theorem for constrained controllability at x 0 t 0 only requires that omega be compact not that it contain zero the results are compared to those available in the literature furthermore numerical aspects of the problem are discussed as is a technique for determining a steering control

constrained approximate boundary controllability in this paper constrained approximate boundary controllability for linear disturbed parameter dynamical systems described by partial differential equations of parabolic type with mixed boundary conditions is considered using methods of spectral analysis for linear self adjoint operators and general constrained controllability results given by son 1990 necessary and sufficient conditions for constrained approximate boundary controllability are formulated and proved moreover in illustrative examples the constrained approximate boundary controllability of 1d distributed parameter dynamical systems is investigated the constrained controllability conditions obtained represent an extension of the unconstrained controllability results given by glothin 1974

nonlinear liouville theorems for grushin and tricomi operators the aim of this paper is to study necessary conditions for existence of weak solutions of the inequality l x y dx dy u u q x 1 y 2 x rd y rk where l is a quasi homogeneous differential operator q 1 1 2 r and k d 1 as special cases our results can be applied when l is tricomi or grushin type operators

eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent in this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power like variable reaction term one of the main features of our work is that the differential operator involves partial derivatives with different variable exponents so that the functional analytic framework relies upon anisotropic sobolev and lebesgue spaces existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations

uniqueness result for non negative solutions of semi linear inequalities on riemannian manifolds abstract we consider certain semi linear partial differential inequalities on complete connected riemannian manifolds and provide a simple condition in terms of volume growth for the uniqueness of a non negative solution we also show the sharpness of this condition

sharp sobolev asymptotics for critical anisotropic equations we investigate blow up theory and prove sharp sobolev asymptotics for a general class of anisotropic critical equations in bounded domains of mathbb r n

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

a new critical curve for a class of quasilinear elliptic systems abstract we study a class of systems of quasilinear differential inequalities associated to weakly coercive differential operators and power reaction terms the main model cases are given by the p laplacian operator as well as the mean curvature operator in nonparametric form we prove that if the exponents lie under a certain curve then the system has only the trivial solution these results hold without any restriction provided the possible solutions are more regular the underlying framework is the classical euclidean case as well as the carnot groups setting

liouville type results and a maximum principle for non linear differential operators on the heisenberg group abstract we prove liouville type results for non negative solutions of the differential inequality u f u l 0 u on the heisenberg group under a generalized keller osserman condition the operator u is the laplacian defined by div 0 0 u 1 0 u 0 u and f and l satisfy mild structural conditions in particular l is allowed to vanish at the origin a key tool that can be of independent interest is a strong maximum principle for solutions of such differential inequality

approximate controllability for abstract measure differential systems abstract in this paper we investigate approximate controllability for abstract measure differential systems based on generalizing knowledge for ordinary differential systems we first introduce new concepts of the reachable set and approximate controllability for abstract measure differential systems then based on the nonlinear alternative for condensing mapping in banach space we present sufficient conditions of approximate controllability for a class of abstract measure differential systems our results in dealing with approximate controllability are less conservative than those in the previous literature finally an example is given to illustrate the availability of our results for approximate controllability

on properties of solutions of quasilinear second order elliptic inequalities abstract let be an unbounded open subset of r n n 2 and a r n r n be a function such that c 1 p a x a x c 2 p 1 with some constants c 1 0 c 2 0 and p 1 for almost all x and for all r n we obtain blow up conditions and priori estimates for inequalities of the form div a x d u b x d u q x g u in where p 1 p is a real number and b q and g are non negative functions

constrained approximate controllability in the paper constrained approximate controllability for linear dynamical systems described by abstract differential equations with unbounded control operator is considered using methods of spectral analysis for linear self adjoint operators and general constrained controllability results given by son 1990 necessary and sufficient conditions of the constrained approximate controllability for the piecewise polynomial controls with values in a given cone are formulated and proved moreover as illustrative examples constrained approximate boundary controllability of one dimensional distributed parameter dynamical systems described by partial differential equations of parabolic type with dirichlet and neumann boundary conditions are investigated the constrained controllability conditions obtained in the paper represent an extension of the unconstrained controllability results given by glothin 1974 and klamka 1991

controlling a constrained linear system to an affine target we consider the problem of steering the state of a linear system to an affine target when the admissible controls are required to satisfy magnitude constraints a necessary and sufficient condition for the existence of an admissible control which steers the system to the target from a specified initial condition is presented as well as a necessary condition and a sufficient condition for global controllability to the target the conditions are similar to those available in the literature in that they involve a finite dimensional search here however the affine nature of the target is exploited to reduce the dimensionality of the resulting optimization problem hence the results are more easily applied than those developed in 13

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

liouville theorems for anisotropic quasilinear inequalities abstract the purpose of this paper is to prove some nonexistence results for nonnegative weak solutions of differential inequalities l u u q or l u u q or l u 0 on r n here l is a quasilinear operator belonging to a wide class of operators including the anisotropic pseudo m laplacian operator the m laplacian operator the mean curvature operator acting in euclidean setting as well as their counterparts in carnot groups setting

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c