on the rate of convergence of solutions in singular perturbation problems where the obstacle cp is a given smooth function satisfying that rp 0 on dq using a technique in the theory of variational inequalities j l lions 11 proved that 11 min o p i l cnj ce for some constant c we claim that we can never deduce the pointwise convergence of a from the above estimate on the other hand the notion of viscosity solotions was introduced by m g crandall and p l lions 2 for nonlinear first order pdes and then it was extended for nonlinear second order pdes by p l lions 12 141 recently it appeared that a technique in the theory of viscosity solutions could be applied to derive a pointwise convergence for some singular perturbation problems in fact by the same argument as in 7 we can show that u converges to min 0 cp uniformly on 0 243 0022 247x 91 3 00

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

wavefront propagation for reaction diffusion systems of pde m i freidlin has introduced probabilistic techniques to study propagation for systems of reaction diffusion pde the motivating idea is that should a reaction diffusion system possess only a single unstable and a single stable equilibrium then the solution u of the system will presumably tend to switch for large times from near the former to near the latter state this paper brings to bear purely pde techniques to this problem especially the theory of viscosity solutions on hamilton jacobi equations due to crandall lions

symmetry results for viscosity solutions of fully nonlinear equations in annular and exterior domains abstract in this paper we prove the symmetry and monotonicity results of viscosity solutions for fully nonlinear elliptic equations f d 2 u d u u x 0 and fully nonlinear parabolic equations u t f d 2 u d u u x t 0 in annular domains by using the method of moving planes we also prove the symmetry and monotonicity results of viscosity solutions for fully nonlinear parabolic equations u t f d 2 u d u u t 0 in exterior domains this extends the results of fully nonlinear elliptic equations in exterior domains

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

viscosity solutions for weakly coupled systems of first order partial differential equations abstract we consider viscosity solutions for weakly coupled systems of first order partial differential equations and apply these existence and uniqueness results to examples involving random evolutions

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

stochastic exact solutions of the wick type stochastic nls equation based upon the hermite transformation and the extended sub equation expansion method soliton like solutions of the stochastic schrodinger equation in the white noise environment are obtained via maple

on an investment consumption model with transaction costs this paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and n risky assets whose prices are log normal diffusions we suppose that transactions between the assets incur a cost proportional to the size of the transaction the problem is to maximize the total utility of consumption dynamic programming leads to a variational inequality for the value function existence and uniqueness of a viscosity solution are proved the variational inequality is solved by using a numerical algorithm based on policies iterations and multigrid methods numerical results are displayed for n 1 and n 2

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

on hopf s formulas for solutions of hamilton jacobi equations soit h r n r on considere l equation de hamilton jacobi u t h du 0 dans r n 1 r n 0 on etudie les proprietes des formules de hopf pour les solutions

a viscosity solution approach to the asymptotic analysis of queueing systems we consider a system of several interconnected queues with a single class of customers and model the state xt of the system as a jump markov process the problem of interest is to estimate the large deviations behavior of the rescaled system xt ext s corresponding to large time and large excursions of the original unscaled system the techniques employed are those of the theory of viscosity solutions to hamilton jacobi equations from the point of view of large deviation theory the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty since an abrupt change in the jump measure occurs from the point of view of viscosity solutions the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations much of the paper is devoted to proving uniqueness of viscosity solutions for these equations and these sections are of independent interest while our use of test functions in proving the uniqueness is an adaptation of the usual technique the construction of the test functions themselves via the legendre transform is new we obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem which suggests the correct integrand in the large deviation rate functional since it is the treatment of the effects due to the boundaries that is novel we devote the majority of the paper to the detailed development of a simple two dimensional system that exhibits all the essential new features however the arguments may be applied to nueueing systems that are considerably more general and we attempt to indicate this generality as well

determinants torsion and strings we apply the results of bf1 bf2 on determinants of dirac operators to string theory for the bosonic string we recover the holomorphic factorization of belavin and knizhik witten s global anomaly formula is used to give sufficient conditions for anomaly cancellation in the heterotic string for arbitrary background spacetimes to prove the latter result we develop certain torsion invariants related to characteristic classes of vector bundles and to index theory

radial symmetry for elliptic boundary value problems on exterior domains by the alexandroff serrin method 2 14 of moving hyperplanes we obtain radial symmetry for the domain and the solutions of on an exterior domain subject to the overdetermined boundary conditions on at and in in particular the following conjecture from potential theory due to p gruber cf 11 8 is proved let or be a bounded smooth domain with a constant source distribution on and let be the induced single layer potential if is constant in then is a ball

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

regularisable and minimal orbits for group actions in infinite dimensions we introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like yang mills and string theory and which generalise finite dimensional riemannian principal fibre bundles induced by an isometric action we show that the orbits of regularisable bundles have well defined both heat kernel and zeta function regularised volumes we introduce a notion of minimality for these orbits which extend the finite dimensional one our approach uses heat kernel methods and yields both heat kernel obtained via heat kernel regularisation and zeta function obtained via zeta function regularisation minimality for specific values of the parameter for each of these notions we give an infinite dimensional version of hsiang s theorem which extends the finite dimensional case interpreting minimal orbits as orbits with extremal regularised volume

a stochastic control approach to the pricing of options we construct a stochastic control model of a portfolio in which the investor may invest in stock call options on the stock or risk free bonds he may also borrow funds we assume that the investor is attempting to maximize expected utility of total wealth at expiration of the options where the utility function is given as wealth to the power 0 1 this includes both the risk averse and risk neutral cases we allow the borrowing interest rate r to be distinct from the lending interest rate r we make two fundamental assumptions i the investor cannot achieve infinite expected utility of wealth in finite time stable markets and ii the market will set the option price to minimize the investor s maximal expected utility of wealth i e options are priced using a minimax strategy the market is stable only when r r we prove in the case r r and 1 that the minimax european call option price ft s is given as the solution of the black scholes equation ft i i 2s2fss i s fs i f 0 where i r when r i 2 1 i r when i 2 1 r and i i 2 1 if r i 2 1 r the parameter is the stock return drift and i is the stock return diffusion if r r then the call option price must by stable markets always be the black scholes price with rate r in the risk neutral 1 case we must have r r for a stable market and the option price given by the black scholes equation but with i explicit solutions are presented and the optimal portfolios and returns are also derived comparable results are obtained for american call options which can be exercised at any time up to and including the expiration date in this case the option pricing function is shown to be the solution of an obstacle problem

holder gradient estimates for fully nonlinear elliptic equations in this paper we prove interior and global holder estimates for lipschitz viscosity solutions of second order nonlinear uniformly elliptic equations the smoothness hypotheses on the operators are more general than previously considered for classical solutions so that our estimates are also new in this case and readily extend to embrace obstacle problems in particular isaac s equations of stochastic differential game theory constitute a special case of our results and moreover our techniques in combination with recent existence theorems of ishii lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic isaac s equation

elliptic operators in the functional quantisation for gauge field theories given a gauge theory with gauge groupg acting on a path spacex g andx being both infinite dimensional manifolds modelled on spaces of sections of vector bundles on a compact riemannian manifold without boundary it is shown that when the action ofg onx is smooth free and proper the same ellipticity condition on an operator naturally given by the geometry of the problem yields both the existence of a principal fibre bundle structure induced by the canonical projection x x g and the existence of the faddeev popov determinant arising in the functional quantisation of the gauge theory this holds for certain gauge theories with anomalies like bosonic closed string theory in non critical dimension and also holds for a class of gauge theories which includes yang mills theory

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

the perturbed test function method for viscosity solutions of nonlinear pde the method of viscosity solutions for nonlinear partial differential equations pdes justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions we describe in this paper a perturbed test function device which entails various modifications of the test functions by lower order correctors applications include homogenisation for quasilinear elliptic pdes and approximation of quasilinear parabolic pdes by systems of hamilton jacobi equations

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

fully nonlinear neumann type boundary conditions for first order hamilton jacobi equations we consider problems of the following type h x u u 0 in where is a smooth bounded open set in r n u is a scalar function from into r u denotes its frechet differential at x and h is a given continuous function on xr r n this equation is nothing but a general form of the classical first order hamilton jacobi equation in order to investigate existence and uniqueness questions the equation has to be complemented with a boundary condition on or on a part of we consider here for reasons described below a fully nonlinear boundary condition of the form f x u u 0 on that we describe as being of the neumann type

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

remarks on hamilton jacobi equations with measurable time dependent hamiltonians on considere les equations de hamilton jacobi de la forme suivante u t h x t u u 0 dans r n 0 t ou h x t s p est continu en x s p et mesurable en t on considere une extension de la notion de solution viscosite au cas de h mesurable dependant du temps

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

hamilton jacobi equations with state constraints in the present paper we consider hamilton jacobi equations of the form h x u vu 0 x e q where q is a bounded open subset of rn h is a given continuous real valued function of x s p e q x r x rn and vu is the gradient of the unknown function u we are interested in particular solutions of the above equation which are required to be supersolutions in a suitable weak sense of the same equation up to the boundary of q this requirement plays the role of a boundary condition the main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system as well from related questions in constrained geodesics

a remark on a system of inequalities with bilateral obstacles up x k s up x i up l x k x e sz apup fp if up x k up x up x k x e q apup 5 fp if up x up l x k x e l2 1 1 apup l f if up x up x k x e q u x 0 onaq p 1 2 m where p ranges over 1 m and urn represents u is dealt with by the second author in 5 and the existence of a weak solution of 1 1 is established under the assumption ap s have sufficiently large zeroth order coefficients 1 2 the proof of the existence result in 5 is based on the iv estimate independent of e for the solution u 124 u of the following approximate system apu u p uf l k u k uf f in r 1 3 u 0 0naq p 1 2 m where p ranges over 1 ml p is the so called penalty function and uf represents u the gradient estimate for solutions u of 1 3 in 5 is obtained by using bernstein s method and assumption 1 2 is solely employed to make it possible in addition it should be remarked that standard variants of the bernstein trick like in 2 and 4 which are useful to obtain the iv estimate for solutions of the system of variational inequalities with unilateral constraints without 1 2 do not work in our case

the faddeev popov procedure and application to bosonic strings an infinite dimensional point of view a generalisation of the finite dimensional presentation of the faddeev popov perocedure is derived in an infinite dimensional framework for gauge theories with finite dimensional moduli space using heat kernel regularised determinants it is shown that the infinite dimensional faddeev popov determinant is up to a finite dimensional determinant determined by a choice of a slice canonically determined by the geometrical data defining the gauge theory namely a fibre bundlep p g with structure groupg and the invariance group of a metric structure given on the total spacep the case of closed bosonic string theory is discussed

a regularity result for viscosity solutions of hamilton jacobi equations in one space dimension on demontre que dans le cas particulier d un probleme stationnaire a une dimension sous des hypotheses tres generales les solutions viscosite continues de lipschitz ont des derivees a droite et a gauche en chaque point

representation of solutions of hamilton jacobi equations on etudie le probleme de cauchy pour l equation de hamilton jacobi u t h t x du 0 dans o t r n avec la condition initiale u o x u o x pour x r n on a t 0 hi o t r n r n r et u o r n r sont donnes on represente la solution viscosite comme la fonction valeur d un jeu differentiel approprie

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

viscosity solutions of hamilton jacobi equations with unbounded nonlinear terms abstract the aim of this paper is to introduce a new definition of viscosity solutions for hamilton jacobi equations in infinite dimensions with unbounded nonlinear terms we prove existence and uniqueness results under usual assumptions regarding the bounded part of the hamiltonian

on the bundle of connections and the gauge orbit manifold in yang mills theory in an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations in yang mills theory is ac principal fibration the underlying quotient space the gauge orbit space is seen explicitly to be ac manifold modelled on a hilbert space

neumann type boundary conditions for hamilton jacobi equations abstract in this paper we present a notion of viscosity solutions of hamilton jacobi equations for neumann type boundary conditions or more generally oblique derivative in particular we prove the existence uniqueness stability of such solutions and we show that the vanishing viscosity method yields such solutions next we check that value functions of control problems or differential games problems for reflected dynamical processes are solutions in that sense of the associated bellman or isaacs equations finally we consider the ergodic problems

generalized viscosity solutions for hamilton jacobi equations with time measurable hamiltonians the objective of this paper is to extend the idea of viscosity solution for nonlinear first order hamilton jacobi equations u h t x d u 0 with time continuous h to time measurable hamiltonians we are primarily motivated by the fact that control problems for which the value function is the viscosity solution of a hamilton jacobi equation should not be restric ted to continuity in time in barron and jensen 1 we were confronted with a linear hamilton jacobi equation arising in the proof of the pon tryagin which had time measurable coeflicients the existing theory of viscosity solutions did not apply to even this case moreover important models of controlled first order hamiltonjacobi equations require time measurable controls for example models of controlled traffic flow or queueing processes might be of this type we are about then to extend the definition of viscosity solution to generalized viscosity solution applying to time measurable hamiltonians in this paper we give the definition and derive the corresponding uni queness result with an implicit domain of dependence consequent the fun damental technique used in the proof is the so called blow up method which has also been used elsewhere s although not in the generality discussed here this method effectively freezes the point to x0 under consideration and essentially removes it from the problem of course t is

existence of viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games recently m g crandall and p l lions trans amer math soc 277 1983 1 42 introduced the class of viscosity solutions of these equations and proved uniqueness within this class this paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness

the dynamic programming equation for the time optimal control problem in infinite dimensions the existence and uniqueness of a viscosity solution for the bellman equation associated with the time optimal control problem for a semilinear evolution equation in hilbert space is provided applications to time optimal control problems governed by parabolic equations are given

analytical solutions for the stochastic gardner equation with the help of the hermite transformation some analytical solutions of the stochastic gardner equation in the white noise environment are obtained using the exp function method

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

over determined problems for k hessian equations in ring shaped domains abstract in this paper we firstly use a variant of the moving plane method of alexandroff to obtain radial symmetry of solutions for k hessian equations in annulus type domains which can be regarded as a generalization of gidas ni nirenberg result in 1979 then we consider an over determined problem for k hessian equations in ring shaped domains and prove the radial symmetry of the solutions and the domains

an n soliton solution to the dnls equation based on revised inverse scattering transform based on a revised version of inverse scattering transform for the derivative nonlinear schrodinger dnls equation with vanishing boundary condition vbc the explicit n soliton solution has been derived by some algebra techniques of some special matrices and determinants especially the binet cauchy formula the one and two soliton solutions have been given as the illustration of the general formula of the n soliton solution moreover the asymptotic behaviors of the n soliton solution have been discussed

lax pair backlund transformation and n soliton like solution for a variable coefficient gardner equation from nonlinear lattice plasma physics and ocean dynamics with symbolic computation abstract in this paper a generalized variable coefficient gardner equation arising in nonlinear lattice plasma physics and ocean dynamics is investigated with symbolic computation the lax pair and backlund transformation are explicitly obtained when the coefficient functions obey the painleve integrable conditions meanwhile under the constraint conditions two transformations from such an equation either to the constant coefficient gardner or modified korteweg de vries mkdv equation are proposed via the two transformations the investigations on the variable coefficient gardner equation can be based on the constant coefficient ones the n soliton like solution is presented and discussed through the figures for some sample solutions it is shown in the discussions that the variable coefficient gardner equation possesses the right and left travelling soliton like waves which involve abundant temporally inhomogeneous features

exit time problems in optimal control and vanishing viscosity method the authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first order hamilton jacobi hj equation up to the boundary this equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary when these solutions are equal they can be obtained through the vanishing viscosity method finally when the hj equation has a continuous solution it is proved to be the value function for the first exit time of the domain it is also the vanishing viscosity limit arising in particular in some large deviations problems

an asymptotic formula for solutions of hamilton jacobi bellman equations on demontre que la solution de l equation de hamilton jacobi bellman converge exponentiellement vers 0 quand e 0 avec un taux de decroissance i x 0 1 e et on donne une formule de representation pour i x

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

the pontryagin maximum principle from dynamic programming and viscosity solutions to first order partial differential equations on demontre le principe de maximum de pontryagin pour le probleme de lagrange de commande optimale en utilisant le fait que la fonction valeur du probleme est la solution viscosite de l equation de hamilton jacobi bellman associee

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

the necessary conditions for optimal control in hilbert spaces obtention du principe maximum de pontryagin pour une classe generale de problemes de commande de type lagragien afin qu associe avec un ensemble de conditions necessaires une commande soit satisfaite le principe est demontre en utilisant la programmation dynamique et l equation resultante d hamilton jacobi bellman de dimension finie

viscosity solutions for weakly coupled systems of hamilton jacobi equations we obtain comparison results for viscosity solutions of weakly coupled systems of hamilton jacobi equations perron s method is extended to give existence proofs for systems a stronger definition of sub and supersolutions is introduced leading to existence results for continuous solutions the needed sub and supersolutions are constructed for continuous boundary data crandall s and lions notion of viscosity solutions for first order hamilton jacobi equations can be extended in a componentwise way to systems a key tool in obtaining uniqueness comparison and existence results is the maximum principle we obtain such results for weakly coupled systems of hamilton jacobi equations in which the coupling has such a structure to allow the use of maximum principle techniques to motivate the coupling assumptions consider this linear system n

perturbed dynamical systems with an attracting singularity and weak viscosity limits in hamilton jacobi equations nous donnons une nouvelle preuve par des methodes edp du theoreme de wentzell freidlin concernant les petites perturbations d un systeme dynamique l e u e e 2 u e b u e 0 dans u e sur nous prouvons que si b a un seul point singulier attractif alors u e converge vers une constante uniformement sur tout compact et avec une vitesse exponentielle nous determinons nous traitons aussi le cas de conditions aux limites de neumann pour cela nous faisons l analyse asymptotique d une mesure ergodique intervenant naturellement dans le probleme ce qui revient a etudier la limite par viscosite evanescente dans une equation de hamilton jacobi

generalized one sided estimates for solutions of hamilton jacobi equations and applications on etudie les proprietes locales des solutions de l equation de hamilton jacobi u t h t x u 0 et plus generalement de l equation u t h t x u u 0 ou t o t et x r n

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

the geometrical interpretation of the faddeev popov determinant in polyakov s theory of random surfaces in the context of polyakov s theory the geometrical approach in which the functional measure is defined as a formal volume form is developed the special version of the fubini theorem on manifolds is derived which provides the geometrical interpretation of the faddeev popov method the conformal gauge is studied within this framework as a result the explicit form of functional measure in effective liouville quantum field theory is obtained