a non classical solution to a hessian equation from cartan isoparametric cubic we show how to construct a non smooth solution to a hessian fully nonlinear second order uniformly elliptic equation using the cartan isoparametric cubic in 5 dimensions

homotopy analysis method for the sine gordon equation with initial conditions in this work approximate analytical solution of the sine gordon equation with initial conditions is obtained by the homotopy analysis method ham the ham solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions it is shown that the solutions obtained by the variational iteration method vim are only special cases of the ham solutions

some solutions of the linear and nonlinear klein gordon equations using the optimal homotopy asymptotic method abstract we investigate the effectiveness of the optimal homotopy asymptotic method oham in solving time dependent partial differential equations to this effect we consider the homogeneous non homogeneous linear and nonlinear klein gordon equations with boundary conditions the results reveal that the method is explicit effective and easy to use

control of the stochastic burgers model of turbulence we consider a control problem for a stochastic burgers equation this problem is motivated by a model from the control of turbulence see choi et al j fluid mech 253 1993 pp 509 543 we study a sequence of approximated hamilton jacobi equations by using dynamic programming

a singular critical point for some variational free boundary problem in five dimensions abstract using the cartan isoparametric cubic we construct a singular critical point of a variational free boundary problem in 5 dimensions which is homogeneous of degree 1 the intersection of the free boundary with s 4 consists of certain level sets of the cartan cubic and thus admits a three dimensional automorphism group

series solutions of nano boundary layer flows by means of the homotopy analysis method we present here a similar solution for the nano boundary layer with nonlinear navier boundary condition three types of flows are considered i the flow past a wedge ii the flow in a convergent channel iii the flow driven by an exponentially varying outer flows the resulting differential equations are solved by the homotopy analysis method different from the perturbation methods the present method is independent of small physical parameters so that it is applicable for not only weak but also strong nonlinear flow phenomena numerical results are compared with the available exact results to demonstrate the validity of the present solution the effects of the slip length the index parameters n and m on the velocity profile and the tangential stress are investigated and discussed

a new analytic algorithm of lane emden type equations an reliable ease to use analytic algorithm is provided for lane emden type equation which models many phenomena in mathematical physics and astrophysics this algorithm logically contains the well known adomian decomposition method different from all other analytic techniques this algorithm itself provides us with a convenient way to adjust convergence regions even without pade technique some applications are given to show its validity

the essence of the homotopy analysis method the generalized taylor expansion including a secret auxiliary parameter h which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by liao the secret of h cannot be understood in the frame of the homotopy analysis method this is a serious shortcoming of liao s method we solve the problem through a detailed study of a simple example we show that the generalized taylor expansion is just the usual taylor s expansion at different point t 1 we prove that there is a relationship between h and t 1 which reveals the meaning of h and the essence of the homotopy analysis method as an important example we study the series solution of the blasius equation using the series expansion method at different points we obtain the same result with liao s solution given by the homotopy analysis method

optimal control of a stochastic delay heat equation with boundary noise and boundary control in this paper a controlled stochastic delay heat equation with neumann boundary noise and boundary control is considered the existence and uniqueness of the mild solution for the associated hamilton jacobi bellman equations are obtained by means of the backward stochastic differential equations which is applied to the optimal control problem

control of error in the homotopy analysis of nonlinear klein gordon initial value problems in the present paper we discuss the application of homotopy analysis to general nonlinear klein gordon type equations we first outline the method for general forms of the nonlinearity as well as for general functional forms of the initial conditions in particular we discuss a method of controlling the residual error in approximate solutions which may be found via homotopy analysis through adequate selection of the convergence control parameter with the general problem outlined we apply the method to various equations including the quasilinear cubic klein gordon equation the modified liouville equation the sinh gordon equation and the tanh gordon equation for each of these equations and related initial data we obtain residual error minimizing solutions which demonstrate the qualitative behavior of the true solutions in each case

some results on non linear optimal control problems and hamilton jacobi equations in infinite dimensions abstract the optimal control of a distributed parameter system is connected to the solution of the corresponding hamilton jacobi equation this is a first order equation in infinite dimensions with discontinuous coefficients we study the hamilton jacobi equation of a system governed either by a semilinear or by a monotone dynamics replacing the unbounded terms of this equation by their yosida approximations we prove convergence of the approximate solutions to the value function of the original problem by using the uniqueness result for viscosity solutions examples and applications are included

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

on viscosity solutions of path dependent pdes in this paper we propose a notion of viscosity solutions for path dependent semi linear parabolic pdes this can also be viewed as viscosity solutions of non markovian backward sdes and thus extends the well known nonlinear feynman kac formula to non markovian case we shall prove the existence uniqueness stability and comparison principle for the viscosity solutions the key ingredient of our approach is a functional ito calculus recently introduced by dupire functional ito calculus 2009 preprint

sinh gordon equation painleve property and backlund transformation abstract the painleve property for partial differential equation proposed by weiss et al is studied for the sinh gordon equation we performed a singular point analysis to show that the painleve property is a valuable analytical test for integrability and we found in a remarkably straightforward manner the backlund transformation of the sinh gordon equation

partial regularity of viscosity solutions for a class of kolmogorov equations arising from mathematical finance we study value functions which are viscosity solutions of certain kolmogorov equations using pde techniques we prove that they are c1 regular on special finite dimensional subspaces the problem has origins in hedging derivatives of risky assets in mathematical finance

dynamic programming for optimal control of stochastic mckean vlasov dynamics we study the optimal control of general stochastic mckean vlasov equation such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles players in mean field interaction under common noise our first main result is to state a dynamic programming principle for the value function in the wasserstein space of probability measures which is proved from a flow property of the conditional law of the controlled state process next by relying on the notion of differentiability with respect to probability measures due to p l lions 32 and ito s formula along a flow of conditional measures we derive the dynamic programming hamilton jacobi bellman equation and prove the viscosity property together with a uniqueness result for the value function finally we solve explicitly the linear quadratic stochastic mckean vlasov control problem and give an application to an interbank systemic risk model with common noise

an improved optimal homotopy analysis algorithm for nonlinear differential equations abstract this paper presents an improved optimal homotopy analysis algorithm to deal with nonlinear first order differential equations we discuss the optimal selection of the auxiliary linear operator in the application of the homotopy analysis method that will accelerate the convergence of series solutions then a powerful algorithm that employed the optimal auxiliary linear operator of the homotopy analysis method is presented to facilitate the calculations numerical illustrations are examined to show the efficiency and the pertinent features of the proposed algorithm this algorithm could lead to a promising approach for many nonlinear problems in applied sciences

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

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

partially observed control of markov processes iii let v denote the value function of a partially observed control problem if v is one differentiable in a certain direction b then optimal controls are characterized by a feedback involving the directional derivative b v it is also shown that v satisfies the corresponding bellman equation an infinite dimensional pde on the space of measures in the viscosity sense of crandall and lions

partially observed control of markov processes iv abstract a class of degenerate elliptic nonlinear second order equations is considered these equations arising from a related class of probabilistic variational problems are defined on the space of probability measures on rd the existence of c1 1 viscosity solutions is established as well as the existence uniqueness and feedback characterization of the minima of the variational problems

optimal control of semilinear stochastic evolution equations the aim of this paper is to initiate a semigroup theory based approach to characterization of optimal markov controls for controlled semilinear stochastic evolution equations it may be recalled that markov controls are those that depend only on the current state at each time for finite dimensional controlled stochastic differential equations with a nondegenerate diffusion matrix this task is traditionally achieved through the hamilton jacobi bellman equation of dynamic programming associated with the problem and an accompanying verification theorem the latter states that an optimal markov control can be explicitly obtained by the pointwise minimization of a hamiltonian derivable from the solution of the hjb equation moreover any optimal markov control is obtainable in this manner

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

a family of nonlinear klein gordon equations and their solutions various forms of the nonlinear klein gordon equation are seen to have exact solitonlike solutions when separation of variables is postulated the family for which these exact solutions are found includes the sine gordon equation as a special case an interesting conclusion obtained is that both the soliton antisoliton and breather solutions hitherto known for the sine gordon equations result from a broader class of klein gordon equations the method can be extended to other equations

mild solutions of semilinear elliptic equations in hilbert spaces abstract this paper extends the theory of regular solutions c 1 in a suitable sense for a class of semilinear elliptic equations in hilbert spaces the notion of regularity is based on the concept of g derivative which is introduced and discussed a result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property that is it maps continuous functions into g differentiable ones the validity of this smoothing assumption is fully discussed for the case of the ornstein uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature it is shown that the results apply to hamilton jacobi bellman hjb equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that in particular they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

separation of variables in the nonlinear wave equation we develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations using it we obtain a number of new two dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions

hamilton jacobi equations in infinite dimensions iii on etudie des equations de la forme f x u du 0 dans ou est un sous ensemble ouvert d un espace de banach v la fonction inconnue u r est continue du x est la derivee de frechet de u en x et f est une fonction non lineaire qui est une application continue de r v dans r on introduit la notion de solution viscosite stricte afin d eliminer l hypothese que v a la propriete de radon nikodym

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

regularity results for the minimum time function of a class of semilinear evolution equations of parabolic type under suitable controllability and smoothness assumptions the minimum time function t x of a semilinear control system is proved to be locally lipschitz continuous and semiconcave on the controllable set these properties are then applied to derive optimality conditions relating optimal trajectories to the superdifferential of t

generalized solutions of hjb equations applied to stochastic control on hilbert space abstract in this paper we prove the existence of borel measurable optimal control laws for infinite dimensional nonstationary stochastic control problems on hilbert space this is based on some recent results on existence of generalized solutions of hjb equations on hilbert space l2 h where is a suitable reference measure on h for illustration some examples are presented

optimal control problems for stochastic reaction diffusion systems with non lipschitz coefficients by using the dynamic programming approach we study a control problem for a class of stochastic reaction diffusion systems with coefficients having polynomial growth in the cost functional a non lipschitz term appears and this allows us to treat the quadratic case which is of interest in the applications the corresponding hamilton jacobi bellman equation is first resolved by a fixed point argument in a small time interval and then is extended to arbitrary time intervals by suitable a priori estimates the main ingredient in the proof is the smoothing effect of the transition semigroup associated with the uncontrolled system

viscosity solutions of hamilton jacobi equations in infinite dimensions iv hamiltonians with unbounded linear terms in this work we show how to interpret s and e in the viscosity sense when the solutions may be expected to be weakly sequentially continuous in particular we introduce notions which allow us to handle the terms and for which we can obtain existence and uniqueness results in a variety of circumstances

regularity for obstacle problems in infinite dimensional hilbert spaces in this paper we study fully nonlinear obstacle type problems in hilbert spaces we introduce the notion of q elliptic equation and prove existence uniqueness and regularity of viscosity solutions of q elliptic obstacle problems in particular we show that solutions of concave problems with semiconvex obstacles are in the space

infinite dimensional hamilton jacobi equations with large zeroth order coefficient abstract we establish the existence and uniqueness of the c 1 1 solution of an infinite dimensional hamilton jacobi equation defined on the space of probability measures on r d when the zeroth order coefficient is sufficiently large

regular solutions of second order stationary hamilton jacobi equations abstract we study a second order stationary hamilton jacobi equation in infinite dimension this equation is nonlinear and convex with respect to the first order term we use properties of the transition semigroup associated to the linear equation to write the hamilton jacobi equation in integral form and we prove existence uniqueness and regularity of a solution by the theory of maximal monotone operators we also prove that this solution is the pointwise limit of a uniformly bounded sequence of classical solutions of approximating problems finally the solution is the value function of the associated optimal stochastic control problem some examples are given

analytical solutions to a generalized drinfel d sokolov equation related to dssh and kdv6 abstract analytical solutions to the generalized drinfel d sokolov gds equations u t 1 uu x 1 u xxx v x 0 and v t 2 uv x 2 v xxx 0 are obtained for various values of the model parameters in particular we provide perturbation solutions to illustrate the strong influence of the parameters 1 and 2 on the behavior of the solutions we then consider a miura type transform which reduces the gds equations into a sixth order nonlinear differential equation under the assumption that 1 under such a transform the gds reduces to the sixth order drinfel d sokolov satsuma hirota dssh equation also known as kdv6 in the very special case 1 2 the method of homotopy analysis is applied in order to obtain analytical solutions to the resulting equation for arbitrary 1 and 2 an error analysis of the obtained approximate analytical solutions is provided

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

optimal control problems for stochastic delay evolution equations in banach spaces we consider the optimal control for a banach space valued stochastic delay evolution equation the existence and uniqueness of the mild solution for the associated hamilton jacobi bellman equations are obtained by means of backward stochastic differential equations an application to optimal control of stochastic delay partial differential equations is also given

hamilton jacobi equations in infinite dimensions ii existence of viscosity solutions on demontre des resultats d existence generaux pour 2 problemes 1 le probleme de cauchy u t h x t u du 0 dans v 0 t u x 0 x dans v et 2 le probleme stationnaire u h x u du 0 dans v ou h et sont donnees

solving the klein gordon equation by means of the homotopy analysis method an analytic technique namely the homotopy analysis method is applied to solve the nonlinear travelling waves governed by the klein gordon equation the phase speed and the solution which are dependent on the amplitude a are given and valid in the whole region 0

viscosity solutions of hamilton jacobi equations in infinite dimensions v unbounded linear terms and b continuous solutions we continue the systematic study of first order hamilton jacobi equations in infinite dimensions begun in parts i iv of this series 121 our approach relies on as before an appropriate interpretation of the notion of viscosity solutions as introduced in 11 lo in the interim the strong development of viscosity solution theory in many directions has continued unabated in addition to the many references given in earlier papers in this series which we will not repeat here see 12 we would like to point out that a complete theory of second order fully nonlinear and possibly degenerate elliptic equations in finite dimensions has been made possible by the use of viscosity solutions s 9 18 20 22 19 32 331 for example and there is now a substantial body of work concerning second order equations in infinite dimensions 25 271 however in this work we come back to the particular situation studied in 12 part iv namely we come back to hamilton jacobi equations in

hamilton jacobi bellman equations for the optimal control of the duncan mortensen zakai equation abstract we study a class of hamilton jacobi bellman hjb equations associated to stochastic optimal control of the duncan mortensen zakai equation the equations are investigated in weighted l 2 spaces we introduce an appropriate notion of weak viscosity solution of such equations and prove that the value function is the unique solution of the hjb equation we apply the results to stochastic optimal control problems with partial observation and correlated noise

explicit solutions for continuous time mean variance portfolio selection with nonlinear wealth equations abstract this paper concerns the continuous time mean variance portfolio selection problem with a special nonlinear wealth equation this nonlinear wealth equation has a nonsmooth coefficient and the dual method developed in ji 2010 does not work we invoke the hjb equation of this problem and give an explicit viscosity solution of the hjb equation furthermore via this explicit viscosity solution we obtain explicitly the efficient portfolio strategy and efficient frontier for this problem finally we show that our nonlinear wealth equation can cover three important cases

the existence and uniqueness of the solution for nonlinear kolmogorov equations abstract by means of backward stochastic differential equations the existence and uniqueness of the mild solution are obtained for the nonlinear kolmogorov equations associated with stochastic delay evolution equations applications to optimal control are also given

stationary hamilton jacobi equations in hilbert spaces and applications to a stochastic optimal control problem we study an infinite horizon stochastic control problem associated with a class of stochastic reaction diffusion systems with coefficients having polynomial growth the hamiltonian is assumed to be only locally lipschitz continuous so that the quadratic case can be covered we prove that the value function v corresponding to the control problem is given by the solution of the stationary hamilton jacobi equation associated with the state system to this purpose we write the hamilton jacobi equation in integral form and by using the smoothing properties of the transition semigroup relative to the state system and the theory of m dissipative operators we show that it admits a unique solution moreover the value function v is obtained as the limit of minima for some approximating control problems which admit unique optimal controls and states

on existence and uniqueness of solutions of hamilton jacobi equations abstract the theory of scalar first order nonlinear partial differential equations has been enjoying a rapid development in the last few years this development occurred because the authors established uniqueness criteria for generalized solutions called viscosity solutions which correctly identify the solutions sought in areas of application including control theory differential games and the calculus of variations the concept of viscosity solutions is relatively easy to work with and many formally heuristic or difficult proofs have been made rigorous or simple using this concept a feedback process has begun and the experience recently gained in working with viscosity solution has suggested new existence and uniqueness results the current paper continues this interaction by establishing new existence ane uniqueness results in a natural generality suggested by earlier proofs it is also felt that the presentation of the comparison results which imply uniqueness continuous dependence and are used to estimate moduli of continuity has something to offer over earlier presentations in special cases author

optimal control problem for stochastic evolution equations in hilbert spaces in this article we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in hilbert spaces and the cost functional has a quadratic growth the existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value as an application an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

jacobi elliptic function solutions of nonlinear wave equations via the new sinh gordon equation expansion method in this paper based on the well known sinh gordon equation a new sinh gordon equation expansion method is developed this method transforms the problem of solving nonlinear partial differential equations into the problem of solving the corresponding systems of algebraic equations with the aid of symbolic computation the procedure can be carried out by computer many nonlinear wave equations in mathematical physics are chosen to illustrate the method such as the kdv mkdv equation 2 1 dimensional coupled davey stewartson equation the new integrable davey stewartson type equation the modified boussinesq equation 2 1 dimensional mkp equation and 2 1 dimensional generalized kdv equation as a consequence many new doubly periodic jacobian elliptic function solutions are obtained when the modulus m 1 or 0 the corresponding solitary wave solutions and singly periodic solutions are also found this approach can also be applied to solve other nonlinear differential equations

comparison of viscosity solutions of semi linear path dependent pdes this paper provides a probabilistic proof of the comparison result for viscosity solutions of path dependent semilinear pdes we consider the notion of viscosity solutions introduced in 8 which considers as test functions all those smooth processes which are tangent in mean when restricted to the markovian case this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in 1 2 our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison this is most remarkable in the context of the linear path dependent heat equation as a key ingredient for our methodology we introduce a notion of punctual differentiation similar to the corresponding concept in the standard viscosity solutions 3 and we prove that semimartingales are almost everywhere punctually differentiable this smoothness result can be viewed as the counterpart of the aleksandroff smoothness result for convex functions a similar comparison result was established earlier in 8 the result of this paper is more general and more importantly the arguments that we develop do not rely on any representation of the solution

smooth fit principle for impulse control of multidimensional diffusion processes value functions of impulse control problems are known to satisfy quasi variational inequalities qvis a bensoussan and j l lions impulse control and quasivariational inequalities heyden son philadelphia 1984 translation of controle impulsionnel et inequations quasi variationnelles gauthier villars paris 1982 this paper proves the smooth fit c 1 property of the value function for multidimensional controlled diffusions using a viscosity solution approach we show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions

implicit differential equation arising in the steady flow of a sisko fluid consideration is given to the thin film flow of a sisko fluid on a moving belt using the implicit function theorem an existence result for the solution of the resulting non linear differential equation is established also the homotopy analysis method is used to obtain approximate analytical solution of the problem for non integer values of the power index the numerical results thus obtained are presented graphically and the salient features of the solution are discussed for various values of the power index parameter the results obtained reveal many interesting behaviors that warrant further study of the equations related to non newtonian fluid phenomena especially the shear thinning phenomena shear thinning reduces the wall shear stress

nonlinear semigroups associated with optimal stopping of controlled diffusions under partial observation abstract we treat optimal stopping problems of controlled diffusions under partial observation two kinds of admissibility strict sense and wide sense are considered using the girsanov transformation the unnormalized conditional probability turns out the state process subject to the zakai equation and we can derive a nonlinear semigroup which presents the bellman principle in our problems in the diffusion case an optimal stopping time is given appealing to a snell envelope