effect of symmetry to the structure of positive solutions in nonlinear elliptic problems ii abstract we consider the problem u hu f u 0 in r u 0 on r u 0 in r where r x r n r 1

positive radial solutions and non radial bifurcation for semilinear elliptic equations in annular domains abstract we discuss the existence and multiplicity of positive radial solutions and the non radial bifurcation of u f u 0 in and u 0 on where is an annular domain of r n n 2 we prove that if f u 0 for u 0 and lim u f u u then there exists 0 such that there are at least two positive radial solutions for each 0 at least one for and none for if f 0 0 lim u 0 f u u 1 and uf u 1 e f u for u 0 e 0 then there exists a variational solution for 0 1 where 1 is the least eigenvalue of if f 0 0 lim u 0 f u u 0 and lim u f u u then there exists at least one positive radial solution for any 0 we obtain some precise multiplicity results for narrow annulus and show that the non radial bifurcation occurs if the growth of f u is rapid enough as u

renormalized entropy solutions of scalar conservation laws with boundary condition we study an initial boundary value problem for a scalar conservation law u t div phi u f on a bounded domain existence and uniqueness of a renormalized entropy solution is established for general l 1 data phi is an element ofc r r n

symmetry breaking for δu 2δe u 0 on a disk with general boundary conditions we discuss the radially symmetric solutions and the symmetry breaking of the equation u 2 e u 0 in d and u b u n 0 on d where d is the unit disk in 2 0 and b is a constant we prove that for any b 0 such that there are exactly two radially symmetric solutions for 0 one for and none for b for where m is a positive integer there are b k 1 m such that the equation has symmetry breaking at k b on the lower branch of radially symmetric solutions

partial information state based optimization of partially observable markov decision processes and the separation principle we propose a partial information state based approach to the optimization of the long run average performance in a partially observable markov decision process pomdp in this approach the information history is summarized at least partially by a or a few statistic s not necessary sufficient called a partial information state and actions depend on the partial information state rather than system states we first propose the single policy based comparison principle under which we derive an hjb type of optimality equation and policy iteration for the optimal policy in the partial information state based policy space we then introduce the q sufficient statistics and show that if the partial information state is q sufficient then the optimal policy in the partial information state based policy space is optimal in the space of all feasible information state based policies we show that with some further conditions the well known separation principle holds the results are obtained by applying the direct comparison based approach initially developed for discrete event dynamic systems

entropy formulation for parabolic degenerate equations with general dirichlet boundary conditions and application to the convergence of fv methods this paper is devoted to the analysis and the approximation of parabolic hyperbolic degenerate problems defined on bounded domains with nonhomogeneous boundary conditions it consists of two parts the first part is devoted to the definition of an original notion of entropy solutions to the continuous problem which can be adapted to define a notion of measure valued solutions or entropy process solutions the uniqueness of such solutions is established in the second part the convergence of the finite volume method is proved this result relies on weak estimates and on the theorem of uniqueness of the first part it also entails the existence of a solution to the continuous problem

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

well posedness of general boundary value problems for scalar conservation laws in this paper we investigate well posedness for the problem u t div ph u f on 0 t times om om subset r n with initial condition u 0 cdot u 0 on om and with general dissipative boundary conditions varphi u cdot nu in beta t x u on 0 t times ptl om here for a e t x in 0 t times ptl om beta t x cdot is a maximal monotone graph on r this includes as particular cases dirichlet neumann robin obstacle boundary conditions and their piecewise combinations as for the well studied case of the dirichlet condition one has to interprete the it formal boundary condition given by beta by replacing it with the adequate it effective boundary condition such effective condition can be obtained through a study of the boundary layer appearing in approximation processes such as the vanishing viscosity approximation we claim that the formal boundary condition given by beta should be interpreted as the effective boundary condition given by another monotone graph tilde beta which is defined from beta by the projection procedure we describe we give several equivalent definitions of entropy solutions associated with tilde beta and thus also with beta for the notion of solution defined in this way we prove existence uniqueness and l 1 contraction monotone and continuous dependence on the graph beta convergence of approximation procedures and stability of the notion of entropy solution are illustrated by several results

optimal average value convergence in nonhomogeneous markov decision processes abstract we address the undiscounted nonhomogeneous markov decision process with average reward criterion and prove two structural results first we establish equivalence of this problem to a discounted markov decision process by means of an ergodic coefficient embedded in the original problem second we prove for the original problem that the optimal finite horizon average values converge to the infinite horizon optimal average value under an ergodic condition

uniqueness of positive ground state solutions of the logarithmic schrodinger equation we prove the uniqueness of positive ground state solutions of the problem frac d 2 u dr 2 frac n 1 r frac du dr u ln u 0 u r 0 forall r ge 0 and u r u r to 0 0 as r to infty this equation is derived from the logarithmic schrodinger equation rm i psi t delta psi u ln left u 2 right and also from the classical equation frac partial u partial t delta u u left u p 1 right u for each n ge 1 a positive ground state solution is u 0 r exp left frac r 2 4 frac n 2 right 0 le r infty we combine u 0 r with energy estimates and associated ricatti equation estimates to prove that for each n in left 1 9 right u 0 r is the only positive ground state we also investigate the stability of u 0 r several open problems are stated

global bifurcations of solutions of emden fowler type equation δu x λf u x on an annulus in rn n 3 abstract in this article we study global and symmetry breaking bifurcations of solutions of emden fowler type elliptic differential equations as the main tool we use a topological invariant which is suitable for study of global bifurcations and continuations of critical points of equivariant functionals namely we will apply degree for s 1 equivariant orthogonal maps defined in rybicki 1994 nonlinear anal tma 23 83 102

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

event based optimization of markov systems recent research indicates that markov decision processes mdps and perturbation analysis pa based optimization can be derived easily from two fundamental performance sensitivity formulas with this sensitivity point of view an event based optimization approach including event based sensitivity analysis and event based policy iteration was proposed via an example by x r cao discrete event dyn syst theory appl vol 15 pp 169 197 2005 this approach utilizes the special feature of a system and illustrates how the potentials can be aggregated using the special feature the approach applies to many practical problems that do not fit well the standard mdp formulation this note provides a mathematical formulation and proves the main results for this approach

optimization of average rewards of time nonhomogeneous markov chains we study the optimization of average rewards of discrete time nonhomogeneous markov chains in which the state spaces transition probabilities and reward functions depend on time the analysis encounters a few major difficulties 1 notions crucial to homogeneous markov chains such as ergodicity stationarity periodicity and connectivity no longer apply 2 the average reward criterion is under selective i e it does not depend on the decisions in any finite period and thus dynamic programming is not amenable and 3 because of the under selectivity an optimal average reward policy may not be the best in any finite period

on the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems it is the purpose of this paper to discuss the equality sign in d and in related inequalities in particular we shall give an answer to the following question does the equality in d imply that u u this question has been raised by g polya g szego in their book on isoperimetric inequalities l l p 186 they state that their method of proving d unfortunately leaves little hope for discussing the sign of equality another comment on this question can be found in 8 where e lieu conjectures that probably a strict version of d is true as it turns out the doubts of g polya g szeg were indeed justified because there are functions u for which equality holds in d while u does not coincide with u in section 1 of this paper i give an example of such a function on the other hand the conjecture of e lieb is also reasonable since generically i can give a positive answer to the question above there is a dense subset of nonnegative functions in wol 2 1 1 for which the equality in d implies

existence of many positive solutions of semilinear elliptic equations on annulus abstract we study the existence of many nonradial positive solutions in an annulus of rn it is an improvement of results in c v coffman a nonlinear boundary value problem with many positive solutions j differential equations 54 1984 429 437 an annulus is invariant under many group actions and these group actions act naturaly on the functional and leave the functional invariant therefore when we look for critical points of the functional we can restrict the functional to the manifold consisting of all fixed points under the group action we are going to use different group actions to obtain critical points and furthermore we can distinguish them

stability of non isolated asymptotic profiles for fast diffusion the stability of asymptotic profiles of solutions to the cauchy dirichlet problem for fast diffusion equation fde for short is discussed the main result of the present paper is the stability of any asymptotic profiles of least energy it is noteworthy that this result can cover non isolated profiles e g those for thin annular domain cases the method of proof is based on the ojasiewicz simon inequality which is usually used to prove the convergence of solutions to prescribed limits as well as a uniform extinction estimate for solutions to fde besides local minimizers of an energy functional associated with this issue are characterized furthermore the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the ojasiewicz simon inequality in a different way

on non radially symmetric bifurcation in the annulus we discuss the radially symmetric solutions and the non radially symmetric bifurcation of the semilinear elliptic equation u 2 eu 0 in and u 0 on where x r2 a2 0 a the upper branch of radial solutions has a non radially symmetric bifurcation symmetry breaking at each k a k 1 as a 0 the radial solutions will tend to the radial solutions on the disk and 0 a 1 the critical number on the disk

a morse index formula for radial solutions of lane emden problems abstract we consider the semilinear lane emden problem e p u u p 1 u in b u 0 on b where b is the unit ball of r n n 3 centered at the origin and 1 p p s p s n 2 n 2 we prove that for any radial solution u p of e p with m nodal domains its morse index m u p is given by the formula m u p m n m 1 if p is sufficiently close to p s

on nonlinear diffusion problems with strong degeneracy in this paper we study the triply degenerate problem b v t g v div v f on q 0 t b v 0 b v0 on and g v g a on some part of the boundary 0 t in the case of continuous nonhomogeneous and nonstationary boundary data a the functions b g are assumed to be continuous locally lipschitz nondecreasing and to verify the normalization condition b 0 g 0 0 and the range condition r b g r using monotonicity and penalization methods we prove existence of a weak renormalized entropy solution in the spirit of k ammar j carrillo p wittbold scalar conservation laws with general boundary condition and continuous flux function j differential equations 228 2006 111 139

stability of the 1d ibvp for a non autonomous scalar conservation law we prove the stability with respect to the flux of solutions to initial boundary value problems for scalar non autonomous conservation laws in one space dimension key estimates are obtained through a careful construction of the solutions

existence of clustering high dimensional bump solutions of superlinear elliptic problems on expanding annuli abstract we consider the nonlinear elliptic problem u u p in r u 0 in r u 0 in r where p 1 and r x r n r x r 1 with n 3 it is known that as r the number of nonequivalent solutions of the above problem goes to when p 1 n 2 n 2 n 3 here we prove the same phenomenon for any p 1 by finding o n 1 symmetric clustering bump solutions which concentrate near the set x 1 x n r x n 0 for large r 0

existence of positive nonradial solutions for nonlinear elliptic equations in annular domains we study the existence of positive nonradial solutions of equation a f u 0 in sia u 0 on d la where s x 6 w a 2 and is positive and superlinear at both 0 and oo we use a bifurcation method to show that there is a nonradial bifurcation with mode k at ak 0 1 for any positive integer k if is subcritical and for large k if is supercritical when is subcritical then a nehari type variational method can be used to prove that there exists a e 0 1 such that for any a e a 1 the equation has a nonradial solution on 0 a

symmetry breaking bifurcation for the one dimensional liouville type equation abstract the two point boundary value problem for the one dimensional liouville type equation u x l e u 0 x 1 1 u 1 u 1 0 is considered where 0 and l 0 in this paper a symmetry breaking result is obtained by using the morse index the problem u x l u 1 p 0 x 1 1 u 1 u 1 0 is also considered where 0 l 0 p 1 and p 1 l 4

solutions of superlinear elliptic equations and their morse indices we consider here solutions of semilinear second order elliptic equations with superlinear nonlinearities and we present some relationships between their morse indices and some qualitative properties in particular we show that for uo subcritical nonlinearities bounds on solutions are equivalent to bounds on their morse indices

renormalized entropy solutions of stochastic scalar conservation laws with boundary condition abstract this paper is concerned with the renormalized stochastic entropy solutions of stochastic scalar conservation law forced by a multiplicative noise on a bounded domain with a non homogeneous boundary condition we first introduce a notion of renormalized stochastic entropy solutions and then establish the existence and uniqueness of the renormalized stochastic entropy solutions for a general l 1 data our results allow us to give a positive answer to an open problem posed by bauzet vallet and wittbold in 4

a theory of l 1 dissipative solvers for scalar conservation laws with discontinuous flux we propose a general framework for the study of l1 contractive semigroups of solutions to conservation laws with discontinuous flux u t mathfrak f x u x 0 qquad mathfrak f x u left begin array ll f l u x 0 end array right quad quad quad rm cl where the fluxes fl fr are mainly assumed to be continuous developing the ideas of a number of preceding works baiti and jenssen in j differ equ 140 1 161 185 1997 towers in siam j numer anal 38 2 681 698 2000 towers in siam j numer anal 39 4 1197 1218 2001 towers et al in skr k nor vidensk selsk 3 1 49 2003 adimurthi et al in j math kyoto university 43 1 27 70 2003 adimurthi et al in j hyperbolic differ equ 2 4 783 837 2005 audusse and perthame in proc roy soc edinburgh a 135 2 253 265 2005 garavello et al in netw heterog media 2 159 179 2007 burger et al in siam j numer anal 47 1684 1712 2009 we claim that the whole admissibility issue is reduced to the selection of a family of elementary solutions which are piecewise constant weak solutions of the form c x c l11 left x 0 right we refer to such a family as a germ it is well known that cl admits many different l1 contractive semigroups some of which reflect different physical applications we revisit a number of the existing admissibility or entropy conditions and identify the germs that underly these conditions we devote specific attention to the vanishing viscosity germ which is a way of expressing the condition of diehl j hyperbolic differ equ 6 1 127 159 2009 for any given germ we formulate germ based admissibility conditions in the form of a trace condition on the flux discontinuity line x 0 in the spirit of vol pert math ussr sbornik 2 2 225 267 1967 and in the form of a family of global entropy inequalities following kruzhkov math ussr sbornik 10 2 217 243 1970 and carrillo arch ration mech anal 147 4 269 361 1999 we characterize those germs that lead to the l1 contraction property for the associated admissible solutions our approach offers a streamlined and unifying perspective on many of the known entropy conditions making it possible to recover earlier uniqueness results under weaker conditions than before and to provide new results for other less studied problems several strategies for proving the existence of admissible solutions are discussed and existence results are given for fluxes satisfying some additional conditions these are based on convergence results either for the vanishing viscosity method with standard viscosity or with specific viscosities adapted to the choice of a germ or for specific germ adapted finite volume schemes

maximal entropy solutions under prescribed mass and energy abstract we consider a non local elliptic equation with exponential nonlinearity closely related to the mean field liouville equation the motivation for this equation is a variational entropy maximization problem under prescribed mass and energy we provide an unconditional existence proof in case of electrostatic repulsive self interaction and conditional existence and uniqueness in dimension two in the case of gravitational attractive self interaction

positive solutions for the p laplacian in annuli we consider the dirichlet problem for the equation p u q x f u in an annulus r n n 1 where p u div u p 2 u is the p laplacian operator p 1 with no assumption on the behaviour of the nonlinearity f either at zero or at infinity we prove existence and localization of positive radial solutions for this problem by applying schauder s fixed point theorem precisely we show the existence of at least one such solution each time the graph of f passes through an appropriate tunnel so it is easy to exhibit multiple or even infinitely many positive solutions moreover upper and lower bounds for the maximum value of the solution are obtained our results are easily extended to the exterior of a ball when n p

infinitely many non radial solutions to a critical equation on annulus abstract in this paper we build infinitely many non radial sign changing solutions to the critical problem p u u 4 n 2 u in u 0 on on the annulus x r n a x b n 3 in particular for any integer k large enough we build a non radial solution which look like the unique positive solution u 0 to p crowned by k negative bubbles arranged on a regular polygon with radius r 0 such that r 0 n 2 2 u 0 r 0 max a r b r n 2 2 u 0 r

state classification of time nonhomogeneous markov chains and average reward optimization of multi chains in a discrete time nonhomogeneous markov chain tnhmc the states spaces transition probabilities and reward functions at different times may be different in this paper with the confluencity previously introduced we show that the states of a tnhmc can be classified into the branching states and a number of classes of confluent states versus the transient and recurrent states in the time homogeneous case the optimization of average reward in tnhmc s consisting of a single confluent class uni chain have been addressed in a previous paper by the author in this paper we show that with confluencity and the state classification and under some bound conditions we can obtain the necessary and sufficient conditions for optimal policies of the average reward of tnhmcs consisting of multiple confluent classes multi chains just like in the uni chain tnhmc case the sufficient condition does not need to hold in any zero frequently visited time sequence this under selectivity makes the problem not amenable to dynamic programming a direct comparison based approach is used to prove the results the results enhance our understanding of state classification and performance optimization with the notion of confluencity

scalar conservation laws with general boundary condition and continuous flux function abstract we introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition u t div u f on q 0 t u 0 u 0 on and u a on some part of the boundary 0 t existence and uniqueness of the entropy solution is established for any c r r n u 0 l f l q a l 0 t in the l 1 setting a corresponding result is proved for the more general notion of renormalised entropy solution

optimality conditions for long run average rewards with underselectivity and nonsmooth features we study three existing issues associated with optimization of long run average rewards of time nonhomogeneous markov processes in continuous time with continuous state spaces 1 the underselectivity i e the optimal policies do not depend on their actions in any finite time period 2 its related issue the bias optimality i e policies that optimize both long run average and transient total rewards and 3 the effects of a nonsmooth point of a value function on performance optimization these issues require considerations of the performance in the entire period with an infinite horizon and therefore are not easily solvable by dynamic programming which works backwards in time and takes a local view at a particular time instant in this paper we take a different approach called the relative optimization theory which is based on a direct comparison of the performance measures of any two policies we derive tight necessary and sufficient optimality conditions that take the underselectivity into consideration we derive bias optimality conditions for both long run average and transient rewards and we show that the effect of a wide class of nonsmooth points called semismooth points of a value function on the long run average performance is zero and can be ignored

nonradial sign changing solutions to lane emden problem in an annulus abstract in this paper we prove the existence of continua of nonradial solutions for the lane emden equation in the annulus in a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones next using the fixed point index in cone we produce nonradial solutions with a new type of symmetry this result also applies to solutions with fixed signed showing that the set of solutions to the lane emden problem has a very rich and complex structure

nonhomogeneous dirichlet problems for degenerate parabolic hyperbolic equations the aim of the paper is to give a formulation for the initial boundary value problem of parabolic hyperbolic type in the case of nonhomogeneous boundary data a 0 here u u x t with x t q 0 t where is a bounded domain in n with smooth boundary and t 0 the function b is assumed to be nondecreasing allowing degeneration zones where b is constant is locally lipschitz continuous and g l 0 b after introducing the definition of an entropy solution to the above problem in the spirit of otto 14 we prove uniqueness of the solution in the proposed setting moreover we prove that the entropy solution previously defined can be obtained as the limit of solutions of regularized equations of nondegenerate parabolic type specifically the diffusion function b is approximated by functions b that are strictly increasing

nonradial maximizers for a henon type problem and symmetry breaking bifurcations for a liouville gel fand problem with a vanishing coefficient we study the radiality and nonradiality of the maximizers of begin aligned sup left int d x alpha e u 1 dx u in h 0 1 d int d left nabla u right 2dx beta 2 right end aligned from a viewpoint of the bifurcation diagram of the euler lagrange equation begin aligned delta u beta 2 frac x alpha e u int d x alpha ue udx 0 hbox in d qquad u 0 hbox on partial d end aligned where d x in mathbb r 2 x 1 we show that the maximizers are nonradial if begin aligned beta sqrt 8 pi left alpha 2 log frac 2 alpha 4 alpha frac alpha 4 2 right end aligned and that the maximizers are radial if alpha 0 or beta 0 is small we also give the bifurcation diagrams of the liouville gel fand problem with a vanishing coefficient begin aligned delta u lambda x alpha e u 0 hbox in d qquad u 0 hbox on partial d end aligned and of the associated mean field equation the analysis is based on exact solutions and explicit calculations

a minimax method for finding multiple critical points and its applications to semilinear pdes most minimax theorems in critical point theory require one to solve a two level global optimization problem and therefore are not for algorithm implementation the objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way in this paper inspired by the numerical works of choi mckenna and ding costa chen and the idea to define a solution submanifold some local minimax theorems are established which require us to solve only a two level local optimization problem based on the local theory a new local numerical minimax method for finding multiple saddle points is developed the local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex non star shaped and multiconnected domains numerical solutions are illustrated by their graphics for visualization in a subsequent paper y li and j zhou convergence results of a minimax method for finding critical points in review we establish some convergence results for the algorithm

superlinear problems without ambrosetti and rabinowitz growth condition superlinear elliptic boundary value problems without ambrosetti and rabinowitz growth condition are considered existence of nontrivial solution result is established by combining some arguments used by struwe and tarantello and schechter and zou also by wang and wei firstly by using the mountain pass theorem due to ambrosetti and rabinowitz is constructed a solution for almost every parameter by varying the parameter then it is considered the continuation of the solutions

symmetry breaking for solutions of semilinear elliptic equations with general boundary conditions we study the bifurcation of radially symmetric solutions of f u 0 onn balls into asymmetric ones we show that ifu satisfies homogeneous neumann boundary conditions the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension for general boundary conditions we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions and on the structure of the kernels of the linearized operators

on the existence of positive radial solutions for nonlinear elliptic equations in annular domains abstract we study the existence of positive radially symmetric solutions of u g x f u 0 in annulus with dirichlet dirichlet neumann boundary conditions we show that the equation has a positive radial solution on any annulus if f and f are positive and f is superlinear at 0 and

asymptotically radial solutions in expanding annular domains in this paper we consider the problem left begin array ll delta u u p quad rm in omega r u 0 quad quad quad rm on partial omega r quad quad quad 0 1 end array right where p 1 and r is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as r longrightarrow infty the main goal of the paper is to prove for large r the existence of a positive solution to 0 1 which is close to the positive radial solution in the corresponding diffeomorphic annulus the proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions

monotonicity of the first eigenvalue and the global bifurcation diagram for the branch of interior peak solutions abstract let b r n n 1 be a unit ball and d r m be a bounded domain we study the global branch consisting of interior single peak solutions of the elliptic neumann problem u u u p 0 in b u 0 on b and the monotonicity of the first eigenvalue along the branch for large where 1 p p s n 2 n 2 n 3 n 1 2 when the domain is replaced with b d using this monotonicity we show that the branch of solutions concentrating on 0 d has secondary bifurcation points for n 1 and p 2 an integer we determine the global bifurcation diagram by showing the monotonicity of the time map the period function the monotonicity of the first eigenvalue along the whole branch is also proved for n 1 and p 3

a new optimality criterion for nonhomogeneous markov decision processes we propose a new definition of optimality for nonhomogeneous markov decision processes called periodic forecast horizon pfh optimality using measures of discounting and ergodicity we establish conditions under which pfh optimal strategies exist and pfh optimality implies the more conventional notions of optimality and average optimality finally we use this definition of optimality as the basis for a direct development of forecast horizon results for discounted and undiscounted nonhomogeneous markov decision processes

initial boundary value problems for conservation laws with source terms and the degasperis procesi equation we consider conservation laws with source terms in a bounded domain with dirichlet boundary conditions we first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition equipped with this formulation we go on to establish the well posedness of entropy solutions to the initial boundary value problem the proof utilizes the kinetic formulation and the averaging lemma finally we make use of these results to demonstrate the well posedness in a class of discontinuous solutions to the initial boundary value problem for the degasperis procesi shallow water equation which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term

the n th order bias optimality for multichain markov decision processes in this paper we propose a new approach to the theory of finite multichain markov decision processes mdps with different performance optimization criteria we first propose the concept of nth order bias then using the average reward and bias difference formulas derived in this paper we develop an optimization theory for finite mdps that covers a complete spectrum from average optimality bias optimality to all high order bias optimality in a unified way the approach is simple direct natural and intuitive it depends neither on laurent series expansion nor on discounted mdps we also propose one phase policy iteration algorithms for bias and high order bias optimal policies which are more efficient than the two phase algorithms in the literature furthermore we derive high order bias optimality equations this research is a part of our effort in developing sensitivity based learning and optimization theory

least energy solutions of the emden fowler equation in hollow thin symmetric domains abstract in this paper the emden fowler equation is studied in a hollow thin domain which is invariant under the action of a closed subgroup of the orthogonal group then it is proved that if the domain is thin enough a least energy solution is not group invariant

a probabilistic analysis of bias optimality in unichain markov decision processes focuses on bias optimality in unichain finite state and action space markov decision processes using relative value functions we present methods for evaluating optimal bias this leads to a probabilistic analysis which transforms the original reward problem into a minimum average cost problem the result is an explanation of how and why bias implicitly discounts future rewards

stochastic feedback control with one dimensional degenerate diffusions and nonsmooth value functions we study the stochastic control problems with degenerate semimartingales and nonsmooth value functions we first derive the optimality conditions for finite horizon problems with nonsmooth value functions and singular control of infinite horizon discounted problems then we show that the local time at the degenerate point is zero so if the value function has nonsmooth first order derivatives at the degenerate points of all policies the nonsmoothness can be ignored and the optimality equation for stochastic control including optimal stopping is simply the classical hamilton jacobi bellman equation at the smooth points furthermore we model the singular control by the reflecting force in the skorokhod problem and show that there are two classes of degenerate points in the first class singular control at a degenerate point can be treated in the same way as regular control and in the second class the density of the singular control force may be of the order sqrt dt 1 gamma 0 with gamma 0 for nondegenerate points we apply the direct comparison based approach to derive all the results in all the analysis viscosity solution is not needed

solutions in spectral gaps for a nonlinear equation of schrodinger type abstract in this paper we study the existence of a nontrivial h2 r n solution for an equation of the form u x p x u x f x u x u x x r n r where p l r n is a periodic function we assume that the operator p h2 r n l2 r n l2 r n is strongly indefinite and invertible and that x r r is odd and satisfies some superlinear but subcritical growth conditions we extend the class of nonlinearities which has been studied up to now in particular under standard technical restrictions the existence of a solution is derived when lim x f x s f s 0 exists for all s r if we assume that f x s f s for all s r and a e on r n

existence and nonexistence of positive radial solutions for some semilinear elliptic problems in annular domains au f x u 0 in s u 0 on xj when n is some annulus and the above conditions in 4 29 are not necessarily satisfied in fact in our main result theorem 2 4 f is not necessarily c lim f t u u may not exist or may be u rn finite and f t u could be zero for u in some finite interval 0 p we also allow lim f t u u 0 corollary 2 6 and lim f t u u may not exist in example 2 3 below u o u 0 neither lim f u u nor lim f u u exists on the other hand we show that u 0 u m

uniqueness of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic paraboic problems associated with the equation b v t div a v dv f using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for renormalized solutions

the existence and uniqueness of bound state solutions of a semi linear equation we investigate the existence and uniqueness of bounded radially symmetric solutions in r3 of uf u 0 where f u is continuous piecewise linear and has three distinct real zeros we give a detail

elliptic equations of order 2m in annular domains in this paper we study the existence of positive radial solutions for some semilinear elliptic problems of order 2m in an annulus with dirichlet boundary conditions we consider a nonlinearity which is either sublinear or the sum of a sublinear and a superlinear term

bifurcation in lp rn for a semilinear elliptic equation keywords lsp p bifurcation constrained minimization sobolev spaces nonlinear eigenvalue problem semilinear spectrum linearisation variational structure reference ana article 1988 001doi 10 1112 plms s3 57 3 511 record created on 2008 12 10 modified on 2016 08 08

symmetry breaking for positive solutions of semilinear elliptic equations in a recent interesting paper gidas ni and nirenberg 2 proved that positive solutions of the dirichlet problem for second order semi linear elliptic equations on balls must themselves be spherically symmetric functions here we consider the bifurcation problem for such solutions specifically we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution when this happens we say that the symmetry breaks to carry out this program we rely on certain results in 6 where we studied the kernel of the associated linearized operator this enables us to give some necessary conditions for symmetry to break we also find a class of functionsf where these conditions are also sufficient see equation 18 below the problem is of course to show that when zero comes into the spectrum of the linearized operator and our conditions are fulfilled bifurcation into the non radial direction actually occurs this is done by showing first that the bifurcation curve is a smooth manifold near the bifurcation point and then appealing to a theorem of vanderbauwhede 8 which gives sufficient conditions for bifurcation to occur in the presence of symmetries

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

perturbation realization potentials and sensitivity analysis of markov processes two fundamental concepts and quantities realization factors and performance potentials are introduced for markov processes the relations among these two quantities and the group inverse of the infinitesimal generator are studied it is shown that the sensitivity of the steady state performance with respect to the change of the infinitesimal generator can be easily calculated by using either of these three quantities and that these quantities can be estimated by analyzing a single sample path of a markov process based on these results algorithms for estimating performance sensitivities on a single sample path of a markov process can be proposed the potentials in this paper are defined through realization factors and are shown to be the same as those defined by poisson equations the results provide a uniform framework of perturbation realization for infinitesimal perturbation analysis ipa and non ipa approaches to the sensitivity analysis of steady state performance they also provide a theoretical background for the pa algorithms developed in recent years

instability analysis of saddle points by a local minimax method the objective of this work is to develop some tools for local instability analysis of multiple critical points which can be computationally carried out the morse index can be used to measure local instability of a nondegenerate saddle point however it is very expensive to compute numerically and is ineffective for degenerate critical points a local weak linking index can also be defined to measure local instability of a degenerate saddle point but it is still too difficult to compute in this paper a local instability index called a local minimax index is defined by using a local minimax method this new instability index is known beforehand and can help in finding a saddle point numerically relations between the local minimax index and other local instability indices are established those relations also provide ways to numerically compute the morse local linking indices in particular the local minimax index can be used to define a local instability index of a saddle point relative to a reference trivial critical point even in a banach space while others failed to do so

denumerable state nonhomogeneous markov decision processes we consider denumerable state nonhomogeneous markov decision processes and extend results from both denumerable state homogeneous and finite state nonhomogeneous problems we show that under weak ergodicity accumulation points of finite horizon optima termed algorithmic optima are average cost optimal we also establish the existence of solution horizons finally an algorithm is presented to solve problems of this class for the case where there is a unique algorithmic optimum

existence of radial solutions for nonlinear elliptic equations with gradient terms in annular domains abstract this paper is concerned with the problem of elliptic equations in annular domains involving derivative terms using nonlinear analysis methods some results regarding existence of solutions are established more precisely at least one radial solution is obtained basing on schauder s fixed point theorem and contraction mapping theorem respectively some iterative technique is also applied which allows us to overcome difficulties that the nonlinearity is related to derivative terms

uniqueness for inhomogeneous dirichlet problem for elliptic parabolic equations we show the l 1 contraction and comparison principle for weak and more generally renormalized solutions of the elliptic parabolic problem j v t div w f w f t x w phi v in 0 t omega ir ir n with inhomogeneous dirichlet boundary datum g l 2 0 t w 1 2 omega for w which is taken in the sense w g l 2 0 t h 1 0 omega and initial datum j o l 1 omega for j v here phi j are nondecreasing and we assume f just continuous our proof consists in doubling of variables in the interior of omega as introduced by j carrillo 9 arch rational mech anal vol 147 1999 and in a careful treatment of the flux term near the boundary of omega for this last argument the result is restricted to the linear dependence on w of the diffusion term the proof allows for a wide class of domains omega including e g weakly lipschitz domains with lipschitz cracks we obtain the corresponding results for the associated stationary problem and discuss on generalization of our technique to the case of nonlinear diffusion operators

well posedness results for triply nonlinear degenerate parabolic equations we study well posedness of triply nonlinear degenerate elliptic parabolic hyperbolic problems of the kind b u t div a u u u f u t 0 u0 in a bounded domain with homogeneous dirichlet boundary conditions the nonlinearities b and are supposed to be continuous non decreasing and the nonlinearity a falls within the leray lions framework some restrictions are imposed on the dependence of a u u on u and also on the set where degenerates a model case is a u u f b u u u k u a0 u with a nonlinearity which is strictly increasing except on a locally finite number of segments and the nonlinearity a0 which is of the leray lions kind we are interested in existence uniqueness and stability of l entropy solutions for the parabolic hyperbolic equation b id we obtain a general continuous dependence result on data u0 f and nonlinearities b a similar result is shown for the degenerate elliptic problem which corresponds to the case of b 0 and general non decreasing surjective existence uniqueness and continuous dependence on data u0 f are shown in more generality for instance the assumptions b r r and the continuity of b 1 permit to achieve the well posedness result for bounded entropy solutions of this triply nonlinear evolution problem