the neumann problem for the laplacian and the monge kantorovich mass transfer problem abstract we consider the natural neumann boundary condition for the laplacian we study the limit as p of solutions of p u p 0 in a domain with d u p p 2 u p g on we obtain a natural minimization problem that is verified by a limit point of u p and a limit problem that is satisfied in the viscosity sense it turns out that the limit variational problem is related to the monge kantorovich mass transfer problems when the measures are supported on

limits as p of p laplacian concave convex problems abstract we study the behavior as p of the sequel of positive weak solutions of the concave convex problem p div u p 2 u u q p u r p in u 0 in u 0 on where r n is a bounded domain 0 and the exponents q r satisfy lim p q p p 1 q lim p r p p 1 r with 0 q 1 r we characterize any positive uniform limit of a sequence of weak solutions of p as a viscosity solution of min u max u q u r u 0 in notice that the limit process decouples the nonlinearity we obtain existence non existence and global multiplicity of positive viscosity solutions of the limit problem in terms of the parameter

role of fundamental solutions for optimal lipschitz extensions on hyperbolic space in this paper we consider the problem of finding the relation between absolutely minimizing lipschitz extension of a given function defined over a subset of the hyperbolic space and the viscosity solution of the pde that appears from the associated variational problem here we have shown that the absolute minimizers can be fully characterized by a comparison principle comparison with cones with the fundamental solutions of the associated pde we have finally proved that the three properties i comparison with cones ii absolutely minimizing lipschitz extension and iii viscosity solution of associated pde are equivalent

estimates for p harmonic functions vanishing on a flat we study p harmonic functions in a domain crn near an m dimensional plane an m flat m where 0 m n 1 in particular let u be a positive p harmonic function with n p vanishing on a porti

the infinity laplacian with a transport term abstract we consider the following problem given a bounded domain r n and a vector field r n find a solution to u d u 0 in u f on where is the 1 homogeneous infinity laplace operator that is formally given by u d 2 u d u d u d u d u and f a lipschitz boundary datum if we assume that is a continuous gradient vector field then we obtain the existence and uniqueness of a viscosity solution by an l p approximation procedure also we prove the stability of the unique solution with respect to in addition when is more regular lipschitz continuous but not necessarily a gradient using tug of war games we prove that this problem has a solution

the aronsson euler equation for absolutely minimizing lipschitz extensions with respect to carnot caratheodory metrics we derive the euler lagrange equation also known in this setting as the aronsson euler equation for absolute minimizers of the l variational problem formula math where c g is an open subset of a carnot group 0 u denotes the horizontal gradient of u r and the lipschitz class is defined in relation to the carnot caratheodory metric in particular we show that absolute minimizers are infinite harmonic in the viscosity sense as a corollary we obtain the uniqueness of absolute minimizers in a large class of groups this result extends previous work of jensen and of crandall evans and gariepy we also derive the aronsson euler equation for more regular absolutely minimizing lipschitz extensions corresponding to those carnot caratheodory metrics which are associated to free systems of vector fields

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

solutions to an inhomogeneous equation involving infinity laplacian abstract we obtain existence and uniqueness results of viscosity solutions to the dirichlet boundary value problem for a nonlinear highly degenerate elliptic equation of the form 1 d u h u f where 0 h 2 and u denotes the so called infinity laplacian operator given by u i j 1 n u x i u x j 2 u x i x j we also give an asymptotic behavior of the viscosity solutions of two kinds of inhomogeneous infinity laplace equations near an isolated singular point

a stochastic differential game for the inhomogeneous laplace equation given a bounded c 2 domain gr m and functions g 2 c g r and h 2 c g r 0 let u denote the unique viscosity solution to the equation 2 u h in g with boundary data g we provide a representation for u as the value of a two player zero sum stochastic differential game ams 2000 subject classifications 91a15 91a23 35j70 49l20 g r 0 be given we study a two player zero sum stochastic differential game sdg defined in terms of an m dimensional state process that is driven by a one dimensional brow nian motion played until the state exits the domain the functions g and h serve as terminal and respectively running payoffs the players controls enter in a diffusion coefficient and in an unbounded drift coefficient of the state process the dynamics are degenerate in that it is possible for the players to completely switch off the brownian motion we show that the game has value and characterize the value function as the unique viscosity solution u uniqueness of solutions is known from 9 of the equation 2 u h in g

l variational problems associated to measurable finsler structures abstract we study l variational problems associated to measurable finsler structures in euclidean spaces we obtain existence and uniqueness results for the absolute minimizers

intrinsic geometry and analysis of diffusion processes and l variational problems the aim of this paper is twofold first we obtain a better understanding of the intrinsic distance of diffusion processes precisely a for all n 1 the diffusion matrix a is weak upper semicontinuous on if and only if the intrinsic differential and the local intrinsic distance structures coincide b if n 1 or if n 2 and a is weak upper semicontinuous on the intrinsic distance and differential structures always coincide c if n 2 and a fails to be weak upper semicontinuous on the non coincidence of the intrinsic distance and differential structures depend on the geometry of the non weak upper semicontinuity set of a second for an arbitrary diffusion matrix a we show that the intrinsic distance completely determines the absolute minimizer of the corresponding l variational problem and then obtain the existence and uniqueness for given boundary data we also give an example of a diffusion matrix a for which there is an absolute minimizer that is not of class c1 when a is continuous we also obtain the linear approximation property of the absolute minimizer

new properties of convex functions in the heisenberg group we prove some new properties of the weakly h convex functions recently introduced by danielli garofalo and nhieu as an interesting application of our results we prove a theorem of busemann feller alexandrov type in the heisenberg groups h n 1 2

on the dirichlet and serrin problems for the inhomogeneous infinity laplacian in convex domains regularity and geometric results given an open bounded subset of mathbb r n which is convex and satisfies an interior sphere condition we consider the pde delta infty u 1 in subject to the homogeneous boundary condition u 0 on we prove that the unique solution to this dirichlet problem is power concave precisely 3 4 concave and it is of class c1 we then investigate the overdetermined serrin type problem formerly considered in buttazzo and kawohl int math res not pp 237 247 2011 obtained by adding the extra boundary condition nabla u a on by using a suitable p function we prove that if satisfies the same assumptions as above and in addition contains a ball which touches at two diametral points then the existence of a solution to this serrin type problem implies that necessarily the cut locus and the high ridge of coincide in turn in dimension n 2 this entails that must be a stadium like domain and in particular it must be a ball in case its boundary is of class c2

boundary hemivariational inequality of parabolic type abstract in this paper we consider a parabolic hemivariational inequality with a nonmonotone multivalued boundary condition and with a time dependent pseudomonotone operator we establish the existence of solutions for such inequality our proof is based on a known surjectivity result for operators of pseudomonotone type we present an example of semipermeability heat conduction problem to which the result applies

boundary behavior for the solutions to dirichlet problems involving the infinity laplacian abstract in this paper by constructing suitable comparison functions we mainly analyze the exact boundary behavior for the unique solution near the boundary to the singular dirichlet problem u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 g is decreasing on 0 with lim s 0 g s g is normalized regularly varying at zero with index 1 and b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary

solvability and continuous dependence results for second order nonlinear evolution inclusions with a volterra type operator abstract the paper deals with second order nonlinear evolution inclusions and their applications we study evolution inclusions involving a volterra type integral operator which are considered within the framework of an evolution triple of spaces first we deliver a result on the unique solvability of the cauchy problem for the inclusion by combining a surjectivity result for multivalued pseudomonotone operators and the banach contraction principle next we provide a theorem on the continuous dependence of the solution to the inclusion with respect to the operators involved in the problem finally we consider a dynamic frictional contact problem of viscoelasticity for materials with long memory and indicate how the result on evolution inclusion is applicable to the model of the contact problem

weak fubini property and infinity harmonic functions in riemannian and sub riemannian manifolds we examine the relationship between infinity harmonic functions absolutely minimizing lipschitz extensions strong absolutely minimizing lipschitz extensions and absolutely gradient minimizing extensions in carnot carath eodory spaces using the weak fubini property we show that absolutely minimizing lipschitz extensions are infinity harmonic in any sub riemannian manifold

absolutely minimal lipschitz extension of tree valued mappings we prove that every lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal lipschitz extension amle we relate these extensions to a stochastic game called politics a generalization of a game called tug of war that has been used in peres et al j am math soc 22 1 167 210 2009 to study real valued amles

a multiphase level set framework for image segmentation using the mumford and shah model we propose a new multiphase level set framework for image segmentation using the mumford and shah model for piecewise constant and piecewise smooth optimal approximations the proposed method is also a generalization of an active contour model without edges based 2 phase segmentation developed by the authors earlier in t chan and l vese 1999 in scale space 99 m nilsen et al eds lncs vol 1682 pp 141 151 and t chan and l vese 2001 ieee ip 10 2 266 277 the multiphase level set formulation is new and of interest on its own by construction it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies including triple junctionss in the piecewise smooth case only two level set functions formally suffice to represent any partition based on the four color theorem finally we validate the proposed models by numerical results for signal and image denoising and segmentation implemented using the osher and sethian level set method

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

boundary differentiability of infinity harmonic functions abstract we prove that an infinity harmonic function u x c is differentiable at a boundary point x 0 if both and the boundary data g x are differentiable at x 0 this work improved a former result of changyou wang and yifeng yu 2012 10 they obtained the boundary differentiability of u x with the c 1 assumption on and g x

coercive and semicoercive hemivariational inequalities variational inequalities have a concrete mechanical meaning they are expressions of the principle of virtual work or power in its inequality form in the case of lack of monotonicity of the underlying stress strain or reaction displacement conditions or equivalently if the corresponding energy functionals are nonconvex the variational expressions are no longer variational inequalities another type of inequality expression arises as variational formulation of the problem these expressions have been introduced in mechanics by the author and are called hemivariational inequalities the aim of this paper is the formulation of an abstract mathematical theory for coercive and semicoercive hemivariational inequalities

the vibration of a non prismatic beam on an inertial elastic half plane abstract the problem of vibration of a non prismatic beam resting on an inertial elastic half plane described by classical elastokinetics equations is solved using chebyshev series approximation as a result closed analytical formulae for the sought solution s coefficients the passive foundation pressure function and the system displacement function were obtained the method was applied to solve the problem of harmonically excited vibration

comparison principles for infinity laplace equations in finsler metrics abstract in this paper we study comparison principles for normalized finsler infinity laplace operators with nonhomogeneous terms that depend on solutions and their gradients this is achieved by using a combination of sup convolution methods to approximate viscosity solutions by semiconvex functions and finite difference approximation schemes characterizations of viscosity subsolutions and supersolutions of equations with constant nonhomogeneous terms through comparison with quadratic finsler cone are also discussed

a tour of the theory of absolutely minimizing functions a detailed analysis of the class of absolutely minimizing functions in euclidean spaces and the relationship to the infinity laplace equation

asymptotic estimates of boundary blow up solutions to the infinity laplace equations abstract in this paper we study the asymptotic behavior of boundary blow up solutions to the equation u b x f u in where is the laplacian the nonlinearity f is a positive increasing function in 0 and the weighted function b c is positive in and may vanish on the boundary we first establish the exact boundary blow up estimates with the first expansion when f is regularly varying at infinity with index p 3 and the weighted function b is controlled on the boundary in some manner furthermore for the case of f s s p 1 c g s with the function g normalized regularly varying with index q 0 we obtain the second expansion of solutions near the boundary it is interesting that the second term in the asymptotic expansion of boundary blow up solutions to the infinity laplace equation is independent of the geometry of the domain quite different from the boundary blow up problems involving the classical laplacian

existence of anti periodic solutions for quasilinear parabolic hemivariational inequalities abstract in this paper we consider the problem of the existence of anti periodic solutions for quasilinear parabolic hemivariational inequalities with a pseudomonotone operator using regularized approximating and first eigenfunction methods and the surjectivity result we prove the existence of solutions for quasilinear parabolic hemivariational inequalities

slope estimate and boundary differentiability for inhomogeneous infinity laplace equation on convex domains abstract we study the boundary differentiability for inhomogeneous infinity laplace equations on convex domains with the inhomogeneous term f c l and differentiable boundary data g at a flat point the boundary point where the blow up of the domain is the half space u is differentiable due to a previous result of the second author in hong 2014 at a corner point the boundary point where the blow up of the domain is not the half space we establish a slope estimate for u and provide a necessary and sufficient condition for the boundary differentiability of u in this paper

an axiomatic approach to image interpolation we discuss possible algorithms for interpolating data given in a set of curves and or points in the plane we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolute minimal lipschitz extension model amle is singled out and studied in more detail we show experiments suggesting a possible application the restoration of images with poor dynamic range

derivation of the aronsson equation for c1 hamiltonians it is proved herein that any absolute minimizer u for a suitable hamiltonian h c r r u is a viscosity solution of the aronsson equation hp du u x h du u x x 0 in u the primary advance is to weaken the assumption that h c used by previous authors to the natural condition that h c

the second order expansion of boundary blow up solutions for infinity laplacian equations abstract in this paper by using karamata regular variation theory and a perturbed argument we study the second order expansion of viscosity solutions to boundary blow up elliptic problem u b x f u x u where is a bounded domain with c 2 boundary in r n b c is non negative and non trivial in f c 1 0 is a normalized regularly varying function at infinity with index 3

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

blow up rates of large solutions for infinity laplace equations in this paper by constructing suitable comparison functions we mainly give the boundary behavior of solutions to boundary blow up elliptic problems u b x f u x u where is a bounded domain with smooth boundary in r n the operator is the laplacian b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary and the nonlinear term f is a varying function at infinity whose variation at infinity is not regular

a general theorem of existence of quasi absolutely minimal lipschitz extensions in this paper we consider a wide class of generalized lipschitz extension problems and the corresponding problem of finding absolutely minimal lipschitz extensions we prove that if a minimal lipschitz extension exists then under certain other mild conditions a quasi absolutely minimal lipschitz extension must exist as well here we use the qualifier quasi to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal lipschitz extension up to several factors that can be made arbitrarily small

relaxation of hamilton jacobi equations we study the relaxation of hamilton jacobi equations the relaxation in our terminology is the following phenomenon the pointwise supremum over a certain collection of subsolutions in the almost everywhere sense of a hamilton jacobi equation yields a viscosity solution of the convexified hamilton jacobi equation this phenomenon has recently been observed in 13 in eikonal equations we show in this paper that this relaxation is a common phenomenon for a wide range of hamilton jacobi equations

convexity criteria and uniqueness of absolutely minimizing functions we show that an absolutely minimizing function with respect to a convex hamiltonian h mathbb r n rightarrow mathbb r is uniquely determined by its boundary values under minimal assumptions on h along the way we extend the known equivalences between comparison with cones convexity criteria and absolutely minimizing properties to this generality these results perfect a long development in the uniqueness existence theory of the archetypal problem of the calculus of variations in l

sup inf explicit formulas for minimal lipschitz extensions for 1 fields on r n abstract we study the relationship between the lipschitz constant of 1 field introduced in 12 and the lipschitz constant of the gradient canonically associated with this 1 field moreover we produce two explicit formulas which are two extremal minimal lipschitz extensions for 1 fields as a consequence of the previous results for the problem of minimal extension by lipschitz continuous functions from r m to r n we produce explicit formulas similar to those of bauschke and wang see 7 finally we show that wells s extensions see 24 of 1 fields are absolutely minimal lipschitz extensions when the domain of 1 field to expand is finite we provide a counter example showing that this result is false in general

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

application of chebyshev series to solution of non prismatic beam vibration problems the problem of the vibration of a non prismatic beam resting on a two parameter elastic foundation has been solved by applying the approximation by chebyshev series as a result closed analytical formulas defining the coefficients of the sought solutions were obtained the method was used to solve the eigenproblem for a simply supported beam and a cantilever beam the obtained results were compared with the results reported by other authors

dynamic stability problem of a non prismatic rod a dynamic stiffness matrix for a non prismatic rod finite element resting on a two parameter non homogenous elastic foundation has been determined to obtain the solution the shape function was approximated by chebyshev series this yielded closed analytical formulae for the coefficients of the matrices sought the finite element obtained was used to solve the dynamic stability problem for a non prismatic cantilever column the results were compared with those reported by other authors

the limit as p of the first eigenvalue for the p laplacian with mixed dirichlet and robin boundary conditions abstract we analyze the behavior as p of the first eigenvalue of the p laplacian with mixed boundary conditions of dirichlet robin type we find a nontrivial limit that we associate to a variational principle involving l norms moreover we provide a geometrical characterization of the limit value as well as a description of it using optimal mass transportation techniques our results interpolate between the pure dirichlet case and the mixed dirichlet neumann case

existence and asymptotic stability for the semilinear wave equation with boundary damping and source term in this paper we consider the semilinear wave equation with boundary conditions this work is devoted to prove the existence of solutions and uniform decay rates of the wave equation with boundary damping and source term

a weighted eigenvalue problem of the degenerate operator associated with infinity laplacian abstract in this paper we study a weighted eigenvalue problem of the degenerate operator associated with infinity laplacian h u a x u h 1 u 0 in u 0 on where h 1 h u d u h 3 u is the h homogeneous infinity laplacian and a x is a positive continuous bounded function in we prove the existence of the principal eigenvalue and a corresponding positive eigenfunction the approach to the weighted eigenvalue problem is based on the maximum principle and when a parameter is less than the principal eigenvalue some existence and uniqueness results related to this problem are established

anti periodic solutions to a class of nonlinear differential equations in hilbert space abstract a broad class of nonlinear non monotone anti periodic boundary value problems in a hilbert space is considered as a preliminary step the existence uniqueness and continuous dependence upon data of anti periodic solutions to some first and second order evolution equations associated to odd noncoercive monotone operators is established applications of the theory to nonlinear partial differential equations are also discussed

stochastic homogenization of l variational problems abstract we present a homogenization result for l variational problems in general stationary ergodic random environments by introducing a generalized notion of distance function a special solution of an associated eikonal equation and demonstrating a connection to absolute minimizers of the variational problem we obtain the homogenization result as a consequence of the fact that the latter homogenizes in random environments

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

asymptotic boundary estimates to infinity laplace equations with γ varying nonlinearity abstract in a previous paper of the authors wang et al 2014 40 the asymptotic estimates of boundary blow up solutions were established to the infinity laplace equation u b x f u in r n with the nonlinearity 0 f c 0 regularly varying at and the weighted function b c positive in and vanishing on the boundary the present paper gives a further investigation on the asymptotic behavior of boundary blow up solutions to the same equation by assuming f to be varying note that a varying function grows faster than any regularly varying function we first quantitatively determine the boundary blow up estimates with the first expansion relying on the decay rate of b near the boundary and the growth rate of f at infinity and further characterize these results via examples possessing various decay rates for b and growth rates for f in particular we pay attention to the second order estimates of boundary blow up solutions it was observed in our previous work that the second expansion of solutions to the infinity laplace equation is independent of the geometry of the domain quite different from the classical laplacian the second expansion obtained in this paper furthermore shows a substantial difference on the asymptotic behavior of boundary blow up solutions between the infinity laplacian and the classical laplacian

counterexample to c1 boundary regularity of infinity harmonic functions abstract in this short note we construct a planar infinity harmonic function u in a c 1 domain with smooth boundary data g however d u is not continuous on the boundary changyou wang and yifeng yu 2012 proved the c 1 boundary regularity of planar infinity harmonic functions provided that and g are c 2 they asked the question whether the result holds when and g are assumed to be c 1 our counterexample answered this question negatively our construction of the counterexample is strongly inspired by the counterexample of dongsheng li and lihe wang 2006 for uniformly elliptic equations

existence of minimizers for nonlevel convex supremal functionals the paper is devoted to determining necessary and sufficient conditions for existence of solutions to the problem inf mathop rm ess sup x in omega f nabla u x u in u 0 w 1 infty 0 omega when the supremand f is not necessarily level convex these conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum several conditions of convexity for the supremand f are also investigated

infinity laplacian equation with strong absorptions abstract we study regularity properties of solutions to reaction diffusion equations ruled by the infinity laplacian operator we focus our analysis in models presenting plateaus i e regions where a non negative solution vanishes identically we obtain sharp geometric regularity estimates for solutions along the boundary of plateaus sets in particular we show that the n hausdorff measure of the plateaus boundary is finite for a universal number 0

slope estimate and boundary differentiability of infinity harmonic functions on convex domains abstract we study the boundary differentiability of infinity harmonic functions with given differentiable boundary data on convex domains at a flat point the boundary point where the blow up of the domain is the half space the infinity harmonic function u is differentiable due to a previous result of the first author in hong 2013 at a corner point the boundary point where the blow up of the domain is not the half space an example shows that u is not necessarily differentiable in this paper we establish a slope estimate for u at corner points and provide a necessary and sufficient condition for the differentiability of u at corner points

l variational problems and aronsson equations in this paper we prove that viscosity solutions of aronsson equations are absolute minimizers in certain l variational problems

exact analysis of beams on two parameter elastic foundations abstract efficient beams on two parameter elastic foundation finite elements have recently been developed the stiffness matrix and nodal loud vector of these elements have been derived on the basis of the exact displacement function obtained from the solution of the governing differential equation most of the existing elements are however either limited to certain combinations of beam and foundation parameters or provide only the solution of the homogeneous form of the governing equation in this paper a new finite element is derived which eliminates these limitations the stiffness matrix nodal load vector and shape function of the clement are derived using the differential equation of a beam on a two parameter elastic foundation the complete solution of the equation corresponding to the most common types of load is also presented this permits the determination of the deflections and internal forces anywhere along a simple or continuous beam on two parameter foundations

exact static and dynamic stiffness matrices for general variable cross section members this paper concerns the formulation of a new finite element method for the solution of beams with variable cross section using only one element it is possible to derive the exact static and dynamic stiffness matrices up to the accuracy of the computer for any polynomial variation of axial torsional and bending stiffnesses along the beam examples are given for the accuracy and efficiency of the method

another way to say harmonic it is known that solutions of u n i j 1 u xi u xj u xjxj 0 that is the harmonic functions are exactly those functions having a comparison property with respect to the family of translates of the radial solutions g x a x we establish a more difficult linear result a function in r n is harmonic if it has the comparison property with respect to sums of n translates of the radial harmonic functions g x a x 2 n for n 2 and g x b ln x for n 2 an attempt to generalize these results for u 0 p and u 0 p 2 to the general p laplacian leads to the fascinating discovery that certain sums of translates of radial p superharmonic functions are again p superharmonic mystery remains the class of p superharmonic functions so constructed for p 2 does not suffice to characterize p subharmonic functions

inhomogeneous infinity laplace equation abstract we present the theory of the viscosity solutions of the inhomogeneous infinity laplace equation x i u x j u x i x j 2 u f in domains in r n we show existence and uniqueness of a viscosity solution of the dirichlet problem under the intrinsic condition f does not change its sign we also discover a characteristic property which we call the comparison with standard functions property of the viscosity sub and super solutions of the equation with constant right hand side applying these results and properties we prove the stability of the inhomogeneous infinity laplace equation with nonvanishing right hand side which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right hand side and boundary data are perturbed in the end we prove the stability of the well known homogeneous infinity laplace equation x i u x j u x i x j 2 u 0 which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right hand side and boundary data are perturbed simultaneously

the eigenvalue problem the euler lagrange equation of the nonlinear rayleigh quotient left int omega nabla u p dx right bigg left int omega u p dx right is div left nabla u p 2 nabla u right lambda p p u p 2 u where lambda p p is the minimum value of the quotient the limit as p to infty of these equations is found to be max left lambda infty frac nabla u x u x delta infty u x right 0 where the constant lambda infty lim p to infty lambda p is the reciprocal of the maximum of the distance to the boundary of the domain

the application of chebyshev polynomials to the solution of the nonprismatic timoshenko beam vibration problem abstract chebyshev series approximation is applied to solve the problem of vibration of the nonprismatic timoshenko beam resting on a two parameter elastic foundation as a result a system of equations whose coefficients have a closed form for calculating the coefficients of the sought solution is obtained the method is used to solve the free vibration problem for simple supported and clamped free nonprismatic beams the results are compared with the results obtained by other authors also the nonprismatic beam stability problem is solved and the results are compared with those obtained for euler beams to demonstrate the method s applicability to more complex systems the problem of stability of a frame system is solved

weakly coupled systems of the infinity laplace equations we derive the weakly coupled systems of the infinity laplace equations via a tug of war game introduced by peres schramm sheffield and wilson 2009 we establish existence uniqueness results of the solutions and introduce a new notion of generalized cones for systems by using generalized cones we analyze blow up limits of solutions

on boundary blow up solutions to equations involving the laplacian abstract given a non negative continuous function h on r such that h x 0 0 for all x h x t 0 in 0 and h x t non decreasing in t for each x we study the boundary value problem u h x u in u on where r n n 2 is a bounded domain and is the laplacian a degenerate elliptic operator we provide sufficient conditions on h under which the above problem admits a solution or fails to admit a solution a necessary and sufficient condition on f is given for a solution to exist in the special case when h x t b x f t in the latter case an asymptotic boundary behavior of solutions will be studied as an application a sufficient condition on f will be given to ensure the uniqueness of solutions in case b is a constant

on the evolution governed by the infinity laplacian we investigate the basic properties of the degenerate and singular evolution equation ut d 2 u du du du du which is a parabolic version of the increasingly popular infinity laplace equa tion we prove existence and uniqueness results for both dirichlet and cauchy problems establish interior and boundary lipschitz estimates and a harnack inequality and also provide numerous explicit solutions

reaction diffusion equations for the infinity laplacian abstract we derive sharp regularity for viscosity solutions of an inhomogeneous infinity laplace equation across the free boundary when the right hand side does not change sign and satisfies a certain growth condition we prove geometric regularity estimates for solutions and conclude that once the source term is comparable to a homogeneous function then the free boundary is a porous set and hence has zero lebesgue measure additionally we derive a liouville type theorem when near the origin the right hand side grows not faster than third degree homogeneous function we show that if a non negative viscosity solution vanishes at a point then it has to vanish everywhere

uniqueness of lipschitz extensions minimizing the sup norm of the gradient in this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values geometrically this can be interpreted as finding a minimal lipschitz extension due to the weak convexity of the functional associated to this problem solutions are generally nonunique by adopting g aronsson s notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation in fact we actually prove a weak maximum principle for this partial differential equation which in some sense is the euler equation for the minimization problem this is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity to overcome this difficulty we use the weak solutions of m g crandall and p l lions also known as viscosity solutions in conjunction with some arguments using integration by parts

existence results for evolution hemivariational inequalities of second order abstract in this paper we consider evolution hemivariational inequalities of second order with a time dependent pseudomonotone operator and nonmonotone multivalued perturbations we present the existence of solutions for such inequality the proof profits from a result on the surjectivity of operators of pseudomonotone type we discuss some examples which indicate the practical importance of our theoretical findings

vector valued optimal lipschitz extensions consider a bounded open set and a lipschitz function does this function always have a canonical optimal lipschitz extension to all of u we propose a notion of optimal lipschitz extension and address existence and uniqueness in some special cases in the case n m 2 we show that smooth solutions have two phases in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans we also prove existence and uniqueness for the extension problem on finite graphs 2011 wiley periodicals inc

uniqueness and nonuniqueness of viscosity solutions to aronsson s equation for a bounded domain omega subset mathbb r n and g in c delta omega assume that h in c 2 mathbb r n is convex and coercive and that p in mathbb r n h p rm min mathbb r n h has no interior points then we establish the uniqueness of viscosity solutions to the dirichlet problem of aronsson s equation begin aligned mathcal a u h p nabla u otimes h p nabla u nabla 2 u 0 rm in omega u g rm on delta omega end aligned for h h p x depending on x we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to aronsson s equation and that of the hamilton jacobi equation h nabla u x c

on a family of torsional creep problems involving rapidly growing operators in divergence form let n n 2 be a bounded domain with smooth boundary and p n be a sequence of real numbers converging to as n for each integer n 1 we define the function varphi n t p n vert t vert p n 2 te vert t vert p n for all t and we prove the existence of a unique nonnegative variational solution for the problem div n u x u x u x n 1 when x subject to the homogeneous dirichlet boundary condition next we establish the uniform convergence in of the sequence of solutions for the above family of equations to the distance function to the boundary of our result complements the earlier developments on the topic obtained by payne and philippin 26 kawohl 21 bhattacharya dibenedetto and manfredi 2 perez llanos and rossi 27 and bocea and mih ilescu 4

an optimal pointwise morrey sobolev inequality abstract let be a bounded smooth domain of r n n 1 for each p n we study the optimal function s s p in the pointwise inequality v x s x v l p x v w 0 1 p we show that s p c 0 0 1 n p and that s p converges pointwise to the distance function to the boundary as p moreover we prove that if is convex then s p is concave and has a unique maximum point

an axiomatic approach to scalar data interpolation on surfaces we discuss possible algorithms for interpolating data given on a set of curves in a surface of 3 we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolutely minimizing lipschitz extension model amle is singled out and studied in more detail we study the correctness of our numerical approach and we show experiments illustrating the interpolation of data on some simple test surfaces like the sphere and the torus

principal eigenvalue of a very badly degenerate operator and applications abstract in this paper we define and investigate the properties of the principal eigenvalue of the singular infinity laplace operator u d 2 u d u d u d u d u this operator arises from the optimal lipschitz extension problem and it plays the same fundamental role in the calculus of variations of l functionals as the usual laplacian does in the calculus of variations of l 2 functionals our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of berestycki nirenberg and varadhan h berestycki l nirenberg s r s varadhan the principal eigenvalue and maximum principle for second order elliptic operators in general domains comm pure appl math 47 1 1994 47 92 in the case of uniformly elliptic linear operators as an application we obtain existence and uniqueness results for certain related non homogeneous problems and decay estimates for the solutions of the evolution problem associated to the infinity laplacian

nonlocal tug of war and the infinity fractional laplacian motivated by the tug of war game studied by peres et al in 2009 we consider a nonlocal version of the game that goes as follows at every step two players pick respectively a direction and then instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount e 0 as is done in the classical case it is an s stable levy process that chooses at the same time both the direction and the distance to travel starting from this game we heuristically derive a deterministic nonlocal integrodifferential equation that we call the infinity fractional laplacian we study existence uniqueness and regularity both for the dirichlet problem and for a double obstacle problem both problems having a natural interpretation as tug of war games 2011 wiley periodicals inc

a sharp uniqueness result for a class of variational problems solved by a distance function we consider the minimization problem for an integral functional j possibly nonconvex and non coercive in w 1 1 0 where r n is a bounded smooth set we prove su cient conditions in order to guarantee that a suitable minkowski distance is a minimizer of j the main result is a necessary and su cient condition in order to have the uniqueness of the minimizer we show some application to the uniqueness of solution of a system of pdes of monge kantorovich type arising in problems of mass transfer theory

existence of antiperiodic solutions for hemivariational inequalities in this paper we consider the problem of existence of antiperiodic solutions for first order and second order hemivariational inequalities with a pseudomonotone operator we first give the surjectivity result and then prove a existence of antiperiodic solutions for hemivariational inequalities with the surjectivity result

on the behaviour of harmonic functions near isolated points abstract in this work we study the behaviour of harmonic functions near isolated exceptional points we discuss two situations i when points are in the interior and ii when points are on a flat portion of the boundary on which a solution vanishes we consider both bounded and unbounded solutions under the assumptions of boundedness solutions are well behaved in either case we show that unbounded solutions in i change sign and provide some lower bounds for growth rates valid for both i and ii in ii if the solution has one sign then we can also find an upper bound we also look at the case of the ball