spectral convergence and nonlinear dynamics of reaction diffusion equations under perturbations of the domain abstract in this paper we obtain the continuity of attractors for semilinear parabolic problems with neumann boundary conditions relatively to perturbations of the domain we show that if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the neumann laplacian is granted then we obtain the upper semicontinuity of the attractors if moreover every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors we also give necessary and sufficient conditions for the spectral convergence of neumann problems under perturbations of the domain

regular random attractors for non autonomous stochastic evolution equations with time varying delays on thin domains this paper deals with the limiting dynamical behavior of non autonomous stochastic reaction diffusion equations with time varying delays on thin domains first we prove the existence and uniqueness of the regular random attractor then we prove the upper semicontinuity of the regular random attractors for the equations on a family of n 1 dimensional thin domains that collapses onto an n dimensional domain

the effect of domain squeezing upon the dynamics of reaction diffusion equations abstract let be an arbitrary smooth bounded domain in r m r n and e 0 be arbitrary write x y for a generic point of r m r n squeeze by the factor e in the y direction to obtain the squeezed domain e x ey x y consider the following reaction diffusion equation on e formula here e is the exterior normal vector field on e and f r r is a nonlinearity satisfying some growth and dissipativeness conditions assuring such that ee generates a semiflow on h1 e with a global attractor a e we prove that in some strong sense the equations ee have a limiting equationu au f u e0 as e 0 this limiting equation is an abstract semilinear parabolic equation which defines a semiflow on a closed linear subspace of h1 we show that has a global attractor a 0 and the family of attractors a e e 0 is upper semicontinuous at e 0 if m n 1 and satisfies some natural additional assumptions then the limiting equation e0 is equivalent to a parabolic boundary value problem defined on a finite graph the results of this paper extend previous results obtained by hale and raugel for domains which are ordinate sets of a positive function

the neumann problem in a 2 d exterior domain with cuts and singularities at the tips abstract the neumann problem for the harmonic functions in an exterior connected plane region with cuts is studied the problem is considered with different conditions at infinity which lead to different theorems on uniqueness and solvability the existence of a classical solution is proved by potential theory the problem is reduced to a fredholm equation of the second kind which is uniquely solvable explicit formulas for singularities of a gradient of the solution at the tips of the cuts are obtained the results of the paper can be used to model the flow of an ideal fluid over several obstacles including wings

asymptotic behavior of stochastic fitzhugh nagumo systems on unbounded thin domains abstract this article is concerned with the asymptotic behavior of solutions of a class of non autonomous stochastic fitzhugh nagumo systems driven by multiplicative white noise on unbounded thin domains the existence and uniqueness of tempered pullback random attractors are proved for the systems defined on n 1 dimensional unbounded narrow domains we also establish the upper semicontinuity of these attractors when a family of n 1 dimensional unbounded thin domains collapses onto an n dimensional unbounded domain a cut off method and a decomposition technique are employed to derive the pullback asymptotic compactness of solutions in order to overcome the difficulties caused by the non compactness of sobolev embeddings on unbounded domains as well as the lack of regularity of one component of the solutions

on the dynamics of the error floor behavior in regular ldpc codes it is shown that dominant trapping sets of regular low density parity check ldpc codes so called absorption sets undergo a two phased dynamic behavior in the iterative message passing decoding algorithm using a linear dynamic model for the iteration behavior of these sets it is shown that they undergo an initial geometric growth phase which stabilizes in a final bit flipping behavior where the algorithm reaches a fixed point this analysis is shown to lead to very accurate numerical calculations of the error floor bit error rates down to error rates that are inaccessible by simulation the topology of the dominant absorption sets of an example code the ieee 802 3an 2048 1723 regular ldpc code are identified and tabulated using topological relationships in combination with search algorithms

upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation abstract for a smooth bounded domain r n n 3 and e a real parameter consider the hyperbolic equation eu tt u t u u g in with dirichlet boundary conditions under certain conditions on this equation has a compact attractor a e in h 0 1 l 2 for e 0 the parabolic equation also has a compact attractor which can be naturally embedded into a compact set a 0 in h 0 1 l 2 it is shown that for any neighborhood u of a 0 the set a e u for e small

limiting behavior of non autonomous stochastic reaction diffusion equations on thin domains abstract this paper deals with the limiting behavior of stochastic reaction diffusion equations driven by multiplicative noise and deterministic non autonomous terms defined on thin domains we first prove the existence uniqueness and periodicity of pullback tempered random attractors for the equations in an n 1 dimensional narrow domain and then establish the upper semicontinuity of these attractors when a family of n 1 dimensional thin domains collapses onto an n dimensional domain

signal space characterization of iterative decoding by tracing the flow of computations in the iterative decoders for low density parity check codes we formulate a signal space view for a finite number of iterations in a finite length code on a gaussian channel maximum a posteriori map codeword decoding or maximum likelihood decoding decodes to the codeword signal that is closest to the channel output in euclidean distance in contrast we show that iterative decoding decodes to the pseudosignal that has highest correlation with the channel output the set of pseudosignals corresponds to pseudocodewords only a vanishingly small number of which correspond to codewords we show that some pseudocodewords cause decoding errors but that there are also pseudocodewords that frequently correct the deleterious effects of other pseudocodewords

the global attractor of g navier stokes equations with linear dampness on r2 in this paper the long time behaviors of g navier stokes equations with linear dampness on r 2 were investigated by using the energy equation method the existence of the global attractor for the equations was proved without the restriction of the forcing term belonging to some weighted sobolev space moreover the estimation of the hausdorff and fractal dimensions of such attractors were also obtained

semilinear parabolic problems in thin domains with a highly oscillatory boundary abstract in this paper we study the behavior of the solutions of nonlinear parabolic problems posed in a domain that degenerates into a line segment thin domain which has an oscillating boundary we combine methods from linear homogenization theory for reticulated structures and from the theory on nonlinear dynamics of dissipative systems to obtain the limit problem for the elliptic and parabolic problems and analyze the convergence properties of the solutions and attractors of the evolutionary equations

error floor approximation for ldpc codes in the awgn channel this paper addresses the prediction of error floors of low density parity check codes transmitted over the additive white gaussian noise channel using a linear state space model to estimate the behavior of the sum product algorithm spa decoder in the vicinity of trapping sets tss we study the performance of the spa decoder in the log likelihood ratio llr domain as a function of the llr saturation level when applied to several widely studied codes the model accurately predicts a significant decrease in the error floor as the saturation level is allowed to increase for nonsaturating decoders however we find that the state space model breaks down after a small number of iterations due to the strong correlation of llr messages we then revisit richardson s importance sampling methodology for estimating error floors due to tss when those floors are too low for monte carlo simulation we propose modifications that account for the behavior of a nonsaturating decoder and present the resulting error floor estimates for the margulis code these estimates are much lower significantly steeper and more sensitive to iteration count than those previously reported

asymptotic analysis of the primitive equations under the small depth assumption abstract in this article we study the asymptotic behavior of solutions of the primitive equations pes as the depth of the domain goes to zero we prove that the solutions of the pes can be expanded as a sum of barotropic flow and baroclinic flow up to a uniformly bounded in time and space initial time layer the barotropic flow is solution of the 2d navier stokes equations with coriolis force coupled with density by employing a comparison theorem the baroclinic flow can be approximated by a quasi stationary nonlinear gfd stokes problem this article presents a mathematically rigorous justification that the barotropic flow dominates the baroclinic flow in the motion of the atmosphere and ocean

ldpc block and convolutional codes based on circulant matrices a class of algebraically structured quasi cyclic qc low density parity check ldpc codes and their convolutional counterparts is presented the qc codes are described by sparse parity check matrices comprised of blocks of circulant matrices the sparse parity check representation allows for practical graph based iterative message passing decoding based on the algebraic structure bounds on the girth and minimum distance of the codes are found and several possible encoding techniques are described the performance of the qc ldpc block codes compares favorably with that of randomly constructed ldpc codes for short to moderate block lengths the performance of the ldpc convolutional codes is superior to that of the qc codes on which they are based this performance is the limiting performance obtained by increasing the circulant size of the base qc code finally a continuous decoding procedure for the ldpc convolutional codes is described

analysis of absorbing sets and fully absorbing sets of array based ldpc codes the class of low density parity check ldpc codes is attractive since such codes can be decoded using practical message passing algorithms and their performance is known to approach the shannon limits for suitably large block lengths for the intermediate block lengths relevant in applications however many ldpc codes exhibit a so called error floor corresponding to a significant flattening in the curve that relates signal to noise ratio snr to the bit error rate ber level previous work has linked this behavior to combinatorial substructures within the tanner graph associated with an ldpc code known as fully absorbing sets these fully absorbing sets correspond to a particular type of near codewords or trapping sets that are stable under bit flipping operations and exert the dominant effect on the low ber behavior of structured ldpc codes this paper provides a detailed theoretical analysis of these fully absorbing sets for the class of cp array based ldpc codes including the characterization of all minimal fully absorbing sets for the array based ldpc codes for 2 3 4 and moreover it provides the development of techniques to enumerate them exactly theoretical results of this type provide a foundation for predicting and extrapolating the error floor behavior of ldpc codes

controlling the error floor in ldpc decoding the error floor of ldpc is revisited as an effect of dynamic message behavior in the so called absorbing sets of the code it is shown that if the signal growth in the absorbing sets is properly balanced by the growth of set external messages the error floor can be lowered to essentially arbitrarily low levels importance sampling techniques are discussed and used to verify the analysis as well as to discuss the impact of iterations and message quantization on the code performance in the ultra low ber error floor regime

limiting dynamics for stochastic reaction diffusion equations on the sobolev space with thin domains abstract this paper is devoted to bi spatial random attractors of the stochastic reaction diffusion equation when the terminate space is the sobolev space on a thin domain where the nonlinearity can be decomposed into two functions with p q growth exponents by means of a computation method of induction it is shown that the difference of solutions near the initial time is integrable of k p 2 k 2 order this higher order integrability shows continuity of the solution operator from the square lebesgue space to k p 2 k 2 order lebesgue spaces in particular the 2 p 2 order integrability shows continuity of the solution operator from the square lebesgue space to the sobolev space which further shows existence of a random attractor in the sobolev space when the initial space is the square lebesgue space moreover the higher order integrability can be uniform with respect to all thin domains which provides uniformly asymptotic compactness of the random dynamical systems as a conclusion the upper semi continuity of attractors under the sobolev norm is established when the narrow domain degenerates onto a lower dimensional domain

semilinear elliptic equations in thin domains with reaction terms concentrating on boundary abstract in this paper we analyze the behavior of a family of steady state solutions of a semilinear reaction diffusion equation with homogeneous neumann boundary condition posed in a two dimensional thin domain with reaction terms concentrated in a narrow oscillating neighborhood of the boundary we assume that the domain and therefore the oscillating boundary neighborhood degenerates into an interval as a small parameter goes to zero our main result is that this family of solutions converges to the solution of a one dimensional limit equation capturing the geometry and oscillatory behavior of the open sets where the problem is established

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

analysis of saturated belief propagation decoding of low density parity check codes we consider the effect of log likelihood ratio saturation on the belief propagation decoding of low density parity check codes saturation is commonly done in practice and is known to have a significant effect on the error floor performance our focus is on threshold analysis and the stability of density evolution we analyze the decoder for standard low density parity check code ensembles and show that belief propagation decoding generally degrades gracefully with saturation stability of density evolution is on the other hand rather strongly affected by saturation and the asymptotic qualitative effect of saturation is similar to reduction by one of variable node degree we also describe conditions under which the block error threshold for saturated belief propagation decoding equals the bit error threshold

some results on the navier stokes equations in thin 3d domains abstract we consider the navier stokes equations on thin 3d domains qe 0 e supplemented mainly with purely periodic boundary conditions or with periodic boundary conditions in the thin direction and homogeneous dirichlet conditions on the lateral boundary we prove global existence and uniqueness of solutions for initial data and forcing terms which are larger and less regular than in previous works on thin domains an important tool in the proofs are some sobolev embeddings into anisotropic lp type spaces better results are proved in the purely periodic case where the conservation of enstrophy property is used for example when the forcing term vanishes we prove global existence and uniqueness of solutions if i m u0 h1 2 qe exp c 1e 1 s mu0 2 sl2 qe c for both boundary conditions or mu0 h1 qe ce mu0 3 l2 qe ce i m u0 h1 2 qe ce1 4 2 for purely periodic boundary conditions where 1 2

dynamics in dumbbell domains i continuity of the set of equilibria we analyze the dynamics of a reaction diffusion equation with homogeneous neumann boundary conditions in a dumbbell domain we provide an appropriate functional setting to treat this problem and as a first step we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds

persistence of the bifurcation structure for a semilinear elliptic problem on thin domains we study a semilinear elliptic problem on thin domains with a bifurcation parameter it is shown that the set of solutions is upper semicontinuous as the thickness of a domain tends to 0 and that solution branches including bifurcation points persist near those of a one dimensional limiting equation

smooth convergence of random center manifolds for spdes in varying phase spaces abstract in this paper we study the limiting behavior of center manifolds for a class of singularly perturbed stochastic partial differential equations in terms of the phase spaces we first prove the existence and smoothness of random center manifolds for these equations then we establish the smooth convergence of these random center manifolds as the phase spaces approach to their singular limit

invariant manifolds for parabolic equations under perturbation of the domain abstract we study the effect of domain perturbation on invariant manifolds for semilinear parabolic equations subject to the dirichlet boundary condition under the mosco convergence assumption on the domains we prove the upper and lower semicontinuity of both the local unstable invariant manifold and the local stable invariant manifold near a hyperbolic equilibrium the continuity results are obtained by keeping track of the construction of invariant manifolds in p w bates c k r t jones invariant manifolds for semilinear partial differential equations in dynamics reported vol 2 in dynam report ser dynam systems appl vol 2 wiley chichester 1989 pp 1 38

interface boundary value problem for the navier stokes equations in thin two layer domains abstract we study a system of 3d navier stokes equations in a two layer parallelepiped like domain with an interface coupling of the velocities and mixed free periodic boundary condition on the external boundary the system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere in case when our domain is thin of order e we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms roughly of order e 2 3 we also give some results concerning the large time dynamics of the solutions in particular we prove a spatial regularity of the global weak attractor

attractors for reaction diffusion equations on thin domains whose linear part is non self adjoint abstract for a bounded smooth domain r n x n y let 0 be a family of domains squeezed in y r n y direction on we consider a reaction diffusion equation with nonsymmetrical linear part we show that under natural conditions on the nonlinearity the generated semi flows have global attractors which in a certain sense have limits as 0

quasicyclic low density parity check codes from circulant permutation matrices in this correspondence the construction of low density parity check ldpc codes from circulant permutation matrices is investigated it is shown that such codes cannot have a tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived these results suggest that families of ldpc codes with such girth values are relatively easy to obtain and consequently additional parameters such as the minimum distance or the number of redundant check sums should be considered to this end a necessary condition for the codes investigated to reach their maximum possible minimum hamming distance is proposed

wave diffraction by a half plane with an obstacle perpendicular to the boundary abstract we prove the unique existence of solutions for different types of boundary value problems of wave diffraction by a half plane with a screen or a crack perpendicular to the boundary representations of the solutions are also obtained upon the consideration of some associated operators this is done in a bessel potential spaces framework and for complex non real wave numbers the investigation is mostly based on the construction of explicit operator relations the potential method and a factorization technique for a certain class of oscillating fourier symbols which naturally arise from the problems

image normalization of wiener hopf operators and boundary transmission value problems for a junction of two half planes abstract the present paper deals with an application of the image normalization technique for certain classes of wiener hopf operators whos associated to ill posed boundary transmission value problems we briefly describe the method of normalization and then apply it to boundary transmission value problems issued from diffraction problems for a junction of two half planes which are relevant in mathematical physics applications for each boundary transmission value problem we analyze the conditions under which the associated operator and the equivalent who are not normally solvable and define the corresponding image normalized operators

quasi cyclic representation and vector representation of rs ldpc codes rs ldpc codes constructed based on the codewords of reed solomon rs codes with two information symbols are an important class of ldpc codes in this paper we present two representations namely quasi cyclic qc representation and vector representation for rs ldpc codes under the first representation most part of the parity check matrix of a full length rs ldpc code consists of circulant permutation matrices and zero matrices as a result the class of codes can enjoy the advantages in hardware implementation as qc ldpc codes in addition the base matrix under the qc representation of an rs ldpc code can be explicitly given such that the rank of its parity check matrix can be analyzed combinatorially under the second representation each permutation matrix in the parity check matrix of an rs ldpc code is defined by a nonbinary vector whose entries are a permutation of entries in the field from which the rs code is constructed then the affine invariance property is proved for full length rs ldpc codes which can facilitate the structural analysis of the codes

navier stokes equations in 3d thin domains with navier friction boundary condition abstract in this article we study the 3d navier stokes equations with navier friction boundary condition in thin domains we prove the global existence of strong solutions to the 3d navier stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small we generalize the techniques developed to study the 3d navier stokes equations in thin domains see g raugel g sell navier stokes equations on thin 3d domains i global attractors and global regularity of solutions j amer math soc 6 1993 503 568 g raugel g sell navier stokes equations on thin 3d domains ii global regularity of spatially periodic conditions in nonlinear partial differential equations and their application college de france seminar vol xi longman harlow 1994 pp 205 247 r temam m ziane navier stokes equations in three dimensional thin domains with various boundary conditions adv differential equations 1 1996 499 546 r temam m ziane navier stokes equations in thin spherical shells in optimization methods in partial differential equations in contemp math vol 209 amer math soc providence ri 1996 pp 281 314 to the navier friction boundary condition by introducing a new average operator m e in the thin direction according to the spectral decomposition of the stokes operator a e our analysis hinges on the refined investigation of the eigenvalue problem corresponding to the stokes operator a e with navier friction boundary condition

distance of attractors of reaction diffusion equations in thin domains abstract in this work we consider a dissipative reaction diffusion equation in a d dimensional thin domain shrinking to a one dimensional segment and obtain good rates for the convergence of the attractors to accomplish this we use estimates on the convergence of inertial manifolds as developed previously in 6 and shadowing theory

new characterization and efficient exhaustive search algorithm for leafless elementary trapping sets of variable regular ldpc codes in this paper we propose a new characterization for leafless elementary trapping sets letss of variable regular low density parity check codes recently karimi and banihashemi proposed a characterization of letss which was based on viewing an lets as a layered superset lss of a short cycle in the code s tanner graph a notable advantage of lss characterization is that it corresponds to a simple lss based search algorithm expansion technique that starts from short cycles of the graph and finds the letss with lss structure efficiently compared with the lss based characterization of karimi and banihashemi which is based on a single lss expansion technique the new characterization involves two additional expansion techniques the introduction of the new techniques mitigates two problems that lss based characterization search suffers from 1 exhaustiveness not every lets structure is an lss of a cycle and 2 search efficiency lss based search algorithm often requires the enumeration of cycles with length much larger than the girth of the graph where the multiplicity of such cycles increases rapidly with their length we prove that using the three expansion techniques any lets structure can be obtained starting from a simple cycle no matter how large the size of the structure a or the number of its unsatisfied check nodes b are i e the characterization is exhaustive we also demonstrate that for the proposed characterization search to exhaustively cover all the lets structures within the a b classes with a leq a max and b leq b max for any value of a max and b max the length of the short cycles required to be enumerated is less than that of the lss based characterization search we in fact show that such a length for the proposed search algorithm is minimal we also prove that the three expansion techniques proposed here are the only expansions needed for characterization of lets structures starting from simple cycles in the graph if one requires each and every intermediate sub structure to be a lets as well extensive simulation results are provided to show that compared with lss based search significant improvement in search speed and memory requirements can be achieved

a damped hyperbolic equation on thin domains for a damped hyperbolic equation in a thin domain in r3 over a bounded smooth domain in r2 it is proved that the global attractors are upper semicontinuous it is shown also that a global attractor exists in the case of the critical sobolev exponent

robin type boundary conditions in transition from reaction diffusion equations in 3d domains to equations in 2d domains abstract we consider a singular limit of diffusion equations in 3d domains of thickness converging to zero in the 2d limit the resulting reaction diffusion equation has a source term resulting from the robin type boundary conditions imposed on boundaries of the original 3d domain the proposed approach can be applied to constructing approximate solutions of diffusion problems in thin planar cylindrical or spherical layers between two membranes as an example we refer to the problem of activation of b lymphocytes which typically have large nuclei and a thin cytoplasmic layer which can be considered as a spherical shell for this example assuming additionally axial symmetry we provide a rigorous convergence theorem in the language of semigroups of operators

on the construction of structured ldpc codes free of small trapping sets we present a method to construct low density parity check ldpc codes with low error floors on the binary symmetric channel codes are constructed so that their tanner graphs are free of certain small trapping sets these trapping sets are selected from the trapping set ontology for the gallager a b decoder they are selected based on their relative harmfulness for a given decoding algorithm we evaluate the relative harmfulness of different trapping sets for the sum product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets we apply this method to construct structured ldpc codes to facilitate the discussion we give a new description of structured ldpc codes whose parity check matrices are arrays of permutation matrices this description uses latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the tanner graph of a code to be free of four cycles

finding the exhaustive list of small fully absorbing sets and designing the corresponding low error floor decoder this work provides an efficient exhaustive search algorithm for finding all small fully absorbing sets fass of any arbitrary low density parity check ldpc code the proposed algorithm is based on the branch bound principle for solving np complete problems in particular given any ldpc code the problem of finding all fass of size less than t is formulated as an integer programming problem for which a new branch bound algorithm is devised with new node selection and tree trimming mechanisms the resulting algorithm is capable of finding all fass of size 7 for ldpc codes of length 1000 when limiting the fass of interest to those with the number of violated parity check nodes 3 the proposed algorithm is capable of finding all such fass of size 14 for ldpc codes of lengths 1000 the resulting exhaustive list of small fass is then used to devise a new efficient post processing low error floor ldpc decoder the numerical results show that by exploiting the exhaustive list of small fass the proposed post processing decoder can significantly lower the error floor performance of a given ldpc code for various example codes of length 3000 the proposed post processing decoder lowers the error floor by a couple of orders of magnitude when compared to the standard belief propagation decoder and by an order of magnitude when compared to other existing low error floor decoders

limiting behavior of non autonomous stochastic reaction diffusion equations on unbounded thin domains this article is concerned with the limiting behavior of dynamics of a class of nonautonomous stochastic partial differential equations driven by multiplicative white noise on unbounded thin domains we first prove the existence of tempered pullback random attractors for the equations defined on n 1 dimensional unbounded thin domains then we show the upper semicontinuity of these attractors when the n 1 dimensional unbounded thin domains collapse onto the n dimensional space rn here the tail estimates are utilized to deal with the noncompactness of sobolev embeddings on unbounded domains

eliminating trapping sets in low density parity check codes by using tanner graph covers we discuss error floor asympotics and present a method for improving the performance of low density parity check ldpc codes in the high snr error floor region the method is based on tanner graph covers that do not have trapping sets from the original code the advantages of the method are that it is universal as it can be applied to any ldpc code channel decoding algorithm and it improves performance at the expense of increasing the code length without losing the code regularity without changing the decoding algorithm and under certain conditions without lowering the code rate the proposed method can be modified to construct convolutional ldpc codes also the method is illustrated by modifying tanner mackay and margulis codes to improve performance on the binary symmetric channel bsc under the gallager b decoding algorithm decoding results on awgn channel are also presented to illustrate that optimizing codes for one channel decoding algorithm can lead to performance improvement on other channels

the mixed harmonic problem in an exterior cracked domain with dirichlet condition on cracks the mixed dirichlet neumann problem for the laplace equation in an unboundedconnected plane domain with cuts cracks is studied the neumann condition is given on closed curves making up the boundary of the domain while the dirichlet condition is specified on the cuts the existence of a classical solution is proved by means of potential theory and the boundary integral equation method the integral representation for a solution is obtained in the form of potentials the density in potentials satisfies the uniquely solvable fredholm integral equation of the second kind and index zero singularities of the gradient of the solution at the tips of cuts are investigated

on the regularity of the navier stokes equation in a thin periodic domain abstract we address the global regularity of solutions of the navier stokes equations in a thin domain 0 l 1 0 l 2 0 with periodic boundary conditions where l 1 l 2 0 and 0 1 2 we prove that if u 0 l 2 c l 1 l 2 1 2 log 3 2 where u 0 is the initial datum then there exists a unique global smooth solution with the initial datum u 0 also if u 0 h 1 2 c l 1 l 2 log 1 4 the global regularity of the corresponding solution holds

semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries abstract in this work we study the behavior of a family of solutions of a semilinear elliptic equation with homogeneous neumann boundary condition posed in a two dimensional oscillating thin region with reaction terms concentrated in a neighborhood of the oscillatory boundary our main result is concerned with the upper and lower semicontinuity of the set of solutions we show that the solutions of our perturbed equation can be approximated with one of a one dimensional equation which also captures the effects of all relevant physical processes that take place in the original problem