the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

smooth approximation of plurisubharmonic functions on almost complex manifolds this note establishes smooth approximation from above for j plurisubharmonic functions on an almost complex manifold x j the following theorem is proved suppose x is j pseudoconvex i e x admits a smooth strictly j plurisubharmonic exhaustion function let u be an upper semi continuous j plurisubharmonic function on x then there exists a sequence u j in c infty x of smooth strictly j plurisubharmonic functions point wise decreasing down to u in any almost complex manifold x j each point has a fundamental neighborhood system of j pseudoconvex domains and so the theorem above establishes local smooth approximation on x this result was proved in complex dimension 2 by the third author who also showed that the result would hold in general dimensions if a parallel result for continuous approximation were known this paper establishes the required step by solving the obstacle problem

convergence of weak kahler ricci flows on minimal models of positive kodaira dimension studying the behavior of the kahler ricci flow on mildly singular varieties one is naturally lead to study weak solutions of degenerate parabolic complex monge ampere equations

solvability of the quaternionic monge ampere equation on compact manifolds with a flat hyperkahler metric abstract a quaternionic version of the calabi problem was formulated by alesker and verbitsky 2010 6 it conjectures a solvability of a quaternionic monge ampere equation on a compact hkt manifold hkt stays for hyperkahler with torsion in this paper this problem is solved under the extra assumption that the manifold admits a flat hyperkahler metric compatible with the underlying hypercomplex structure the proof uses the continuity method and a priori estimates

dirichlet duality and the nonlinear dirichlet problem we study the dirichlet problem for fully nonlinear degenerate elliptic equations of the form f hess u 0 on a smoothly bounded domain n in our approach the equation is replaced by a subset f sym2 n of the symmetric n n matrices with f f 0 we establish the existence and uniqueness of continuous solutions under an explicit geometric f convexity assumption on the boundary we also study the topological structure of f convex domains and prove a theorem of andreotti frankel type two key ingredients in the analysis are the use of subaffine functions and dirichlet duality associated to f is a dirichlet dual set f that gives a dual dirichlet problem this pairing is a true duality in that the dual of f is f and in the analysis the roles of f and f are interchangeable the duality also clarifies many features of the problem including the appropriate conditions on the boundary many interesting examples are covered by these results including all branches of the homogeneous monge ampere equation over and equations appearing naturally in calibrated geometry lagrangian geometry and p convex riemannian geometry and all branches of the special lagrangian potential equation 2008 wiley periodicals inc

pseudoconvex classes of functions iii characterization of dual pseudoconvex classes on complex homogeneous spaces invariant classes of functions on complex homogeneous spaces with properties similar to those of the class of plurisubharmonic functions are studied the main tool is a regularization method for these classes and the main theorem characterizes dual classes of functions where duality is defined in terms of the local maximum property these results are crucial in proving a duality theorem for complex interpolation of normed spaces which is given elsewhere

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

levinson s theorem for non local interactions in two dimensions in the light of the sturm liouville theorem the levinson theorem for the schrodinger equation with both local and non local cylindrically symmetric potentials is studied it is proved that the two dimensional levinson theorem holds for the case with both local and non local cylindrically symmetric cut off potentials which is not necessarily separable in addition the problems related to the positive energy bound states and the physically redundant state are also discussed

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

viscosity solutions to degenerate complex monge ampere equations we develop an alternative approach to degenerate complex monge ampere equations on compact kahler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones we generalize to the kahler case a theorem due to dinew and zhang in the projective case to the effect that their pluripotential solutions constructed previously by the authors are continuous

a generalization of pde s from a krylov point of view abstract we introduce and investigate the notion of a generalized equation which extends nonlinear elliptic equations and which is based on the notions of subequations and dirichlet duality precisely a subset h sym 2 r n is a generalized equation if it is an intersection h e g where e and g are subequations and g is the subequation dual to g we utilize a viscosity definition of solution to h the mirror of h is defined by h g e one of the main results theorem 2 6 concerns the dirichlet problem on arbitrary bounded domains r n for solutions to h with prescribed boundary function c we prove that a uniqueness holds h has no interior and b existence holds h has no interior for b the appropriate boundary convexity of must be assumed many examples of generalized equations are discussed including the constrained laplacian the twisted monge ampere equation and the c 1 1 equation the closed sets h sym 2 r n which can be written as generalized equations are intrinsically characterized for such an h the set of subequation pairs e g with h e g is partially ordered see 1 10 if e g e g then any solution for the first is also a solution for the second furthermore in this ordered set there is a canonical least element contained in all others a general form of the main theorem which holds on any manifold is also established

an introduction to potential theory in calibrated geometry in this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry these functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties moreover they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce a number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds this paper introduces and investigates questions of pseudo convexity in the context of a general calibrated manifold x phi analogues of totally real submanifolds are introduced and used to construct enormous families of strictly phi convex spaces with every topological type allowed by morse theory specific calibrations are used as examples throughout in a sequel the duality between phi pluri sub harmonic functions and phi positive currents is investigated this study involves boundaries generalized jensen measures and other geometric objects on a calibrated manifold

weak solutions to degenerate complex monge ampere flows i the authors apply the method of viscosity solutions to the parabolic version of the complex monge amp ere equation in a strictly pseudoconvex domain omega that is exp partial t varphi t f t z varphi t mu z dd c varphi t n 0 text in 0 t times omega with f continuous and nondecreasing in the last variable mu a bounded continuous nonnegative volume form and varphi t varphi t cdot the unknown function first they review the results of carandall ishii lions and others in classical pde viscosity theory then the following results are shown any bounded subsolution is plurisubharmonic on space slices the parabolic comparison principle is valid under certain assumptions on the initial data satisfied when mu 0 the solution of the cauchy dirichlet problem exists for all time and it is an upper envelope of all subsolutions finally with the same assumptions and f independent of time varphi t converges to the solution of the related elliptic equation as t to infty

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

cyclic and quasi cyclic ldpc codes on constrained parity check matrices and their trapping sets this paper is concerned with construction and structural analysis of both cyclic and quasi cyclic codes particularly low density parity check ldpc codes it consists of three parts the first part shows that a cyclic code given by a parity check matrix in circulant form can be decomposed into descendant cyclic and quasi cyclic codes of various lengths and rates some fundamental structural properties of these descendant codes are developed including the characterization of the roots of the generator polynomial of a cyclic descendant code the second part of the paper shows that cyclic and quasi cyclic descendant ldpc codes can be derived from cyclic finite geometry ldpc codes using the results developed in the first part of the paper this enlarges the repertoire of cyclic ldpc codes the third part of the paper analyzes the trapping set structure of regular ldpc codes whose parity check matrices satisfy a certain constraint on their rows and columns several classes of finite geometry and finite field cyclic and quasi cyclic ldpc codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances consequently their error floor performances are dominated by their minimum distances

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

boundary behavior of k convex solutions for singular k hessian equations abstract we discuss the existence and boundary behavior of k convex solution to the singular k hessian problem s k d 2 u x b x f u x x u x 0 x where s k d 2 u k 1 2 n is the k hessian operator r n n 2 is a smooth bounded strictly convex domain here the weight function b x is not necessarily bounded on another interest is that f u as u 0 our approach mainly relies on karamata s regular variation theory and the construction of suitable sub and super solutions

boundary blow up solutions to the k hessian equation with a weakly superlinear nonlinearity abstract by constructing new sub and super solutions we are concerned with determining values of for which there exist k convex solutions to the boundary blow up k hessian problem s k d 2 u x h x u x k ln u x 0 for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator 0 and is a smooth bounded strictly convex domain in r n n 2 we suppose that the nonlinearity behaves like u k ln u as u which is more complex and difficult to deal with than the nonlinearity grows like u p with p k or faster at infinity further several new results of nonexistence global estimates and estimates near the boundary for the solutions are also given

the dirichlet problem with prescribed interior singularities abstract in this paper we solve the nonlinear dirichlet problem uniquely for functions with prescribed asymptotic singularities at a finite number of points and with arbitrary continuous boundary data on a domain in r n the main results apply in particular to subequations with a riesz characteristic p 2 it is shown that without requiring uniform ellipticity the dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the riesz kernel j k p x x j where k p x 1 x p 2 for 2 p log x if p 2 at any prescribed finite set of points x 1 x k in the domain and any finite set of positive real numbers 1 k this sharpens a previous result of the authors concerning the discreteness of high density sets of subsolutions uniqueness and existence results are also established for finite type singularities such as j x x j 2 p for 1 p 2 the main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of armstrong sirakov smart in the uniformly elliptic case

removable singularities for degenerate elliptic pucci operator on the heisenberg group abstract in this paper we study viscosity solutions to a class of degenerate elliptic pucci operators modelled on the heisenberg group where the second order term is obtained by a composition of degenerate elliptic pucci operator with the degenerate heisenberg hessian matrix we study and answer the question which compact sets have the property that each viscosity subsolution outside this set which is bounded below can be extended to a viscosity subsolution on this set

design of ldpc decoders for improved low error rate performance quantization and algorithm choices many classes of high performance low density parity check ldpc codes are based on parity check matrices composed of permutation submatrices we describe the design of a parallel serial decoder architecture that can be used to map any ldpc code with such a structure to a hardware emulation platform high throughput emulation allows for the exploration of the low bit error rate ber region and provides statistics of the error traces which illuminate the causes of the error floors of the 2048 1723 reed solomon based ldpc rs ldpc code and the 2209 1978 array based ldpc code two classes of error events are observed oscillatory behavior and convergence to a class of non codewords termed absorbing sets the influence of absorbing sets can be exacerbated by message quantization and decoder implementation in particular quantization and the log tanh function approximation in sum product decoders strongly affect which absorbing sets dominate in the errorfloor region we show that conventional sum product decoder implementations of the 2209 1978 array based ldpc code allow low weight absorbing sets to have a strong effect and as a result elevate the error floor dually quantized sum product decoders and approximate sum product decoders alleviate the effects of low weight absorbing sets thereby lowering the error floor

the cycle consistency matrix approach to absorbing sets in separable circulant based ldpc codes for low density parity check ldpc codes operating over additive white gaussian noise channels and decoded using message passing decoders with limited precision absorbing sets have been shown to be a key factor in error floor behavior focusing on this scenario this paper introduces the cycle consistency matrix ccm as a powerful analytical tool for characterizing and avoiding absorbing sets in separable circulant based scb ldpc codes scb codes include a wide variety of regular ldpc codes such as array based ldpc codes as well as many common quasi cyclic codes as a consequence of its cycle structure each potential absorbing set in an scb ldpc code has a ccm and an absorbing set can be present in an scb ldpc code only if the associated ccm has a nontrivial null space ccm based analysis can determine the multiplicity of an absorbing set in an scb code and ccm based constructions avoid certain small absorbing sets completely while these techniques can be applied to an scb code of any rate lower rate scb codes can usually avoid small absorbing sets because of their higher variable node degree this paper focuses attention on the high rate scenario in which the ccm constructions provide the most benefit simulation results demonstrate that under limited precision decoding the new codes have steeper error floor slopes and can provide one order of magnitude of improvement in the low frame error rate region

singular solutions to special lagrangian equations with subcritical phases and minimal surface systems we construct singular solutions to special lagrangian equations with subcritical phases and minimal surface systems a priori estimate breaking families of smooth solutions are also produced correspondingly a priori estimates for special lagrangian equations with certain convexity are largely known by now

duality of positive currents and plurisubharmonic functions in calibrated geometry recently the authors showed that there is a robust potential theory attached to any calibrated manifold x phi in particular on x there exist phi plurisubharmonic functions phi convex domains phi convex boundaries etc all inter related and having a number of good properties in this paper we show that in a strong sense the plurisubharmonic functions are the polar duals of the phi submanifolds or more generally the phi currents studied in the original paper on calibrations in particular we establish an analogue of duval sibony duality which characterizes points in the phi convex hull of a compact set k subset x in terms of phi positive green s currents on x and jensen measures on k we also characterize boundaries of phi currents entirely in terms of phi plurisubharmonic functions specific calibrations are used as examples throughout analogues of the hodge conjecture in calibrated geometry are considered

a viscosity approach to the dirichlet problem for complex monge ampere equations we present a viscosity approach to the dirichlet problem for the complex monge ampere equation det u bar j k f x u our approach differs from previous viscosity approaches to this equation in several ways it is based on contact set techniques the alexandrov bakelman pucci estimate on extensive applications of sup inf convolutions and on a relation between real and complex hessians more specifically this paper includes a notion of viscosity solutions a comparison principle and a solvability theorem the equivalence between viscosity and pluripotential solutions an estimate of the modulus of continuity of a solution in terms of that of a given subsolution and of the right hand side f and an alexandrov bakelman pucci type of l estimate

quasicyclic low density parity check codes from circulant permutation matrices in this correspondence the construction of low density parity check ldpc codes from circulant permutation matrices is investigated it is shown that such codes cannot have a tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived these results suggest that families of ldpc codes with such girth values are relatively easy to obtain and consequently additional parameters such as the minimum distance or the number of redundant check sums should be considered to this end a necessary condition for the codes investigated to reach their maximum possible minimum hamming distance is proposed

weak solutions to degenerate complex monge ampere flows ii abstract studying the long term behavior of the kahler ricci flow on mildly singular varieties one is naturally led to study weak solutions of degenerate parabolic complex monge ampere equations the purpose of this article the second of a series on this subject is to develop a viscosity theory for degenerate complex monge ampere flows on compact kahler manifolds our general theory allows in particular to define and study the normalized kahler ricci flow on varieties with canonical singularities generalizing results of song and tian

the existence and asymptotic behavior of boundary blow up solutions to the k hessian equation abstract in this paper we consider the existence and asymptotic behavior of k convex solution to the boundary blow up k hessian problem s k d 2 u h x f u for x u x as dist x 0 where k 1 2 n s k d 2 u is the k hessian operator is a smooth bounded strictly convex domain in r n n 2 h c is positive in but is not necessarily bounded on and f is a smooth positive function that satisfies the so called keller osserman condition further results are obtained for the special case that is a ball our approach to show the existence and asymptotic behavior exploits the method of sub and super solutions and karamata regular variation theory

a uniform estimate for general quaternionic calabi problem with appendix by daniel barlet abstract we prove a c 0 a priori estimate on a solution of the quaternionic calabi problem on an arbitrary compact connected hkt manifold this generalizes earlier results 9 and 7 where this result was proven under certain extra assumptions on the manifold

the degenerate special lagrangian equation abstract this article introduces the degenerate special lagrangian equation dsl and develops the basic analytic tools to construct and study its solutions the dsl governs geodesics in the space of positive graph lagrangians in c n existence of geodesics in the space of positive lagrangians is an important step in a program for proving existence and uniqueness of special lagrangians moreover it would imply certain cases of the strong arnold conjecture from hamiltonian dynamics we show the dsl is degenerate elliptic we introduce a space time lagrangian angle for one parameter families of graph lagrangians and construct its regularized lift the superlevel sets of the regularized lift define subequations for the dsl in the sense of harvey lawson we extend the existence theory of harvey lawson for subequations to the setting of domains with corners and thus obtain solutions to the dirichlet problem for the dsl in all branches moreover we introduce the calibration measure which plays a role similar to that of the monge ampere measure in convex and complex geometry the existence of this measure and regularity estimates allow us to prove that the solutions we obtain in the outer branches of the dsl have a well defined length in the space of positive lagrangians

on a k hessian equation with a weakly superlinear nonlinearity and singular weights abstract consider the existence nonexistence and global estimates of k convex solutions to the boundary blow up k hessian problem s k d 2 u x h x u x k ln u x 0 for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator 0 is a smooth bounded strictly convex domain in r n n 2 and h x is a positive weight function which is singular near the boundary we first give the existence and nonexistence results of k convex solution to the above boundary blow up problem on a larger range of h and then we show that there is a k convex solution provided that h x grows fast near and u k ln u grows slow at it turns out that this case is more difficult to handle than the case in which h x grows slow near and u k ln u grows fast at this needs some new ingredients in the arguments

viscosity solutions to quaternionic monge ampere equations abstract quaternionic monge ampere equations have recently been studied intensively using methods from pluripotential theory we present an alternative approach by using the viscosity methods we study the viscosity solutions to the dirichlet problem for quaternionic monge ampere equations d e t f f q f with boundary value f g on here is a bounded domain on the quaternionic space h n g c and f q t is a continuous function on r r which is non decreasing in the second variable we prove a viscosity comparison principle and a solvability theorem moreover the equivalence between viscosity and pluripotential solutions is shown

geometric plurisubharmonicity and convexity an introduction this is an essay on potential theory for geometric plurisubharmonic functions it begins with a given closed subset g of the grassmann bundle g p tx of tangent p planes to a riemannian manifold x this determines a nonlinear partial differential equation which is convex but never uniformly elliptic p dimx a surprising number of results in complex analysis carry over to this more general setting the notions of a g submanifold an upper semi continuous g plurisubharmonic function a g convex domain a g harmonic function and a g free submanifold are defined results include a restriction theorem as well as the existence and uniqueness of solutions to the dirichlet problem for g harmonic functions on g convex domains