symmetries in special classes of passive state signal systems abstract this article is devoted to a further development of the passive linear continuous time invariant s s state signal systems theory the main focus is on the connections between certain symmetry properties such as reality and reciprocity of the external characteristics of a s s system and the respective symmetry of the evolution of the inner state of the system these connections are investigated for the following classes of passive s s systems simple conservative controllable energy preserving observable co energy preserving optimal optimal and minimal balanced out of which the last three are introduced and studied here for the first time in each of these six classes a s s system is defined by its external characteristics up to unitary similarity our results are connected with the respective results in the input state output systems theory where the external characteristics of a system are scattering impedance or transmission matrices

initial trace of solutions of some quasilinear parabolic equations with absorption we study the existence of an initial trace of nonnegative solutions of the problem tu u p 2 u uq 0in qt 0 t we prove that the initial trace is an outer regular borel measure which may not be locally bounded for some values of the parameters p and q we study also the corresponding cauchy problems with a given generalized borel measure as initial data

canonical conservative state signal shift realizations of passive discrete time behaviors a passive linear discrete time invariant s s state signal system v x w consists of a hilbert state space x a kre n signal space w a maximal nonnegative generating subspace v of the kre n space k x x w the sets of trajectories x w generated by v on the discrete time intervals i z are defined by x n 1 x n w n v n i this system is forward conservative or backward conservative or conservative if v v v v or v v respectively the set w of all signal components w of trajectories x w of on i z with x 0 0 and w l2 z w is called the future time domain behavior of the fourier transform w of w is called the future frequency domain behavior of this set is a maximal nonnegative right shift invariant subspace in the kre n space k2 d w that as a topological vector space coincides with the usual hardy space h2 d w but has the indefinite kre n space inner product inherited from w a subspace of k2 d w with the above properties is called a passive future frequency domain behavior on w it has been shown earlier by the present authors that every passive future frequency domain behavior w on w may be realized as the future frequency domain behavior of some passive s s system and that it is possible to require in addition that is a controllable and forward conservative b observable and backward conservative or c simple and conservative these three types of realizations are determined by w up to unitary similarity canonical functional shift realizations of the types a and b have been obtained earlier by the present authors and their connection to the classical de branges rovnyak models have been discussed here we present analogous results for a realization of the type c

numerical range circumscribed by two polygons abstract we show that for any 2 n 2 distinct points a 1 a 1 a 2 a 2 a n 1 a n 1 in this order on the unit circle there is an n by n matrix a unique up to unitary equivalence which has norm one and satisfies the conditions that it has all its eigenvalues in the open unit disc i n a a has rank one and its numerical range is circumscribed by the two n 1 gons a 1 a 2 a n 1 and a 1 a 2 a n 1 this generalizes the classical result of the existence of a conical curve circumscribed by two triangles which are already inscribed on another conical curve

the periodic logistic equation with spatial and temporal degeneracies in this article we study the degenerate periodic logistic equation with homogeneous neumann boundary conditions tu u au b x t u in 0 u 0 on 0 u x 0 u0 x 6 0 in where r n 2 is a bounded domain with smooth boundary a and p 1 are constants the function b c r 0 1 is t periodic in t nonnegative and vanishes i e has a degeneracy in some subdomain of r we examine the effects of various natural spatial and temporal degeneracies of b x t on the long time dynamical behavior of the positive solutions our analysis leads to a new eigenvalue problem for periodic parabolic operators over a varying cylinder and certain parabolic boundary blow up problems not known before the investigation in this paper shows that the temporal degeneracy causes a fundamental change of the dynamical behavior of the equation only when spatial degeneracy also exists but in sharp contrast whether or not temporal degeneracy appears in the equation the spatial degeneracy always induces fundamental changes of the behavior of the equation though such changes differ significantly according to whether or not there is temporal degeneracy

on a class of semilinear fractional elliptic equations involving outside dirac data abstract the purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation 0 1 u u p 0 in b 1 e n u 0 in r n b 1 e n where p 0 with 0 1 denotes the fractional laplacian operator in the principle value sense b 1 e n is the unit ball centered at e n 0 0 1 in r n with n 2 and 0 is the dirac mass concentrated at the origin we prove that problem eq 0 1 admits a unique weak solution when p 1 2 n moreover if in addition p n 2 n 2 the weak solution vanishes as 1 we also show that problem eq 0 1 does not have any weak solution when p 0 1 2 n these results are very surprising since they are in total contradiction with the classical setting i e u u p 0 in b 1 e n u 0 in r n b 1 e n for which it has been proved that there are no solutions for p n 1 n 1

semilinear fractional elliptic equations involving measures abstract we study the existence of weak solutions to e u g u in a bounded regular domain in r n n 2 which vanish in r n where denotes the fractional laplacian with 0 1 is a radon measure and g is a nondecreasing function satisfying some extra hypotheses when g satisfies a subcritical integrability condition we prove the existence and uniqueness of weak solution for problem e for any measure in the case where is a dirac measure we characterize the asymptotic behavior of the solution when g r r k 1 r with k supercritical we show that a condition of absolute continuity of the measure with respect to some bessel capacity is a necessary and sufficient condition in order e to be solved

boundary singularities of solutions to elliptic viscous hamilton jacobi equations abstract we study the boundary value problem with measures for e1 u g u 0 in a bounded domain in r n satisfying e2 u on and prove that if g l 1 1 t 2 n 1 n d t is nondecreasing e1 e2 can be solved with any positive bounded measure when g r r q with q 1 we prove that any positive function satisfying e1 admits a boundary trace which is an outer regular borel measure not necessarily bounded when g r r q with 1 q q c n 1 n we prove the existence of a positive solution with a general outer regular borel measure as boundary trace and characterize the boundary isolated singularities of positive solutions when g r r q with q c q 2 we prove that a necessary condition for solvability is that must be absolutely continuous with respect to the bessel capacity c 2 q q q we also characterize boundary removable sets for moderate and sigma moderate solutions

a diffusive logistic equation with a free boundary and sign changing coefficient in time periodic environment abstract this paper concerns a diffusive logistic equation with a free boundary and sign changing intrinsic growth rate in heterogeneous time periodic environment in which the variable intrinsic growth rate may be very negative in a suitable large region see conditions h1 h2 4 3 such a model can be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front in the case of higher space dimensions with radial symmetry and when the intrinsic growth rate has a positive lower bound this problem has been studied by du guo peng 11 they established a spreading vanishing dichotomy the sharp criteria for spreading and vanishing and estimate of the asymptotic spreading speed in the present paper we show that the above results are retained for our problem

reduced limits for nonlinear equations with measures we consider equations e delta u g u mu in smooth bounded domains omega subset of r n where g is a continuous nondecreasing function and mu is a finite measure in omega given a bounded sequence of measures mu k assume that for each k 1 there exists a solution u k of e with datum mu k and zero boundary data we show that if u k u to in l 1 omega then u is a solution of e relative to some finite measure mu we call mu the reduced limit of mu k this reduced limit has the remarkable property that it does not depend on the boundary data but only on mu k and on g for power nonlinearities g t vertical bar t vertical bar q 1 t for all t is an element of r we show that if mu k is nonnegative and bounded in w 2 w q omega then mu and mu are absolutely continuous with respect to each other we then produce an example where mu not equal mu c 2009 elsevier inc all rights reserved

semilinear fractional elliptic equations with gradient nonlinearity involving measures we study the existence of solutions to the fractional elliptic equation e1 delta alpha u epsilon g nabla u nu in a bounded regular domain omega of r n n ge2 subject to the condition e2 u 0 in omega c where epsilon 1 or 1 delta alpha denotes the fractional laplacian with alpha in 1 2 1 nu is a radon measure and g r mapsto r is a continuous function we prove the existence of weak solutions for problem e1 e2 when g is subcritical furthermore the asymptotic behavior and uniqueness of solutions are described when nu is dirac mass g s s p p geq 1 and epsilon 1

on the equation δu eu 1 0 with measures as boundary data if is a bounded domain in mathbb r n we study conditions on a radon measure on for solving the equation u e u 1 0 in with u on the conditions are expressed in terms of orlicz capacities

a class of boundary traces for solutions of the equation lu ψ u our subject is the class u of all positive solutions of a semilinear equation lu u in e where l is a second order elliptic differential operator e is a domain in rd and belongs to a convex class of c1 functions which contains functions u u with 1 a special role is played by a class u0 of solutions which we call moderate a solution u is moderate if there exists h u such that lh 0 in e we say that u u is moderate if u is the limit of an increasing sequence of moderate solutions in e b dynkin s e kuznetsov fine topology and fine trace on the boundary associated with a class of quasilinear differential equations comm pure appl math 51 1998 897 936 all moderate solutions were classified by using their fine boundary traces 2 in m marcus l veron the precise boundary trace of positive solutions of the equation u uq in the supercritical case in perspectives in nonlinear partial differential equations in honor of haim brezis contemp math amer math soc providence ri 2007 arxiv org math 0610102 marcus and veron introduced a different characteristic called the precise trace for solutions of the equation u u with 1 in a bounded c2 domain in the present paper we develop a general scheme covering both approaches and we prove the equivalence in a certain sense of the fine and precise traces an implication of this equivalence is a wiener type criterion for vanishing of the poisson kernel of the equation lu x a x u x

semilinear parabolic equations diffusions and superdiffusions the semilinear equationu l0u u wherel0is a second order elliptic differential operator without zero order term and 1 1 related problems are treated with the help of diffusion processes this approach is useful even in the linear theory for instance the first boundary value problem for equationu lu fcan be investigated for a class of domains described in probabilistic terms which is substantially larger than the class considered in the literature on pdes

the boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption in the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation u g x u 0 in a smooth domain n for a general class of positive nonlinearities this class includes every space independent monotone increasing g which satisfies the keller osserman condition as well as degenerate nonlinearities g q of the form g q x u d x u q 1u with 2 and q 1 the boundary trace is given by a positive regular borel measure which may blow up on compact sets in the second part we concentrate on the family of nonlinearities g q determine the critical value of the exponent q for fixed 2 and discuss a positive solutions with an isolated singularity for subcritical nonlinearities and b the boundary value problem for u g q x u 0 with boundary data given by a positive regular borel measure possibly unbounded we show that in the subcritical case the problem possesses a unique solution for every such measure 2003 wiley periodicals inc

boundary layer and spike layer solutions for a bistable elliptic problem with generalized boundary conditions abstract we show that for large 0 the generalized boundary value problem u u u a x 1 u in u where 1 x behaves like the special case 1 in particular we show the existence of solutions with sharp boundary layers and interior spikes and determine the location of the spikes

on the existence of weak solutions of semilinear elliptic equations and systems with hardy potentials abstract let r n n 3 be a bounded c 2 domain and x dist x put l 2 with 0 in this paper we provide various necessary and sufficient conditions for the existence of weak solutions to l u u p in u on where 0 p 0 and are measures on and respectively we then establish existence results for the system l u v p in l v u p in u v on where 1 p 0 p 0 and are measures on and are measures on we also deal with elliptic systems where the nonlinearities are more general

stochastic boundary values and boundary singularities for solutions of the equationlu uα abstract we investigate positive solutions of a nonlinear equationlu u wherelis a second order elliptic differential operator in a riemannian manifoldeand 1 2 we establish a 1 1 correspondence between the set u of positive solutions and a class z of functionals ofxwhich we call linear boundary functionals they depend only on the behavior ofxnear the martin boundarye the class z is a closed convex cone andu u is a subadditive function ofz z special roles belong to moderate solutions corresponding tozwith finite mathematical expectations and to a family of solutions determined by the range ofx a new formula is deduced connectingu zandl diffusions conditioned to hit the boundarye at a given pointy a concept of a singular boundary point foruis introduced in terms of the conditioned diffusion

a kreĭn space coordinate free version of the de branges complementary space let z be a maximal nonnegative subspace of a kreu space x a nd letx z be the quotient of x modulo z d ef ine

solutions of nonlinear differential equations on a riemannian manifold and their trace on the martin boundary let l be a second order elliptic differential operator on a riemannian manifold e with no zero order terms we say that a function h is l harmonic if lh 0 every positive l harmonic function has a unique representation

large solutions for yamabe and similar problems on domains in riemannian manifolds we present a unifled approach to study large positive solutions i e u x 1 as x of the equation u hu i k u if in an arbitrary domain we assume u is convex and grows su ciently fast as u 1 equations of this type arise in geometry yamabe problem two dimensional curvature equation probability superdifiusion we prove that both existence and uniqueness are local properties of points of the boundary i e they depend only on properties of in arbitrary small neighborhood of each boundary point we also flnd several new necessary and su cient conditions for existence and uniqueness of large solutions including an existence theorem on domains with fractal boundaries

reduced measures on the boundary we study the existence of solutions of the nonlinear problem delta u g u 0 in omega u mu on partial derivative omega where mu is a and g r r is a nondecreasing continuous function with g t 0 for all t 0 problem 0 1 admits a solution for every mu epsilon l 1 partial derivative omega but this need not be the case when p is a general bounded measure we introduce a concept of reduced measure mu in the spirit of brezis et al ann math stud to appear this is the closest measure to mu for which 0 1 admits a solution

trace on the boundary for solutions of nonlinear differential equations let l be a second order elliptic differential operator in rd with no zero order terms and let e be a bounded domain in rd with smooth boundary e we say that a function h is l harmonic if lh 0 in e every positive l harmonic function has a unique representation h x j x y i dy e where k is the poisson kernel for l and z is a finite measure on e we call z the trace of h on me our objective is to investigate positive solutions of a nonlinear equation lu u in e for 1 2 we associate with every solution u a pair if i where f is a closed subset of e and zis a radon measure on 0 e f we call f zi the trace of u on me f is empty if and only if u is dominated by an l harmonic function we call such solutions moderate a moderate solution is determined uniquely by its trace in general many solutions can have the same trace we establish necessary and sufficient conditions for a pair f z to be a trace and we give a probabilistic formula for the maximal solution with a given trace

exact singular behavior of positive solutions to nonlinear elliptic equations with a hardy potential abstract in this paper we study the singular behavior at x 0 of positive solutions to the equation u x 2 u x u p x 0 where r n n 3 is a bounded domain with 0 and p 1 2 are given constants for the case n 2 2 4 the singular behavior of all the positive solutions is completely classified in the recent paper 5 here we determine the exact singular behavior of all the positive solutions for the remaining case n 2 2 4 in sharp contrast to the case n 2 2 4 where several converging blow up rates of u x are possible as x 0 we show that when n 2 2 4 every positive solution u x blows up in the same fashion lim x 0 x 2 p 1 u x 2 p 1 2 p 1 2 n 1 p 1

two canonical passive state signal shift realizations of passive discrete time behaviors a discrete time invariant linear state signal systemwith a hilbert state space x and a kreuon signal space w has trajectories x w that are solutions of the

optimal input output stabilization of infinite dimensional discrete time invariant linear systems by output injection we study the optimal input output stabilization of discrete time invariant linear systems in hilbert spaces by output injection we show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a left factorization over h infinity another equivalent condition is that the filter riccati equation of an arbitrary realization has a solution in general unbounded and even nondensely defined we further show that after renorming the state space in terms of the inverse of the smallest solution of the filter riccati equation the closed loop system is not only input output stable but also strongly internally stable

semilinear elliptic equations admitting similarity transformations in this paper we study the equation u 2 h u 0 in a smooth bounded domain where x dist x 0 and h is a nondecreasing function which satisfies keller osserman condition we introduce a condition on h which implies that the equation is subcritical i e the corresponding boundary value problem is well posed with respect to data given by finite measures under additional assumptions on h we show that this condition is necessary as well as sufficient we also discuss b v problems with data given by positive unbounded measures our results extend results of 13 treating equations of the form u uq 0 with q 1 2