adaptive surface related multiple elimination the major amount of multiple energy in seismic data is related to the large reflectivity of the surface a method is proposed for the elimination of all surface related multiples by means of a process that removes the influence of the surface reflectivity from the data an important property of the proposed multiple elimination process is that no knowledge of the subsurface is required on the other hand the source signature and the surface reflectivity do need to be provided as a consequence the proposed process has been implemented adaptively meaning that multiple elimination is designed as an inversion process where the source and surface reflectivity properties are estimated and where the multiple free data equals the inversion residue results on simulated data and field data show that the proposed multiple elimination process should be considered as one of the key inversion steps in stepwise seismic inversion

the neumann problem for the laplacian and the monge kantorovich mass transfer problem abstract we consider the natural neumann boundary condition for the laplacian we study the limit as p of solutions of p u p 0 in a domain with d u p p 2 u p g on we obtain a natural minimization problem that is verified by a limit point of u p and a limit problem that is satisfied in the viscosity sense it turns out that the limit variational problem is related to the monge kantorovich mass transfer problems when the measures are supported on

γ convergence of power law functionals with variable exponents abstract convergence results for power law functionals with variable exponents are obtained the main motivation comes from the study of first failure dielectric breakdown some connections with the generalization of the laplace equation to the variable exponent setting are also explored

the aronsson euler equation for absolutely minimizing lipschitz extensions with respect to carnot caratheodory metrics we derive the euler lagrange equation also known in this setting as the aronsson euler equation for absolute minimizers of the l variational problem formula math where c g is an open subset of a carnot group 0 u denotes the horizontal gradient of u r and the lipschitz class is defined in relation to the carnot caratheodory metric in particular we show that absolute minimizers are infinite harmonic in the viscosity sense as a corollary we obtain the uniqueness of absolute minimizers in a large class of groups this result extends previous work of jensen and of crandall evans and gariepy we also derive the aronsson euler equation for more regular absolutely minimizing lipschitz extensions corresponding to those carnot caratheodory metrics which are associated to free systems of vector fields

l variational problems associated to measurable finsler structures abstract we study l variational problems associated to measurable finsler structures in euclidean spaces we obtain existence and uniqueness results for the absolute minimizers

intrinsic geometry and analysis of diffusion processes and l variational problems the aim of this paper is twofold first we obtain a better understanding of the intrinsic distance of diffusion processes precisely a for all n 1 the diffusion matrix a is weak upper semicontinuous on if and only if the intrinsic differential and the local intrinsic distance structures coincide b if n 1 or if n 2 and a is weak upper semicontinuous on the intrinsic distance and differential structures always coincide c if n 2 and a fails to be weak upper semicontinuous on the non coincidence of the intrinsic distance and differential structures depend on the geometry of the non weak upper semicontinuity set of a second for an arbitrary diffusion matrix a we show that the intrinsic distance completely determines the absolute minimizer of the corresponding l variational problem and then obtain the existence and uniqueness for given boundary data we also give an example of a diffusion matrix a for which there is an absolute minimizer that is not of class c1 when a is continuous we also obtain the linear approximation property of the absolute minimizer

on the dirichlet and serrin problems for the inhomogeneous infinity laplacian in convex domains regularity and geometric results given an open bounded subset of mathbb r n which is convex and satisfies an interior sphere condition we consider the pde delta infty u 1 in subject to the homogeneous boundary condition u 0 on we prove that the unique solution to this dirichlet problem is power concave precisely 3 4 concave and it is of class c1 we then investigate the overdetermined serrin type problem formerly considered in buttazzo and kawohl int math res not pp 237 247 2011 obtained by adding the extra boundary condition nabla u a on by using a suitable p function we prove that if satisfies the same assumptions as above and in addition contains a ball which touches at two diametral points then the existence of a solution to this serrin type problem implies that necessarily the cut locus and the high ridge of coincide in turn in dimension n 2 this entails that must be a stadium like domain and in particular it must be a ball in case its boundary is of class c2

solutions to an inhomogeneous equation involving infinity laplacian abstract we obtain existence and uniqueness results of viscosity solutions to the dirichlet boundary value problem for a nonlinear highly degenerate elliptic equation of the form 1 d u h u f where 0 h 2 and u denotes the so called infinity laplacian operator given by u i j 1 n u x i u x j 2 u x i x j we also give an asymptotic behavior of the viscosity solutions of two kinds of inhomogeneous infinity laplace equations near an isolated singular point

boundary behavior for the solutions to dirichlet problems involving the infinity laplacian abstract in this paper by constructing suitable comparison functions we mainly analyze the exact boundary behavior for the unique solution near the boundary to the singular dirichlet problem u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 g is decreasing on 0 with lim s 0 g s g is normalized regularly varying at zero with index 1 and b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary

estimation of multiple scattering by iterative inversion part i theoretical considerations a review has been given of the surface related multiple problem by making use of the so called feedback model from the resulting equations it has been concluded that the proposed solution does not require any properties of the subsurface however source detector and reflectivity properties of the surface need be specified those properties have been quantified in a surface operator and this operator is estimated as part of the multiple removal problem the surface related multiple removal algorithm has been formulated in terms of a neumann series and in terms of an iterative equation the neumann formulation requires a nonlinear optimization process for the surface operator while the iterative formulation needs a number of linear optimizations the iterative formulation also has the advantage that it can be integrated easily with another multiple removal method an algorithm for the removal of internal multiples has been proposed as well this algorithm is an extension of the surface related method removal of internal multiples requires knowledge of the macro velocity model between the surface and the upper boundary of the multiple generating layer in part ii also published in this issue the success of the proposed algorithms has been demonstrated on numerical experiments and field data examples

weak fubini property and infinity harmonic functions in riemannian and sub riemannian manifolds we examine the relationship between infinity harmonic functions absolutely minimizing lipschitz extensions strong absolutely minimizing lipschitz extensions and absolutely gradient minimizing extensions in carnot carath eodory spaces using the weak fubini property we show that absolutely minimizing lipschitz extensions are infinity harmonic in any sub riemannian manifold

absolutely minimal lipschitz extension of tree valued mappings we prove that every lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal lipschitz extension amle we relate these extensions to a stochastic game called politics a generalization of a game called tug of war that has been used in peres et al j am math soc 22 1 167 210 2009 to study real valued amles

a tour of the theory of absolutely minimizing functions a detailed analysis of the class of absolutely minimizing functions in euclidean spaces and the relationship to the infinity laplace equation

the behaviour of the p x laplacian eigenvalue problem as p x abstract in this paper we study the behaviour of the solutions to the eigenvalue problem corresponding to the p x laplacian operator div u p x 2 u p x u p x 2 u in u 0 on as p x we consider a sequence of functions p n x that goes to infinity uniformly in under adequate hypotheses on the sequence p n namely that the limits ln p n x x and p n n x q x exist we prove that the corresponding eigenvalues p n and eigenfunctions u p n verify that p n 1 n u p n u uniformly in where u is a nontrivial viscosity solution of the following problem min u u 2 log u u u q u q 0 in u 0 on

asymptotic estimates of boundary blow up solutions to the infinity laplace equations abstract in this paper we study the asymptotic behavior of boundary blow up solutions to the equation u b x f u in where is the laplacian the nonlinearity f is a positive increasing function in 0 and the weighted function b c is positive in and may vanish on the boundary we first establish the exact boundary blow up estimates with the first expansion when f is regularly varying at infinity with index p 3 and the weighted function b is controlled on the boundary in some manner furthermore for the case of f s s p 1 c g s with the function g normalized regularly varying with index q 0 we obtain the second expansion of solutions near the boundary it is interesting that the second term in the asymptotic expansion of boundary blow up solutions to the infinity laplace equation is independent of the geometry of the domain quite different from the boundary blow up problems involving the classical laplacian

an axiomatic approach to image interpolation we discuss possible algorithms for interpolating data given in a set of curves and or points in the plane we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolute minimal lipschitz extension model amle is singled out and studied in more detail we show experiments suggesting a possible application the restoration of images with poor dynamic range

exact travelling wave solutions for the n 1 dimensional double sine and sinh gordon equations for the n 1 dimensional double sine and sinh gordon equations by using the approach of dynamical systems to the travelling wave solutions in different regions of the parameter space all possible bounded solutions solitary wave solutions kink and anti kink wave solutions and periodic wave solutions et al are obtained explicit exact parametric representations are given

derivation of the aronsson equation for c1 hamiltonians it is proved herein that any absolute minimizer u for a suitable hamiltonian h c r r u is a viscosity solution of the aronsson equation hp du u x h du u x x 0 in u the primary advance is to weaken the assumption that h c used by previous authors to the natural condition that h c

the second order expansion of boundary blow up solutions for infinity laplacian equations abstract in this paper by using karamata regular variation theory and a perturbed argument we study the second order expansion of viscosity solutions to boundary blow up elliptic problem u b x f u x u where is a bounded domain with c 2 boundary in r n b c is non negative and non trivial in f c 1 0 is a normalized regularly varying function at infinity with index 3

an efficient derivation of the aronsson equation for 1 p the equation which characterizes minima of the functional u u du p dx subject to fixed values of u on u is p u 0 here p is the well known p laplacian when p the corresponding functional is u du 2 l u a new feature arises in that minima are no longer unique unless u is allowed to vary leading to the idea of absolute minimizers aronsson showed that then the appropriate equation is u 0 that is u is infinity harmonic as explained below jensen showed that infinity harmonic functions understood in the viscosity sense are precisely the absolute minimizers here we advance results of barron jensen and wang concerning more general functionals u f x u du l u by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the dirichlet problem for the convex envelope the convex envelope of a given function was recently characterized as the solution of a fully nonlinear partial differential equation pde in this article we study a modified problem the dirichlet problem for the underlying pde the main result is an optimal regularity result differentiability c1 regularity of the boundary data implies the corresponding result for the solution in the interior despite the fact that the solution need not be continuous up to the boundary secondary results are the characterization of the convex envelope as i the value function of a stochastic control problem and ii the optimal underestimator for a class of nonlinear elliptic pdes

a general theorem of existence of quasi absolutely minimal lipschitz extensions in this paper we consider a wide class of generalized lipschitz extension problems and the corresponding problem of finding absolutely minimal lipschitz extensions we prove that if a minimal lipschitz extension exists then under certain other mild conditions a quasi absolutely minimal lipschitz extension must exist as well here we use the qualifier quasi to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal lipschitz extension up to several factors that can be made arbitrarily small

interpolation and extrapolation using a high resolution discrete fourier transform we present an iterative nonparametric approach to spectral estimation that is particularly suitable for estimation of line spectra this approach minimizes a cost function derived from bayes theorem the method is suitable for line spectra since a long tailed distribution is used to model the prior distribution of spectral amplitudes since the data themselves are used as constraints phase information can also be recovered and used to extend the data outside the original window the objective function is formulated in terms of hyperparameters that control the degree of fit and spectral resolution noise rejection can also be achieved by truncating the number of iterations spectral resolution and extrapolation length are controlled by a single parameter when this parameter is large compared with the spectral powers the algorithm leads to zero extrapolation of the data and the estimated fourier transform yields the periodogram when the data are sampled at a constant rate the algorithm uses one levinson recursion per iteration for irregular sampling the algorithm uses one cholesky decomposition per iteration the performance of the algorithm is illustrated with three different problems that arise in geophysical data 1 harmonic retrieval from a time series contaminated with noise 2 linear event detection from a finite aperture array of receivers 3 interpolation extrapolation of gapped data the performance of the algorithm as a spectral estimator is tested with the kay and marple 1981 data set

convexity criteria and uniqueness of absolutely minimizing functions we show that an absolutely minimizing function with respect to a convex hamiltonian h mathbb r n rightarrow mathbb r is uniquely determined by its boundary values under minimal assumptions on h along the way we extend the known equivalences between comparison with cones convexity criteria and absolutely minimizing properties to this generality these results perfect a long development in the uniqueness existence theory of the archetypal problem of the calculus of variations in l

high resolution velocity gathers and offset space reconstruction we present a high resolution procedure to reconstruct common midpoint cmp gathers first we describe the forward and inverse transformations between offset and velocity space then we formulate an underdetermined linear inverse problem in which the target is the artifacts free aperture compensated velocity gather we show that a sparse inversion leads to a solution that resembles the infinite aperture velocity gather the latter is the velocity gather that should have been estimated with a simple conjugate operator designed from an infinite aperture seismic array this high resolution velocity gather is then used to reconstruct the offset space the algorithm is formally derived using two basic principles first we use the principle of maximum entropy to translate prior information about the unknown parameters into a probabilistic framework in other words to assign a probability density function to our model second we apply bayes s rule to relate the a priori probability density function pdf with the pdf corresponding to the experimental uncertainties likelihood function to construct the a posteriori distribution of the unknown parameters finally the model is evaluated by maximizing the a posteriori distribution when the problem is correctly regularized the algorithm converges to a solution characterized by different degrees of sparseness depending on the required resolution the solutions exhibit minimum entropy when the entropy is measured in terms of burg s definition we emphasize two crucial differences in our approach with the familiar burg method of maximum entropy spectral analysis first burg s entropy is minimized rather than maximized which is equivalent to inferring as much as possible about the model from the data second our approach uses the data as constraints in contrast with the classic maximum entropy spectral analysis approach where the autocorrelation function is the constraint this implies that we recover not only amplitude information but also phase information which serves to extrapolate the data outside the original aperture of the array the tradeoff is controlled by a single parameter that under asymptotic conditions reduces the method to a damped least squares solution finally the high resolution or aperture compensated velocity gather is used to extrapolate near and far offset traces

removal of surface related wave phenomena the marine case removal of the effects of the free surface from seismic reflection data is an essential preprocessing step before prestack migration the problem can be formulated by means of rayleigh s reciprocity theorem which leads to an integral equation of the second kind for the desired pressure field that does not include these free surface effects this integral equation can be solved numerically both in the spatial domain and in the double radon domain solving the integral equation in the double radon domain has the advantage of reducing the computation time significantly since the kernel of the integral equation becomes dominant diagonally two methods are proposed to solve the integral equation direct matrix inversion and a recursive subtraction of the free surface multiples using a neumann series both methods have been developed and tested on a synthetic data set which was computed with the help of an independent forward modeling scheme

the euler equation and absolute minimizers of l functionals the aronsson euler equation for the functional on w g 1 m i e w 1 with boundary data g is this equation has been derived for smooth absolute minimizers i e a function which minimizes f on every subdomain we prove in this paper that for m 1 n 1 or n 1 m 1 an absolute minimizer of f exists in w g 1 m and for m 1 n 1 any absolute minimizer of f must be a viscosity solution of the aronsson euler equation

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

solutions to an inhomogeneous equation involving aronsson operator abstract let f r n 0 be a convex function of class c 2 r n 0 which is positively homogeneous of degree 1 we assume further that f p 0 for any p 0 and hess f 2 is positive definite in r n 0 in this paper for a bounded domain r n f c with inf f x 0 and g c we obtain existence and uniqueness results of viscosity solutions to the dirichlet boundary value problem for a nonlinearly highly degenerate elliptic equation of the form 1 f d u h a u f in u g on where a denotes the aronsson operator given by a u i j 1 n 2 u x i x j 1 2 f 2 p i d u 1 2 f 2 p j d u and 0 h 2

adaptive multiple subtraction based on 3d blind separation of convolved mixtures abstractmultiple removal is one of the important preprocessing steps in a seismic data processing sequence for an accurate multiple prediction a full 3d method is required however the application of such methods is often limited by practical and economic constraints therefore in practical situations 2d prediction methods are used such as 2d surface related multiple elimination however the resulting predicted multiples may have significant temporal shift spatial mismatch and amplitude inconsistency compared with true 3d multiples adaptive multiple subtraction based on 2d blind separation of convolved mixtures bscm has been proposed to estimate the 2d matching filter in a single gather to improve the flexibility of the adaptive multiple subtraction for the inconsistencies between the 2d predicted multiples and true 3d multiples we evaluated the adaptive multiple subtraction as a problem of 3d bscm in the proposed method the predicted multiples were modeled as the convolution of the true

the limit as p of the first eigenvalue for the p laplacian with mixed dirichlet and robin boundary conditions abstract we analyze the behavior as p of the first eigenvalue of the p laplacian with mixed boundary conditions of dirichlet robin type we find a nontrivial limit that we associate to a variational principle involving l norms moreover we provide a geometrical characterization of the limit value as well as a description of it using optimal mass transportation techniques our results interpolate between the pure dirichlet case and the mixed dirichlet neumann case

a weighted eigenvalue problem of the degenerate operator associated with infinity laplacian abstract in this paper we study a weighted eigenvalue problem of the degenerate operator associated with infinity laplacian h u a x u h 1 u 0 in u 0 on where h 1 h u d u h 3 u is the h homogeneous infinity laplacian and a x is a positive continuous bounded function in we prove the existence of the principal eigenvalue and a corresponding positive eigenfunction the approach to the weighted eigenvalue problem is based on the maximum principle and when a parameter is less than the principal eigenvalue some existence and uniqueness results related to this problem are established

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

stochastic homogenization of l variational problems abstract we present a homogenization result for l variational problems in general stationary ergodic random environments by introducing a generalized notion of distance function a special solution of an associated eikonal equation and demonstrating a connection to absolute minimizers of the variational problem we obtain the homogenization result as a consequence of the fact that the latter homogenizes in random environments

l variational problems and aronsson equations in this paper we prove that viscosity solutions of aronsson equations are absolute minimizers in certain l variational problems

absolutely minimizing lipschitz extensions and infinity harmonic functions on the sierpinski gasket abstract aim of this note is to study the infinity laplace operator and the corresponding absolutely minimizing lipschitz extension problem on the sierpinski gasket in the spirit of the classical construction of kigami for the laplacian we introduce a notion of infinity harmonic functions on pre fractal sets and we show that these functions solve a lipschitz extension problem in the discrete setting then we prove that the limit of the infinity harmonic functions on the pre fractal sets solves the absolutely minimizing lipschitz extension problem on the sierpinski gasket

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

a singular boundary value problem for a degenerate elliptic pde abstract we consider a singular boundary value problem associated with the laplacian and having nonseparable inhomogeneous term we establish the existence of ground state solutions to such a problem in bounded domains as well as in the whole euclidean space the investigation is based on subsolution and supersolution methods and existence of principal eigenfunctions to the eigenvalue problem of dirichlet type corresponding to the laplacian

3d surface related multiple prediction a sparse inversion approach the theory of iterative surface related multiple elimination holds for 2d as well as 3d wavefields the 3d prediction of surface multiples however requires a dense and extended distribution of sources and receivers at the surface since current 3d marine acquisition geometries are very sparsely sampled in the crossline direction the direct fresnel summation of the multiple contributions calculated for those surface positions at which a source and a receiver are present cannot be applied without introducing severe aliasing effects in this newly proposed method the regular fresnel summation is applied to the contributions in the densely sampled inline direction but the crossline fresnel summation is replaced with a sparse parametric inversion with this procedure 3d multiples can be predicted using the available input data the proposed method is demonstrated on a 3d synthetic data set as well as on a 3d marine data set from offshore norway

a study of the dual problem of the one dimensional l infinity optimal transport problem with applications the monge kantorovich problem for the infinite wasserstein distance presents several peculiarities among them the lack of convexity and then of a direct duality we study in dimension 1 the dual problem introduced by barron bocea and jensen we construct a couple of kantorovich potentials which is as less trivial as possible more precisely we build a potential which is non constant around any point that the plan which is locally optimal moves at maximal distance as an application we show that the set of points which are displaced to maximal distance by a locally optimal transport plan is minimal

exact explicit traveling wave solutions for the cdf equation abstract the traveling wave solutions of the calogero degasperis fokas cdf equation are studied by using the approach of dynamical systems and the theory of bifurcations with the aid of maple all bifurcations and phase portraits in the parametric space are obtained all possible explicit parametric representations of the bounded traveling wave solutions solitary wave solutions kink and anti kink wave solutions periodic wave solutions and breaking wave solutions are given moreover the reason for appearance of breaking waves and for persistence of smoothness of smooth traveling waves is explained

another way to say harmonic it is known that solutions of u n i j 1 u xi u xj u xjxj 0 that is the harmonic functions are exactly those functions having a comparison property with respect to the family of translates of the radial solutions g x a x we establish a more difficult linear result a function in r n is harmonic if it has the comparison property with respect to sums of n translates of the radial harmonic functions g x a x 2 n for n 2 and g x b ln x for n 2 an attempt to generalize these results for u 0 p and u 0 p 2 to the general p laplacian leads to the fascinating discovery that certain sums of translates of radial p superharmonic functions are again p superharmonic mystery remains the class of p superharmonic functions so constructed for p 2 does not suffice to characterize p subharmonic functions

inhomogeneous infinity laplace equation abstract we present the theory of the viscosity solutions of the inhomogeneous infinity laplace equation x i u x j u x i x j 2 u f in domains in r n we show existence and uniqueness of a viscosity solution of the dirichlet problem under the intrinsic condition f does not change its sign we also discover a characteristic property which we call the comparison with standard functions property of the viscosity sub and super solutions of the equation with constant right hand side applying these results and properties we prove the stability of the inhomogeneous infinity laplace equation with nonvanishing right hand side which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right hand side and boundary data are perturbed in the end we prove the stability of the well known homogeneous infinity laplace equation x i u x j u x i x j 2 u 0 which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right hand side and boundary data are perturbed simultaneously

on the calogero degasperis fokas equation in 2 1 dimensions abstract in this paper we study a 2 1 dimensional integrable calogero degasperis fokas equation derivable by using a method proposed by calogero a catalogue of classical symmetry reductions are given these reductions to partial differential equations in 1 1 admit symmetries which lead to further reductions i e to second order ordinary differential equations these odes provide several classes of solutions all of them are expressible in terms of known functions some of them are expressible in terms of the second and third painleve trascendents the corresponding solutions of the 2 1 dimensional equation involve up to three arbitrary smooth functions consequently the solutions exhibit a rich variety of qualitative behaviour indeed by making appropriate choices for the arbitrary functions we exhibit solitary waves coherent structures and bound states

random walks and random tug of war in the heisenberg group we study mean value properties of mathbf p harmonic functions on the first heisenberg group mathbb h in connection to the dynamic programming principles of certain stochastic processes we implement the approach of peres and sheffield duke math j 145 1 91 120 2008 to provide a game theoretical interpretation of the sub elliptic mathbf p laplacian and of manfredi et al proc am math soc 138 3 881 889 2010 to characterize its viscosity solutions via asymptotic mean value expansions

the eigenvalue problem the euler lagrange equation of the nonlinear rayleigh quotient left int omega nabla u p dx right bigg left int omega u p dx right is div left nabla u p 2 nabla u right lambda p p u p 2 u where lambda p p is the minimum value of the quotient the limit as p to infty of these equations is found to be max left lambda infty frac nabla u x u x delta infty u x right 0 where the constant lambda infty lim p to infty lambda p is the reciprocal of the maximum of the distance to the boundary of the domain

3d surface related multiple elimination using parabolic sparse inversion the basic theory of surface related multiple elimination srme can be formulated easily for 3d seismic data however because standard 3d seismic acquisition geometries violate the requirements of the method the practical implementation for 3d seismic data is far from trivial a major problem is to perform the crossline summation step of 3d srme which becomes aliased because of the large separation between receiver cables and between source lines a solution to this problem based on hyperbolic sparse inversion has been presented previously this method is an alternative to extensive interpolation and extrapolation of data the hyperbolic sparse inversion is formulated in the time domain and leads to few but large systems of equations in this paper we propose an alternative formulation using parabolic sparse inversion based on an efficient weighted minimum norm solution that can be computed in the angular frequency domain the main advantage of the new method is numerical efficiency because solving many small systems of equations often is faster than solving a few big ones the method is demonstrated on 3d synthetic and real data with reflected and diffracted multiples numerical results show that the proposed method gives improved results compared to 2d srme for typical seismic acquisition geometries the numerical cost running on 50 processors is 0 05 s per output trace this makes production scale processing of 3d seismic data feasible on current linux clusters

finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method the vanishing moment method was introduced by the authors in feng and neilan 2009 as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations pdes it is based on the simple idea of approximating a hard to handle fully nonlinear second order pde by a family parameterized by a small parameter e of easy to handle quasilinear fourth order pdes the primary objective of this article is to present a comprehensive finite element analysis for the vanishing moment approximation of general fully nonlinear second order elliptic pdes which fulfill some structure conditions abstract methodological and convergence analysis frameworks of conforming finite element methods are first developed for fully nonlinear second order pdes in a general setting the abstract framework is then applied to three prototypical nonlinear equations namely the monge ampere equation the equation of prescribed gauss curvature and the infinity laplacian equation numerical experiments are presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology

on the evolution governed by the infinity laplacian we investigate the basic properties of the degenerate and singular evolution equation ut d 2 u du du du du which is a parabolic version of the increasingly popular infinity laplace equa tion we prove existence and uniqueness results for both dirichlet and cauchy problems establish interior and boundary lipschitz estimates and a harnack inequality and also provide numerous explicit solutions

on boundary blow up solutions to equations involving the laplacian abstract given a non negative continuous function h on r such that h x 0 0 for all x h x t 0 in 0 and h x t non decreasing in t for each x we study the boundary value problem u h x u in u on where r n n 2 is a bounded domain and is the laplacian a degenerate elliptic operator we provide sufficient conditions on h under which the above problem admits a solution or fails to admit a solution a necessary and sufficient condition on f is given for a solution to exist in the special case when h x t b x f t in the latter case an asymptotic boundary behavior of solutions will be studied as an application a sufficient condition on f will be given to ensure the uniqueness of solutions in case b is a constant

reduction technique for matrix nonlinear evolution equations solvable by the spectral transform the main purpose of this paper is to describe a technique of reduction whereby from the class of evolution equations for matrices of order n solvable via the spectral transform associated to the matrix linear schrodinger eigenvalue problem one derives subclasses of nonlinear evolution equations involving less than n2 fields to illustrate the method from the equations for matrices of order 2 two subclasses of equations for 2 fields rather than 4 are obtained the first class coincides or rather includes that solvable via the spectral transform associated to the generalized zakharov shabat spectral problem further reduction to nonlinear evolution equations for a single field reproduces a number of well known equations but also yields a novel one highly nonlinear the second class also yields highly nonlinear equations some examples are given including another novel evolution equation for a single field

specializations of integrable systems and affine lie algebras we analyze a number of restrictions on the evolution systems associated with the zero curvature equations corresponding to the extended dynkin diagram a 1 n 1 the resulting specialized evolution systems contain exponential terms like a third order differential equation previously derived in ref 5

uniqueness of lipschitz extensions minimizing the sup norm of the gradient in this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values geometrically this can be interpreted as finding a minimal lipschitz extension due to the weak convexity of the functional associated to this problem solutions are generally nonunique by adopting g aronsson s notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation in fact we actually prove a weak maximum principle for this partial differential equation which in some sense is the euler equation for the minimization problem this is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity to overcome this difficulty we use the weak solutions of m g crandall and p l lions also known as viscosity solutions in conjunction with some arguments using integration by parts

uniqueness and nonuniqueness of viscosity solutions to aronsson s equation for a bounded domain omega subset mathbb r n and g in c delta omega assume that h in c 2 mathbb r n is convex and coercive and that p in mathbb r n h p rm min mathbb r n h has no interior points then we establish the uniqueness of viscosity solutions to the dirichlet problem of aronsson s equation begin aligned mathcal a u h p nabla u otimes h p nabla u nabla 2 u 0 rm in omega u g rm on delta omega end aligned for h h p x depending on x we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to aronsson s equation and that of the hamilton jacobi equation h nabla u x c

an axiomatic approach to scalar data interpolation on surfaces we discuss possible algorithms for interpolating data given on a set of curves in a surface of 3 we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolutely minimizing lipschitz extension model amle is singled out and studied in more detail we study the correctness of our numerical approach and we show experiments illustrating the interpolation of data on some simple test surfaces like the sphere and the torus

principal eigenvalue of a very badly degenerate operator and applications abstract in this paper we define and investigate the properties of the principal eigenvalue of the singular infinity laplace operator u d 2 u d u d u d u d u this operator arises from the optimal lipschitz extension problem and it plays the same fundamental role in the calculus of variations of l functionals as the usual laplacian does in the calculus of variations of l 2 functionals our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of berestycki nirenberg and varadhan h berestycki l nirenberg s r s varadhan the principal eigenvalue and maximum principle for second order elliptic operators in general domains comm pure appl math 47 1 1994 47 92 in the case of uniformly elliptic linear operators as an application we obtain existence and uniqueness results for certain related non homogeneous problems and decay estimates for the solutions of the evolution problem associated to the infinity laplacian

a boundary value problem for normalized finsler infinity laplacian equations with singular nonhomogeneous terms abstract we study boundary value problems of finsler infinity laplacian equations with nonhomogeneous terms that may exhibit singularity when solutions vanish on the boundary

an optimization problem with volume constraint with applications to optimal mass transport in this manuscript we study the following optimization problem with volume constraint min left frac 1 p int omega nabla v pdx int partial omega gv ds colon v in w 1 p left omega right text and v 0 leq alpha right here g is acontinuous function and alpha is a fixed constant such that 0 0 we prove that a minimizer exists and satisfies left begin array ccl delta p u p 0 text in u p 0 cup u p 0 5pt nabla u p p 2 frac partial u p partial nu g text on partial omega cap partial u p 0 cup u p 0 5pt u p 0 alpha end array right next we analyze the limit as p to infty we obtain that any sequence of weak solutions converges up to a subsequence lim limits p j to infty u p j x u infty x uniformly in overline omega and uniform limits u infty are solutions to the maximization problem with volume constraint max left int partial omega gv ds colon v in w 1 infty left omega right nabla v l infty omega leq 1 text and v 0 leq alpha right furthermore we obtain the limit equation that is verified by u infty in the viscosity sense finally it turns out that such a limit variational problem is connected to the monge kantorovich mass transfer problem with the involved measures are supported on partial omega and along the limiting free boundary partial u infty neq 0

parabolic radon transform sampling and efficiency a good choice of the sampling in the transform domain is essential for a successful application of the parabolic radon transform the parabolic radon transform is computed for each temporal frequency and is essentially equivalent to the nonuniform fourier transform this leads to new and useful insights in the parabolic radon transform using nonuniform fourier theory we derive a minimum sampling interval for the curvature parameter and a maximum curvature range for which stability is guaranteed for general irregular sampling a significantly smaller sampling interval requires stabilization if diagonal stabilization is used no gain in resolution is obtained in contrast to conventional implementations the curvature sampling interval is proposed to be inversely proportional to the temporal frequency this results in improved quality of the transform and yields significant savings in computation time

fast slow diffusion and growing sandpiles abstract we study a coupled system of ode introduced by the first author insiam j appl math 22 1972 437 458 for the heights of growing interacting sand cones we show that these ode correspond to the evolution inl2generated by the subdifferential of the convex functional which vanishes on functions whose gradient has length less than or equal to one and is infinity otherwise additionally we explain how the ode arise from evolutions governed by thep laplacian in the infinitely fast infinitely slow diffusion limit asp

l variational problems and weak kam theory in the first part of this paper we extend several results in crandall evans and gariepy 11 and crandall and evans 10 to absolute minimizers of more general l functionals in the second part we present some interesting connections between l variational problems and weak kam theory as an application we will advance the main result in fathi and siconolfi 17 i e the existence of a c1 subsolution of the hamilton jacobi equation moreover we will propose a possible approximation of the projected aubrey set by a variational approach that was first used by evans in 14 2007 wiley periodicals inc

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

simultaneous source separation using a robust radon transform abstractwe adopted the robust radon transform to eliminate erratic incoherent noise that arises in common receiver gathers when simultaneous source data are acquired the proposed robust radon transform was posed as an inverse problem using an l1 misfit that is not sensitive to erratic noise the latter permitted us to design radon algorithms that are capable of eliminating incoherent noise in common receiver gathers we also compared nonrobust and robust radon transforms that are implemented via a quadratic l2 or a sparse l1 penalty term in the cost function the results demonstrated the importance of incorporating a robust misfit functional in the radon transform to cope with simultaneous source interferences synthetic and real data examples proved that the robust radon transform produces more accurate data estimates than least squares and sparse radon transforms

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159