the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

relationship between juxtaposed overlapping and fractal representations of multimodal spatial variability 1 it has been shown by neuman 1990 di federico and neuman 1997 and di federico et al 1999 that the power variogram of a statistically isotropic or anisotropic fractal field can be constructed as a weighted integral from zero to infinity of exponential or gaussian variograms of overlapping homogeneous random fields modes having mutually uncorrelated increments and variance proportional to a power 2h of the integral spatial correlation scale where h is the hurst coefficient low and high frequency cutoffs are related to length scales of the sampling window domain and data support sample volume respectively intermediate cutoffs account for lacunarity due to gaps in the multiscale hierarchy created by a hiatus of modes associated with discrete ranges of scales alternative mathematical representations of multimodal spatial variability were formulated by various authors in which space is filled by a discrete number of juxtaposed mutually exclusive materials or categories each having its own architecture and attributes the spatial distribution of categories is characterized by indicator random variables and their attributes by random fields this paper focuses on expressions developed by lu and zhang 2002 in which the indicator variables and their attributes are mutually uncorrelated while each is autocorrelated and cross correlated within and between categories upon rewriting their expressions for statistically homogeneous and anisotropic media it is demonstrated mathematically that in the limit as the categories stretch to occupy each point in space overlap in a way that preserves their local architecture their attributes become mutually uncorrelated categories are said to overlap if they are found in fixed proportions within a representative sampling volume centered about any mathematical point throughout space the idea is analogous to the well known dual continuum concept in which two categories most commonly fractures and porous blocks are considered to overlap in reality the categories do not overlap but are considered to preserve their relative architectural arrangement throughout space the variogram of an attribute sampled jointly over overlapping statistically homogeneous and anisotropic categories is the sum of the variograms of statistically homogeneous and anisotropic components of this attribute sampled over individual categories weighted by their volumetric proportions it is further demonstrated that when the overlapping categories represent an infinite or truncated hierarchy of modes whose attributes have exponential or gaussian variograms such that the product of category probability density function pdf and attribute variance is proportional to a power 2h 1 of the attribute integral scale they represent a fractal field characterized by a power or truncated power variogram as in the work of di federico and neuman any category pdf that satisfies this condition and any attribute pdf having a variance that satisfies the same condition are admissible neither of them need to be gaussian to form fractional brownian motion or levy like to approximate fractional levy motion for the attribute to form a random fractal intermediate cutoffs render the latter model discrete as is that of lu and zhang hence the two multimodal models are mutually consistent and complementary

a planar network between short and long range behaviour abstract we study the projection of lucena network pln a planar network whose nodes coincide with the sites of the square lattice the pln has one internal parameter for 1 the network resembles a symmetric regular lattice with almost none highly connected node while 0 is more asymmetric and there are several highly connected nodes we estimate p r the probability of a node be connected to another node at distance r the pln follows the fractal scaling p r r with 3 4 according to for 1 we have close to 4 which is a signature of short range interactions while 0 shows 3 and has long range interactions in addition the fractal dimension d f of the pln behaves as for 1 d f 2 the dimension of the embedding space while for 0 d f increases and the pln shares some similarities with a small world graph

nonlocal product rules for percolation despite original claims of a first order transition in the product rule model proposed by achlioptas et al science 323 1453 2009 recent studies indicate that this percolation model in fact displays a continuous transition the distinctive scaling properties of the model at criticality however strongly suggest that it should belong to a different universality class than ordinary percolation here we introduce a generalization of the product rule that reveals the effect of nonlocality on the critical behavior of the percolation process precisely pairs of unoccupied bonds are chosen according to a probability that decays as a power law of their manhattan distance and only that bond connecting clusters whose product of their sizes is the smallest becomes occupied interestingly our results for two dimensional lattices at criticality shows that the power law exponent of the product rule has a significant influence on the finite size scaling exponents for the spanning cluster the conducting backbone and the cutting bonds of the system in all three cases we observe a clear transition from ordinary to nonlocal explosive percolation exponents

multifractal anisotropic scaling of the hydraulic conductivity 1 we analyzed the scaling properties of the hydraulic conductivity k at three sites in northern america made borden and cape cod we found that k at all sites exhibits multifractality fractal and multiscaling in both the vertical and horizontal directions though the multiscaling was within a range smaller than that of the maximum distance between measurements in the vertical direction the k data for made borden and cape cod were multiscaling from 0 15 to 1 35 m 0 05 to 0 5 m and 0 15 to 0 9 m respectively they were multiscaling in the horizontal direction from 9 to 45 m 1 to 10 m and 1 to 17 m respectively the multiscaling was also anisotropic evidence of scaling was poorest for the horizontal direction of the made site and it spanned half an order of magnitude such results compel one to treat multifractality in the horizontal direction of made as supported by heuristic arguments rather than by pure statistical evaluation we fitted a multifractal model to the data and estimated its parameters we found the underlying statistics of all data to be non gaussian and the model capable of reproducing the probability distribution of k data especially the negative skewness of log k we also generated two dimensional isotropic multifractal fields illustrating the role of the parameters of the selected multifractal model

bounded solutions of a k hessian equation involving a weighted nonlinear source abstract we consider the problem 1 s k d 2 u x 1 u q in b u 0 in b u 0 on b where b denotes the unit ball in r n n 2 k k n 0 q k and 0 we study the existence multiplicity and uniqueness of radially symmetric bounded solutions of 1 the methodology for obtaining our results is based on a dynamical systems approach for this we introduce a new transformation which reduces problem 1 to an autonomous two dimensional generalized lotka volterra system

flow in multiscale log conductivity fields with truncated power variograms in a previous paper we offered an interpretation for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with the size of the region window being sampled we did so by demonstrating that such behavior is typical of any random field with a truncated power semi variogram and that this field can be viewed as a truncated hierarchy of mutually uncorrelated homogeneous fields with either exponential or gaussian spatial autocovariance structures the low and high frequency cutoff scales l and u are related to the length scales of the sampling window and data support respectively we showed how this allows the use of truncated power variograms to bridge information about a multiscale random field across windows of different sizes either at a given locale or between different locales in this paper we investigate mean uniform steady state groundwater flow in unbounded domains where the log hydraulic conductivity forms a truncated multiscale hierarchy of gaussian fields each associated with an exponential autocovariance we start by deriving an expression for effective hydraulic conductivity as a function of the hurst coefficient h and the cutoff scales in one two and three dimensional domains which is qualitatively consistent with experimental data we then develop leading order analytical expressions for two and three dimensional autocovariance and cross covariance functions of hydraulic head velocity and log hydraulic conductivity versus h l and u examine their behavior and compare them with those corresponding to an exponential log hydraulic conductivity autocovariance our results suggest that it should be possible to bridge information about hydraulic heads and groundwater velocities across windows of disparate scales in particular when l u the variance of head is infinite in two dimensions and grows in proportion to l2 2h in three dimensions while the variance and longitudinal integral scale of velocity grow in proportion to l2h and l respectively in both cases

analysis on the evolution process of bfw like model with discontinuous percolation of multiple giant components abstract recently the modified bfw model on random graphs w chen r m d souza phys rev lett 106 2011 115701 which shows a discontinuous percolation transition with multiple giant components has attracted much attention from physicists and statisticians in this paper by establishing the evolution equations on the modified bfw model the evolution process and steady states on both random graphs and finite dimensional lattices are analyzed on a random graph by varying the edge accepted rate the system stabilizes in a steady state with different numbers of giant components moreover a close correspondence is built between the values of and the number of giant components in steady states the efficiency of which is verified by the numerical simulations then the sizes of giant components for different evolution strategies can be obtained by solving some constraints derived from the evolution equations meanwhile a similar analysis is expanded to finite dimensional lattices and we find the bfw model on a finite dimensional lattice has different steady states from those on a random graph but they have the same evolution mechanism the analysis of the evolution process and steady state is of great help to explain the properties of discontinuous percolation and the role of nonlocality

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

existence and uniqueness of nontrivial radial solutions for k hessian equations abstract we prove existence and uniqueness of nontrivial radial solutions to the k hessian problem s k d 2 u f u in u 0 on where s k d 2 u is the k hessian operator of u is the open unit ball in r n f is a continuous function and may have k sublinear growth at with possible k superlinear growth at 0 and is a large parameter

weighted eigenvalue problems for hessian equations abstract in this paper we consider weighted eigenvalue problems for fully nonlinear elliptic equations involving hessian operators in particular we consider a singular weight which behaves like a hardy potential and we prove the existence of weak eigenfunctions

transport in multiscale log conductivity fields with truncated power variograms di federico and neuman this issue investigated mean uniform steady state groundwater flow in an unbounded domain with log hydraulic conductivity that forms a truncated multiscale hierarchy of statistically homogeneous and isotropic gaussian fields each associated with an exponential autocovariance here we present leading order expressions for the displacement covariance and dispersion coefficient of an ensemble of solute particles advected through such a flow field in two or three dimensions both quantities are functions of the mean travel distance s the hurst coefficient h and the low and high frequency cutoff integral scales l and u the latter two are related to the length scales of the sampling window region under investigation and sample volume data support respectively if one considers transport to be affected by a finite domain much larger than the mean travel distance so that s l then an early preasymptotic regime develops during which longitudinal and transverse dispersivities grow linearly with s if one considers transport to be affected by a domain which increases in proportion to s then l and s are of similar order and a preasymptotic regime never develops instead transport occurs under a regime that is perpetually close to asymptotic under the control of an evolving scale l s we show that if additionally u l then the corresponding longitudinal dispersivity grows in proportion to l1 2h or equivalently s1 2h both these preasymptotic and asymptotic theoretical growth rates are shown to be consistent with the observed variation of apparent longitudinal fickian dispersivities with scale we conclude by investigating the effect of variable separations between cutoff scales on dispersion

boundary behavior of k convex solutions for singular k hessian equations abstract we discuss the existence and boundary behavior of k convex solution to the singular k hessian problem s k d 2 u x b x f u x x u x 0 x where s k d 2 u k 1 2 n is the k hessian operator r n n 2 is a smooth bounded strictly convex domain here the weight function b x is not necessarily bounded on another interest is that f u as u 0 our approach mainly relies on karamata s regular variation theory and the construction of suitable sub and super solutions

scaling of random fields by means of truncated power variograms and associated spectra an interpretation is offered for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with domain size we first demonstrate that the power semi variogram and associated spectra of random fields having homogeneous isotropic increments can be constructed as weighted integrals from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated homogeneous isotropic fields modes we then analyze the effect of filtering out truncating high and low frequency modes from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous with an autocovariance function that varies monotonically with separation distance in a manner not too dissimilar than that of its constituent modes the integral scales of the lowest and highest frequency modes cutoffs are related respectively to the length scales of the sampling window domain and data support sample volume taking each relationship to be one of proportionality renders our expressions for the integral scale and variance of a truncated field dependent on window and support scales in a manner consistent with observations the traditional approach of truncating power spectral densities yields autocovariance functions that oscillate about zero with finite in one and two dimensions or vanishing in one dimension integral scales our hierarchical theory allows bridging across scales at a specific locale by calibrating a truncated variogram model to data observed on a given support in one domain and predicting the autocovariance structure of the corresponding multiscale field in domains that are either smaller or larger one may also venture we suspect with less predictive power to bridge across both domain scales and locales by adopting generalized variogram parameters derived on the basis of juxtaposed hydraulic and tracer data from many sites

a variational theory of the hessian equation by studying a negative gradient flow of certain hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous hessian equations to achieve this we derive uniform first and second order a priori estimates for the elliptic and parabolic hessian equations our results generalize well known results for semilinear elliptic equations and the monge ampere equation c 2001 john wiley sons inc

anisotropic elliptic equations with general growth in the gradient and hardy type potentials abstract in this paper we give existence and regularity results for the solutions of problems whose prototype is q v v h d v q h o x p v p 2 v f x in v 0 on with bounded domain of r n n 2 0 p 1 q p n is a nonnegative continuous function and 0 moreover h is a general norm of r n h o is its polar and q v i 1 n x i h d v p 1 h i d v

anisotropy lacunarity and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms it has been shown by di federico and neuman 1997 1998a b that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework which views hydraulic conductivity as a random fractal characterized by a power variogram such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments di federico and neuman 1997 have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated fields modes that are homogeneous and isotropic we show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic provided the ratios between principal integral spatial correlation scales are the same for all modes we then analyze the effect of filtering out truncating modes of low high and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous the integral scales of the lowest and highest frequency cutoff modes are related to length scales of the sampling window domain and data support sample volume respectively taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale in a manner previously shown to be consistent with observations in the isotropic case it also allows in principle bridging across scales at a specific locale as well as among locales by adopting either site specific or generalized variogram parameters the introduction of intermediate cutoffs allows us to account in a straightforward manner for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales for example where textural and structural features are associated with distinct ranges of scale such as fractures having discrete ranges of trace length and density which dissect the rock into matrix blocks having corresponding ranges of sizes we explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale variance covariance and spectra of a truncated fractal field we then develop an expression for the equivalent hydraulic conductivity of a box shaped porous block embedded within a multiscale log hydraulic conductivity field under mean uniform flow the block is larger than the support scale of the field but is smaller than a surrounding sampling window consequently its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function conditioned on a known mean value of support scale conductivity across the window are given explicitly by our multiscale theory

multifractal analyses of hydraulic conductivity distributions the concept of a universal multifractal a generalization of a monofractal is a recently developed scaling model for natural phenomena characterized by irregularity presented herein is an effort to use universal multifractal concepts to deal with spatial variations of hydraulic conductivity k which have a significant effect on contaminant transport in the subsurface structure function analyses of four k data sets show that some vertical variations of k display multifractal structures while others are consistent with monofractal behavior in order to make multifractal concepts more useful multifractal noise is introduced and defined as the increments of a multifractal it is concluded that the multifractal formalism of schertzer and lovejoy 1987 has provided a rather general approach for modeling in k variations in the vertical with the exception of horizontal variations of the borden data all results fell within the domain of universal multifractal behavior which includes the monofractal case parameters were well defined in an empirical sense and easy to calculate indicating a robust formalism results were consistent with the recent finding of liu and molz 1997 also non gaussian and scale variant behavior in hydraulic conductivity distributions submitted to water resources research 1997 hereinafter referred to as submitted paper that k variations display increasing heterogeneity at decreasing scales

hessian measures ii in our previous paper on this topic we introduced the notion of k hessian measure associated with a continuous k convex function in a domain q in euclidean n space k 1 n and proved a weak continuity result with respect to local uniform convergence in this paper we consider k convex functions not necessarily continuous and prove the weak continuity of the associated k hessian measure with respect to convergence in measure the proof depends upon local integral estimates for the gradients of k convex functions

evidence for non gaussian scaling behavior in heterogeneous sedimentary formations vertical and horizontal fluctuations in permeability and porosity in sedimentary formations are analyzed and are found to be consistent with scaling models based on levy stable probability distributions the approach avoids the assumption of gaussian behavior and is supported by evidence from horizontal and vertical well logs and from permeability measurements on a sandstone outcrop and segments of core from a heterogeneous formation the incremental values in these measurement sequences are accurately modeled as having levy stable distributions the width of the distribution of increments depends on the spatial scale in a manner consistent with scaling behavior the width of the distribution is smaller for horizontal increments than for vertical increments reflecting the reduced variability in the horizontal direction the scaling parameters are in the range associated with antipersistence and are roughly the same magnitude in the vertical and horizontal directions the relationships between different physical properties are briefly studied and it is suggested that they be quantified through off diagonal terms in a multivariate levy width matrix simulations designed to reproduce the observed statistical features are also described these results have some fundamental implications as levy stable distributions require a different set of statistical tools and theoretical methods compared to finite variance distributions

optimal transport exponent in spatially embedded networks ij w here is a variable exponent it has been shown that by introducing a cost constraint on the total length of the additional links regardless of the strategy usedbythetraveler independentofwhetheritisbasedonlocalorglobalknowledgeofthenetworkstructure the best transportation condition is obtained with an exponent d 1 wheredis the dimension of the underlying lattice here we present further support through a high performance real time algorithm on the validity of this conjecture in three dimensional regular as well as in two dimensional critical percolation clusters our results clearly indicate that cost constraint in the navigation problem provides a proper theoretical framework to justify the evolving topologies of real complex network structures as recently demonstrated for the networks of the us airports and the human brain activity doi 10 1103 physreve 87 042810

critical dimension of a hessian equation involving critical exponent and a related asymptotic result abstract let s k 2 u 1 k n 2 be the k th hessian operator we study the problem formula where k is the critical exponent for s k and is a ball results generalizing those obtained by brezis nirenberg for the special case k 1 are established a discussion 8 on the asymptotic behavior of solutions of the problem as 0 is also included

a general fractal model of flow and solute transport in randomly heterogeneous porous media 1 one of the challenges of hydrology is modeling flow and solute transport through media that are not uniformly porous this paper proposes a general fractal model of flow and solute transport in randomly heterogeneous porous media we describe the random field using fractional levy motion flm which more effectively represents a field with a high degree of variability understanding flow and solute transport in the flm field expands the applicability of stochastic theory following the work of v di federico and s p neuman 1997 and v di federico et al 1999 we derive truncated isotropic and anisotropic power variograms for the flm fields the results are analogous to but more general than those of fractional brownian motion fbm v di federico and s p neuman 1997 and v di federico et al 1999 for a random field when the levy index in flm is replaced by a constant of 2 fbm is recovered this study employs first order perturbation analyses of flow and solute transport in two dimensional flm fields for the stationary fbm and flm fields macrodispersion coefficients are compared the plume is found to grow slowest in the flm field but will show highest concentration the proposed fractal model is general in the sense that when the levy index of flm equals 2 the results of flow and solute transport for the commonly used fbm model are recovered when the upper cutoff of the truncated power variogram is close to the lower cutoff the results for the stationary exponential or gaussian model can be well approximated since the proposed general fractal model has broader applications than the stationary and fbm models it simulates flow and solute transport in more different scenarios to yield more accurate modeling results