convergent finite difference solvers for viscosity solutions of the elliptic monge ampere equation in dimensions two and higher the elliptic monge ampere equation is a fully nonlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail novel solution methods are required for stability and convergence to the weak viscosity solution in this article we build a wide stencil finite difference discretization for the monge ampere equation the scheme is monotone so the barles souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation solutions of the scheme are found using a damped newton s method we prove convergence of newton s method and provide a systematic method to determine a starting point for the newton iteration computational results are presented in two and three dimensions which demonstrates the speed and accuracy of the method on a number of exact solutions which range in regularity from smooth to nondifferentiable

a partial differential equation for the rank one convex envelope a partial differential equation pde for the rank one convex envelope is introduced the existence and uniqueness of viscosity solutions to the pde is established elliptic finite difference schemes are constructed and convergence of finite difference solutions to the viscosity solution of the pde is proven computational results are presented and laminates are computed from the envelopes results include the kohn strang example the classical four gradient example and an example with eight gradients which produces nontrivial laminates

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

on the inequality f x d 2u ge f u g u du q we consider fully nonlinear degenerate elliptic equations with zero and first order terms we provide a priori upper bounds and characterize the existence of entire subsolutions under growth conditions on the lower order coefficients which extend the classical keller osserman condition for semilinear equations

regularity for an obstacle problem of hessian equations on riemannian manifolds abstract in this paper we study the regularity for solutions to an obstacle problem of hessian type fully nonlinear equations on riemannian manifolds as an application the existence of a c 1 1 solution is proved

on the smoothness of convex envelopes we examine differentiability properties of the convex envelope conv e of a given function e rn cc cc in terms of properties of e it is shown that c1 as well as optimal c1 a regularity results 0 a 1 can be obtained under general conditions

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

on positive solutions of fully nonlinear degenerate lane emden type equations abstract we prove existence and uniqueness results of positive viscosity solutions of fully nonlinear degenerate elliptic equations with power like zero order perturbations in bounded domains the principal part of such equations is either p k d 2 u or p k d 2 u some sort of truncated laplacians given respectively by the smallest and the largest partial sum of k eigenvalues of the hessian matrix new phenomena with respect to the semilinear case occur moreover for p k we explicitly find the critical exponent p of the power nonlinearity that separates the existence and nonexistence range of nontrivial solutions with zero dirichlet boundary condition

existence through convexity for the truncated laplacians we study the dirichlet problem on a bounded convex domain of mathbb r n with zero boundary data for truncated laplacians mathcal p k pm which are degenerate elliptic operators for k n defined by the upper and respectively lower partial sum of k eigenvalues of the hessian matrix we establish a necessary and sufficient condition theorem 1 in terms of the flatness of domains for existence of a solution for general inhomogeneous term this result in particular shows that the strict convexity of the domain is sufficient for the solvability of the dirichlet problem the result and related ideas are applied to the solvability of the dirichlet problem for the operator mathcal p k with lower order term when the domain is strictly convex and the existence of principal eigenfunctions for the operator mathcal p 1 an existence theorem is presented with regard to the principal eigenvalue for the dirichlet problem with zero th order term for the operator mathcal p 1 a nonexistence result is established for the operator mathcal p k with first order term when the domain has a boundary portion which is nearly flat furthermore when the domain is a ball we study the dirichlet problem with a constant inhomogeneous term and a possibly sign changing first order term and the associated eigenvalue problem

the dirichlet problem for the convex envelope the convex envelope of a given function was recently characterized as the solution of a fully nonlinear partial differential equation pde in this article we study a modified problem the dirichlet problem for the underlying pde the main result is an optimal regularity result differentiability c1 regularity of the boundary data implies the corresponding result for the solution in the interior despite the fact that the solution need not be continuous up to the boundary secondary results are the characterization of the convex envelope as i the value function of a stochastic control problem and ii the optimal underestimator for a class of nonlinear elliptic pdes

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

unimodular eigenvalues uniformly distributed sequences and linear dynamics we study increasing sequences of positive integers nk k 1 with the following property every bounded linear operator t acting on a separable banach or hilbert space with supk 1 tnk has a countable set of unimodular eigenvalues whether this property holds or not depends on the distribution modulo one of sequences nk k 1 r or on the growth of nk 1 nk counterexamples to some conjectures in linear dynamics are given for instance a hilbert space operator which is frequently hypercyclic chaotic but not topologically mixing is constructed the situation of c0 semigroups is also discussed

invariant measures for frequently hypercyclic operators abstract we investigate frequently hypercyclic and chaotic linear operators from a measure theoretic point of view among other things we show that any frequently hypercyclic operator t acting on a reflexive banach space admits an invariant probability measure with full support which may be required to vanish on the set of all periodic vectors for t that there exist frequently hypercyclic operators on the sequence space c 0 admitting no ergodic measure with full support and that if an operator admits an ergodic measure with full support then it has a comeager set of distributionally irregular vectors we also give some necessary and sufficient conditions which are satisfied by all the known chaotic operators for an operator t to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures finally we give a baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support

an existence result for a class of shape optimization problems given a bounded open subset of rn we prove the existence of a minimum point for a functional f defined on the family a of all quasiopen subsets of under the assumption that f is decreasing with respect to set inclusion and that f is lower semicontinuous on a with respect to a suitable topology related to the resolvents of the laplace operator with dirichlet boundary condition applications are given to the existence of sets of prescribed volume with minimal kth eigenvalue or with minimal capacity with respect to a given elliptic operator

invariant gaussian measures for operators on banach spaces and linear dynamics we give conditions for an operator t on a complex separable banach space x with sufficiently many eigenvectors associated to eigenvalues of modulus 1 to admit a non degenerate invariant gaussian measure with respect to which it is weak mixing the existence of such a measure depends on the geometry of the banach space and on the possibility of parametrizing the mathbb t eigenvector fields of t in a regular way we also investigate the connection with frequent hypercyclicity

shape optimization problems on metric measure spaces abstract we consider shape optimization problems of the form min j x m c where x is a metric measure space and j is a suitable shape functional we adapt the notions of convergence and weak convergence to this new general abstract setting to prove the existence of an optimal domain several examples are pointed out and discussed

dynamics of non convolution operators and holomorphy types abstract in this article we study the hypercyclic behavior of non convolution operators defined on spaces of analytic functions of different holomorphy types over banach spaces the operators in the family we analyze are a composition of differentiation and composition operators and are extensions of operators in h c studied by aron and markose in 2004 the dynamics of this class of operators in the context of one and several complex variables was further investigated by many authors it turns out that the situation is somewhat different and that some purely infinite dimensional difficulties appear for example in contrast to the several complex variable case it may happen that the symbol of the composition operator has no fixed points and still the operator is not hypercyclic we also prove a runge type theorem for holomorphy types on banach spaces

optimal regularity for the convex envelope and semiconvex functions related to supersolutions of fully nonlinear elliptic equations in this paper we prove optimal regularity for the convex envelope of supersolutions to general fully nonlinear elliptic equations with unbounded coefficients more precisely we deal with coefficients and right hand sides rhs in lq with q geq n this extends the result of caffarelli on the c loc 1 1 regularity of the convex envelope of supersolutions of fully nonlinear elliptic equations with bounded rhs moreover we also provide a regularity result with estimates for omega semiconvex functions that are supersolutions to the same type of equations with unbounded rhs i e rhs in l q q geq n by a completely different method our results here extend the recent regularity results obtained by braga et al adv math 334 184 242 2018 for q n as far as fully nonlinear pdes are concerned these results include in particular the apriori estimate obtained by caffarelli et al commun pure appl math 38 2 209 252 1985 on the modulus of continuity of the gradient of omega semiconvex supersolutions for linear equations and bounded rhs that have a holder modulus of semiconvexity

hypercyclicity and unimodular point spectrum abstract we show that an operator on a separable complex banach space with sufficiently many eigenvectors associated to eigenvalues of modulus 1 is hypercyclic we apply this result to construct hypercyclic operators with prescribed k unimodular point spectrum we show how eigenvectors associated to unimodular eigenvalues can be used to exhibit common hypercyclic vectors for uncountable families of operators and prove that the family of composition operators c on h 2 d where is a disk automorphism having 1 as attractive fixed point has a residual set of common hypercyclic vectors

chaoticity and invariant measures for a cell population model abstract we present a structured model of a cell reproduction system given by a partial differential equations with a nonlocal division term this equation generates semiflows acting on some subspaces of locally integrable functions we show that these semiflows possess invariant mixing measures positive on open sets from this it follows that the system is chaotic i e it has dense trajectories and each trajectory is unstable we also show the chaoticity of this system in the sense of devaney

frequently hypercyclic operators we investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators t on separable complex f spaces t is frequently hypercyclic if there exists a vector x such that for every nonempty open subset u of x the set of integers n such that t n x belongs to u has positive lower density we give several criteria for frequent hypercyclicity and this leads us in particular to study linear transformations from the point of view of ergodic theory several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting

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

strong mixing measures for linear operators and frequent hypercyclicity abstract we construct strongly mixing invariant measures with full support for operators on f spaces which satisfy the frequent hypercyclicity criterion for unilateral backward shifts on sequence spaces a slight modification shows that one can even obtain exact invariant measures

a numerical approach to variational problems subject to convexity constraint we describe an algorithm to approximate the minimizer of an elliptic functional in the form int omega j x u nabla u on the set cal c of convex functions u in an appropriate functional space x such problems arise for instance in mathematical economics 4 a special case gives the convex envelope u 0 of a given function u 0 let t n be any quasiuniform sequence of meshes whose diameter goes to zero and i n the corresponding affine interpolation operators we prove that the minimizer over cal c is the limit of the sequence u n where u n minimizes the functional over i n cal c we give an implementable characterization of i n cal c then the finite dimensional problem turns out to be a minimization problem with linear constraints

fast finite difference solvers for singular solutions of the elliptic monge ampere equation the elliptic monge ampere equation is a fully nonlinear partial differential equation which originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail in this article we build a finite difference solver for the monge ampere equation which converges even for singular solutions regularity results are used to select a priori between a stable provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain this allows singular solutions to be computed using a stable method and regular solutions to be computed more accurately the resulting nonlinear equations are then solved by newton s method computational results in two and three dimensions validate the claims of accuracy and solution speed a computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities

chaotic semigroups from second order partial differential equations abstract we give general conditions on given parameters to ensure devaney and distributional chaos for the solution c 0 semigroup corresponding to a class of second order partial differential equations we also provide a critical parameter that led us to distinguish between stability and chaos for these semigroups in the case of chaos we prove that the c 0 semigroup admits a strongly mixing measure with full support we also give concrete examples of partial differential equations such as the telegraph equation whose solutions satisfy these properties

shape optimization for dirichlet problems relaxed solutions and optimality conditions we study an optimal design problem for the domain of an elliptic equation with dirichlet boundary conditions we introduce a relaxed formulation of the problem which always admits a solution and we prove some necessary conditions for optimality both for the relaxed and for the original problem