on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

asymptotic behavior of large solutions to p laplacian of bieberbach rademacher type abstract by the karamata regular variation theory and the method of lower and upper solutions we establish the asymptotic behavior of boundary blow up solutions of the quasilinear elliptic equation div u p 2 u b x f u in a bounded r n subject to the singular boundary condition u x where the weight b x is non negative and non trivial in which may be vanishing on the boundary or go to unbounded the nonlinear term f is a varying function at infinity whose variation at infinity is not regular

existence of solutions for weighted p r laplacian system boundary value problems this paper investigates the existence of solutions for weighted p r laplacian ordinary system boundary value problems via leray schauder degree the sufficient conditions for the existence of solutions be given as an application we obtain the existence of the radial solutions for the p x laplacian partial differential system dirichlet problems

integral action in robust adaptive control conditions are studied under which integral action in robust adaptive control is soundly based the method of analysis amounts to verifying that the key conditions needed for convergence are satisfied a simple strategy is described for choosing the integral constant and global convergence is established for the resultant algorithm this illustrates a proof paradigm which could be similarly applied to other problems the principal assumptions used are that the plant when augmented by an integrator has a known degree of controllability and that an overbounding function is known for the unmodeled system response the continuous time case is treated but the corresponding discrete time results follow mutatis mutandis by simply replacing p d dt by delta q 1 delta l sub 2 by l sub 2 integral by sigma and so on

eigenvalues of p x laplacian dirichlet problem abstract this paper studies the eigenvalues of the p x laplacian dirichlet problem div u p x 2 u u p x 2 u in u 0 on where is a bounded domain in rn and p x is a continuous function on such that p x 1 we show that the set of eigenvalues is a nonempty infinite set such that sup we present some sufficient conditions for inf 0 and for inf 0 respectively

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

besov spaces with variable smoothness and integrability abstract in this article we introduce besov spaces with variable smoothness and integrability indices we prove independence of the choice of basis functions as well as several other basic properties we also give sobolev type embeddings and show that our scale contains variable order holder zygmund spaces as special cases we provide an alternative characterization of the besov space using approximations by analytic functions

boundary blow up solutions for a class of elliptic equations on a bounded domain this paper deals with existence of solutions of the quasilinear elliptic equation du f x u u in w satisfying the boundary blow up condition u x x email protected where email protected r n is a bounded domain with smooth boundary email protected our main result applies to the existence of blow up solutions of a class of equations which appear in stochastic control theory namely du a x g u email protected u email protected x in w where a x g u j x are suitable functions and l s are nonnegative constants our approach employs an approximation procedure combined with both a non monotone iteration variant of the method of lower and upper solutions and fixed point arguments

a robust indirect adaptive control approach this paper considers the robust design of an indirect adaptive control approach which is applicable when the unknown parameters of a linear time invariant plant lie in a known convex set throughout which no unstable pole zero cancellation occurs in order to achieve the robustness the use of a relative dead zone in the adaptive law is proposed it is shown that with a suitably designed relative dead zone the adaptive control system is globally stable even in the presence of small unmodeled plant uncertainties

existence and uniqueness of positive solutions to the boundary blow up problem for an elliptic system abstract an elliptic system is considered in a smooth bounded domain subject to dirichlet boundary conditions of three different types based on the construction of certain upper and sub solutions we obtain some conditions on the parameters a i b i c i i 1 2 and the exponents m n p q to ensure the existence of positive solutions furthermore uniqueness and boundary behavior of positive solutions is also discussed

fdtd dispersive modeling with high order rational constitutive parameters in this work we present a dispersive finite difference time domain fdtd algorithm using a four pole complex rational function crf for the sake of a better curve fitting of the four pole crf dispersion model we use a particle swarm optimization technique we also discuss an efficient memory storage strategy using a state space approach the numerical aspects of four pole crf fdtd the numerical accuracy and the numerical stability are investigated in detail numerical examples are used to validate four pole crf fdtd and numerical stability issues are discussed in detail we also discuss the computational accuracy and the computational efficiency of an arbitrary n pole crf fdtd

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

robust model reference adaptive control we propose a new model reference adaptive control algorithm and show that it provides the robust stability of the resulting closed loop adaptive control system with respect to unmodeled plant uncertainties the robustness is achieved by using a relative error signal in combination with a dead zone and a projection in the adaptive law the extra a priori information needed to design the adaptive law are bounds on the plant parameters and an exponential bound on the impulse response of the inverse plant transfer function

general uniqueness results and variation speed for blow up solutions of elliptic equations let omega be a smooth bounded domain in r n we prove general uniqueness results for equations of the form delta u au b x f u in omega subject to u infty on partial omega our uniqueness theorem is established in a setting involving karamata s theory on regularly varying functions which is used to relate the blow up behavior of u x with f u and b x where b equiv 0 on partial omega and a certain ratio involving b is bounded near partial omega a key step in our proof of uniqueness uses a modification of an iteration technique due to safonov

symmetry of large solutions of nonlinear elliptic equations in a ball let g be a locally lipschitz continuous real valued function which satisfies the keller osserman condition and is convex at infinity then any large solution of u g u 0 in a ball is radially symmetric

boundary blow up and degenerate equations let r2 be a bounded domain of class c2 0 1 we show that if u is the solution of u 4exp 2u which tends to as x y then the hyperbolic radius v exp u is also of class c2 up to the boundary the proof relies on new schauder estimates for degenerate elliptic equations of fuchsian type

harnack s inequality for p harmonic functions with unbounded exponent p abstract we study properties of the function u lim u where u is the solution of the min p laplacian dirichlet problem with bounded sobolev boundary function here p n is a variable exponent such that 1 p is lipschitz continuous we derive bloch type estimates and using them we prove harnack s inequality in cases of unbounded but finite exponent

exact multiplicity for boundary blow up solutions abstract the singularly perturbed boundary blow up problem e 2 u u u a 1 u u 0 in b u on b is studied in the unit ball b r n n 2 a 1 2 1 is a constant it is shown that for e 0 sufficiently small there exist exactly three positive solutions for the problem and all of them are radially symmetric solutions

existence and asymptotic behavior of blow up solutions to a class of p x laplacian problems abstract this paper investigates the problem p x u f x u 0 in u x as d x 0 where p x u div u p x 2 u is called p x laplacian the existence of blow up solutions is discussed and the singularity of blow up solutions is given

existence of strong solutions of a p x laplacian dirichlet problem without the ambrosetti rabinowitz condition abstract in this paper we consider the existence of strong solutions of the following p x laplacian dirichlet problem via critical point theory d i v u p x 2 u f x u in u 0 on we give a new growth condition under which we use a new method to check the cerami compactness condition hence we prove the existence of strong solutions of the problem as above without the growth condition of the well known ambrosetti rabinowitz type and also give some results about multiplicity of the solutions

on the sub supersolution method for p x laplacian equations abstract this paper deals with the sub supersolution method for the p x laplacian equations a sub supersolution principle for the dirichlet problems involving the p x laplacian is established it is proved that the local minimizers in the c 1 topology are also local minimizers in the w 1 p x topology for given energy functionals a strong comparison theorem for the p x laplacian equations is presented some applications of the abstract theorems obtained in this paper to the eigenvalue problems for the p x laplacian equations are given

numerical dispersion analysis of the unconditionally stable three dimensional lod fdtd method the numerical dispersion characteristics of the recently developed three dimensional unconditionally stable locally one dimensional finite difference time domain lod fdtd method are derived analytically the effect of grid size and the courant friedrich lewy cfl limits on dispersion are studied in detail the lod fdtd method allows larger time steps as compared to the conventional fdtd method cfl limit the analysis shows that the unconditionally stable three dimensional lod fdtd method has an advantage over the conventional fdtd method when modeling structures that require fine grids the lod fdtd method allows larger cfl numbers as long as the dispersion error remains in acceptable range

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

a necessary and sufficient condition for existence of large solutions to semilinear elliptic equations we consider the semilinear equation u p x f u on a domain rn n 3 where f is a nonnegative nondecreasing continuous function which vanishes at the origin and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in such that p is positive on its boundary for bounded we show that a nonnegative solution u satisfying u x as x exists if and only if the function s s0 f t dt satisfies 1 s 1 2 ds for unbounded including rn we show that a similar result holds where u x as x within and u x as x if p x decays to zero rapidly as x

dispersion analysis of the three dimensional complex envelope adi fdtd method in this paper numerical dispersion properties of the three dimensional complex envelope ce alternate direction implicit finite difference time domain adi fdtd method are studied the variations of dispersion errors with propagation direction ratio of carrier to envelope frequencies and spatial and temporal steps are presented it is found that the ce adi fdtd scheme have much better accuracy and efficiency over the adi fdtd especially with a higher ratio of carrier to envelope frequencies therefore the ce adi fdtd is recommended for use in efficient narrow bandwidth electromagnetic modeling

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

examination clarification and simplification of stability and dispersion analysis for adi fdtd and cnss fdtd schemes we describe a rigorous analysis of unconditional stability for the alternating direction implicit finite difference time domain adi fdtd and crank nicholson split step cnss fdtd schemes avoiding use of the von neumann spectral criterion the proof is performed in the spectral domain and uses skew hermitivity of matrix terms in the adi and cnss total amplification matrices a bound for the total adi amplification matrix is provided while the cn and cnss fdtd amplification matrices are unitary the bound on the adi fdtd depends on the courant number importantly we have found that the adi fdtd amplification matrix is not normal i e the unit spectral radius alone cannot be used to prove the adi fdtd unconditional stability the paper also shows that the adi fdtd and cnss fdtd schemes share the same dispersion equation by a similarity of their total amplification matrices and the two schemes have identical numerical dispersion in the frame of plane waves

fundamental schemes for efficient unconditionally stable implicit finite difference time domain methods generalized formulations of fundamental schemes for efficient unconditionally stable implicit finite difference time domain fdtd methods are presented the fundamental schemes constitute a family of implicit schemes that feature similar fundamental updating structures which are in simplest forms with most efficient right hand sides the formulations of fundamental schemes are presented in terms of generalized matrix operator equations pertaining to some classical splitting formulae including those of alternating direction implicit locally one dimensional and split step schemes to provide further insights into the implications and significance of fundamental schemes the analyses are also extended to many other schemes with distinctive splitting formulae detailed algorithms are described for new efficient implementations of the unconditionally stable implicit fdtd methods based on the fundamental schemes a comparative study of various implicit schemes in their original and new implementations is carried out which includes comparisons of their computation costs and efficiency gains

a strong maximum principle for differential equations with nonstandard p x growth conditions abstract in this paper with some special technics we give a strong maximum principle for the equations with nonstandard p x growth conditions div x u u d x f x u 0 in where x s d x f x u satisfy some conditions

existence results to elliptic systems with nonstandard growth conditions abstract nonstandard growth conditions in partial differential equations have been the subject of recent developments in elastic mechanics and electrorheological fluid dynamics lecture notes in mathematics vol 1748 2000 c r acad sci paris ser i math 329 1999 393 398 math ussr izv 29 1987 33 66 in this work elliptic systems with nonstandard growth conditions are studied existence and multiplicity results under growth conditions on the reaction terms are established

analysis of the divergence properties for the three dimensional leapfrog adi fdtd method analysis of the divergence properties for the three dimensional 3 d leapfrog alternating direction implicit finite difference time domain adi fdtd method is presented in this paper the analysis of divergence properties is based on the fourier domain plane wave method for dispersion analysis but extended with additional procedure to determine the homogeneous solutions the singular value decomposition svd is used to obtain the solutions of the homogeneous system in a reliable manner there are two linearly independent null modes in the homogeneous system corresponding to the repeated zero singular values to provide more physical insights proper linear combinations of the two null modes are determined as te and tm modes with respect to the direction of discrete numerical space wavevector the divergence of e fields is analyzed by examining the tm mode while the divergence of h fields is analyzed by examining the te mode numerical results for the divergence properties of the leapfrog adi fdtd method are presented and compared with the conventional adi fdtd method along with numerical experiments to illustrate the existence of divergent waves the leapfrog adi fdtd method is not divergence free in the source free region as is the case for the conventional adi fdtd method furthermore both methods have different divergence properties in general

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

on numerical artifacts of the complex envelope adi fdtd method we examine two spurious numerical artifacts of the complex envelope ce alternating direction implicit finite difference time domain adi fdtd method viz spurious charges and anomalous wave propagation modes with positive phase velocity and negative group velocity these artifacts are also present in the conventional adi fdtd however the spurious charges in ce adi fdtd have a fundamental distinction from those of adi fdtd they are static in adi fdtd and implicitly time harmonic in ce adi fdtd spurious charges are particularly detrimental to ce adi fdtd simulations because they produce secondary radiation we also show that spurious charges can be reduced by a fixed point iterative correction in ce adi fdtd

unconditionally stable fundamental lod fdtd method with second order temporal accuracy and complying divergence an unconditionally stable fundamental locally one dimensional lod finite difference time domain fdtd method with second order temporal accuracy and complying divergence cd denoted as lod2 cd fdtd is presented for three dimensional 3 d maxwell s equations while the conventional lod fdtd method does not have complying divergence the lod2 cd fdtd method has complying divergence in a manner analogous to the conventional explicit fdtd method the update procedures for a family of lod fdtd methods that employ similar splitting matrix operators are presented by extending the previous concept of achieving second order temporal accuracy for the lod2 fdtd method via implicit output processing we hereby propose novel explicit output processing that not only retains second order temporal accuracy but also complying divergence for the lod2 cd fdtd method the current source implementation for the lod2 cd fdtd method involves source incorporation in only the first procedure to further enhance efficiency the lod2 cd fdtd method is formulated into the fundamental lod2 cd fdtd method with efficient matrix operator free right hand sides subsequently detailed implementation for the fundamental lod2 cd fdtd method is presented analytical proof is provided to ascertain the second order temporal accuracy of the lod2 cd fdtd method numerical results and examples are also presented to validate the divergence complying property of the lod2 cd fdtd method

boundary blow up elliptic problems of bieberbach and rademacher type with nonlinear gradient terms abstract by a perturbation method and constructing comparison functions we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x e u x u where is a bounded domain with smooth boundary in r n q 0 b is non negative and non trivial in which may be vanishing on the boundary

on positive weak solutions for a class of quasilinear elliptic systems abstract in this paper we study the existence and nonexistence of positive weak solution for the quasilinear elliptic system p u u v q v u v with zero dirichlet boundary condition in bounded domain r n a critical exponent is obtained for existence and nonexistence of positive weak solution

a multiplicity theorem for problems with the p laplacian we consider a nonlinear elliptic problem driven by the p laplacian with a parameter r and a nonlinearity exhibiting a superlinear behavior both at zero and at infinity we show that if the parameter is bigger than 2 the second eigenvalue of p w01 p z then the problem has at least three nontrivial solutions our approach combines the method of upper lower solutions with variational techniques involving the second deformation theorem the multiplicity result that we prove extends an earlier semilinear i e p 2 result due to struwe m struwe variational methods springer verlag berlin 1990

existence and blow up rate of large solutions of p x laplacian equations with gradient terms abstract in this paper we investigate boundary blow up solutions of the problem p x u f x u k x u m x in u x as d x 0 where p x u div u p x 2 u is called the p x laplacian our results extend the previous work 25 of y liang q h zhang and c s zhao from the radial case to the non radial setting and 46 due to q h zhang and d motreanu from the assumption that k x u x m x is a small perturbation to the case in which k x u m x is a large perturbation we provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of d x and in terms of the growth of the exponents furthermore the comparison principle is no longer applicable in our context since f x is not assumed to be monotone in this paper

the boundary blow up rate of large solutions abstract in this paper we ascertain the blow up rate of the large solutions of a class of sublinear elliptic boundary value problems with a weight function in front of the nonlinearity that vanishes on the boundary of the underlying domain at different rates according to the point of the boundary x all previous results in the literature assumed the decay rate of the underlying weight function to be the same at any point of this hypothesis substantially simplified the mathematical analysis of the problem as it allowed constructing global sub and supersolutions in an open neighborhood of obtaining general results requires localizing at each particular point of the boundary making particularly involved the mathematical analysis of the problem

gradient estimates for the p x laplacean system we prove calderon and zygmund type estimates for a class of elliptic problems whose model is the non homogeneous p x laplacean system div jduj p x 2 du div jf j p x 2 f

symmetric source implementation for the adi fdtd method the alternating direction implicit finite difference time domain adi fdtd method has been introduced to overcome the courant limit present in the standard fdtd scheme however the adi fdtd scheme may produce small asymmetries in the field distributions according to the source implementation in this communication we compare different source implementation strategies and show that adi fdtd scheme has no asymmetry errors up to the numerical noise level and can be implemented most accurately if the excitation is directly incorporated within the tridiagonal matrix and if the time discretization of the source is done appropriately within each full time step

on the boundary blow up solutions of p x laplacian equations with singular coefficient abstract this paper investigates the problem p x u x f x u 0 in u x as d x 0 where p x u div u p x 2 u is called the p x laplacian x is a singular coefficient the existence and nonexistence of boundary blow up solutions are discussed and the singularity of the boundary blow up solution is given