pulse distortion in depth migration when migrating seismic primary reflections obtained from arbitrary source receiver configurations e g common shot or constant offset into depth a pulse distortion occurs along the reflector this distortion exists even if the migration was performed using the correct velocity model regardless of the migration algorithm this distortion is a consequence of varying reflection angle reflector dip and or velocity variation the relationship between the original time pulse and the depth pulse after migration can be explained and quantified in terms of a prestack kirchhoff type diffraction stack migration theory

extrapolation of the dirichlet problem for elliptic equations with complex coefficients abstract in this paper we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as p ellipticity specifically let be a chord arc domain in r n and the operator l i a i j x j b i x i be elliptic with b i x k x 1 for a small k let p 0 sup p 1 a is p elliptic we establish that if the l q dirichlet problem is solvable for l for some 1 q p 0 n 1 n 2 then the l p dirichlet problem is solvable for all p in the range q p 0 n 1 n 2 in particular if the matrix a is real or n 2 the l p dirichlet problem is solvable for p in the range q

the lp neumann problem for the heat equation in non cylindrical domains we consider the lp neumann problem for the heat equation when 1 p 2 in a time varying domain using a perturbation technique and previous l2 neumann results the authors obtain a generalization of earlier work in caloric cylinders

on the imaging of reflectors in the earth in this paper i present a modification of the beylkin inversion operator this modification accounts for the band limited nature of the data and makes the role of discontinuities in the sound speed more precise the inversion presented here partially dispenses with the small parameter constraint of the born approximation this is shown by applying the proposed inversion operator to upward scattered data represented by the kirchhoff approximation using the angularly dependent geometrical optics reflection coefficient a fully nonlinear estimate of the jump in sound speed may be extracted from the output of this algorithm interpreted in the context of these kirchhoff approximate data for the forward problem the inversion of these data involves integration over the source receiver surface the reflecting surface and frequency the spatial integrals are computed by the method of stationary phase the output is asymptotically a scaled singular function of the reflecting surface the singular function of a

true amplitude weight functions in 3d limited aperture migration revisited the true amplitude weight function in 3d limited aperture migration is obtained by extending its formula at an actual reflection point to any arbitrary subsurface point this implies that the recorded seismic signal is a delta impulse when the weight function is used in depth migration it results in an amplitude distortion depending on the vertical distance from the target reflector this distortion exists even if the correct velocity model is used if the image point lies at a depth shallower than the half offset the distortion cannot be ignored even for a spatial wavelet having a short length using paraxial ray theory i find a formula for the true amplitude weight function causing no amplitude distortion under the condition that the earth s surface is smoothly curved however the formula is reflector dependent as a result amplitude distortion in parallel with pulse distortion is an intrinsic effect in depth migration and true amplitude migration without amplitude distortion is possible only

uniqueness results for inverse robin problems with bounded coefficient in this paper we address the uniqueness issue in the classical robin inverse problem on a lipschitz domain omega subset rr n with l infty robin coefficient l 2 neumann data and isotropic conductivity of class w 1 r omega r n we show that uniqueness of the robin coefficient on a subpart of the boundary given cauchy data on the complementary part does hold in dimension n 2 but needs not hold in higher dimension we also raise on open issue on harmonic gradients which is of interest in this context

a b cyclic submodules abstract an a b cyclic submodule m is generated by the states of one single trajectory of a linear control system whose parameters come from a commutative ring m is finite when it is generated by the states of a deadbeat control process motivations and basic properties of such modules are given and among several further results it is shown that the family of finite a b cyclic submodules is an invariant which e g over polynomials can be determined by an appropriate grobner basis computation

existence and uniqueness for a semilinear elliptic problem on lipschitz domains in riemannian manifolds ii abstract for a general class of semilinear equations in a lipschitz domain where m is a smooth compact riemannian manifold we establish the existence and uniqueness of solutions to dirichlet and neumann boundary problems the main contribution of this paper is that we consider rough boundary data that are typically just functions results of this type for the linear equation are typically obtained using singular integral techniques

the dirichlet problem in non smooth domains dirichlet problem for an elliptic operator 2s laj jaj with smooth coefficients aij in domains d in rm m 3 with non smooth boundaries the domains we treat are given locally in some cm coordinate system by the graph of a continuous function qs with vq5 e li for some p 1 p oo such domains are called li domains see section 1 for the precise definitions note that l domains are usually called lipschitz domains if d is an li domain and x e d we study the elliptic measure a7 associated with s and d at x thus for f a continuous function on ad the formula u x hf x f doh gives the perron wiener brelot generalized soluid

geometrical ray theory rays and traveltimes in seismic systems second order approximations of the traveltimes in this paper i establish general laws that represent global properties for the reflected and transmitted rays in seismic systems the method used for establishing these laws is based on hamilton s original ideas the laws do not depend on the specific seismic system being used the reflected and transmitted rays as well as their traveltimes are described by the same four 2 x 2 matrices these matrices are interrelated by three equations that provide the basis for the general laws and also provide relations between reflected and transmitted rays that eventually will be exploitable for downward continuation and migration the traveltimes of reflected and transmitted rays are summed up in single functions giving the traveltime for any pair of source and receiver positions in a seismic system and corresponding to hamilton s point characteristics in optics the traveltimes of the gathers commonly used in seismic exploration are special cases of the general point characteristics the consequences from the derived laws are directly available as a priori knowledge before any actual ray tracing and computation of traveltimes are done in any particular seismic system the laws are also available as a general foundation for future treatments of 1 downward continuation 2 amplitudes of reflected and transmitted events 3 true amplitude migration 4 examples of nonuniqueness of seismic interpretations 5 focus phenomena and 6 higher order approximations

a criterion for the absolute continuity of the harmonic measure associated with an elliptic operator for some a o and for all x e q and 4 e r n we also assume a a for such operators the dirichlet problem is solvable in q if and only if it is solvable for the laplace operator according to a theorem of littman stampacchia and weinberger 1 this means that if q c r n is a sufficiently nice bounded region the unit ball b is an example and f is a given continuous function on the boundary of q then there exists a unique function u continuous on q so that l u 0 in q and u f on aq let us assume for convenience that the origin belongs to q then the mapping f c f aq u o is a positive linear functional so there exists a unique nonnegative measure w on aq such that for every f e aq

solutions of the divergence operator on john domains if rn is a bounded domain the existence of solutions u w01 p of divu f for f lp with vanishing mean value and 1 p is a basic result in the analysis of the stokes equations it is known that the result holds when is a lipschitz domain and that it is not valid for domains with external cusps in this paper we prove that the result holds for john domains our proof is constructive the solution u is given by an explicit integral operator acting on f to prove that u w01 p we make use of the calderon zygmund singular integral operator theory and the hardy littlewood maximal function for domains satisfying the separation property introduced in s buckley p koskela sobolev poincare implies john math res lett 2 5 1995 577 593 and 1 p n we also prove a converse result thus characterizing in this case the domains for which a continuous right inverse of the divergence exists in particular our result applies to simply connected planar domains because they satisfy the separation property

homogenization of elliptic boundary value problems in lipschitz domains in this paper we study the lp boundary value problems for mathcal l u 0 in mathbb r d 1 where mathcal l rm div a nabla is a second order elliptic operator with real and symmetric coefficients assume that a is periodic in xd 1 and satisfies some minimal smoothness condition in the xd 1 variable we show that the lp neumann and regularity problems are uniquely solvable for 1 p 2 we also present a new proof of dahlberg s theorem on the lp dirichlet problem for 2 p dahlberg s original unpublished proof is given in the appendix as the periodic and smoothness conditions are imposed only on the xd 1 variable these results extend directly from mathbb r d 1 to regions above lipschitz graphs consequently by localization techniques we obtain uniform lp estimates for the dirichlet neumann and regularity problems on bounded lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization the results on the neumann and regularity problems are new even for smooth domains

uniform regularity estimates in homogenization theory of elliptic system with lower order terms abstract in this paper we extend the uniform regularity estimates obtained by m avellaneda and f lin in 3 6 to the more general second order elliptic systems in divergence form l e e 0 with rapidly oscillating periodic coefficients we establish not only sharp w 1 p estimates holder estimates lipschitz estimates and non tangential maximal function estimates for the dirichlet problem on a bounded c 1 domain but also a sharp o e convergence rate in h 0 1 by virtue of the dirichlet correctors moreover we define the green s matrix associated with l e and obtain its decay estimates we remark that the well known compactness methods are not employed here instead we construct the transformations 1 11 to make full use of the results in 3 6

analyticity of layer potentials and l 2 solvability of boundary value problems for divergence form elliptic equations with complex l infty coefficients we consider divergence form elliptic operators of the form l dv a x nabla defined in r n 1 x t in r n times r n geq 2 where the l infty coefficient matrix a is n 1

the effective calculation of all assignable polynomials to a single input system over a bezout domain the set of all feedback assignable polynomials to a given single input system over a bezout domain is obtained the general case is reduced to the case of weakly reachable systems for these systems a canonical form for the algebraic equivalence is obtained algorithms and examples over a euclidean domain and in particular over the ring z of rational integers are also given the problem of deciding and feedback assignment of a given polynomial to a given non reachable n dimensional linear system is solved effectively up to a cost of o n4 arithmetic operations

layer potential methods for elliptic homogenization problems in this paper we use the method of layer potentials to study l 2 boundary value problems in a bounded lipschitz domain for a family of second order elliptic systems with rapidly oscillating periodic coefficients defining l e div a e 1 x under the assumption that a x is elliptic symmetric periodic and holder continuous we establish the solvability of the l 2 dirichlet regularity and neumann problems for l e u e 0 in with optimal estimates uniform in e 0

a new approach to absolute continuity of elliptic measure with applications to non symmetric equations in the late 1950s and early 1960s the work of de giorgi degi and nash n and then moser mo initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients weak solutions in a domain 0 a priori only in a sobolev space w 1 loc 0 were shown to be ho lder continuous of some order depending just on ellipticity and maximum principles and harnack inequalities were established the dirichlet problem for such operators with continuous data on the boundary was established in lsw this in turn paved the way for a more systematic and detailed study of the properties of the elliptic measures d l associated to l div a on a domain 0 the classical properties of existence of non tangential limits of solutions fatou type theorems and comparison principles appeared in cfms but owed a great deal to the earlier work of carleson ca and hunt and wheeden h w on harmonic functions in lipschitz domains all the results mentioned above were carried out for elliptic operators l div a where the matrix a aij has bounded measurable coefficients and is symmetric however it turns out that the symmetry of the matrix is not needed to get these results morrey mor first observed this in connection with the de giorgi nash moser theory for the results in cfms this fact has not been formally observed until now with appropriate reformulation in terms of adjoint operators and adjoint green s functions the results of cfms are valid without the symmetry assumption see section 1 doi 10 1006 aima 1999 1899 available online at http www idealibrary com on

a new slant on seismic imaging migration and integral geometry a new approach to seismic migration formalizes the classical diffraction or common tangent stack by relating it to linearized seismic inversion and the generalized radon transform this approach recasts migration as the problem of reconstructing the earth s acoustic scattering potential from its integrals over isochron surfaces the theory rests on a solution of the wave equation with the geometrical optics green function and an approximate inversion formula for the generalized radon transform the method can handle both complex velocity models and nearly arbitrary configurations of sources and receivers in this general case the method can be implemented as a weighted diffraction stack with the weights determined by tracing rays from image points to the experiment s sources and receivers when tested on a finite difference simulation of a deviated well vertical seismic profile a hybrid experiment which is difficult to treat with conventional wave equation methods the algorithm accurately reconstructed faulted earth models analytical reconstruction formulas are derived from the general formula for zero offset and fixed offset surface experiments in which the background velocity is constant the zero offset inversion formula resembles standard kirchhoff migration our analysis provides a direct connection between the experimental setup source and receiver positions source wavelet background velocity and the spatial resolution of the reconstruction synthetic examples illustrate that the lateral resolution in seismic images is described well by the theory and is improved greatly by combining surface data and borehole data the best resolution is obtained from a zero offset experiment that surrounds the region to be imaged

on c 1 c 2 c2 and weak type 1 1 estimates for linear elliptic operators part ii we extend and improve the results in dong and kim commun partial differ equ 42 3 417 435 2017 showing that weak solutions to full elliptic equations in divergence form with zero dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have dini mean oscillation and the lower order coefficients verify certain conditions similar results are obtained for non divergence form equations we extend the weak type 1 1 estimates in dong and kim commun partial differ equ 42 3 417 435 2017 and escauriaza duke math j 74 1 177 201 1994 up to the boundary and derive a harnack inequality for non negative adjoint solutions to non divergence form elliptic equations when the leading coefficients have dini mean oscillation

limited aperture migration i establish a general theory on the true amplitude migration with limited aperture the key point of this theory is to find the true amplitude weight function eliminating the contribution from the boundary of the migration aperture or edge effect i show that the true amplitude weight function is a singular generalized function defined by the migration aperture if it is used without regularization the singular weight function reduces a diffraction stack to the distribution of the amplitude at a given traveltime to every point along a corresponding isochron after regularization i obtain a formula for the true amplitude weight function as a weak solution having the same global properties as the fundamental solution to the governing equation derived here in comparison with other published weight functions that disregard migration aperture this theory contains a taper function that makes the obtained weight function suitable for the migration with limited aperture if the migration aperture goes to in

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

the neumann problem for elliptic equations with non smooth coefficients in this paper we study the neumann and regularity problems for divergence form symmetric elliptic operators l div av where a aifix is a matrix of bounded measurable functions and ellipticity is the existence of a 0 such that 2 114 2 ais ir 21412 for all i r we assume that l is defined in the unit ball b but the methods of proof are sufficiently general to allow b to be a bounded starlike lipschitz domain the neumann problem lu 0 in b avun q f

computation of maximal pre controllability submodules over a noetherian ring for dynamical systems with coefficient in a ring it is not always possible to compute by a finite procedure the maximal a b invariant submodule contained in the kernel of the output map c namely v kerc this difficulty prevents one from checking necessary and sufficient geometric conditions for the solvability of noninteracting control problems in this paper we will prove that for dynamical systems with coefficients in a noetherian ring it is always possible to compute by a finite procedure the maximal pre controllability submodule contained in the kernel of the output map c namely r kerc this result is useful in dealing with the block decoupling problem or the disturbance rejection problem

parabolic regularity and dirichlet boundary value problems abstract we study the relationship between the regularity and dirichlet boundary value problems for parabolic equations of the form l u div a u u t 0 in lip 1 1 2 time varying cylinders where the coefficient matrix a a i j x t is uniformly elliptic and bounded we show that if the regularity problem r p for the equation l u 0 is solvable for some 1 p then the dirichlet problem d p for the adjoint equation l v 0 is also solvable where p p p 1 this result is an analogue of the result established in the elliptic case by kenig and pipher 1993 in the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by hofmann and lewis 1996 and nystrom 2006 for scalar parabolic systems in comparison our result is abstract with no assumption on the coefficients beyond the ellipticity condition and is valid in more general class of domains

computation of maximal reachability submodules abstract a new and conceptually simple procedure is derived for the computation of the maximal reachability submodule of a given submodule of the state space of a linear discrete time system over a noethenian ring r the procedure is effective if r is effective and if kernels and intersections can be computed the procedure is compared with a rather different procedure by assan e a published recently

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

3 d crosswell transmissions paraxial ray solutions and reciprocity paradox transmission of seismic waves through a 3 d earth model is of fundamental importance in seismology if the model consists of many layers separated by curved interfaces the only feasible solution to the transmitted waves is the one given by the geometrical optics approximation transmitted rays transmitted wavefield and the first fresnel zone associated with a transmission point can be expressed by four 2 x 2 constant matrices constituting the 4 x 4 linearized ray transformation matrix generally the ray transformation matrix can be constructed by dynamic ray tracing however if the layers are homogeneous it can be formulated in closed form by using elementary vector calculus and coordinate transformations using the symplecticity of the ray transformation matrix transmissions in opposite directions can be formulated by the ray transformation matrix for only one direction and a reciprocity relation can be established after the decomposition theorem for the ray transformation matrix the transmitted wavefield in a model with many curved interfaces can be computed in a cascaded way using the b matrix decomposition theorem the normalized geometrical spreading factor can be expressed by means of the area of the first fresnel zone of a transmission point if seismic waves propagate through a locally spherical interface the reciprocity relation may not hold using ray theory this fact is shown by formulating the transmitted wavefield with the two principal radii of curvature of the transmitted wavefront at the transmission point under consideration using wave theory this fact is shown by analyzing the debye integral with the method of stationary phase

layer potentials and boundary value problems for elliptic equations with complex l coefficients satisfying the small carleson measure norm condition abstract we consider divergence form elliptic equations l u a u 0 in the half space r n 1 x t r n 0 whose coefficient matrix a is complex elliptic bounded and measurable in addition we suppose that a satisfies some additional regularity in the direction transverse to the boundary namely that the discrepancy a x t a x 0 satisfies a carleson measure condition of fefferman kenig pipher type with small carleson norm under these conditions we establish a full range of boundedness results for double and single layer potentials in l p hardy sobolev bmo and holder spaces furthermore we prove solvability of the dirichlet problem for l with data in l p r n bmo r n and c r n and solvability of the neumann and regularity problems with data in the spaces l p r n h p r n and l 1 p r n h 1 p r n respectively with the appropriate restrictions on indices assuming invertibility of layer potentials for the t independent operator l 0 a 0

elliptic operators conormal derivatives and positive parts of functions with an appendix by haim brezis haim brezis and augusto ponce introduced and studied several extensions of kato s inequality in particular kato s inequalities up to the boundary involving the laplacian and the normal derivative of the positive part of a w1 1 function in a smooth domain h brezis a c ponce kato s inequality when u is a measure c r acad sci paris ser i 338 2004 599 604 h brezis a c ponce kato s inequality up to the boundary commun contemp math 10 2008 1217 1241 using potential theoretic methods we answer here some questions raised in h brezis a c ponce kato s inequality up to the boundary commun contemp math 10 2008 1217 1241 about the relations between the normal derivative of a function u and the normal derivative of its positive part u the results apply to a large class of domains and elliptic operators in divergence form and finally an expression of the normal derivative of a function of u is given in the final appendix h brezis solves an old question of j serrin about pathological solutions of certain elliptic equations j serrin pathological solutions of elliptic differential equations ann sc norm super pisa 3 18 1964 385 387 this is used in the paper to extend the first version of our main result

functional calculus of dirac operators and complex perturbations of neumann and dirichlet problems we prove that the neumann dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in l 2 for small complex l infty perturbations of a coefficient matrix which is either real symmetric of block form or constant all matrices are assumed to be independent of the transversal coordinate we solve the neumann dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order dirac type operator we establish quadratic estimates for this dirac operator which implies that the associated hardy projection operators are bounded and depend continuously on the coefficient matrix we also prove that certain transmission problems for k forms are well posed for small perturbations of block matrices

the regularity problem for second order elliptic operators with complex valued bounded measurable coefficients the present paper establishes a certain duality between the dirichlet and regularity problems for elliptic operators with t independent complex bounded measurable coefficients t being the transversal direction to the boundary to be precise we show that the dirichlet boundary value problem is solvable in l p subject to the square function and non tangential maximal function estimates if and only if the corresponding regularity problem is solvable in l p moreover the solutions admit layer potential representations in particular we prove that for any elliptic operator with t independent real possibly non symmetric coefficients there exists a p 1 such that the regularity problem is well posed in l p

potential theory on lipschitz domains in riemannian manifolds the case of dini metric tensors we study the applicability of the method of layer potentials in the treatment of boundary value problems for the laplace beltrami operator on lipschitz sub domains of riemannian manifolds in the case when the metric tensor g jk dx j desk has low regularity under the assumption that g jk x g jk y c x y where the modulus of continuity satisfies a dini type condition we prove the well posedness of the classical dirichlet and neumann problems with l p boundary data for sharp ranges of p s and with optimal nontangential maximal function estimates

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