asymptotic spatial patterns on the complex time dependent ginzburg landau equation asymptotic spatial patterns wavefront solutions on time dependent ginzburg landau tdgl equations with complex coefficients are discussed the condition for the existence of a spatial limit cycle solution is found to be given in terms of the propagation velocity of the wavefront solution and the stability criterion for the spatial limit cycle is obtained the amplitude oscillation of an asymptotic spatial pattern on the tdgl equation non linear schrodinger equation with purely imaginary coefficients is expressed in terms of jacobi s elliptic function while an exact solution of the asymptotic spatial pattern on the tdgl equation force free duffing equation with purely real coefficients is aperiodic and has a unique propagation velocity

eliminating leading corrections to scaling in the three dimensional o n symmetric φ 4 model n 3 and 4 we study corrections to scaling in the o 3 and o 4 symmetric 4 model on the three dimensional simple cubic lattice with nearest neighbour interactions for this purpose we use monte carlo simulations in connection with a finite size scaling method we find that a finite value of the coupling exists for both values of n where leading corrections to scaling vanish as a first application we compute the critical exponents 0 710 2 and 0 0380 10 for n 3 and 0 749 2 and 0 0365 10 for n 4

25th order high temperature expansion results for three dimensional ising like systems on the simple cubic lattice 25th order high temperature series are computed for a general nearest neighbor three dimensional ising model with arbitrary potential on the simple cubic lattice in particular we consider three improved potentials characterized by suppressed leading scaling corrections critical exponents are extracted from high temperature series specialized to improved potentials obtaining gamma 1 2373 2 nu 0 63012 16 alpha 0 1096 5 eta 0 03639 15 beta 0 32653 10 delta 4 7893 8 moreover biased analyses of the 25th order series of the standard ising model provide the estimate delta 0 52 3 for the exponent associated with the leading scaling corrections by the same technique we study the small magnetization expansion of the helmholtz free energy the results are then applied to the construction of parametric representations of the critical equation of state using a systematic approach based on a global stationarity condition accurate estimates of several universal amplitude ratios are also presented

holder estimate for non uniform parabolic equations in highly heterogeneous media abstract uniform bound for the solutions of non uniform parabolic equations in highly heterogeneous media is concerned the media considered are periodic and they consist of a connected high permeability sub region and a disconnected matrix block subset with low permeability parabolic equations with diffusion depending on the permeability of the media have fast diffusion in the high permeability sub region and slow diffusion in the low permeability subset and they form non uniform parabolic equations each medium is associated with a positive number denoting the size ratio of matrix blocks to the whole domain of the medium let the permeability ratio of the matrix block subset to the connected high permeability sub region be of the order 2 for 0 1 it is proved that the holder norm of the solutions of the above non uniform parabolic equations in the connected high permeability sub region are bounded uniformly in one example also shows that the holder norm of the solutions in the disconnected subset may not be bounded uniformly in

boundary value problems in morrey spaces for elliptic systems on lipschitz domains let q be a bounded lipschitz domain in rn n 3 let be a second order elliptic system with constant coefficients satisfying the legendre hadamard condition we consider the dirichlet problem u 0 in q u f on 4 where e 0 depends on q and 0 a 2 for n 3 we obtain existence and uniqueness results with nontangential maximal function estimate u 2 a q c f 2 a 0q if c satisfies the strong elliptic condition and 0 a min n 1 2 e we show that the neumann type problem u 0 in q g g 2 a dq on 9q vu wfp idci has a unique solution here h2 x dq is an atomic space with the property h2 x dq l2 a 9q the invertibility of layer potentials on l2 x dq and h2 x dq is also obtained finally we study the dirichlet problem for the biharmonic equation we establish a similar estimate in l2 x for the biharmonic equation in which case the range 0 a 2 is sharp for n 4 or 5

a new kind of periodic stationary solution of the cubic ginzburg landau equation abstract we analyze the long scale dynamics of the modulationally unstable plane wave in the cubic complex ginzburg landau equation without a diffusion term our numerical studies reveal that when the dissipative effects dominate over the conservative effects a new kind of temporally stationary but spatially periodic solution can be found

criticality of o n symmetric models in the presence of discrete gauge symmetries we investigate the critical properties of the three dimensional antiferromagnetic rp n 1 model which is characterized by a global o n symmetry and a discrete z 2 gauge symmetry we perform a field theoretical analysis using the landau ginzburg wilson lgw approach and a numerical monte carlo study the lgw field theoretical results are obtained by high order perturbative analyses of the renormalization group flow of the most general 4 theory with the same global symmetry as the model assuming a gauge invariant order parameter field for n 4 no stable fixed point is found implying that any transition must necessarily be of first order this is contradicted by the numerical results that provide strong evidence for a continuous transition this suggests that gauge modes are not always irrelevant as assumed by the lgw approach but they may play an important role to determine the actual critical dynamics at the phase transition of o n symmetric models with a discrete z 2 gauge symmetry

transmission problems and spectral theory for singular integral operators on lipschitz domains we prove the well posedness of the transmission problem for the laplacian across a lipschitz interface with optimal non tangential maximal function estimates for data in lebesgue and hardy spaces on the boundary as a corollary we show that the spectral radius of the adjoint harmonic double layer potential k in lp0 is less than 12 whenever is a bounded convex domain and 1 p 2

a monte carlo study of leading order scaling corrections of phi 4 theory on a three dimensional lattice we present a monte carlo study of the one component 4 model on a cubic lattice in three dimensions leading order scaling corrections are studied using the finite size scaling method we compute the corrections to scaling exponent with high precision we determine the value of the coupling at which leading order corrections to scaling vanish using this result we obtain estimates for critical exponents that are more precise than those obtained with field theoretic methods

stable and chaotic solutions of the complex ginzburg landau equation with periodic boundary conditions abstract we study analytically and numerically the dynamical behavior of the solutions of the complex ginzburg landau equation with diffraction but without diffusion which governs the spatial evolution of the field in an active nonlinear laser cavity accordingly the solutions are subject to periodic boundary conditions the analysis reveals regions of stable stationary solutions in the model s parameter space and a wide range of oscillatory and chaotic behaviors close to the first bifurcation destabilizing the spatially uniform solution a stationary single humped solution is found in an asymptotic analytical form which turns out to be in very good agreement with the numerical results simulations reveal a series of stable stationary multi humped solutions

regularity properties of solutions to transmission problems we show that the gradients of solutions to certain elliptic and parabolic transmission problems with internal lipschitz boundary and constant co efficients at each side of the internal boundary are square integrable along the internal boundary

critical exponents of the n vector model recently the series for two renormalization group functions corresponding to the anomalous dimensions of the fields and of the three dimensional field theory have been extended to next order seven loops by murray and nickel we examine the influence of these additional terms on the estimates of critical exponents of the n vector model using some new ideas in the context of the borel summation techniques the estimates have slightly changed but remain within the errors of the previous evaluation exponents such as related to the field anomalous dimension which were poorly determined in the previous evaluation of le guillou zinn justin have seen their apparent errors significantly decrease more importantly perhaps summation errors are better determined the change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values finally because an error has been discovered in the last order of the published expansions order we have also re analysed the determination of exponents from the expansion the conclusion is that the general agreement between expansion and three dimensional series has improved with respect to le guillou zinn justin

the transmission problem on a three dimensional wedge we consider the transmission problem for the laplace equation on an infinite three dimensional wedge determining the complex parameters for which the problem is well posed and characterizing the infinite multiplicity nature of the spectrum this is carried out in two formulations leading to rather different spectral pictures one formulation is in terms of square integrable boundary data the other is in terms of finite energy solutions we use the layer potential method which requires the harmonic analysis of a non commutative non unimodular group associated with the wedge

non analyticity of the callan symanzik β function of two dimensional o n models we discuss the analytic properties of the callan symanzik function g associated with the zero momentum four point coupling g in the two dimensional 4 model with o n symmetry using renormalization group arguments we derive the asymptotic behaviour of g at the fixed point g we argue that g g o g g 1 7 for n 1 and g g o 1 log g g for n 3 our claim is supported by an explicit calculation in the ising lattice model and by a 1 n calculation for the two dimensional 4 theory we discuss how these non analytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g

transmission problem for the laplace equation and the integral equation method abstract we shall study a weak solution in the sobolev space of the transmission problem for the laplace equation using the integral equation method first we use the indirect integral equation method we look for a solution in the form of the sum of the double layer potential corresponding to the skip of traces on the interface and a single layer potential with an unknown density we get an integral equation on the boundary we prove that this equation has a form i m f where m is a contractive operator so we can obtain a solution of this equation using the successive approximation method moreover we are able to estimate the norm of the operator m and control how quickly this process converges then we study the direct integral equation method we obtain the same integral equation like for the indirect integral equation method so we can again calculate a solution using the successive approximation method

stability results on interpolation scales of quasi banach spaces and applications we investigate the stability of fredholm properties on interpolation scales of quasi banach spaces this analysis is motivated by problems arising in pde s and several applications are presented

pattern competition and the decay of unstable patterns in quasi one dimensional systems abstract the dynamics of quasi one dimensional pattern forming systems with simple periodic initial conditions and additional noise or with almost periodic initial conditions is studied analytically and numerically using the lowest order amplitude equation in this way the decay of eckhaus unstable patterns and subsequent formation of stable ones as observed recently and the competition of patterns with different wavenumbers is studied

a stability criterion for two fluid interfaces and applications we derive a new stability criterion for two fluid interfaces that ensures the existence of stable local solutions that do not break down too fast due to kelvin helmholtz instabilities it can be seen both as a two fluid generalization of the rayleigh taylor criterion and as a nonlinear version of the kelvin stability condition we show that gravity can control the inertial effects of the shear up to frequencies that are high enough for the surface tension to play a relevant role this explains why surface tension is a necessary condition for well posedness while the low frequency main dynamics of interfacial waves are unaffected by it in order to derive a practical version of this criterion we work with a nondimensionalized version of the equations and allow for the possibility of various asymptotic regimes such as the shallow water limit this limit being singular we have to derive a new symbolic analysis of the dirichlet neumann operator that includes an infinitely smoothing tail accounting for the contribution of the bottom we then validate our criterion by comparison with experimental data in two important settings air water interfaces and internal waves the good agreement we observe allows us to discuss the scenario of wave breaking and the behavior of water brine interfaces and to propose a formula for the maximal amplitude of interfacial waves we also show how to rigorously justify two fluid asymptotic models used for applications and how to relate some of their properties to kelvin helmholtz instabilities

the xy model and the three state antiferromagnetic potts model in three dimensions critical properties from fluctuating boundary conditions we present the results of a monte carlo study of the three dimensionalxy model and the three dimensional antiferromagnetic three state potts model in both cases we compute the difference of the free energies of a system with periodic and a system with antiperiodic boundary conditions in a neighborhood of the critical coupling from the finite size scaling behaviour of this quantity we extract values for the critical temperature and the critical exponentv that are compatible with recent high statistics monte carlo studies of the models the results for the free energy difference at the critical temperature and for the exponentv confirm that both models belong to the same universality class

existence results for a nonlinear transmission problem abstract let o and i be open bounded regular subsets of r n such that the closure of i is contained in o let f o be a regular function on o and let f and g be continuous functions from i r to r by exploiting an argument based on potential theory and on the leray schauder principle we show that under suitable and completely explicit conditions on f and g there exists at least one pair of continuous functions u o u i such that u o 0 in o cl i u i 0 in i u o x f o x for all x o u o x f x u i x for all x i i u o x i u i x g x u i x for all x i where the last equality is attained in certain weak sense a simple example shows that such a pair of functions u o u i is in general neither unique nor locally unique if instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition then we prove the existence of at least one classical solution which is in addition locally unique

a nonlinear fokker planck equation modelling the approach to thermal equilibrium in a homogeneous plasma this work deals with the problem consisting in the equation 1 partial derivative f partial derivative t 1 x 2 partial derivative partial derivative x x 4 partial derivative f partial derivative x f f 2 when x is an element of 0 infinity t 0 together with no flux conditions at x 0 and x infinity i e 2 x 4 partial derivative f partial derivative x f f 2 0 as x 0 or x infinity such a problem arises as a kinetic approximation to describe the evolution of the radiation distribution f x t in a homogeneous plasma when radiation interacts with matter via compton scattering we shall prove that there exist solutions of 1 2 which develop singularities near x 0 in a finite time regardless of how small the initial number of photons n 0 integral 0 infinity x 2 f x 0 dx is the nature of such singularities is then analyzed in detail in particular we show that the flux condition 2 is lost at x 0 when the singularity unfolds the corresponding blow up pattern is shown to be asymptotically of a shock wave type in rescaled variables it consists in an imploding travelling wave solution of the burgers equation near x 0 that matches a suitable diffusive profile away from the shock finally we also show that on replacing 2 near x 0 as determined by the manner of blow up such solutions can be continued for all times after the onset of the singularity

new spectral results for the electrostatic integral operator abstract in this paper two new fundamental results are established concerning the spectra of both the double layer potential and the electrostatic integral operators in r 3 while it is known that 0 lies in the spectrum of each operator its specific nature is not well understood for some geometries e g a sphere and a prolate spheroid it is known that 0 lies in their continuous spectra in this paper however specific examples are produced where 0 lies in the point spectrum of each operator thus it is established that the spectral nature of 0 depends upon the geometry of the underlying surface the second new result given is an important generalization of a recent paper concerning the possible refinement of a classical result of plemelj

two point correlation function of three dimensional o n models the critical limit and anisotropy in three dimensional o n models we investigate the low momentum behavior of the two point green s function g x in the critical region of the symmetric phase we consider physical systems whose criticality is characterized by a rotationally invariant fixed point several approaches are exploited such as strong coupling expansion of lattice n vector model and 1 n expansion field theoretical methods within the ensuremath varphi 4 continuum formulation non gaussian corrections to the universal low momentum behavior of g x are evaluated and found to be very small in nonrotationally invariant physical systems with o n invariant interactions the vanishing of the spatial anisotropy approaching the rotationally invariant fixed point is described by a critical exponent ensuremath rho which is universal and is related to the leading irrelevant operator breaking rotational invariance at n ensuremath infty one finds ensuremath rho 2 we show that for all values of ng 0 ensuremath rho ensuremath simeq 2

high precision monte carlo study of the 3d xy universality class we present a monte carlo study of the two component 4 model on the simple cubic lattice in three dimensions by suitable tuning of the coupling constant we eliminate leading order corrections to scaling high statistics simulations using finite size scaling techniques yield 0 6723 3 8 and 0 0381 2 2 where the statistical and systematical errors are given in the first and second bracket respectively these results are more precise than any previous theoretical estimate of the critical exponents for the 3d xy universality class

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy