oscillation of certain neutral delay parabolic equations abstract sufficient conditions are given to ensure the solutions of a class of neutral parabolic equations with continuous distributed deviating arguments to oscillate

some oscillation theorems for systems of even order quasilinear partial differential equations in this paper new sufficient conditions are obtained for oscillation for the solutions of the systems of a class even order quasilinear partial functional differential equations of neutral type these results are illustrated by some examples

scaling properties of driven interfaces in disordered media we perform a systematic study of several models that have been proposed for the purpose of understanding the motion of driven interfaces in disordered media we identify two distinct universality classes i one of these referred to as directed percolation depinning dpd can be described by a langevin equation similar to the kardar parisi zhang equation but with quenched disorder ii the other referred to as quenched edwards wilkinson qew can be described by a langevin equation similar to the edwards wilkinson equation but with quenched disorder we find that for the dpd universality class the coefficient ensuremath lambda of the nonlinear term diverges at the depinning transition while for the qew universality class either ensuremath lambda 0 or ensuremath lambda ensuremath rightarrow 0 as the depinning transition is approached the identification of the two universality classes allows us to better understand many of the results previously obtained experimentally and numerically however we find that some results cannot be understood in terms of the exponents obtained for the two universality classes at the depinning transition in order to understand these remaining disagreements we investigate the scaling properties of models in each of the two universality classes above the depinning transition for the dpd universality class we find for the roughness exponent mathrm ensuremath alpha mathit p 0 63 ifmmode pm else textpm fi 0 03 for the pinned phase and mathrm ensuremath alpha mathit m 0 75 ifmmode pm else textpm fi 0 05 for the moving phase for the growth exponent we find mathrm ensuremath beta mathit p 0 67 ifmmode pm else textpm fi 0 05 for the pinned phase and mathrm ensuremath beta mathit m 0 74 ifmmode pm else textpm fi 0 06 for the moving phase furthermore we find an anomalous scaling of the prefactor of the width on the driving force a new exponent mathit cphi mathit m 0 12 ifmmode pm else textpm fi 0 06 characterizing the scaling of this prefactor is shown to relate the values of the roughness exponents on both sides of the depinning transition for the qew universality class we find that mathrm ensuremath alpha mathit p ensuremath approxeq mathrm ensuremath alpha mathit m 0 92 ifmmode pm else textpm fi 0 04 and mathrm ensuremath beta mathit p ensuremath approxeq mathrm ensuremath beta mathit m 0 86 ifmmode pm else textpm fi 0 03 are roughly the same for both the pinned and moving phases moreover we again find a dependence of the prefactor of the width on the driving force for this universality class the exponent mathit cphi mathit m 0 44 ifmmode pm else textpm fi 0 05 is found to relate the different values of the local mathrm ensuremath alpha mathit p and global roughness exponent mathrm ensuremath alpha mathit g ensuremath approxeq 1 23 ifmmode pm else textpm fi 0 04 at the depinning transition these results provide us with a more consistent understanding of the scaling properties of the two universality classes both at and above the depinning transition we compare our results with all the relevant experiments

oscillation of parabolic equations of neutral type in this paper we obtain new sufficient conditions for the oscillation of a class of neutral parabolic differential equations satisfying two kinds of boundary value conditions

integral average method for oscillation of second order partial differential equations with delays using a generalized riccati transformation some new oscillation criteria for second order partial differential equations with delays are found through the method of integral average these results can be considered as generalizations and improvements of the results due to kamenev in ordinary differential equations cases

self organized growth model for the quenched herring mullin equation we introduce a simple self organized growth model mimicking the dynamics of a driven tensionless interface in a random medium near the depinning threshold the roughness and growth exponents for the model are obtained as ensuremath zeta ensuremath simeq 1 93 and ensuremath beta ensuremath simeq 0 96 respectively we discuss the possible continuum equation describing the motion of a driven interface in our model

on pinned fields interlacements and random walk on mathbb z n mathbb z 2 we define two families of poissonian soups of bidirectional trajectories on mathbb z 2 which can be seen to adequately describe the local picture of the trace left by a random walk on the two dimensional torus mathbb z n mathbb z 2 started from the uniform distribution run up to a time of order n log n 2 and forced to avoid a fixed point the local limit of the latter was recently established in comets et al commun math phys 343 129 164 2016 our construction proceeds by considering somewhat in the spirit of statistical mechanics a sequence of finite volume approximations consisting of random walks avoiding the origin and killed at spatial scale n either using dirichlet boundary conditions or by means of a suitably adjusted mass by tuning the intensity u of such walks with n the occupation field can be seen to have a nontrivial limit corresponding to that of the actual random walk our construction thus yields a two dimensional analogue of the random interlacements model introduced in sznitman ann math 171 3 2039 2087 2010 in the transient case it also links it to the pinned free field in mathbb z 2 by means of a pinned ray knight type isomorphism theorem

transforming fixed length self avoiding walks into radial sle8 3 we conjecture a relationship between the scaling limit of the fixed length ensemble of self avoiding walks in the upper half plane and radial sle8 3 in this half plane from 0 to i the relationship is that if we take a curve from the fixed length scaling limit of the saw weight it by a suitable power of the distance to the endpoint of the curve and apply the conformal map of the half plane that takes the endpoint to i then we get the same probability measure on curves as radial sle8 3 in addition to a non rigorous derivation of this conjecture we support it with monte carlo simulations of the saw using the conjectured relationship between the saw and radial sle8 3 our simulations give estimates for both the interior and boundary scaling exponents the values we obtain are within a few hundredths of a percent of the conjectured values

toward a conformal field theory for schramm loewner evolutions we discuss the partition function point of view for chordal schramm loewner evolutions and their relationship with correlation functions in conformal field theory both are closely related to crossing probabilities and interfaces in critical models in two dimensional statistical mechanics we gather and supplement previous results with different perspectives point out remaining difficulties and suggest directions for future studies

oscillation of the solutions of parabolic differential equations of neutral type sufficient conditions are obtained for the oscillation of the solutions of linear parabolic differential equations of neutral type of the form t u si 1m l i t u x t t i a t u si 1sa i ft u x t p i p x t u si 1kp i x t u x t s i x t e wx 0 where w is a bounded domain in r n with a piecewise smooth boundary

gaussian chaos and sample path properties of additive functionals of symmetric markov processes let x be a strongly symmetric hunt process with potential density u x y let g 2 x y 2 d x d y and let l t denote the continuous additive functional with revuz measure for a set of positive measures m g 2 subject to some additional regulariy conditions we consider families of continuous in time additive functionals l l t t r x m of x and a second order gaussian chaos h h m which is associated with l by an isomorphism theorem of dynkin a general theorem is obtained which shows that with some additional regularity conditions depending on x and m if h has a continuous version on m almost surely then so does l and furthermore that moduli of continuity for h are also moduli of continuity for l special attention is given to levy processes in r n and t n the n dimensional torus with m taken to be the set of translates of a fixed measure many concrete examples are given especially when x is brownian motion in r n and t n for n 2 and 3 for certain measures on t n and processes including brownian motion in t 3 necessary and sufficient conditions are given for the continuity of l t t r x m where m is the set of all translates of

on the smoothness of the partition function for multiple schramm loewner evolutions we consider the measure on multiple chordal schramm loewner evolution textit sle kappa curves we establish a derivative estimate and use it to give a direct proof that the partition function is c 2 if kappa 4

on conformally invariant cle explorations we study some conformally invariant dynamic ways to construct the conformal loop ensembles with simple loops introduced in earlier papers by sheffield and by sheffield and werner one outcome is a conformally invariant way to measure a distance of a cle4 loop to the boundary within the cle4 when one identifies all points of each loop

markov processes as a tool in field theory abstract in a recent paper brydges frohlich and spencer have successfully applied markov chains to classical spin systems using probabilistic methods in a more systematic way we improve some of their results the methods can be extended to fields with continuous parameter such as p fields

forced oscillation of solutions of certain hyperbolic equations of neutral type some oscillation properties of solutions of certain hyperbolic equations of neutral type are studied and sufficient conditions for the oscillation of the solutions of a boundary value problem in a cylindrical domain are given

a ray knight theorem for symmetric markov processes let x be a strongly symmetric recurrent markov process with state space s and let l t x denote the local time of x at x in s for a fixed element 0 in the state space s let tau t inf s l 0 s t the 0 potential density u 0 x y of the process x killed at t 0 inf s x s 0 is symmetric and positive definite let eta eta x x in s be a mean zero gaussian process with covariance e eta eta x eta y u 0 x y the main result of this paper is the following generalization of the classical second ray knight theorem for any b in r and t 0 l x tau t frac 1 2 eta x b 2 x in s frac 1 2 eta x sqrt 2 b 2 2 x in s text in law a version of this theorem is also given when x is transient

random walk loop soup the brownian loop soup introduced by lawler and werner 2004 is a poissonian realization from a finite measure on unrooted loops this measure satisfies both conformal invariance and a restriction property in this paper we define a random walk loop soup and show that it converges to the brownian loop soup in fact we give a strong approximation result making use of the strong approximation result of komlos major and tusnady to make the paper self contained we include a proof of the approximation result that we need

oscillation of solutions of nonlinear parabolic equations via comparison method in the present paper sufficient conditions are found for oscillation of the solutions of nonlinear parabolic equations subject to boundary conditions of neumann or dirichlet type the results are obtained via the method of comparison of solutions of nonlinear parabolic equations by oscillatory solutions of certain ordinary differential equations

oscillation of solutions of neutral partial functional differential equations abstract sufficient conditions are established for the oscillation of solutions of neutral partial functional differential equations of the form t p t t u x t i 1 l i t u x t i a t u x t k 1 s a k t u x t k t q x t u x t j 1 m q j x t f j u x t j x t 0 g where is a bounded domain inrnwith a piecewise smooth boundary and is the laplacian in the euclideann spacern

gaussian and non gaussian random fields associated with markov processes to every markov process with a symmetric transition density there correspond two random fields over the state space a gaussian field the free field and the occupation field t which describes amount of time the particle spends at each state a relation between these two random fields is established which is useful both for the field theory and theory of markov processes

a class of even order neutral differential inequalities and its applications the properties of a class of even order neutral differential inequalities with distributed type deviating arguments are obtained using these results we established some criteria for the oscillation of even order neutral partial functional differential equations

self organized interface growth with the negative nonlinearity in a random medium we introduce two self organized growth models that describe the motion of the driven interfaces in random media including the kardar parisi zhang kpz nonlinearity one model follows the quenched kpz equation with a positive nonlinear term while the other model follows the quenched kpz equation with a negative nonlinear term by obtaining the critical exponents for two models we confirm that the sign of the kpz nonlinear term does not affect the universality class

forced oscillation properties for certain systems of partial functional differential equations some sufficient conditions are established for the forced oscillation of certain systems of partial functional differential equations with deviating arguments these results are illustrated by some examples

from loop clusters and random interlacements to the free field it was shown by le jan that the occupation field of a poisson ensemble of markov loops loop soup of parameter one half associated to a transient symmetric markov jump process on a network is half the square of the gaussian free field on this network we construct a coupling between these loops and the free field such that an additional constraint holds the sign of the free field is constant on each cluster of loops as a consequence of our coupling we deduce that the loop clusters of parameter one half do not percolate on periodic lattices we also construct a coupling between the random interlacement on mathbb z d d geq 3 introduced by sznitman and the gaussian free field on the lattice such that the set of vertices visited by the interlacement is contained in a level set of the free field we deduce an inequality between the critical level for the percolation by level sets of the free field and the critical parameter for the percolation of the vacant set of the random interlacement both in the case of loops and of the random interlacement the couplings are constructed by replacing discrete graphs by metric graphs le jan s and sznitman s isomorphism theorems between the gaussian free field and the occupation field of trajectories can be extended to the metric graph setting on which the intermediate value principle for continuous fields holds

the nested simple conformal loop ensembles in the riemann sphere simple conformal loop ensembles cle are random collections of simple non intersecting loops that are of particular interest in the study of conformally invariant systems among other things related to these cles we prove the invariance in distribution of their nested full plane versions under the inversion z mapsto 1 z

some oscillation problems for impulsive hyperbolic differential systems with several delays this paper investigates the oscillatory properties of impulsive hyperbolic differential systems with several delays and establishes sufficient conditions for oscillation of the systems moreover some necessary and sufficient conditions for oscillation of the solutions of robin boundary problems of impulsive delay hyperbolic systems are obtained

oscillation of solutions of systems of neutral type partial functional differential equations abstract sufficient conditions are established for the oscillation of solutions of systems of neutral type partial functional differential equations of the form where is a bounded domain in r n with a piecewise smooth boundary and is the laplacian in euclidean n space r n the obtained results are illustrated by some interesting examples

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

oscillation of solutions of parabolic differential equations of neutral type in this paper we investigate a class of parabolic differential equations of neutral type and obtain some sufficient conditions of the oscillation for such equations satisfying two kinds of boundary value conditions

interval oscillation criteria for second order partial differential equations with delays in this paper we present new interval oscillation criteria related to integral averaging technique for second order partial differential equations with delays that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on whole half line our results are sharper than some previous results and handles the cases which are not covered by known criteria

h oscillation of solutions of certain vector hyperbolic differential equations with deviating arguments in this paper sufficient conditions are established for h oscillation of solutions of a class of vector hyperbolic differential equations with deviating arguments where h is a unit vector in r m the main results are illustrated by two examples

the random walk representation of classical spin systems and correlation inequalities ii the skeleton inequalities on utilise la representation marche aleatoire pour prouver les premieres d une nouvelle famille d inegalites de correlation pour des modeles reticules ferromagnetiques 4 ces inegalites etablissent que les sommes partielles finies du developpement en perturbation resomme du propagateur pour la fonction a 4 points forment un ensemble alterne de limites superieures et inferieures rigoureuses pour la fonction a 4 points exactes

necessary and sufficient conditions for oscillation of neutral delay parabolic differential equations necessary and sufficient conditions are established for the oscillation of neutral delay parabolic differential equations of the form t u x t pu x t t a t du x t k 1 sa k t du x t r k t j 1 mq j t u x t s j t x t wx 0 g where w is a bounded domain in r n with a piecewise smooth boundary w and d is the laplacian in euclidean n space r n

forced oscillation of a class of neutral hyperbolic differential equations in this paper we study the following boundary value problem for a class of neutral hyperbolic differential equations 2 t2 u c t u x t a0 t u a1 t u x t ab q x t f u x g t d g x t x t r g u n v x t u 0 x t r a number of theorems for oscillatory solutions of the problem under two different cases are developed

the first passage sets of the 2d gaussian free field convergence and isomorphisms in a previous article we introduced the first passage set fps of constant level a of the two dimensional continuum gaussian free field gff on finitely connected domains informally it is the set of points in the domain that can be connected to the boundary by a path along which the gff is greater than or equal to a this description can be taken as a definition of the fps for the metric graph gff and in the current article we prove that the metric graph fps converges towards the continuum fps in the hausdorff distance we also draw numerous consequences in particular we obtain a relatively simple proof of the fact that certain natural interfaces of the metric graph gff converge to hbox sle 4 level lines these results improve our understanding of the continuum gff by strengthening its relationship with the critical brownian loop soup indeed a new construction of the fps using clusters of brownian loops and excursions helps to strengthen the known gff isomorphism theorems and allows us to use brownian loop soup techniques to prove technical results on the geometry of the gff we also obtain a new representation of brownian loop soup clusters and as a consequence we prove that the clusters of a critical brownian loop soup admit a non trivial minkowski content in the gauge r mapsto log r 1 2 r 2

markov loops and renormalization we study the poissonnian ensembles of markov loops and the associated renormalized self intersection local times

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

permanental fields loop soups and continuous additive functionals a permanental field psi psi nu nu in mathcal v is a particular stochastic process indexed by a space of measures on a set s it is determined by a kernel u x y x y in s that need not be symmetric and is allowed to be infinite on the diagonal we show that these fields exist when u x y is a potential density of a transient markov process x in s a permanental field psi can be realized as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of x which we carefully construct a dynkin type isomorphism theorem is obtained that relates psi to continuous additive functionals of x continuous in t l l t nu nu t in mathcal v times r sufficient conditions are obtained for the continuity of l on mathcal v times r the metric on mathcal v is given by a proper norm

forced oscillation for certain systems of hyperbolic differential equations in this paper sufficient conditions are established for the forced oscillation of certain systems of hyperbolic differential equations with deviating arguments the main result is illustrated by an example

multi point green s functions for sle and an estimate of beffara in this paper we define and prove of the existence of the multi point green s function for sle a normalized limit of the probability that an sle curve passes near to a pair of marked points in the interior of a domain when 8 this probability is nontrivial and an expression can be written in terms two sided radial sle one of the main components to our proof is a refinement of a bound first provided by beffara ann probab 36 2008 1421 1452 this work contains a proof of this bound independent from the original

necessary and sufficient conditions for oscillation of robin boundary problems of neutral functional parabolic equations some necessary and sufficient conditions for oscillation of the solutions of robin boundary problems of neutral parabolic equations with several deviating arguments are obtained

universality in the 2d ising model and conformal invariance of fermionic observables it is widely believed that the celebrated 2d ising model at criticality has a universal and conformally invariant scaling limit which is used in deriving many of its properties however no mathematical proof has ever been given and even physics arguments support a priori weaker mobius invariance we introduce discrete holomorphic fermions for the 2d ising model at criticality on a large family of planar graphs we show that on bounded domains with appropriate boundary conditions those have universal and conformally invariant scaling limits thus proving the universality and conformal invariance conjectures

phase transition in loop percolation we are interested in the clusters formed by a poisson ensemble of markovian loops on infinite graphs this model was introduced and studied in le jan c r math acad sci paris 350 13 14 643 646 2012 ill j math 57 2 525 558 2013 it is a model with long range correlations with two parameters alpha and kappa the non negative parameter alpha measures the amount of loops and kappa plays the role of killing on vertices penalizing kappa 0 or favoring kappa 0 appearance of large loops it was shown in le jan ill j math 57 2 525 558 2013 that for any fixed kappa and large enough alpha there exists an infinite cluster in the loop percolation on mathbb z d in the present article we show a non trivial phase transition on the integer lattice mathbb z d d ge 3 for kappa 0 more precisely we show that there is no loop percolation for kappa 0 and alpha small enough interestingly we observe a critical like behavior on the whole sub critical domain of alpha namely for kappa 0 and any sub critical value of alpha the probability of one arm event decays at most polynomially for d ge 5 we prove that there exists a non trivial threshold for the finiteness of the expected cluster size for alpha below this threshold we calculate up to a constant factor the decay of the probability of one arm event two point function and the tail distribution of the cluster size these rates are comparable with the ones obtained from a single large loop and only depend on the dimension for d 3 or 4 we give better lower bounds on the decay of the probability of one arm event which show importance of small loops for long connections in addition we show that the one arm exponent in dimension 3 depends on the intensity alpha

partition functions loop measure and versions of sle we discuss the partition function view of the schramm loewner evolution after reviewing a number of known results in the framework of brownian loop measures and scaling rules for partition functions we give some speculation about multiply connected domains

the fixed irreducible bridge ensemble for self avoiding walks we define a new ensemble for self avoiding walks in the upper half plane the fixed irredicible bridge ensemble by considering self avoiding walks in the upper half plane up to their n th bridge height y n and scaling the walk by 1 y n to obtain a curve in the unit strip and then taking n rightarrow infty we then conjecture a relationship between this ensemble to sle 8 3 in the unit strip from 0 to a fixed point along the upper boundary of the strip integrated over the conjectured exit density of self avoiding walk spanning a strip in the scaling limit we conjecture that there exists a positive constant sigma such that n sigma y n converges in distribution to that of a stable random variable as n rightarrow infty then the conjectured relationship between the fixed irreducible bridge scaling limit and sle 8 3 can be described as follows if one takes a saw considered up to y n and scales by 1 y n and then weights the walk by y n to an appropriate power then in the limit n rightarrow infty one should obtain a curve from the scaling limit of the self avoiding walk spanning the unit strip in addition to a heuristic derivation we provide numerical evidence to support the conjecture and give estimates for the boundary scaling exponent

extreme nesting in the conformal loop ensemble the conformal loop ensemble cle with parameter 8 3 8 is the canonical conformally invariant measure on countably infinite collections of noncrossing loops in a simply connected domain given and we compute the almost sure hausdorff dimension of the set of points z for which the number of cle loops surrounding the disk of radius e centered at z has asymptotic growth log 1 e as e 0 by extending these results to a setting in which the loops are given i i d weights we give a cle based treatment of the extremes of the gaussian free field

percolation for the vacant set of random interlacements we investigate random interlacements on d d 3 this model recently introduced in 8 corresponds to a poisson cloud on the space of doubly infinite trajectories modulo time shift tending to infinity at positive and negative infinite times a nonnegative parameter u measures how many trajectories enter the picture our main interest lies in the percolative properties of the vacant set left by random interlacements at level u we show that for all d 3 the vacant set at level u percolates when u is small this solves an open problem of 8 where this fact has only been established when d 7 it also completes the proof of the nondegeneracy in all dimensions d 3 of the critical parameter u of 8 2008 wiley periodicals inc

scaling behavior of roughness in the two dimensional kardar parisi zhang growth abstract we investigate the time evolution of the roughness of kadar parisi zhang kpz equation in 2 1 dimensions scaling behavior of the roughness is analyzed from scale transformations and the scaling function is obtained from numerical simulations

vacant set of random interlacements and percolation we introduce a model of random interlacements made of a countable collection of doubly infinite paths on d d 3 a nonnegative parameter u measures how many trajectories enter the picture this model describes in the large n limit the microscopic structure in the bulk which arises when considering the disconnection time of a discrete cylinder n d 1 x by simple random walk or the set of points visited by simple random walk on the discrete torus n d at times of order u n d in particular we study the percolative properties of the vacant set left by the interlacement at level u which is an infinite connected translation invariant random subset of d we introduce a critical value u such that the vacant set percolates for u u our main results show that u is finite when d 3 and strictly positive when d 7

discrete complex analysis on isoradial graphs abstract we study discrete complex analysis and potential theory on a large family of planar graphs the so called isoradial ones along with discrete analogues of several classical results we prove uniform convergence of discrete harmonic measures green s functions and poisson kernels to their continuous counterparts among other applications the results can be used to establish universality of the critical ising and other lattice models

the random pseudo metric on a graph defined via the zero set of the gaussian free field on its metric graph we further investigate properties of the gaussian free field gff on the metric graph associated to a discrete weighted graph where the edges of the latter are replaced by continuous line segments of appropriate length that has been introduced by the first author on such a metric graph the gff is a random continuous function that generalises one dimensional brownian bridges so that one dimensional techniques can be used in the present paper we define and study the pseudo metric defined on the metric graph and therefore also on the discrete graph itself where the length of a path on the metric graph is defined to be the local time at level zero accumulated by the gaussian free field along this path we first derive a pathwise transformation that relates the gff on the metric graph with the reflected gff on the metric graph via the pseudo distance defined by the latter this is a generalisation of paul levy s result relating the local time at zero of brownian motion to the supremum of another brownian motion we also compute explicitly the distribution of certain functionals of this pseudo metric and of the gff in particular we point out that when the boundary consists of just two points the law of the pseudo distance between them depends solely on the resistance of the network between them we then discuss questions related to the scaling limit of this pseudo metric in the two dimensional case which should be the conformally invariant way to measure distances between cle 4 loops introduced and studied by the second author with wu and by sheffield watson and wu our explicit laws on metric graphs also lead to new conjectures for related functionals of the continuum gff on fairly general riemann surfaces

oscillation properties for systems of hyperbolic differential equations of neutral type abstract sufficient conditions are established for the oscillations of systems of hyperbolic differential equations of the form 2 t 2 p t u i x t r 1 d r t u i x t r a i t u i x t j 1 m k 1 s a ijk t u j x k t q i x t u i x t j 1 m h 1 l q ijh x t u j x h t x t 0 g i 1 2 m where is a bounded domain in rn with a piecewise smooth boundary and is the laplacian in euclidean n space rn