existence and regularity to an elliptic equation with logarithmic nonlinearity abstract we study the nonlinear elliptic problem u u 0 log u f x u in r n with u 0 on the function f 0 0 is nondecreasing sublinear and f u is continuous for every 0 we obtain a maximal solution u 0 and prove its global regularity c 1 there is a constant such that u vanishes on a set of positive measure for 0 and u 0 for if f is concave for we characterize u by its stability

fractional analytic index for a finite rank projective bundle over a compact manifold so associated to a torsion dixmier douady 3 class w on the manifold we define the ring of dierential operators acting on sections of the projective bundle in a formal sense in particular any oriented even dimensional manifold carries a projective spin dirac operator in this sense more generally the correspond ing space of pseudodierential operators is defined with supports suciently close to the diagonal i e the identity relation for such elliptic operators we define the numerical index in an essentially analytic way as the trace of the commutator of the operator and a parametrix and show that this is homotopy invariant using the heat kernel method for the twisted projective spin dirac operator we show that this index is given by the usual formula now in terms of the twisted chern character of the symbol which in this case defines an element of k theory twisted by w hence the index is a rational number but in general it is not an integer a genus is a rational number but not necessarily an integer in this paper we show that in the oriented even dimensional case one can nevertheless define a projective spin dirac operator with an analytic index valued in the rational numbers and prove the analogue of the atiyah singer index theorem for this operator twisted by a general projective bundle in fact we establish the analogue of the atiyah singer index theorem for a general projective elliptic pseudodierential operator in a subsequent paper the families case will be discussed for a compact manifold z and vector bundles e and f over z the schwartz ker nel theorem gives a one to one correspondence between continuous linear operators 1

center problem for several differential equations via cherkas method cherkas method characterizes centers for analytic lienard differential equations we extend his method to degenerate lienard differential equations and we apply this extension to solve the center problem for several families of polynomial differential equations in particular we give all centers for some differential equations given by a vector field which is the sum of two quasi homogeneous ones

weakly parametric pseudodifferential operators and atiyah patodi singer boundary problems this paper introduces a class of pseudodifferential operators depending on a parameter in a particular way the main application is a complete expansion of the trace of the resolvent of a dirac type operator with nonlocal boundary conditions of the kind introduced by atiyah patodi and singer aps this extends the partial expansion in g2 to a complete one and extends the complete expansion in gs 1 to the case where the dirac operator does not have a product structure near the boundary a secondary application is to obtain a complete expansion of the resolvent of a bdo on a compact manifold essentially reproving a result of agranovich agr the resolvent expansion yields immediately an expansion of the trace of the heat kernel and determines the singularities of the zeta function moreover a pseudodifferential factor can be allowed a major motive for these expansions is to obtain index formulas for elliptic operators there are many such applications in the physics and geometry literature the index formula comes from one particular term in the expansion but each term is a spectral invariant and they have been used for other purposes as well as for the index in particular branson and gilkey have a number of papers e g bg and gi analyzing these invariants and drawing geometric consequences interest in the asymptotic behavior of the resolvent goes back to carleman c more recently agmon agm developed it extensively for analytic applications he introduced the fundamental idea of treating the resolvent parameter essentially as another cotangent variable this idea was developed in s1 to analyze the singularities of the zeta function of an elliptic odo on a compact manifold and in 3 to analyze the resolvent of a differential operator with differential boundary conditions the technique works smoothly for differential operators producing so called local invariants integrals over the underlying

on global asymptotic stability of systems of lienard type we give new necessary and sufficient conditions under which the zero solution of lienard type systems is globally asymptotically stable to this end we examine i whether all trajectories intersect the vertical isocline or not and ii whether all trajectories tend to the origin or not we also apply our results to a pseudolinear system

from tracial anomalies to anomalies in quantum field theory regularized traces resp super traces are defined on a classical pseudo differential operator a by tr q a f p tr a q z vert z 0 quad resp quad str q a f p str a q z vert z 0 where f p refers to the finite part and q is an invertible and admissible elliptic reference operator with positive order they are commonly used in quantum field theory in spite of the fact that unlike ordinary traces on matrices they are neither cyclic nor do they commute with exterior differentiation thus giving rise to tracial anomalies the purpose of this article is to show on two examples how tracial anomalies can lead to anomalous phenomena in quantum field theory

the continuation of solutions for certain nonlinear differential equations and its applications we give some sufficient conditions for the continuation of solutions for some nonlinear differential equations as an application we obtain a new criterion for the oscillation of solutions of the lienard equation

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 c0 b p1 t is the regular value at s 0 of the zeta function tr b p 1 t s 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 p1 t is a realization of an elliptic differential operator p1 having a ray free of eigenvalues relative formulas e g for the difference between the constants with two different choices of p1 t have been known for some time and are local we here determine c0 b p1 t itself with even order p1 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 p1 t since the complex powers of p1 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

positive solutions to a singular neumann problem abstract we show the existence of positive solution for the following class of singular neumann problem u a x u h x u p in b r with u 0 on b r where r 0 0 is a positive parameter 0 p 0 1 b r b r 0 r n a b r r and h b r r are radially symmetric nonnegative c 1 functions using variational methods and sub and supersolutions we obtain a solution for an approximated problem involving mixed boundary conditions the limit of the approximated solutions is a positive solution

trace expansions and the noncommutative residue for manifolds with boundary for a pseudodifferential boundary operator a of order 2 z and class 0 in the boutet de monvel calculus on a compact n dimensional manifold with boundary we consider the function tr ab s where b is an auxiliary system formed of the dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary we prove that tr ab s has a meromorphic extension to c with poles at the half integers s n j 2 j 2 n possibly double for s 0 and we prove that its residue at 0 equals the noncommutative residue of a as defined by fedosov golse leichtnam and schrohe by a different method to achieve this we establish a full asymptotic expansion of tr a b k in powersl 2 and log powersl 2 log where the noncommutative residue equals the coefficient of the highest order log power there is a related expansion of tr ae tb the paper will appear in journal reine angew math crelle s journal then is a holomorphic function of s for large re s we show that it extends to a mero morphic function on the whole complex plane with at most double poles moreover we prove that the noncommutative residue res a of a can be recovered as a residue in this

the noncommutative residue for manifolds with boundary abstract we construct a trace on the algebra of classical elements in boutet de monvel s calculus on a compact manifold with boundary of dimensionn 2 this trace coincides with wodzicki s noncommutative residue if the boundary is reduced to the empty set moreover we show that it is the unique continuous trace on this algebra up to multiplication by a constant

the center conditions and local bifurcation of critical periods for a lienard system in this paper we study the problems of centers and isochronous centers and the local bifurcation of critical periods for a lienard system with forth damping calculating the singular point values and period constants we find all center conditions and isochronous center conditions moreover the numbers of local critical periods bifurcating from centers and isochronous centers is obtained by computing the orders of weak centers

a new proof of weyl s formula on the asymptotic distribution of eigenvalues one of the more perplexing problems of finite particle quantum mechanics is to find the appropriate quantum mechanical description of a given classical mechanical system unfortunately from the mathematical point of view this problem is not terribly well posed suppose the classical system in question is represented by a symplectic manifold x and a classical observable p x r suppose also for simplicity that this system is a confined system in the sense that the sets p 1 are all compact let h be a hilbert space and let p be a self adjoint operator on h representing a quantum observable without specifying anything further about p for instance its relation to other observables there is practically no way of ascertaining whether p is an appropriate quantum counterpart of p at first glance the only requirements this appears to impose on p are that the spectrum of p be discrete and that the number of eigenvalues n a less than 1 be finite for all 1 this is the analogue of the condtion that p 1 be compact by examining the statistical properties of a large aggregate of classical systems each a copy of x p one can in fact say a little bit more about the relation p to p by comparing classical and quantum partition functions one can show that for iz large

a neumann problem with logarithmic nonlinearity in a ball abstract the nonlinear elliptic problem u log u h x u p in a ball b r r n with neumann boundary conditions u 0 on b r were studied by means of a double perturbation argument we obtain a positive radial solution u c 2 b r 0 c b r