existence and uniqueness of solution in sobolev space for an unsteady crystal growth problem with zero surface tension abstract we study an initial value problem for a two dimensional dendritic crystal growth model with zero surface tension if the initial data is in sobolev space h 2 r it is proved that an unique local solution exists in proper sobolev space

analyticity and nonexistence of classical steady hele shaw fingers this paper concerns analyticity of a classical steadily translating nger in a hele shaw cell and nonexistence of solutions when the relative nger width is smaller than 1 it is proven that any classical solution to the nger problem if it exists for sufciently small but nonzero surface tension is close to some saffman taylor zero surface tension solution and satises some algebraic decay conditions at 1 must belong to the analytic function space a0 as dened in section 1 and chosen in a previous study 34 of the existence of nger solutions further it is proven that for any x ed 2 0 1 there can be no classical steadynger solution when surface tension is sufciently small which contradicts a previous conclusion based on numerical simulation c 2002 wiley periodicals inc

green s function for the harmonic potential of the three dimensional wedge transmission problem the point source static potential in a wedge geometry consisting of two homogeneous media is solved via the kontorovich lebedev and fourier transforms inverse transforms enable the solution of laplace s equation to be expressed in terms of image contributions plus residue sums fourier series of toroidal functions as in previous wave equation solutions for isovelocity wedges explicit expressions for the poles that are the site of the residues are exploited when the wedge angle is a rational multiple of spl pi

the diaphanous wedge complete solutions for the scattering by a diaphanous wedge meaning a wedge with identical wavenumbers inside and outside the wedge are presented the results are obtained from an integral equation for the fields on the wedge which is solved by the mellin and kantorovich lebedev transforms in the static and dynamic cases respectively pertinent formulations of gegenbauer s addition theorems play an important part in the derivation of the results which are presented in closed form

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

image solution for poisson s equation in wedge geometry the poisson s equation for a dielectric wedge is solved by deriving a static image that corresponds to the potential contribution of the wedge the problem is studied mainly in three dimensions but the solution for the analog two dimensional problem is also provided it appears that the use of image sources provides numerically a simple and efficient method to compute the potential inside and outside the wedge image sources for the exterior and interior regions consist of a line charge that decays exponentially as a function of complex angle and a set of point charges that can be interpreted as reflection or transmission images of a dielectric plane all known closed form solutions in terms of elementary functions are derived for an electrically and magnetically conducting half plane

the complete extension of the biot tolstoy solution to the density contrast wedge with sample calculations the biot tolstoy exact time domain solution for the three dimensional impulse response of an impenetrable wedge is extended to accommodate the isovelocity or density contrast wedge fourier transformation of the time and axial variables along with a kantorovich lebedev transform applied to the cylindrical radial coordinate leads to a solution in terms of residue series when the wedge angle is a rational fraction of the residue series can be reduced to a finite sum which is evaluated for some special cases the total pressure field consists of geometrical acoustics contributions as predicted by snell s laws plus a modified version of the biot tolstoy diffraction field

general solutions of the narrow strip and slot integral equations analytic solutions of the integral equations for the current induced on a narrow conducting strip are obtained by two methods one method follows from an expansion of the strip current in a series of chebyshev polynomials while the other is based on a special power series expansion in the former fourier type coefficients depending on integrals of the excitation are derived while in the latter the coefficients depend upon derivatives of the excitation from knowledge of the strip current simple expressions for the far zone scattered field are derived explicit results are given for plane wave excitation the solution methods apply to the equations for the narrow slot in a planar conducting screen

discrete point spectrum of linear stability operator for saffman taylor bubbles with nonzero surface tension abstract we will study the linear spectral properties of steady saffman taylor bubbles with surface tension in a hele shaw cell it was known 21 that the saffman taylor exact solution without surface tension is linearly unstable and the point spectrum is a continuum set in this paper we rigorously conclude that the point spectrum of the stability operator of any branch of saffman taylor bubbles is a discrete set when surface tension is not zero

impulse response of density contrast wedge using normal coordinates the exact impulse response of a point source to a rigid wedge was first derived by biot and tolstoy in normal coordinates i tolstoy and c s clay ocean acoustics theory and experiment in underwater sound american institute of physics new york 1987 appendix 5 the solution is compact and includes both reflected and diffracted waves explicitly kinney et al j acoust soc am 73 183 194 1983 introduced the solution of free wedge pressure release which is similar to the rigid case except sign changes for certain terms the solutions have been verified in laboratory experiments however neither model can be applied directly to many actual cases such as ocean floor where the ratio of density of the base sediment to that of the water is neither infinite ridge nor zero pressure release the normal coordinate solution presented here is the impulse response to an isovelocity c c and density contrast wedge it is an extension to the solution given by biot and tolstoy the eigenva

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

scattering by polygonal cross section dielectric cylinders at oblique incidence we analyze the scattering by polygonal cross section lossless dielectric cylinders illuminated by an obliquely incident plane wave the problem is formulated in terms of a system of surface integral equations opportunely devised so as to be valid for objects with edges and incident angles including the total reflection limit angle by means of galerkin s method in the spectral domain with analytically fourier transformable expansion functions factorizing the correct edge behaviors and continuity conditions of the unknowns convergence of exponential type is achieved and the coefficients of the scattering matrix are reduced to single integrals that can be efficiently evaluated numerical results for both near field and far field parameters are presented showing the quick convergence of the method even when applied to composite cylindrical objects

the effect of finiteness in the saffman taylor viscous fingering problem we derive a family of exact time evolving solutions for the the evolution of a finite blob of fluid confined to a channel in a hele shaw cell we show rigorously that for large fluid volume there are solutions for which one of the interfaces approaches the steady saffman taylor finger solution of arbitrary width 0 1 on the basis of this we argue that the far field effects of a displaced second interface do not provide a selection mechanism for the formation of a width frac 1 2 finger when surface tension or any other regularization is ignored

response of a source on top of a vertically stratified half space the solution of the response of a source on top of a horizontally stratified half space is well known however when the half space is vertically stratified the problem can only be solved with numerical methods like the finite element method here a semi analytic approach to solve such a problem is presented the three dimensional variation of the problem is reduced to two dimensional variation by fourier transform in one coordinate variable the remaining two dimensional problem is solved by finding the eigensolution in each of the half spaces the eigensolutions of each region are found from the partial differential equation directly using the same basis set of expansion functions this makes the calculation of the reflection and transmission operators very efficient the reflection and transmission operators account for the mode conversion reflection and transmission of the waves with the reflection and transmission operators the field everywhere can be calculated the solution reduces to that of the sommerfeld s half space problem when the two half spaces are homogeneous

fast analytical methods for macroscopic electrostatic models in biomolecular simulations we review recent developments of fast analytical methods for macroscopic electrostatic calculations in biological applications including the poisson boltzmann pb and the generalized born models for electrostatic solvation energy the focus is on analytical approaches for hybrid solvation models especially the image charge method for a spherical cavity and also the generalized born theory as an approximation to the pb model this review places much emphasis on the mathematical details behind these methods

mellin transform solution for the static line source excitation of a dielectric wedge an integral transform analysis of the static scattering of the two dimensional potential radiated by a line source in the vicinity of a penetrable wedge is presented the mellin transform is used to derive the exact static solution to laplace s equation for the dielectric wedge in the form of a modal series the important dielectric edge condition behavior is explicitly contained in this analytic solution

electrostatic fractional image methods for perfectly conducting wedges and cones engheta 1996 introduced a definition for the electric charge fractional order multipoles using the concept of fractional derivatives and integrals here we utilize that definition to introduce a detailed image theory for the two dimensional 2 d electrostatic potential distributions in front of a perfectly conducting wedge with arbitrary wedge angles and for the three dimensional potential in front of a perfectly conducting cone with arbitrary cone angles we show that the potentials in the presence of these structures can be described equivalently as the electrostatic potentials of sets of equivalent image charge distributions that effectively behave as fractional order multipoles hence the name fractional image methods the fractional orders of these so called fractional images depend on the wedge angle for the wedge problem and on the cone angle for the cone problem special cases where these fractional images behave like the discrete images are discussed and physical justification and insights into these results are given

double layer potentials on three dimensional wedges and pseudodifferential operators on lie groupoids abstract let w be a three dimensional wedge and k be the double layer potential operator associated to w and the laplacian we show that 1 2 k are isomorphisms between suitable weighted sobolev spaces which implies a solvability result in weighted sobolev spaces for the dirichlet problem on w furthermore we show that the double layer potential operator k is an element in c g m 2 c where g is the action transformation groupoid m g with g 1 0 a b a r b r which is a lie group and m is a kind of compactification of g this result can be used to prove the fredholmness of 1 2 k where is a domain with edge singularities and k the double layer potential operator associated to the laplacian and

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

analyticity of classical steady needle crystals abstract this paper is a continuation of our previous work rigorous results in selection of steady needle crystals j differential equations 197 2004 349 426 it concerns analyticity of a classical steadily translating needle crystal it is proved that any classical solution to the needle crystal problem with sufficiently small but nonzero surface tension if its slope deviation is close to some ivantsov zero surface tension solution and if its curvature satisfies some algebraic decay conditions at must belong to the analytic function space a 0 defined in 1 and chosen in the previous study mentioned above the analyticity result implies that there can be no classical steady needle crystal solution when anisotropy is zero

nonexistence of classical steady hele shaw bubble this paper concerns nonexistence of steadily translating bubbles in a hele shaw cell for small but nonzero surface tension it is proved that for any fixed bubble velocity u relative to the fluid velocity at infinity in the interval 2 there can be no classical steady bubble solution

mapping nonlinear integro differential equations to a generalized describing function form an algebraic relationship is derived which enables the generalized amplitude dependent describing function to be expressed in terms of the parameters of a general nonlinear integro differential equation the algorithm is not restricted to a specific input but treats the whole class of harmonically related phase independent sinusoidal inputs and is illustrated by a number of examples

analysis of the effects of time delay in nonlinear systems using generalised frequency response functions abstract a recursive algorithm for computing the generalised frequency response functions gfrfs of nonlinear time delay systems described by nonlinear differential difference equation models is derived using the operator e n and used to analyse the effects of time delay in nonlinear systems the algorithm shows an explicit relationship between the model parameters and the gfrfs such a relationship provides important insight into the behaviour of nonlinear systems the effect of delay on several properties of nonlinear systems such as harmonic generation gain compression expansion and desensitisation is studied by considering the example of a duffing oscillator with retarded damping results of the studies convincingly demonstrate that the system delay has a significant effect on several important properties of nonlinear systems

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

asymptotics beyond all orders asymptotology and borel summation asymptotics superasymptotics hyperasymptotics m berry the geometric model of crystal growth numerical analysis of the geometric model of crystal growth j m hammersley dendritic crystal growth an experimental assessment of continuum models of dendritic growth j p gollub directional solidification of solids directional growth of dilute mixtures and lamellar eutectics k kassner et al flow in a hele shaw cell viscous displacement in a hele shaw cell s tanveer the rapidly forced pendulum exponentially small estimate for separatrix splittings j scheurle et al ordinary differential equations singular perturbations of solitons y pomeau solitary water waves in the presence of small surface tension solitary waves with ripples beyond all orders j t beale problems in optics potpourri 23 additional articles index

analytic theory for the selection of saffman taylor fingers in the presence of thin film effects an analytic theory is presented for the width selection of saffman taylor fingers in the presence of thin film effect in the limit of small capillary number ca and small gap to width ratio such that ca 1 it is found that fingers with relative width 2 1 1 2 k ca 3 2 where the positive constant k depends on the branch of solution and equals 2 776 for the first branch a fully nonlinear analysis is necessary in this problem even to obtain the correct scaling law it is also shown how in principle the selection rule for arbitrary ca can be obtained

shape selection of saffman taylor fingers among all problems of pattern selection the one posed by the saffman taylor finger is yet unsolved although numerics and experiments indicate the very simple property that the relative width of the finger tends to 1 2 in the low surface tension limit we explain this by performing an expansion beyond all orders leading to the formulation of a nonlinear eigenvalue problem with a discrete set of solutions in particular we predict that the relative width of the finger tends to 1 2 as the surface tension to the power 2 3 pacs numbers 47 10 g 03 40 gc 68 10 m

mellin transform and image charge method for dielectric sphere in an electrolyte we revisit the image charge method for the green s function problem of the poisson boltzmann equation for a dielectric sphere immersed in ionic solutions using finite mellin transformation we represent the reaction potential due to a source charge inside the sphere in terms of a one dimensional distribution of image charges the image charges are generically composed of a point image at the kelvin point and a line image extending from the kelvin point to infinity with an oscillatory line charge strength we further develop an efficient and accurate algorithm for discretization of the line image using pade approximation and finite fraction expansion finally we illustrate the power of our method by applying it in a multiscale reaction field monte carlo simulation of monovalent electrolytes

image solution for the 3 d static green s function of a vertically stratified two layer half space the general form of 3 d static green s function for a general structure consisting of n arbitrary adjacent dielectric wedges is derived using the kontorovich lebedev and fourier transforms it consists of several point sources in physical space and a line image source in complex space for each region of the problem geometry while the point sources can be treated conventionally the line image distributions are obtained by extracting a proper asymptotic behavior of the solution which can be numerically computed as an example the case of a vertically stratified two layer half space is investigated and the respective point and line image sources are obtained the efficiency of the proposed solution method is demonstrated by comparing the computational time for a given structure with that required by a commercially available numerical code using the finite integration technique

layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains the symbolic calculus of pseudodifferential operators of mellin type is applied to study layer potentials on a plane domain q whose boundary 9q is a curvilinear polygon a singularity type is a zero of the determinant of the matrix of symbols of the mellin operators and can be used to calculate the bad values of p for which the system is not fredholm on lp 9q using the method of layer potentials we study the singularity types of the system of elastostatics lu uau au vdivu 0 in a plane domain q whose boundary 9k2 is a curvilinear polygon here u 0 and u a oo when a oo the system is the stokes system of hydrostatics for the traction double layer potential we show that all singularity types in the strip 0 re z i lie in the interval 1 1 so that the system of integral equations is a fredholm operator of index 0 on lp aq for all p 2 p oo the explicit dependence of the singularity types on a and the interior angles 0 of aq is calculated the singularity type of each corner is independent of a if the corner is nonconvex introduction recently there has been considerable interest in using layer potentials to solve lp boundary value problems for elliptic operators and systems on a lipschitz domain q in r for the systems of elastostatics dkv and hydrostatics fkv dahlberg fabes kenig and verchota have used rellich type identities to prove that the double layer potential integral equations yield a fredholm operator of index 0 on l 2 q for p 2 only limited information is available on the boundary integral equations for general lipschitz domains in rn the general problem of the notion of symbol on the boundary of a general lipschitz domain is still very much open in this paper we treat a very special case a curvilinear polygonal domain in r in this 2 dimensional case a precise symbolic calculus of pseudodifferential operators of mellin type is available we show that certain double layer boundary integral equations yield operators which for all p 2 p 00 are received by the editors december 14 1988 1980 mathematics subject classification 1985 revision primary 35j25 45e05 research partially supported by the italian cnr 1990 american mathematical society 0002 9947 90 1 00 25 per page

rigorous results in selection of steady needle crystals abstract this paper concerns the existence of solutions to a steady needle crystal growth problem in a one sided model we rigorously prove that for small nonzero anisotropy analytic symmetric needle crystal solutions exist in the limit of surface tension e 2 if only if the stokes constant s for a relatively simple nonlinear differential equation is zero this stokes constant s depends on the parameter 2 9 7 e 8 7 and earlier numerical calculations by a number of investigators have shown this to be zero for a discrete set of values of it is also proven that for 0 there can be no symmetric needle crystal solution in the considered space the methodology consists of two steps first the original problem is reduced to a weak half strip problem for any in a compact set of 0 1 by relaxation of the symmetry condition the weak problem is shown to have a unique solution in the function space considered for any 0 m for some m 0 when a symmetry is invoked the weak problem is shown equivalent to the original needle crystal problem next by considering the behavior of the solution in neighborhood of an appropriate complex turning point for 0 m we extract an exponentially small term in e as e 0 that generally violates the symmetric condition we prove that the symmetry condition is satisfied for small e when the parameter is constrained appropriately

higher order acoustic diffraction by edges of finite thickness generalized temporal and spectral solutions of acoustic diffraction by edges of finite thickness involving higher orders diffraction are presented the solution in the time domain is an extension of medwins double edge second order diffraction solution h medwin et al j acoust soc am 72 1005 1013 1982 which is based on the huygens principle while the solution in the frequency domain is an extension of the asymptotic formulas for the double edge second order diffraction via amplitude and phase matching given by pierce a d pierce j acoust soc am 55 943 955 1974 since both solutions are approximate they have different advantages and disadvantages the contribution from higher order diffraction depends strongly on the thickness and geometry of the edges comparison between the theory and experimental diffraction data demonstrates that for edges of finite thickness the first order diffraction alone under estimates the total diffraction field significantly especially when the flat en

transient image theory for 2 d and 3 d conducting wedge problems in the present paper two and three dimensional transient scattering from a perfectly conducting wedge is studied the exciting sources analyzed are an infinite impulsive line current parallel to edge of the conductor and an arbitrarily oriented electric dipole the analysis presented is based on the time harmonic image solution found for the scattered field the frequency domain image current corresponding to the field contribution from the wedge is an exact function expressed in terms of simple trigonometric functions it is dependent only on the coordinates and the amplitude of the source but it is independent of the observation point it is shown that the transient scattering can be interpreted as arising from an image source by generalizing the time domain green s function to complex source points

mathematical analysis of plasmonic resonances for nanoparticles the full maxwell equations abstract in this paper we use the full maxwell equations for light propagation in order to analyze plasmonic resonances for nanoparticles we mathematically define the notion of plasmonic resonance and analyze its shift and broadening with respect to changes in size shape and arrangement of the nanoparticles using the layer potential techniques associated with the full maxwell equations we present an effective medium theory for resonant plasmonic systems and derive a condition on the volume fraction under which the maxwell garnett theory is valid at plasmonic resonances

two dimensional acoustical diffraction by a penetrable wedge a problem that has consistently defied rigorous analysis is the construction of the time periodic green s function for an acoustic line source in the presence of a penetrable wedge with distinct wave speed this paper begins by noting that in the soft limit density ratio becoming zero the interior field must become null and hence it is sensible to express the required green s function as a series in powers of the density ratio applications of green s theorem then reduce the problem to a sequence of integral equations for the determination of the coefficients in a similar expansion of the normal velocity at the wedge boundary the exterior hard green s function now appears as the kernel while the interior hard green s function appears in the forcing terms to take account of penetration into the wedge after observing that the velocity coefficients can be calculated recursively in terms of the solution of a single integral equation of the first kind the latter is itself transformed to an equatio

existence and uniqueness of analytic solution for unsteady crystals with zero surface tension abstract we study the initial value problem for two dimensional dendritic crystal growth with zero surface tension if the initial data is analytic and close to ivantsov steady solution it is proved that unique analytic solution exists locally in time the analysis is based on a nirenberg theorem on abstract cauchy kovalevsky problem in properly chosen banach spaces

automatic computation of polyharmonic balance equations for non linear differential systems an automated algorithm is presented which enables the harmonic balance equations for any polynomial type pure or cross product non linear differential system to be written down directly in terms of the coefficients of the governing equation and the complex amplitudes of a general harmonic waveform the system frequency response in the form of a multi input amplitude dependent describing function is therefore readily computed the method is illustrated by means of an example and the results validated against detailed numeric simulation

poincare s variational problem in potential theory one of the earliest attempts to rigorously prove the solvability of dirichlet s boundary value problem was based on seeking the solution in the form of a potential of double layer and this leads to an integral equation whose kernel is in general both singular and non symmetric c neumann succeeded with this approach for smoothly bounded convex domains and h poincare by a tremendous tour de force showed how to push through the analysis for domains with sufficiently smooth boundaries but no hypothesis of convexity in this work he was according to his own account guided by consideration of a variational problem involving the partition of energy of an electrostatic field induced by charges placed on the boundary of a domain more precisely the charge distributions which render stationary the energy of the field inside the domain divided by the energy of the field outside the domain unfortunately a rigorous treatment of this problem was not possible with the tools available at that time as poincare was well aware so far as we know the only one to propose a rigorous treatment of poincare s problem was t carleman in the two dimensional case in his doctoral dissertation thanks to later developments especially concerning sobolev spaces and spectral theory of operators on hilbert space we can now give a complete general and rigorous account of poincare s variational problem and that is the main object of the present paper as a by product we refine some technical aspects in the theory of symmetrizable operators and prove in any number of dimensions the basic properties of the analogue of the planar bergman schiffer singular integral equation we interpret poincare s variational principle as a non selfadjoint eigenvalue problem for the angle operator between two distinct pairs of subspaces of potentials we also prove a series of novel spectral analysis facts some of them conjectured by poincare related to the poincare neumann integral operator

moving boundary approximation for curved streamer ionization fronts solvability analysis the minimal density model for negative streamer ionization fronts is investigated an earlier moving boundary approximation for this model consisted of a kinetic undercooling type boundary condition in a laplacian growth problem of hele shaw type here we derive a curvature correction to the moving boundary approximation that resembles surface tension the calculation is based on solvability analysis with unconventional features namely there are three relevant zero modes of the adjoint operator one of them diverging furthermore the inner outer matching ahead of the front must be performed on a line rather than on an extended region and the whole calculation can be performed analytically the analysis reveals a relation between the fields ahead and behind a slowly evolving curved front the curvature and the generated conductivity this relation forces us to give up the ideal conductivity approximation and we suggest to replace it by a charge neutrality approximation this implies that the electric potential in the streamer interior is no longer constant but solves a laplace equation this leads to a muskat type problem

spectral resolution of the neumann poincare operator on intersecting disks and analysis of plasmon resonance the purpose of this paper is to investigate the spectral nature of the neumann poincare operator on the intersecting disks which is a domain with the lipschitz boundary the complete spectral resolution of the operator is derived which shows in particular that it admits only the absolutely continuous spectrum no singularly continuous spectrum and no pure point spectrum we then quantitatively analyze using the spectral resolution the plasmon resonance at the absolutely continuous spectrum

describing functions volterra series and the analysis of non linear systems in the frequency domain a relationship between the higher order frequency response functions of the volterra series model and the well known describing function representation for non linear systems is derived it is shown that a large class of generalized describing functions can be derived from the volterra model and that this approach removes the restriction to specific inputs the concept of a worst case transfer function is introduced and examples are included to illustrate the results

electromagnetic scattering from a dielectric wedge and the single hypersingular integral equation electromagnetic fields scattered by a finite dielectric wedge are computed using a hypersingular integral equation hie the results are compared with those obtained previously using a singular integral equation sie and with the theory that predicts that the fields near the edge of the wedge behave like static fields the hie produces more consistent results than the sie probably because the unknown boundary function tends to a constant near the edge instead of diverging the two numerical methods agree reasonably well and these results agree only in part with the static field behavior

two numerical solutions for the parallel plate fed slot antenna an integrodifferential equation for the electric field in the aperture of a parallel plate waveguide fed slot antenna is numerically solved using galerkin s method with a basis of edge condition weighted chebyshev polynomials and pulse expansion with point matching efficient algorithms are obtained for the elements of the reduced matrix equations for both moment methods the numerical convergence of the techniques is demonstrated by the resultant aperture field distributions and input reflection coefficients for the case of dominant transverse electromagnetic mode excitation

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

a moving boundary problem motivated by electric breakdown i spectrum of linear perturbations abstract an interfacial approximation of the streamer stage in the evolution of sparks and lightning can be written as a laplacian growth model regularized by a kinetic undercooling boundary condition we study the linear stability of uniformly translating circles that solve the problem in two dimensions in a space of smooth perturbations of the circular shape the stability operator is found to have a pure point spectrum except for the eigenvalue 0 0 for infinitesimal translations all eigenvalues are shown to have negative real part therefore perturbations decay exponentially in time we calculate the spectrum through a combination of asymptotic and series evaluation in the limit of vanishing regularization parameter all eigenvalues are found to approach zero in a singular fashion and this asymptotic behavior is worked out in detail a consideration of the eigenfunctions indicates that a strong intermediate growth may occur for generic initial perturbations both the linear and the nonlinear initial value problem are considered in a second paper

existence and selection of steady bubbles in a hele shaw cell abstract this paper concerns the existence of a steadily translating bubble in a hele shaw cell for small but non zero surface tension 2 we rigorously conclude that for bubble velocity u relative to the fluid velocity at infinity in the interval 1 2 analytic symmetric solutions exist in the asymptotic limit of surface tension 2 0 if and only if the stokes constant for a relatively simple nonlinear differential equation is zero this stokes constant s depends on the parameter 0 1 corresponding to bubble size and a u 1 2 2 u 1 2 2 2 1 2 4 3 earlier calculations have shown s to be zero for a discrete set of values of a

rigorous results in steady finger selection in viscous fingering this paper concerns the existence of a steadily translating finger solution in a hele shaw cell for small but non zero surface tension 2 though there are numerous numerical and formal asymptotic results for this problem we know of no mathematically rigorous results that address the selection problem we rigorously conclude that for relative finger width in the range with small analytic symmetric finger solutions exist in the asymptotic limit of surface tension if and only if the stokes constant for a relatively simple nonlinear differential equation is zero this stokes constant s depends on the parameter and earlier calculations by a number of authors have shown it to be zero for a discrete set of values of a the methodology consists of proving the existence and uniqueness of analytic solutions for a weak half strip problem for any in a compact subset of 0 1 the weak problem is shown to be equivalent to the original finger problem in the function space considered provided we invoke a symmetry condition next we consider the behavior of the solution in a neighborhood of an appropriate complex turning point for the restricted case for some this turning point accounts for exponentially small terms in as 0 that generally violate the symmetry condition we prove that the symmetry condition is satisfied for small when the parameter a is constrained appropriately