homogenization of an optimal control problem we consider an optimal control problem in which both the state equation and the cost functional have rapidly oscillating coefficients characterized respectively by matrices a varepsilon and b varepsilon where varepsilon is a small parameter we make no periodicity assumption we study the limit of the problem when varepsilon rightarrow 0 and work in the framework of h convergence we prove that the limit satisfies a problem similar to the original one but with matrices a 0 the h limit of a varepsilon and b sharp which is a perturbation of the h limit b 0 of b varepsilon we also study some particular cases this paper extends former results obtained by kesavan and vanninathan in the periodic case

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

multiscale asymptotic analysis and computation of optimal control for elliptic systems with constraints this paper reports the multiscale analysis and numerical algorithms for a class of distributed elliptic optimal control problems with constraints multiscale asymptotic expansions are presented and an explicit rate of convergence is derived a multiscale numerical method is developed numerical experiments are carried out to validate the predicted convergence results

heterogeneous multiscale method for optimal control problem governed by parabolic equation with highly oscillatory coefficients abstract in this paper we investigate the heterogeneous multiscale method hmm for the optimal control problem governed by the parabolic equation with highly oscillatory coefficients the state variable and adjoint state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements while the control variable is discretized by the piecewise constants by applying the well known lions lemma to the discretized optimal control problem we obtain the necessary and sufficient optimality conditions a priori error estimates are derived for the state co state and the control with uniform bounded constants finally numerical results are presented to illustrate our theoretical findings

factorization of j unitary matrix polynomials on the line and a schur algorithm for generalized nevanlinna functions we prove that a 2 2 matrix polynomial which is j unitary on the real line can be written as a product of normalized elementary j unitary factors and a j unitary constant in the second part we give an algorithm for this factorization using an analog of the schur transformation

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

wavelet based numerical homogenization a numerical homogenization procedure for elliptic differential equations is presented it is based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination of the fine scale contributions if the operator is in divergence form this is preserved by the homogenization procedure for periodic problems results similar to classical effective coefficient theory are proved the procedure can be applied to problems that are not cell periodic

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

spectral problems for generalized jacobi matrices a new class of generalized jacobi matrices is introduced every proper real rational function is proved to be the m function of a unique finite generalized jacobi matrix moreover every generalized nevanlinna function m which is a solution of a determinate indefinite moment problem turns out to be the m function of a unique infinite generalized jacobi matrix the method we use is based on the step by step schur process of solving the indefinite moment problem the convergence of the sequence of subdiagonal pade approximants for the corresponding hamburger series is investigated

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

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

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

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

finite perturbations of orthogonal polynomials in this paper we study orthogonal polynomials pn which arise from a given system of orthogonal polynomials pn by a finite perturbation of the recurrence coefficients i e the recurrence coefficients n n 1 of pn respectively n n 1 of pn are related to each other by n l n m and n 1 l n 1 m for n n where l and m are fixed nonnegative integers closed expressions for the polynomial pn in terms of pn and its associated polynomial pn 1 1 as well as for the orthogonality measure of pn in terms of the orthogonality measure of pn are given in fact the results are presented for the most general case when is an arbitrary not necessarily positive measure

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

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

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

regularization of the indefinite stieltjes moment problem abstract a not regular indefinite stieltjes moment problem is studied in the generalized stieltjes class n k the new class of the sequence s s i i 0 l is defined which is called regular class regularization of the indefinite stieltjes moment problem is obtained the set of solutions of this problem is found in terms of the generalized stieltjes polynomials and s fractions with the shift

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

analysis of the heterogeneous multiscale method for parabolic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

numerical methods for multiscale elliptic problems we present an overview of the recent development on numerical methods for elliptic problems with multiscale coefficients we carry out a thorough study of two representative techniques the heterogeneous multiscale method hmm and the multiscale finite element method msfem for problems with scale separation but without specific assumptions on the particular form of the coefficients analytical and numerical results show that hmm gives comparable accuracy as msfem with much less cost for problems without scale separation our numerical results suggest that hmm performs at least as well as msfem in terms of accuracy and cost even though in this case both methods may fail to converge since the cost of msfem is comparable to that of solving the full fine scale problem one might expect that it does not need scale separation and still retains some accuracy we show that this is not the case specifically we give an example showing that if there exists an intermediate scale comparable to h the size of the macroscale mesh then msfem commits a finite error even with overlapping

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

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

a short and versatile finite element multiscale code for homogenization problems we describe a multiscale finite element fe solver for elliptic or parabolic problems with highly oscillating coefficients based on recent developments of the so called heterogeneous multiscale method hmm the algorithm relies on coupled macro and microsolvers the framework of the hmm allows to design a code whose structure follows the classical finite elements implementation at the macro level to account for the fine scales of the problem elementwise numerical integration is replaced by micro fe methods on sampling domains we discuss a short and flexible fe implementation of the multiscale algorithm which can accommodate simplicial or quadrilateral fe and various coupling conditions for the constrained micro simulations extensive numerical examples including three dimensional and time dependent problems are presented illustrating the efficiency and the versatility of the computational strategy

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

an efficient derivation of the aronsson equation for 1 p the equation which characterizes minima of the functional u u du p dx subject to fixed values of u on u is p u 0 here p is the well known p laplacian when p the corresponding functional is u du 2 l u a new feature arises in that minima are no longer unique unless u is allowed to vary leading to the idea of absolute minimizers aronsson showed that then the appropriate equation is u 0 that is u is infinity harmonic as explained below jensen showed that infinity harmonic functions understood in the viscosity sense are precisely the absolute minimizers here we advance results of barron jensen and wang concerning more general functionals u f x u du l u by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses

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

analysis of the heterogeneous multiscale method for elliptic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

the euler equation and absolute minimizers of l functionals the aronsson euler equation for the functional on w g 1 m i e w 1 with boundary data g is this equation has been derived for smooth absolute minimizers i e a function which minimizes f on every subdomain we prove in this paper that for m 1 n 1 or n 1 m 1 an absolute minimizer of f exists in w g 1 m and for m 1 n 1 any absolute minimizer of f must be a viscosity solution of the aronsson euler equation

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

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

a multiscale approach for optimal control problems of linear parabolic equations this paper discusses multiscale analysis for optimal control problems of linear parabolic equations with rapidly oscillating coefficients that depend on spatial and temporal variables there are mainly three new results in the present paper first we obtain the convergence results with an explicit convergence rate for the multiscale asymptotic expansions of the solution of the optimal control problem in the case without constraints second for a general bounded lipschitz polygonal domain the boundary layer solution is defined and the corresponding convergence results are also derived finally an explicit convergence rate varepsilon 1 2 in the presence of constraint is reported

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

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

nonstandard coarse spaces and schwarz methods for elliptic problems with discontinuous coefficients using non conforming elements two level domain decomposition methods are developed for a simple nonconforming approximation of second order elliptic problems a bound is established for the condition number of these iterative methods that grows only logarithmically with the number of degrees of freedom in each subregion this bound holds for two and three dimensions and is independent of jumps in the value of the coefficients and number of subregions we introduce face coarse spaces and isomorphisms to map between conforming and nonconforming spaces

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

approximate controllability of the semilinear heat equation this article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain when the control acts on any open and nonempty subset of or on a part of the boundary in the case of both an internal and a boundary control the approximate controllability in l p for 1 p l in the case of the interior control we also prove approximate controllability in c 0 the proof combines a variational approach to the controllability problem for linear equations and a fixed point method we also prove that the control can be taken to be of quasi bang bang form

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

l variational problems for maps and the aronsson pde system by employing aronsson s absolute minimizers of l functionals we prove that absolutely minimizing maps u rn rn solve a tangential aronsson pde system by following sheffield and smart 2012 24 we derive with respect to the dual operator norm and show that such maps miss information along a hyperplane when compared to tight maps we recover the lost term which causes non uniqueness and derive the complete aronsson system which has discontinuous coefficients in particular the euclidean laplacian is u du du d2u du 2 du u where du is the projection on the null space of du we demonstrate c solutions having interfaces along which the rank of their gradient is discontinuous and propose a modification with c0 coefficients which admits varifold solutions away from the interfaces aronsson maps satisfy a structural property of local splitting to 2 phases a horizontal and a vertical horizontally they possess gradient flows similar to the scalar case and vertically solve a linear system coupled by a scalar hamilton jacobi pde we also construct singular harmonic local c1 diffeomorphisms and singular aronsson maps

a multiscale finite element method for optimal control problems governed by the elliptic homogenization equations in this article we develop and analyze a priori estimates for optimal control problems with multiscale governed by the elliptic homogenization equations the multiscale finite element is applied to capture the effect of microscale through modification of finite element basis functions without resolving all the small scale features the optimal estimate is derived for elliptic homogenization problems without resonance effect o e h by using an over sampling technique and the boundary layer assumption furthermore the a priori estimate is obtained for the optimal control problems governed by the elliptic homogenization equations

rough polyharmonic splines method for optimal control problem governed by parabolic systems with rough coefficient abstract rough polyharmonic splines rps is a variational method which has recently been developed for linear divergence form operators with arbitrary rough coefficients rps method does not rely on concepts of ergodicity or scale separation but on compactness properties of the solution space in this paper we extend rps approach method for the optimal control problem governed by parabolic systems with rough l coefficients rps method is used for the spatial discretization while the temporal discretization is performed by the finite difference method as the iterative solution of the optimal control problem requires solving the state and co state equations many times with different right hand sides rps method only requires one time pre computation on the fine scale and the following iterations can be done to coarse degrees of freedom first of all we extend an approximation method for the multiscale optimal control problem secondly we obtain the error estimates of the multiscale optimal control problem finally numerical experiments are presented to validate the theoretical analysis

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

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

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

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

the aronsson equation for absolute minimizers of l infty functionals associated with vector fields satisfying hormander s condition given a carnot caratheodory metric space r d x generated by vector fields x i m i 1 satisfying hormander s condition we prove in theorem a that any absolute minimizer u w 1 x to f v q ess sup x f x xv x is a viscosity solution to the aronsson equation x i f x x u x f pi x x u x 0 in under suitable conditions on f in particular any amle is a viscosity solution to the subelliptic oo laplacian equation x u x i ux j ux i x j u 0 in if the carnot caratheodory space is a carnot group g and f is independent of the x variable we establish in theorem c the uniqueness of viscosity solutions to the aronsson equation a xu d 2 u f pi xu f pj xu x i x j u 0 in u on under suitable conditions on f as a consequence the uniqueness of both amle and viscosity solutions to the subelliptic oo laplacian equation is established on any carnot group g

multiscale asymptotic method of optimal control on the boundary for heat equations of composite materials in this paper we study the optimal control on the boundary for parabolic equations with rapidly oscillating coefficients arising from the heat transfer problems and the optimal control on the boundary of composite materials or porous media the multiscale asymptotic expansion of the solution for the problem in the case without any constraints is presented we derive the proofs of all convergence results

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

lie point symmetries and some group invariant solutions of the quasilinear equation involving the infinity laplacian abstract using the classical lie method we obtain the full lie point symmetry group of the aronsson equation in two independent variables some group invariant solutions of this equation are found and a conjecture on the lie point symmetry group of the aronsson equation in r n is presented