construction of indistinguishable conductivity perturbations for the point electrode model in electrical impedance tomography we explain how to build indistinguishable isotropic conductivity perturbations of the unit conductivity in the framework of the point electrode model for two dimensional electrical impedance tomography the theoretical approach based on solving a fixed point problem is constructive and allows the implementation of an algorithm for approximating the invisible perturbations the functionality of the method is demonstrated via numerical examples

determining the first order perturbation of a polyharmonic operator on admissible manifolds abstract we consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator l g x q with x being a w 1 vector field and q being an l function on compact riemannian manifolds with boundary which are conformally embedded in a product of the euclidean line and a simple manifold we show that the knowledge of the dirichlet to neumann map determines x and q uniquely the method is based on the construction of complex geometrical optics solutions using the carleman estimate for the laplace beltrami operator due to dos santos ferreira kenig salo and uhlmann notice that the corresponding uniqueness result does not hold for the first order perturbation of the laplace beltrami operator

the calderon problem with partial data in this paper we improve an earlier result by bukhgeim and uhlmann 1 by showing that in dimension n 3 the knowledge of the cauchy data for the schr nodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential we follow the general strategy of 1 but use a richer set of solutions to the dirichlet problem this implies a similar result for the problem of electrical impedance tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary

an inverse problem for a semi linear elliptic equation in riemannian geometries abstract we study the inverse problem of unique recovery of a complex valued scalar function v m c c defined over a smooth compact riemannian manifold m g with smooth boundary given the dirichlet to neumann map in a suitable sense for the elliptic semi linear equation g u v x u 0 we show that uniqueness holds for a large class of non linearities when the manifold is conformally transversally anisotropic the proof is constructive and is based on a multiple fold linearization of the semi linear equation near complex geometric optic solutions for the linearized operator and the resulting non linear interactions these interactions result in the study of a weighted integral transform along geodesics that we call the jacobi weighted ray transform

on the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology abstract we analyze the sensitivity of a climatological model with respect to small changes in one of the distinguished parameters the solar constant we start by proving the stabilization of solutions of the evolution model when time tends to infinity later we study the stationary problem and obtain uniqueness or a multiplicity of solutions for different values of the solar constant

determining the first order perturbation of a bi harmonic operator on bounded and unbounded domains from partial data abstract in this paper we study inverse boundary value problems with partial data for the bi harmonic operator with first order perturbation we consider two types of subsets of r n n 3 one is an infinite slab the other is a bounded domain in the case of a slab we show that from dirichlet and neumann data given either on the different boundary hyperplanes of the slab or on the same boundary hyperplane one can uniquely determine the magnetic potential and the electric potential in the case of a bounded domain we show the unique determination of the magnetic potential and the electric potential from partial dirichlet and neumann data under two different assumptions the first assumption is that the magnetic and electric potentials are known in a neighborhood of the boundary in this situation we obtain the uniqueness result when the dirichlet and neumann data are only given on two arbitrary open subsets of the boundary the second assumption is that the dirichlet and neumann data are known on the same part of the boundary whose complement is a part of a hyperplane we also establish the unique determination result in this local data case

transport upscaling in heterogeneous aquifers what physical parameters control memory functions 1 power law tailing is often observed in the breakthrough curves btcs of tracer tests tailing is attributed to heterogeneity of aquifer properties and cannot be properly modeled by means of the homogeneous advection dispersion equation mass transfer models e g continuous time random walk method multirate mass transfer or fractional order advection dispersion equations using memory have been widely applied for reproducing observed tails the relationship between memory parameters obtained from btc fitting and the parameters characterizing the heterogeneity of hydraulic properties is still unclear here we investigate the conditions under which heterogeneity produces the type of tailing observed in the field and how memory functions are influenced by measurable heterogeneity parameters e g variance variogram or integral scale of the underlying transmissivity field we find that the slope of a btc in a log log plot is mainly influenced by the connectivity of the underlying permeability field but is insensitive to its variance the slope btc reaches asymptotically 2 as connectivity increases we conclude that an appropriate choice of the memory function allows reproducing the spreading caused by hydraulic heterogeneity but not necessarily the rate of mixing

reconstructing discontinuities using complex geometrical optics solutions in this paper we provide a framework for constructing general complex geometrical optics solutions for several systems of two variables that can be reduced to a system with the laplacian as the leading order term we apply these special solutions to the problem of reconstructing inclusions inside a domain filled with known conductivity from local boundary measurements computational results demonstrate the versatility of these solutions to determine electrical inclusions

pore scale modeling and continuous time random walk analysis of dispersion in porous media 1 we provide a physically based explanation for the complex macroscopic behavior of dispersion in porous media as a function of peclet number pe using a pore scale network model that accurately predicts the experimental dependence of the longitudinal dispersion coefficient dl on pe the asymptotic dispersion coefficient is only reached after the solute has traveled through a large number of pores at high pe this implies that preasymptotic dispersion is the norm even in experiments in statistically homogeneous media interpreting transport as a continuous time random walk we show that 1 the power law dispersion regime is controlled by the variation in average velocity between throats the distribution of local pe giving dl pe with 3 1 2 where is an exponent characterizing the distribution of transit times between pores 2 the crossover to a linear regime dl pe for pe pecrit 400 is due to a transition from a diffusion controlled late time cutoff to transport governed by advective movement and 3 the transverse dispersion coefficient dt pe for all pe 1

validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils sorption processes that occur during reactive solute movement through porous media can be modeled using either an equilibrium or kinetic approach because of the resulting conceptual and mathematical simplification many transport models assume local chemical equilibrium is valid for describing sorption reactions this paper presents quantitative criteria to assess the validity of the local equilibrium assumption for one dimensional steady flow through homogeneous soils a method is described whereby formulas for solute breakthrough curve time moments can be determined without knowledge of the nalytical solution to the mass transport model this method is applied to several commonly used nonequilibrium formulations as well as the standard linear equilibrium model the formulations considered include both the physical nonequilibrium models where the sorption rate is controlled by diffusive solute transfer between mobile and stagnant fluid zones and the chemical nonequilibrium models where the overall sorption rate is governed by the rate of reaction at the soil solution interfaces criteria for local equilibrium to be valid are derived by comparing the time moment formulas for the nonequilibrium and equilibrium models these criteria explicitly show that basic system parameters e g seepage velocity dispersion coefficient distribution coefficient sorption rate boundary conditions have a significant influence on the attainment of local equilibrium

a continuous time random walk approach to transient flow in heterogeneous porous media 1 we propose a new physical interpretation of the diffusion process for the piezometric head h x t in heterogeneous aquifers based on the continuous time random walk ctrw theory for the typical heterogeneities considered in this work we find that a ctrw based diffusion equation for h x t provides better fits to the transient flow simulations than the classical diffusion equation de the de is found to be a special case of the ctrw diffusion equation the results of this work have clear implications for the interpretation of pumping tests and what information can be extracted from them 2 well testing is an essential tool for groundwater aquifer assessment the key idea being that hydraulic properties can be derived by means of transient drawdown and or flux measurements the commonly accepted mathematical model for groundwater flow in porous formations is based on the well known darcy law which relates the fluid flow to the gradient of the piezometric head h x t l together with a mass balance and the assumptions of small compressibility constant porosity and permeability and small head gradients it is possible to derive a partial differential equation of parabolic type diffusion equation for h th x t r a x rh x t

hyperbolic geometry and local dirichlet neumann map abstract we show in three dimensions that if we know the dirichlet to neumann dn map associated to the schrodinger equation with a potential around the intersection of a sphere centered outside the convex hull of a bounded domain with smooth boundary we can determine uniquely the corresponding spherical mean of the potential with an appropriate measure this allows one to determine the potential near the boundary where the dn map is measured similar results are valid for electrical impedance tomography

inverse problem for schrodinger equations with yang mills potentials in a slab abstract in this work we consider the inverse boundary value problem for schrodinger equations with yang mills potentials in the domain of infinite slab type we prove that the potentials can be determined uniquely up to a gauge equivalent class assuming that only partial measurements are known on the boundary hyperplanes

transport behavior of a passive solute in continuous time random walks and multirate mass transfer 1 we consider the transport behavior of a passive solute in a heterogeneous medium modeled by continuous time random walks ctrw and linear multirate mass transfer mrmt within the ctrw framework we formulate a transport model which is formally equivalent to mrmt in both approaches the total concentration is divided into mobile and immobile parts the immobile concentration is given by the convolution in time of the mobile concentration and a memory function the memory function is a functional of the distribution of transition and trapping times for the ctrw and the mrmt approach respectively and determines the transport behavior of the solute based on different expressions for the memory function in the two frameworks we derive conditions for which both approaches describe the same transport behavior we focus on anomalous transport behavior that can arise if the transition and trapping time distributions behave algebraically in a given time regime using an expansion of the laplace transform of the memory function we develop explicit expressions for the time behavior of the flux concentration as well as for the center of mass velocity and the macro dispersion coefficients of the solute distribution we observe anomalous power law as well as normal fickian transport behavior depending on the exponents that dominate the trapping and transition time distributions respectively the results show that the character of the anomalous transport does not depend on the details of the transport model but only on the exponents dominating the transition or trapping time distributions the unified transport framework presented here comprising ctrw and mrmt shows new aspects and opens new perspectives for the modeling of transport in heterogeneous media

stability estimates for the calderon problem with partial data this is a follow up of a previous article where we proved local stability estimates for a potential in a schr odinger equation on an open bounded set in dimension n 3 from the dirichlet to neumann map with partial data the region under control was the penumbra delimited by a source of light outside of the convex hull of the open set these local estimates provided stability of log log type corresponding to the uniqueness results in calder on s inverse problem with partial data proved by kenig sj ostrand and uhlmann in this article we prove the corresponding global estimates in all dimensions higher than three the estimates are based on the construction of solutions of the schr odinger equation by complex geometrical optics developed in the anisotropic setting by dos santos ferreira kenig salo and uhlmann to solve the calder on problem in certain admissible geometries

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

stability in the determination of a time dependent coefficient for wave equations from partial data let be a c2 bounded domain of rn n 2 and fix q 0 t with t 0 we consider the stability in the inverse problem of determining a time dependent coefficient of order zero q appearing in a dirichlet initial boundary value problem for a wave equation t2u xu q t x u 0 in q from partial observations on q the observation is given by a boundary operator associated to the wave equation using suitable geometric optics solutions and carleman estimates we prove a stability estimate in the determination of q from the boundary operator

inverse problems for the anisotropic maxwell equations we prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time harmonic maxwell equations in certain anisotropic settings we give a uniqueness result in the inverse problem for maxwell equations on an admissible riemannian manifold and a uniqueness result for maxwell equations in euclidean space with admissible matrix coefficients the proofs are based on a new fourier analytic construction of complex geometrical optics solutions on admissible manifolds and involve a proper notion of uniqueness for such solutions

on the asymptotic behaviour of solutions of a stochastic energy balance climate model abstract we prove the existence of a random global attractor for the multivalued random dynamical system associated to a nonlinear multivalued parabolic equation with a stochastic term of amplitude of the order of e the equation was initially suggested by north and cahalan following a previous deterministic model proposed by m i budyko for the modeling of some non deterministic variability as for instance the cyclones which can be treated as a fast varying component and are represented as a white noise process in the context of energy balance climate models we also prove the convergence in some sense of the global attractors when e 0 i e the convergence to the global attractor for the associated deterministic case e 0

calderon inverse problem with partial data on riemann surfaces on a fixed smooth compact riemann surface with boundary m0 g we show that for the schrodinger operator v with potential v c m0 for some 0 the dirichlet to neumann map n measured on an open set m0 determines uniquely the potential v we also discuss briefly the corresponding consequences for potential scattering at 0 frequency on riemann surfaces with asymptotically euclidean or asymptotically hyperbolic ends

probing for electrical inclusions with complex spherical waves let a physical body in 2 or 3 be given assume that the electric conductivity distribution inside consists of conductive inclusions in a known smooth background further assume that a subset is available for boundary measurements it is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on more precisely given a ball b with center outside the convex hull of and satisfying b boundary measurements on with explicitly given dirichlet data are enough to determine whether b intersects the inclusion an approximate detection algorithm is introduced based on the theory numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusion free domain near and is robust against measurement noise 2007 wiley periodicals inc

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

the calderon problem with partial data for conductivities with 3 2 derivatives we extend a global uniqueness result for the calderon problem with partial data due to kenig sjostrand uhlmann ann math 2 165 567 591 2007 to the case of less regular conductivities specifically we show that in dimensions n ge 3 the knowledge of the diricihlet to neumann map measured on possibly very small subsets of the boundary determines uniquely a conductivity having essentially 3 2 derivatives in an l2 sense

spatial markov processes for modeling lagrangian particle dynamics in heterogeneous porous media we investigate the representation of lagrangian velocities in heterogeneous porous media as markov processes we use numerical simulations to show that classical descriptions of particle velocities using markov processes in time fail because low velocities are much more strongly correlated in time than high velocities we demonstrate that lagrangian velocities describe a markov process at fixed distances along the particle trajectories i e a spatial markov process this remarkable property has significant implications for modeling effective transport in heterogeneous velocity fields i the spatial lagrangian velocity transition densities are sufficient to fully characterize these complex velocity field organizations ii classical effective transport descriptions that rely on markov processes in time for the particle velocities are not suited for describing transport in heterogeneous porous media and iii an alternative effective transport description derives from the markovian nature of the spatial velocity transitions it expresses particle movements as a random walk in space time characterized by a correlated random temporal increment and thus generalizes the continuous time random walk model to transport in correlated velocity fields

the calderon problem with partial data in two dimensions we prove for a two dimensional bounded domain that the cauchy data for the schrodinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential this implies for the conductivity equation that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset we can determine uniquely the conductivity we use carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results

lagrangian statistical model for transport in highly heterogeneous velocity fields we define an effective lagrangian statistical model in phase space x t v for describing transport in highly heterogeneous velocity fields with complex spatial organizations the spatial markovian nature and temporal non markovian nature of lagrangian velocities leads to an effective transport description that turns out to be a correlated continuous time random walk this model correctly captures the lagrangian velocity correlation properties and is demonstrated to represent a forward model for predicting transport in highly heterogeneous porous media for different types of velocity organizations

theory of anomalous chemical transport in random fracture networks we show that dominant aspects of chemical particle transport in fracture networks char22 non gaussian propagation char22 result from subtle features of the steady flow field distribution through the network this is an outcome of a theory based on a continuous time random walk formalism structured to retain the key space time correlations of particles as they are advected across each fracture segment the approach is designed to treat the complex geometries of a large variety of fracture networks and multiscale interactions monte carlo simulations of steady flow in these networks are used to determine the distribution of velocities in individual fractures as a function of their orientation the geometry and velocity distributions are used in conjunction with particle mixing rules to map the particle movement between fracture intersections onto a joint probability density ensuremath psi mathbf r t the chemical concentration plume and breakthrough curves can then be calculated analytically particle tracking simulations on these networks exhibit the same non gaussian profiles demonstrating quantitative agreement with the theory the analytic plume shapes display the same basic behavior as extensive field observations at the columbus air force base mississippi the quantitative correlation between the time dependence of the mean and standard deviation of the field plumes and their shape is predicted by the theory

inverse problem for the riemannian wave equation with dirichlet data and neumann data on disjoint sets we consider the inverse problem to determine a smooth compact riemannian manifold with boundary m g from a restriction s r of the dirichlet to neumann operator for the wave equation on the manifold here s and r are open sets in m and the restriction s r corresponds to the case where the dirichlet data is supported on r s and the neumann data is measured on r r in the novel case where s r we show that s r determines the manifold m g uniquely assuming that the wave equation is exactly controllable from the set of sources s moreover we show that the exact controllability can be replaced by the hassell tao condition for eigenvalues and eigenfunctions that is j c j 2l2 s j 1 2 where j are the dirichlet eigenvalues and j j 1 is an orthonormal basis of the corresponding eigenfunctions

stability estimates for the radon transform with restricted data and applications in this article we prove a stability estimate going from the radon transform of a function with limited angle distance data to the l p norm of the function itself under some conditions on the support of the function we apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data

integral geometry of tensor fields on a class of non simple riemannian manifolds we study the geodesic x ray transform ig of tensor fields on a compact riemannian manifold m with non necessarily convex boundary and with possible conjugate points we assume that ig is known for geodesics belonging to an open set g with endpoints on the boundary we prove generic s injectivity and a stability estimate under some topological assumptions and under the condition that for any x t m there is a geodesic in g through x normal to without conjugate points

limiting carleman weights and anisotropic inverse problems in this article we consider the anisotropic calderon problem and related inverse problems the approach is based on limiting carleman weights introduced in kenig et al ann math 165 567 591 2007 in the euclidean case we characterize those riemannian manifolds which admit limiting carleman weights and give a complex geometrical optics construction for a class of such manifolds this is used to prove uniqueness results for anisotropic inverse problems via the attenuated geodesic ray transform earlier results in dimension n 3 were restricted to real analytic metrics

a fidelity embedded regularization method for robust electrical impedance tomography electrical impedance tomography eit provides functional images of an electrical conductivity distribution inside the human body since the 1980s many potential clinical applications have arisen using inexpensive portable eit devices eit acquires multiple trans impedance measurements across the body from an array of surface electrodes around a chosen imaging slice the conductivity image reconstruction from the measured data is a fundamentally ill posed inverse problem notoriously vulnerable to measurement noise and artifacts most available methods invert the ill conditioned sensitivity or the jacobian matrix using a regularized least squares data fitting technique their performances rely on the regularization parameter which controls the trade off between fidelity and robustness for clinical applications of eit it would be desirable to develop a method achieving consistent performance over various uncertain data regardless of the choice of the regularization parameter based on the analysis of the structure of the jacobian matrix we propose a fidelity embedded regularization fer method and a motion artifact reduction filter incorporating the jacobian matrix in the regularization process the new fer method with the motion artifact reduction filter offers stable reconstructions of high fidelity images from noisy data by taking a very large regularization parameter value the proposed method showed practical merits in experimental studies of chest eit imaging

reconstruction of a penetrable obstacle by complex spherical waves we consider an inverse acoustic scattering problem for identifying a non convex penetrable obstacle in three dimensions in a homogeneous medium we apply the complex geometrical optics solutions with logarithmic phase which is called complex spherical waves to reconstruction problem the reconstruction schemes will be demonstrated in the last section

local and global carleman estimates for parabolic operators with coefficients with jumps at interfaces in 0 t open subset of n n 2 we consider a parabolic operator p t x t x x where the scalar coefficient t x is piecewise smooth in space yet discontinuous across a smooth interface s we prove a global in time local in space carleman estimate for p in the neighborhood of any point of the interface the observation region can be chosen independently of the sign of the jump of the coefficient at the considered point the derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface in the high frequency regime we use calderon projectors in the low frequency regime we follow a more classical approach because of the parabolic nature of the problem we need to introduce weyl hormander anisotropic metrics symbol classes and pseudo differential operators each frequency regime and the associated technique require a different calculus a global in time and space carleman estimate on 0 t m m a manifold is also derived from the local result

a posteriori error estimate and convergence analysis for conductivity image reconstruction in mreit magnetic resonance electrical impedance tomography mreit takes advantage of internal information to solve its nonlinear inverse problem of recovering a conductivity distribution inside an imaging object when we inject current into the imaging object there occurs a distribution of internal magnetic flux density mathbf b b x b y b z in mreit we utilize a magnetic resonance imaging scanner with its main magnetic field in the z direction to acquire b z data the harmonic b z algorithm was invented in 2001 to reconstruct cross sectional conductivity images from b z data sets subject to multiple injection currents utilizing internal b z data it overcomes the inherent ill posedness in electrical impedance tomography we can set up the inverse problem in mreit as a coefficient identification problem of finding sigma appearing in nabla cdot sigma nabla u 0 from acquired data of the z component of nabla times sigma nabla u the harmonic b z algorithm has shown an excellent perfor

the neumann to dirichlet map in two dimensions abstract for the two dimensional schrodinger equation in a bounded domain we prove uniqueness of the determination of potentials in w p 1 p 2 in the case where we apply all possible neumann data supported on an arbitrarily non empty open set of the boundary and observe the corresponding dirichlet data on an immediate consequence is that one can uniquely determine a conductivity in w p 3 with p 2 by measuring the voltage on an open subset of the boundary corresponding to a current supported in the same set

coupled estimation of flow and solute transport parameters a methodology for solving the coupled linear flow and transport inverse problem is presented it allows the estimation of aquifer parameters transmissivity storativity recharge boundary heads and flows leakage dispersivity molecular diffusion porosity retardation linear decay boundary concentrations using their prior estimates as well as head and concentration measurements as basic information transient flow and transport equations are solved using the finite element method in space and a weighted finite difference scheme in time flow and transport domains can be one two or quasi three dimensional applying linear one dimensional elements linear two dimensional triangles or bilinear two dimensional rectangles mixed at will time regimes can be transient or steady state in any equation flow or transport parameters are estimated on the basis of maximum likelihood theory which allows us to obtain information about parameters uncertainty minimization of the negative log likelihood is performed using marquardt s method the applicability of the method is shown by means of two real examples they illustrate that the proposed approach can successfully address model selection parameter uncertainty and nontrivial identifiability problems estimation of time dependent sources and the role of prior estimates are also addressed in the examples while the formulation of the problem leaves room for both stochastic and deterministic treatment of hydraulic parameters the latter has been adopted in both examples limitations of the approach are also apparent

inverse boundary value problems for the perturbed polyharmonic operator author s krupchyk k lassas m uhlmann g abstract we show that a first order perturbation a x d q x of the polyharmonic operator m m 2 can be determined uniquely from the set of the cauchy data for the perturbed polyharmonic operator on a bounded domain in n n 3 notice that the corresponding result does not hold in general when m 1 2013 american mathematical society

carleman estimates for stratified media abstract we consider anisotropic elliptic and parabolic operators in a bounded stratified media in r n characterized by discontinuities of the coefficients in one direction the surfaces of discontinuities cross the boundary of the domain we prove carleman estimates for these operators with an arbitrary observation region

uniqueness in an inverse boundary problem for a magnetic schrodinger operator with a bounded magnetic potential we show that the knowledge of the set of the cauchy data on the boundary of a bounded open set in mathbb r n n geq 3 for the magnetic schrodinger operator with l magnetic and electric potentials determines the magnetic field and electric potential inside the set uniquely the proof is based on a carleman estimate for the magnetic schrodinger operator with a gain of two derivatives

determining a first order perturbation of the biharmonic operator by partial boundary measurements abstract we consider an operator 2 a x d q x with the navier boundary conditions on a bounded domain in r n n 3 we show that a first order perturbation a x d q can be determined uniquely by measuring the dirichlet to neumann map on possibly very small subsets of the boundary of the domain notice that the corresponding result does not hold in general for a first order perturbation of the laplacian

on the regularization effect of space time white noise on quasi linear parabolic partial differential equations we show that an existence and uniqueness and a comparison theorem hold if we add a space time white noise to a quasi linear parabolic equation in one space dimension even if the nonlinearity is only measurable and not even locally bounded

determining a magnetic schrodinger operator from partial cauchy data in this paper we show in dimension n 3 that knowledge of the cauchy data for the schrodinger equation in the presence of a magnetic potential measured on possibly very small subsets of the boundary determines uniquely the magnetic field and the electric potential we follow the general strategy of 7 using a richer set of solutions to the dirichlet problem that has been used in previous works on this problem

an initial boundary value problem for the maxwell equations abstract in this paper by using methods from complex analysis and quaternionic analysis we investigate an initial boundary value problem for the maxwell equations and obtain the general solutions and solvable conditions of the problem respectively in different cases in addition by using a similar method we also discuss an initial boundary value problem for a hyperbolic complex system of first order equations in r 3

s shaped bifurcation branch in a quasilinear multivalued model arising in climatology abstract in this paper we show the existence of a continuous and unbounded connected s shaped set q u where q is the solar constant and u satisfies a quasilinear eventually multivalued stationary equation on a riemannian manifold without boundary arising as a stationary energy balance model for the earth surface temperature