elliptic problems with nonlinearities indefinite in sign abstract let r n n 3 be a bounded open set with smooth boundary and r we study the dirichlet problem formula with 1 q p and h k l nonnegative functions we prove existence nonexistence and multiplicity results depending on and according to the integrability properties of the ratio k p 1 h q 1

multiple solutions of nonlinear schrodinger equations via flow invariance and morse theory we prove the existence of multiple bound states of the nonlinear schrodinger equation u v x u f u here the linear potential v is continuous and bounded from below and the nonlinearity f is of asymptotically linear type we show that under certain assumptions on the spectrum of the schrodinger operator v and the asymptotic behaviour of f u u the above equation has at least four non trivial solutions two of them sign changing

two a posteriori error estimates for one dimensional scalar conservation laws in this paper we propose a posteriori local error estimates for numerical schemes in the context of one dimensional scalar conservation laws we first consider the schemes for which a consistent in cell entropy inequality can be derived then we extend this result to second order schemes written in viscous form satisfying weak entropy inequalities as an illustration we show several numerical tests on the burgers equation and we propose an adaptive algorithm for the selection of the mesh

on the positive solutions of semilinear equations δ ℎ 0 on the compact manifolds in this paper we study the existence nonexistence and uniqueness of positive solutions of semilinear equations u u hu p 0 on compact riemannian manifolds as well as on bounded smooth domains in r n with homogeneous dirichlet or neumann boundary conditions

a priori error estimates for numerical methods for scalar conservation laws part ii flux splitting monotone schemes on irregular cartesian grids this paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux splitting monotone schemes to the entropy solution we obtain the optimal rate of convergence of x 1 2 in l l 1 for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution we then extend this result to less consistent schemes called p consistent schemes and prove that they converge to the entropy solution with the rate of x min 1 2 p in l l 1 again no regularity property of the approximate solution is used finally we propose a new explanation of the fact that even inconsistent schemes converge with the rate of x 1 2 in l l 1 we show that this well known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution total variation boundedness to compensate for its lack of consistency the nonlinear nature of the problem does not play any role in this mechanism all the above results hold in the multidimensional case provided the grids are cartesian products of one dimensional nonuniform grids

existence and multiplicity of solutions for asymptotically hamiltonian elliptic systems in rn abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h u x u v in r n v v x v h v x u v in r n u x 0 and v x 0 as x where the potential v is bounded below and h is asymptotically linear in z as z with z u v by applying a generalized linking theorem of strongly indefinite functionals we obtain solutions for the above system

trapping and cascading of eigenvalues in the large coupling limit we consider eigenvaluese of the hamiltonianh v w w compactly supported in the limit forw 0 we find monotonic convergence ofe to the eigenvalues of a limiting operatorh associated with an exterior dirichlet problem and we estimate the rate of convergence for 1 dimensional systems in 1 dimensional systems withw 0 or withw changing sign we do not find convergence instead we find a cascade phenomenon in which as each eigenvaluee stays near a dirichlet eigenvalue for a long interval of lengtho sqrt lambda of the scaling range quickly drops to the next lower dirichlet eigenvalue stays there for a long interval drops again and so on as a result for most large values of the discrete spectrum ofh is close to that ofe but when reaches a transition region the entire spectrum quickly shifts down by one we also explore the behavior of several explicit models as

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

the discrete one sided lipschitz condition for convex scalar conservation laws physical solutions to convex scalar conservation laws satisfy a one sided lipschitz condition oslc that enforces both the entropy condition and their total variation boundedness consistency with this condition is therefore desirable for a numerical scheme and was proved for both the godunov and the lax friedrichs scheme also in a weakened version for the roe scheme all of them being only first order accurate a new fully second order scheme is introduced here which is consistent with the oslc the modified equation is considered and shows interesting features another second order scheme is then considered and numerical results are discussed

boundary blow up in nonlinear elliptic equations of bieberbach rademacher type the research of the first author was carried out at victoria university melbourne with the support of the australian government through detya the second author has been supported by grant 2 cex06 11 18 2006

multiple solutions for a semiclassical schrodinger equation abstract this paper is concerned with the semiclassical nonlinear schrodinger equation e 2 u v x u f x u x r n u x 0 as x where e 0 is a small parameter under certain hypotheses on v and a general spectral assumption the existence and multiplicity of semiclassical solutions are obtained for asymptotically linear nonlinearity via variational methods

a smoothness indicator for adaptive algorithms for hyperbolic systems the formation of shock waves in solutions of hyperbolic conservation laws calls for locally adaptive numerical solution algorithms and requires a practical tool for identifying where adaption is needed in this paper a new smoothness indicator si is used to identify rough solution regions and is implemented in locally adaptive algorithms the si is based on the weak local truncation error of the approximate solution it was recently reported in s karni and a kurganov local error analysis for approximate solutions of hyperbolic conservation laws where error analysis and convergence properties were established the present paper is concerned with its implementation in scheme adaption and mesh adaption algorithms the si provides a general framework for adaption and is not restricted to a particular discretization scheme the implementation in this paper uses the central upwind scheme of a kurganov s noelle and g petrova siam j sci comput 23 707 2001 the extension of the si to two space dimensions is given numerical results in one and two space dimensions demonstrate the robustness of the proposed si and its potential in reducing computational costs and improving the resolution of the solution

internal duality for resolution of rings the following example provided the motivation for this paper let f be 8 the field with 8 elements and let f be its group of units which acts on f 8 8 u u via left multiplication the cohomology ring h f i f f 8 8 2 w xzr7 w x f x x x is of much interest to topologists a cs m it is a 2 1 2 3 straightforward exercise to construct minimal sets of homogeneous generators and relations for this ring there are 13 generators and 54 relations the surprising fact is that the minimal relations exhibit a certain internal duality

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

eigenvalue branches of the schrodinger operatorh λw in a gap of σ h the authors study the eigenvalue branches of the schrodinger operatorh w in a gap of h in particular they consider questions of asymptotic distribution of eigenvalues and bounds on the number of branches they also address the completeness problem

transient l1 error estimates for well balanced schemes on non resonant scalar balance laws abstract the ability of well balanced wb schemes to capture very accurately steady state regimes of non resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by greenberg and leroux 1996 15 see also the anterior wb glimm scheme in e 1992 8 this paper aims at showing by means of rigorous c t 0 l x 1 estimates that these schemes deliver an increased accuracy in transient regimes too namely after explaining that for the vast majority of non resonant scalar balance laws the c t 0 l x 1 error of conventional fractional step tang and teng 1995 45 numerical approximations grows exponentially in time like exp max g t x as a consequence of the use of gronwall s lemma it is shown that wb schemes involving an exact riemann solver suffer from a much smaller error amplification thanks to strict hyperbolicity their error grows at most only linearly in time see also layton 1984 30 numerical results on several test cases of increasing difficulty including the classical leveque yee s benchmark problem leveque and yee 1990 34 in the non stiff case confirm the analysis

convergence rate of approximate solutions to conservation laws with initial rarefactions the authors address the question of local convergence rate of conservative textit lip stable approximations u varepsilon x t to the entropy solution u x t of a genuinely nonlinear conservation law this question has been answered in the case of rarefaction free i e textit lip bounded initial data it has been shown that by post processing u varepsilon pointwise values of u and its derivatives may be recovered with an error as close to o varepsilon as desired where varepsilon measures in w 1 1 the truncation error of the approximate solution u varepsilon this paper extends the previous results by including textit lip unbounded initial data specifically it is shown that for arbitrary l infty cap bv initial data u and its derivatives may be recovered with an almost optimal modulo a spurious log factor error of o varepsilon ln varepsilon this analysis relies on obtaining new lip stability estimates for the speed a

a priori error estimates for numerical methods for scalar conservation laws part iii multidimensional flux splitting monotone schemes on non cartesian grids this paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we consider multidimensional flux splitting monotone schemes defined on non cartesian grids we identify those schemes which are consistent and prove that the l infty 0 t l 1 bbb r d norm of the error goes to zero as delta x 1 2 when the discretization parameter delta x goes to zero moreover we show that nonconsistent schemes can converge at optimal rates of delta x 1 2 because i the conservation form of the schemes and ii the so called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency

existence and multiplicity of semiclassical solutions for a schrodinger equation abstract we establish the existence and multiplicity of solutions for the semiclassical nonlinear schrodinger equation e 2 u v x u g x u for x r n u x 0 as x where e 0 is a small parameter and the potential v changes sign and may not be bounded from below and g x u is asymptotically linear in u as u

high resolution nonoscillatory central schemes for hamilton jacobi equations in this paper we construct second order central schemes for multidimensional hamilton jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle thus these schemes provide the first examples of nonoscillatory second order godunov type schemes based on global projection operators numerical experiments are performed l 1 l infty errors and convergence rates are calculated for convex hamiltonians numerical evidence confirms that our central schemes converge with second order rates when measured in the l1 norm advocated in our recent paper numer math to appear the standard l infty norm however fails to detect this second order rate

morse index and multiple solutions for the asymptotically linear schrodinger type equation abstract we study the nonlinear problem u v x u f x u x r n where the schrodinger type operator v is semibounded and has a finite number of eigenvalues below the infimum of the essential spectrum we classify the linear schrodinger equation prove the semi continuity and monotonicity of the index function and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear schrodinger equations based on an application of the classical ambrosetti rabinowitz symmetric mountain pass theorem

indecomposablea module summands inh v which are unstable algebras letp be a prime and denote bya the modp steenrod algebra we determine the indecomposablea module summands ofh p d f p which admit the structure of an unstablea algebra in fact it turns out that this is equivalent to the problem of determining those indecomposablea module summands which arise as the modp cohomology of a space or even a classifying space of a finite group we reduce this problem to one in modular representation theory namely for whichd andp is the projective cover of the trivial one dimensional gl d f p representationf p a permutation module our solution of this latter problem makes use of the classification of subgroups of gl d f p acting transitively on f p d 0 and hence depends on the classification of finite simple groups on feit thompson s odd order theorem ifp 2

nonnegative solutions for the degenerate logistic indefinite sublinear equation the goal of this paper is to study the nonnegative steady states solutions of the degenerate logistic indefinite sublinear problem we combine bifurcation method and linking local subsuper solution technique to show the existence and multiplicity of nonnegative solutions we employ a change of variable already used in a different context and the spectral singular theory to prove uniqueness results

on a class of asymptotically linear elliptic problems in rn under these conditions p is an asymptotically linear problem when this equation is considered in a bounded domain 0 r with say the dirichlet boundary condition there is a large literature on existence and multiplicity results of particular interest is then the case of resonance where s and s is the asymptotic linearization of the problem in other words s d s l 0 l 0 is the operator given by

compactness results for schrodinger equations with asymptotically linear terms abstract we study the nonlinear problem u v x f x u x r n lim x u x 0 where the schrodinger operator v is semibounded and the nonlinearity f is linearly bounded we establish compactness of palais smale sequences and cerami sequences for the associated energy functional under general spectral theoretic assumptions applying these results we obtain existence of three nontrivial solutions if the energy functional has a mountain pass geometry

a priori error estimates for approximate solutions to convex conservation laws summary we introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite difference approximation calculated with the scheme of engquistosher lax friedrichs or godunov this technique is a discrete counterpart of the duality technique introduced by tadmor siam j numer anal 1991 the error is related to the consistency error of cell averages of the entropy weak solution this consistency error can be estimated by exploiting a regularity structure of the entropy weak solution one ends up with optimal error estimates

on the asymptotic distribution of the eigenvalue branches of the schrodinger operator h λw in a spectral gap of h on considere des operateurs de schrodinger h v dans l espace de hilbert l 2 r avec un potentiel v satisfaisant 1 vel r on suppose que l intervalle a b est contenu dans l ensemble resolvant de h on obtient des estimations asymptotiques sur le nombre des branches de valeurs propes dans a b des familles d operateurs h w 0 quand ou w est un potentiel a courte portee non negatif

the fucik spectrum of schrodinger operator and the existence of four solutions of schrodinger equations with jumping nonlinearities abstract this paper contains the existence of four solutions of schrodinger equations with jumping nonlinearities the proof procedure is supported by a lot of new results initially a consequence is rendered as a minimax principle on h 1 r n which allows us to achieve the feasibility verification of the ps condition furthermore the constructions of minimal and maximal curves of fucik spectrum in q l see the introduction for the definition of q l warrant an intensive investigation that we encounter some thorny problems is largely due to the absence of compact embedding and the appearance of essential spectrum based on a nontrivial argument we can compute critical groups of homogeneous functional at zero if a b is free of fucik spectrum and a b q l this together with convexity and concavity offers a detailed description of the two curves by a series of sophisticated tricks additionally we present a new version of morse theory in view of the fact that classical version doesn t work directly for weak smooth functional on h 1 r n finally we prove a weak maximum principle for r n which serves as a tool to get a critical point in positive and negative cone respectively and also compute critical groups of critical points of mountain pass type with the help of above preparations we attain the ultimate aim by morse inequalities and various exact homology sequences

pointwise error estimates for scalar conservation laws with piecewise smooth solutions we introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation and in fact more general approximations to scalar conservation laws with piecewise smooth solutions to this end we derive a transport inequality for an appropriately weighted error function the key ingredient in our approach is a one sided interpolation inequality between classical l1 error estimates and lip stability bounds the one sided interpolation interesting for its own sake enables us to convert a global l1 result into a nonoptimal local estimate this in turn provides the necessary bounds on the coefficients of the above mentioned transport inequality estimates on the weighted error then follow from the maximum principle and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation unlike previous works in this direction our method can deal with finitely many waves with possible collisions moreover in our approach one does not follow the characteristics but instead makes use of the energy method and hence this approach could be extended to other types of approximate solutions

l 1 stability and error estimates for approximate hamilton jacobi solutions we study the l 1 stability and error estimates of general approximate solutions for the cauchy problem associated with multidimensional hamilton jacobi h j equations for strictly convex hamiltonians we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors we then demonstrate this general theory for two types of approximations approximate solutions constructed by the vanishing viscosity method and by godunov type finite difference methods if we let epsilon denote the small scale of such approximations the viscosity amplitude epsilon the spatial grad size delta x etc then our l 1 error estimates are of cal o epsilon and are sharper than the classical l infty results of order one half cal o sqrt epsilon the main building blocks of our theory are the notions of the semi concave stability condition and l 1 measure of the truncation error the whole theory could be viewed as a multidimensional extension of the lip prime stability theory for one dimensional nonlinear conservation laws developed by tadmor et al 34 24 25 in addition we construct new godunov type schemes for h j equations which consist of an exact evolution operator and a global projection operator here we restrict our attention to linear projection operators first order schemes we note however that our convergence theory applies equally well to nonlinear projections used in the context of modern high resolution conservation laws we prove semi concave stability and obtain l 1 bounds on their associated truncation errors l 1 convergence of order one then follows second order central godunov type schemes are also constructed numerical experiments are performed errors and orders are calculated to confirm our l 1 theory

on the regularity of approximate solutions to conservation laws with piecewise smooth solutions in this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted w1 1 space convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise error estimate is given in other words we are able to convert weak pointwise error estimates to optimal error bounds in a weighted w1 1 space for convex conservation laws the assumption of a weak pointwise error estimate is verified by tadmor siam j numer anal 28 1991 pp 891 906 therefore one immediate application of our w1 1 convergence theory is that for convex conservation laws we indeed have w1 1 error bounds for the approximate solutions to conservation laws furthermore the co en pointwise error estimates of tadmor and tang siam j numer anal 36 1999 pp 1739 1758 are recovered by the use of the w1 1 convergence result

existence and multiplicity of solutions for asymptotically linear nonperiodic hamiltonian elliptic system abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h v x u v v v x v h u x u v x r n u x v x 0 as x where z u v r n r r n 3 assuming the potential v is nonperiodic and sign changing h x z is nonperiodic in x and asymptotically quadratic in z u v existence and multiplicity of solutions are obtained via a variational approach

multiple solutions for a semilinear elliptic equation let ql be a bounded smooth domain in rn n 1 we consider the problem of finding nontrivial solutions to the elliptic boundary value problem au au h x iuip lu in q u 0 onoq where h 0 h 0 is holder continuous on q and p 1 a are constants let qlo denote the interior of the set where h vanishes in q we assume h 0 a e on q k1o and consider the eigenvalues ai k2 and ai 0o of the dirichlet problem in q and qlo respectively we prove that no nontrivial solution of the equation exists if a satisfies for some k 1 ak 90 k 1 a satisfies as ql 1 we consider the problem of finding nontrivial solutions to the elliptic boundary value problem 0 1 au au h x iuip u in q 0 2 u 0 on aq where h is a nonnegative not identically zero holder continuous function defined on q and p 1 a are constants ouyang 5 studied existence and uniqueness of positive solutions to problem 0 1 0 2 and also of its corresponding analogue on a compact riemannian manifold that paper mainly focused on studying the effect of vanishing of the received by the editors october 13 1993 and in revised form october 24 1994 1991 mathematics subject classification primary 35j25 35j20

existence and multiplicity of homoclinic orbits for the second order hamiltonian systems in this paper we study the following nonperiodic second order hamiltonian system u t l t u t ur t u t r where l t may not be uniformly positive definite for all t r and r t u is asymptotically quadratic in u as u by applying a generalized linking theorem in bartsch and ding 2006 8 we obtain the existence and multiplicity of homoclinic orbits for the above system

existence of solutions for semilinear elliptic equations with indefinite linear part we study equations on rn lu w x u p 1 ulu f u where l is an indefinite schrodinger operator 0 w x 0 as and subcritical nonlinearities we prove existence of solutions u h1 rn via an inversion to a dual variational problem

existence and multiplicity of semiclassical solutions for asymptotically hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system e 2 u v x u h u x u v in r n e 2 v v x v h v x u v in r n u x 0 and v x 0 as x where e 0 is a small parameter and the potential v is bounded below and h is asymptotically linear in z as z with z u v by applying a generalized linking theorem of strongly indefinite functionals we obtain the existence of multiple semiclassical solutions for the above system

existence and multiplicity of solutions for a non periodic schrodinger equation abstract this paper is concerned with the non periodic schrodinger equation u v x u f x u x r n u x 0 as x under a general spectral assumption the existence and multiplicity of solutions are obtained for asymptotically linear nonlinearity via variational methods

the optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws we are interested in the rate of convergence in l1 of the approximate solution of a conservation law generated by a monotone finite difference scheme kuznetsov has proved that this rate is 1 2 ussr comput math math phys 16 1976 pp 105 119 and topics numer anal iii in proc roy irish acad conf dublin 1976 pp 183 197 and recently teng and zhang have proved this estimate to be sharp for a linear flux siam j numer anal 34 1997 pp 959 978 we prove by constructing appropriate initial data for the cauchy problem that kuznetsov s estimates are sharp for a nonlinear flux as well

the large time behavior of the scalar genuinely nonlinear lax friedrichs scheme the lax friedrichs scheme approximating the scalar genuinely nonlinear conservation law u sub t f sub x u 0 where f u is say strictly convex double dot f dot a sub asterisk 0 is studied the divided differences of the numerical solution at time t do not exceed 2 t dot a sub asterisk to the 1 this one sided lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case in particular it is independent of the initial amplitude in sharp contrast to liner problems it guarantees the entropy compactness of the scheme in this case as well as providing a quantitive insight into the large time behavior of the numerical computation

existence of solutions for a nonhomogeneous semilinear elliptic equation abstract for a bounded domain a bounded caratheodory function g in r p 1 and a nonnegative locally integrable function h in which is strictly positive in a set of positive measure we prove that contrarily with the case h 0 there exists a solution of the semilinear elliptic problem u u g x u h u p 1 u f in u 0 on for every r and f l 2

an asymptotically periodic and asymptotically linear schrodinger equation with indefinite linear part abstract we consider the semilinear schrodinger equation u v x u f x u x r n u h 1 r n where v x is asymptotically periodic and sign changing f x u is asymptotically periodic in x and asymptotically linear as u under some weak assumptions on v and f we prove that the above problem has a nontrivial solution

generalized monotone schemes discrete paths of extrema and discrete entropy conditions solutions of conservation laws satisfy the monotonicity property the number of local extrema is a non increasing function of time and local maximum minimum values decrease increase monotonically in time this paper investigates this property from a numerical standpoint we introduce a class of fully discrete in space and time high order accurate difference schemes called generalized monotone schemes convergence toward the entropy solution is proven via a new technique of proof assuming that the initial data has a finite number of extremum values only and the flux function is strictly convex we define discrete paths of extrema by tracking local extremum values in the approximate solution in the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum as a corollary we obtain a proof of convergence for a muscl type scheme that is second order accurate away from sonic points and extrema

convergence rate of approximate solutions to weakly coupled nonlinear systems we study the convergence rate of approximate solutions to nonlinear hyperbolic systems which are weakly coupled through linear source terms such weakly coupled 2 x 2 systems appear for example in the context of resonant waves in gas dynamics equations this work is an extension of our previous scalar analysis this analysis asserts that a one sided lipschitz condition oslc or lip stability together with w 1 1 consistency imply convergence to the unique entropy solution moreover it provides sharp convergence rate estimates both global quantified in terms of the w s p norms and local we focus our attention on the lip stability of the viscosity regularization associated with such weakly coupled systems we derive sufficient conditions interesting for their own sake under which the viscosity and hence the entropy solutions are lip stable in an appropriate sense equipped with this we may apply the abovementioned convergence rate analysis to approximate solutions that share this type of lip stability

growing up positive solutions of semilinear elliptic boundary value problems this paper is devoted to the study of the existence uniqueness and asymptotic behavior of positive solutions of a class of degenerate boundary value problems for semilinear second order elliptic differential operators which originates from the so called yamabe problem in riemannian geometry our approach is based on the super sub solution method adapted to the degenerate case

the convergence rate of godunov type schemes godunov type schemes form a special class of transport projection methods for the approximate solution of nonlinear hyperbolic conservation laws the authors study the convergence rate of such sche