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

a priori estimate for convex solutions to special lagrangian equations and its application we derive a priori interior hessian estimates for special lagrangian equations when the potential is convex when the phase is very large we show that continuous viscosity solutions are smooth in the interior of the domain c 2008 wiley periodicals inc

small families of complex lines for testing holomorphic extendibility let b be the open unit ball in bbb c 2 this paper deals with the analog of hartogs separate analyticity theorem for cr functions on the sphere bb we prove such a theorem for functions in c infty bb if a b in overline b a not b and if f in c infty bb extends holomorphically into b along any complex line passing through either a or b then f extends holomorphically through b on the other hand for each k in bbb n there is a function f in c k bb which extends holomorphically into b along any complex line passing through either a or b yet f does not extend holomorphically through b more generally in the paper we obtain a fairly complete description of pairs of points a b in bbb c 2 a not b such that if f in c infty bb extends holomorphically into b along every complex line passing through either a or b that meets b then f extends holomorphically through b

liouville type results of curvature operators of fuchsian convex surfaces abstract in this paper we deduce some liouville type results of curvature operators of fuchsian convex surfaces in minkowski space

interior estimates for solutions of the quadratic hessian equations in dimension three abstract we derive a priori c 2 interior estimates in dimension three for viscosity solutions of the quadratic hessian equation 2 d 2 u f x provided that the positive f is bounded and its holder semi norm is small actually we relax the holder continuity condition of f to be of integral sense

parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

regularity of generalized solutions of monge ampere equations on etudie la regularite interieure des solutions generalisees convexes des equations de monge ampere de la forme detd 2 d f x u du dans ou est un domaine de r n f est une fonction positive suffisamment lisse sur r r n

explicit gradient estimates for minimal lagrangian surfaces of dimension two we derive explicit uniform a priori interior hessian and gradient estimates for special lagrangian equations of all phases in dimension two

hessian estimates for special lagrangian equations with critical and supercritical phases in general dimensions we derive a priori interior hessian estimates for special lagrangian equation with critical and supercritical phases in general higher dimensions our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases

hessian and gradient estimates for three dimensional special lagrangian equations with large phase we obtain a priori interior hessian and gradient estimates for special lagrangian equations with phase larger than a critical value in dimension three gradient estimates are also derived for critical and super critical phases in general dimensions

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

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

testing families of complex lines for the unit ball abstract let p 1 and p 2 be two points in c 2 such that the complex line joining them is tangent to the unit sphere b 2 we prove that if a real analytic function f on b 2 extends holomorphically on every complex line through p 1 and p 2 then f extends holomorphically to the whole ball b 2

the exterior dirichlet problem for hessian quotient equations abstract in this paper using a reverse maclaurin inequality we established the existence theorem of the exterior dirichlet problem for hessian quotient equations with a stronger asymptotic behavior at infinity when k n 1 2 we improve the result in l m dai the dirichlet problem for hessian quotient equations in exterior domains j math anal appl 380 2011 87 93 for s k l with k l 3 to the result with any 0 l k

a fast near to far field transformation in body of revolution finite difference time domain method we present an efficient near to far field transformation technique to be used in conjunction with the body of revolution finite difference time domain bor fdtd algorithm the efficiency is gained by factoring out the azimuthal angle spl phi from the two dimensional 2d far field integration using an analytical approach thereby reducing the 2d integral on the surface of the huygens box to a 1d version in the process to illustrate the application of the proposed algorithm the potter horn is analyzed the numerical results show good agreement with those obtained by using either analytical where possible or 2d integration schemes

the dirichlet problem for hessian quotient equations in exterior domains in this paper we obtain the uniqueness and existence of viscosity solutions with prescribed asymptotic behavior at infinity to hessian quotient equations in exterior domains

holomorphic extension from the sphere to the ball abstract real analytic functions on the boundary of the sphere which have separate holomorphic extension along the complex lines through a boundary point have holomorphic extension to the ball this was proved in baracco 2009 4 by an argument of cr geometry we give here an elementary proof based on the expansion in holomorphic and antiholomorphic powers

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

the neumann problem of special lagrangian equations with supercritical phase abstract in this paper we establish the global c 2 estimates of the neumann problem of special lagrangian equations with supercritical phase and the existence theorem by the method of continuity

a compactness approach to hessian estimates for special lagrangian equations with supercritical phase abstract we derive interior hessian estimates for special lagrangian equations with supercritical phase in general dimension through a compactness approach

analyticity of functions analytic on circles abstract let be the open unit disc in c let p b and let f be a continuous function on which extends holomorphically from each circle in centered at the origin and from each circle in which passes through p then f is holomorphic on

aleksandrov bakelman pucci type estimates for integro differential equations in this work we provide an aleksandrov bakelman pucci type estimate for a certain class of fully nonlinear elliptic integro differential equations the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal fractional order setting and on the use of riesz potentials to interpret second derivatives as fractional order operators this result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations furthermore this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the l and ln norms of the right hand side in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions these results appear to be new even for the linear case of the relevant equations

a bernstein problem for special lagrangian equations where is are the eigenvalues of the hessian d2u namely any global convex solution to 1 1 in rn must be a quadratic polynomial recall the classical result any global convex solution in rn to the laplace equation u 1 n c or the monge ampere equation log det d2u log 1 log n c must be quadratic equation 1 1 originates from special lagrangian geometry hl the lagrangian graph x u x rn rn is called special when the argument of the complex number 1 1 1 1 1 n is constant c or u satisfies 1 1 and it is special if and only if x u x is a minimal surface in rn rn hl theorem 2 3 proposition 2 17 in terms of minimal surface our result is the following

on axially symmetric solutions of fully nonlinear elliptic equations we show that in dimension 3 axial symmetric viscosity solutions of uniformly elliptic hessian equations are in fact the classical ones

a bernstein type theorem for parabolic k hessian equations abstract we are concerned with the characterization of entire solutions to the parabolic k hessian equation of the form u t f k d 2 u 1 in r n 0 we prove that for 1 k n any strictly convex monotone solution u u x t c 4 2 r n 0 to u t f k d 2 u 1 in r n 0 must be a linear function of t plus a quadratic polynomial of x under some growth assumptions on u

a special lagrangian type equation for holomorphic line bundles let l be a holomorphic line bundle over a compact kahler manifold x motivated by mirror symmetry we study the deformed hermitian yang mills equation on l which is the line bundle analogue of the special lagrangian equation in the case that x is calabi yau we show that this equation is the euler lagrange equation for a positive functional and that solutions are unique global minimizers we provide a necessary and sufficient criterion for existence in the case that x is a kahler surface for the higher dimensional cases we introduce a line bundle version of the lagrangian mean curvature flow and prove convergence when l is ample and x has non negative orthogonal bisectional curvature

hessian equations on exterior domain abstract in the paper we consider the hessian equation k d 2 u f x where f is a positive function outside a bounded domain of r n n 3 and f x 1 o x for some 2 at infinity using the perron s method we prove the existence and uniqueness for viscosity solutions of exterior dirichlet problem with prescribed asymptotic behavior at infinity there are examples to show that the result is optimal this is an extension of the theorems given by bao li li in 2 for f 1 and bao li zhang in 3 for k n

the obstacle problem for hessian equations on riemannian manifolds abstract in this paper we consider the obstacle problem for a class of fully nonlinear equations on riemannian manifolds the c 1 1 regularity of the greatest viscosity solutions is established

liouville property and regularity of a hessian quotient equation we are concerned with properties of convex solutions to the hessian quotient equation sn c d2 p 1 k n as our first main result we prove some regularity of strong solutions while the second states that for p 1 an entire convex solution with a quadratic growth bound must be a quadratic polynomial

testing analyticity on circles consider a continuous one parameter family of circles in a complex plane that contains two circles lying in the exterior of one another we prove that if a continuous function on the union of the circles extends holomorphically into each circle then the function is holomorphic in the interior of the union of the circles

orthogonal testing families and holomorphic extension from the sphere to the ball let mathbb b 2 denote the open unit ball in mathbb c 2 and let p in mathbb c 2 overline mathbb b 2 we prove that if f is an analytic function on the sphere partial mathbb b 2 that extends holomorphically in each variable separately and along each complex line through p then f is the trace of a holomorphic function in the ball

existence of entire solutions to the monge ampere equation we prove the existence of infinitely many entire convex solutions to the monge amp ere equation rm det d 2 u f in bbb r n assuming that the inhomogeneous term f ge 0 and is of polynomial growth at infinity when f satisfies the doubling condition we show that solution is of polynomial growth as an application we resolve the existence of translating solutions to a class of gauss curvature flow

bifurcation and admissible solutions for the hessian equation abstract we study the following eigenvalue problem of k hessian equation s k d 2 u k f u in b u 0 on b global bifurcation result is established for this problem as applications of the bifurcation result we determine the intervals of for the existence nonexistence uniqueness and multiplicity of radially symmetric k admissible solutions for this problem

on the solution of a class of large body problems with full or partial circular symmetry by using the finite difference time domain fdtd method this paper presents an efficient method to accurately solve large body scattering problems with partial circular symmetry the method effectively reduces the computational domain from three to two dimensions by using the reciprocity theorem it does so by dividing the problem into two parts a larger 3 d region with circular symmetry and a smaller 2 d region without circular symmetry an finite difference time domain fdtd algorithm is used to analyze the circularly symmetric 3 d case while a method of moments mom code is employed for the nonsymmetric part of the structure the results of these simulations are combined via the reciprocity theorem to yield the radiation pattern of the composite system the advantage of this method is that it achieves significant savings in computer storage and run time in performing an equivalent 2 d as opposed to a full 3 d fdtd simulation in addition to enhancing computational efficiency the fdtd algorithm used in this paper also features one improvement over conventional fdtd methods a conformal approach for improved accuracy in modeling curved dielectric and conductive surfaces the accuracy of the method is validated via a comparison of simulated and measured results

the liouville bratu gelfand problem for radial operators abstract we determine precise existence and multiplicity results for radial solutions of the liouville bratu gelfand problem associated with a class of quasilinear radial operators which includes perturbations of k hessian and p laplace operators

the local regularity for strong solutions of the hessian quotient equation abstract by means of the reilly formula and the alexandrov maximum principle we obtain the local c 1 1 estimates of the w 2 p strong solutions to the hessian quotient equations for p sufficiently large and then prove that these solutions are smooth there are counterexamples to show that the integral exponent p is optimal in some cases we modify partially the known result in the hessian case and extend the regularity result in the special lagrangian case to the hessian quotient case

the exterior dirichlet problem for fully nonlinear elliptic equations related to the eigenvalues of the hessian in this paper we establish the existence theorem for the exterior dirichlet problems for a class of fully nonlinear elliptic equations which are related to the eigenvalues of the hessian matrix with prescribed asymptotic behavior at infinity this extends the previous results on monge ampere equation and k hessian equation to more general cases in particular including the special lagrangian equation

a bernstein problem for special lagrangian equations in exterior domains abstract we establish quadratic asymptotics for solutions to special lagrangian equations with supercritical phases or with semiconvexity on solutions in exterior domains the method for convex case is based on an exterior evans krylov or exterior liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded hessian and also certain rotation arguments toward hessian bound our unified approach also leads to quadratic asymptotics for convex solutions to monge ampere equations previously known quadratic hessian equations and inverse harmonic hessian equations over exterior domains the semiconvex case is based on allard almgren s uniqueness of tangent cones

an interior estimate for convex solutions and a rigidity theorem abstract we establish an interior c 2 estimate for k 1 convex solutions to dirichlet problems of k hessian equations we also use such estimate to obtain a rigidity theorem for k 1 convex entire solutions of k hessian equations in euclidean space

a variational theory of the hessian equation by studying a negative gradient flow of certain hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous hessian equations to achieve this we derive uniform first and second order a priori estimates for the elliptic and parabolic hessian equations our results generalize well known results for semilinear elliptic equations and the monge ampere equation c 2001 john wiley sons inc

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

holomorphic extensions from open families of circles for a circle z c z c p write z w z a w a 2 0 z a p a continuous function f on extends holomorphically from into the disc bounded by if and only if the function f z z f z defined on z z z e r has a bounded holomorphic extension into a r in the paper we consider open connected families of circles c write u u e c and assume that a continuous function on u extends holomorphically from each e c we show that this happens if and only if the function f z z f z defined on z z z e u has a bounded holomorphic extension into the domain u r e q for each open family q compactly contained in c this allows us to use known facts from several complex variables in particular we use the edge of the wedge theorem to prove a theorem on real analyticity of such functions

hessian estimates for the sigma 2 equationin dimension 3 we derive a priori interior hessian estimates for the special lagrangian equation 2 d1 in dimension3 c 2008 wiley periodicals inc

the l p l p cr hartogs separate analyticity theorem for convex domains in this note a very general theorem of the cr hartogs type is proved for almost generic strictly convex domains in mathbf c mathbf n with real analytic boundary given such a domain d and given an l p function f on partial d which has holomorphic extensions on the slices of d by complex lines parallel to the coordinate axes f must be cr i e f has a holomorphic extension to d which is in the hardy space h p d this is the first general result of cr hartogs type which is not for the ball or some other domain with symmetries and holds for l 1 functions as a corollary the szeg kernel is shown to be the strong operator limit of pi 1 pi 2 n where pi i is the projection onto z i holomorphically extendible l 2 partial d functions in mathbf c mathbf 2 with a slightly more complicated formula in mathbf c mathbf n

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

continuity estimates for solutions to the prescribed curvature dirichlet problem on etudie des solutions au probleme de dirichlet pour l equation de courbure moyenne prescrite ainsi que pour des equations de weingarten plus generales on etablit des modules d estimation de continuite dependant seulement des valeurs limites

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

anisotropic flow of convex hypersurfaces by the square root of the scalar curvature abstract we show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor when the strictly convex initial hypersurface in euclidean space is suitably pinched we also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere for a general case in dimension two it is shown that with a volume preserving rescaling the limit profile satisfies a soliton equation

the constraint equations of lovelock gravity theories a new σk yamabe problem this paper is devoted to the study of the constraint equations of the lovelock gravity theories in the case of a conformally flat time symmetric and space like manifold we show that the hamiltonian constraint equation becomes a generalisation of the k yamabe problem that is to say the prescription of a linear combination of the k curvatures of the manifold we search solutions in a conformal class for a compact manifold using the existing results on the k yamabe problem we describe some cases in which they can be extended to this new problem this requires to study the concavity of some polynomial we do it in two ways regarding the concavity of a root of this polynomial which is connected to algebraic properties of the polynomial and seeking analytically a concavifying function this gives several cases in which a conformal solution exists finally we show an implicit function theorem in the case of a manifold with negative scalar curvature and find a conformal solution when the lovelock theories are close to general relativity this paper is devoted to the study of the constraint equations of the lovelock gravity theories in the case of a conformally flat time symmetric and space like manifold we show that the hamiltonian constraint equation becomes a generalisation of the k yamabe problem that is to say the prescription of a linear combination of the k curvatures of the manifold we search solutions in a conformal class for a compact manifold using the existing results on the k yamabe problem we describe some cases in which they can be extended to this new problem this requires to study the concavity of some polynomial we do it in two ways regarding the concavity of a root of this polynomial which is connected to algebraic properties of the polynomial and seeking analytically a concavifying function this gives several cases in which a conformal solution exists finally we show an implicit function theorem in the case of a manifold with negative scalar curvature and find a conformal solution when the lovelock t

isolated singularities for some types of curvature equations abstract we consider the removability of isolated singularities for the curvature equations of the form h k u 0 which is determined by the k th elementary symmetric function in an n dimensional domain we prove that for 1 k n 1 isolated singularities of any viscosity solutions to the curvature equations are always removable provided the solution can be extended continuously at the singularities we also consider the class of generalized solutions and prove the removability of isolated singularities

complete affine hypersurfaces part i the completeness of affine metrics on etudie les proprietes des hypersurfaces invariantes sous le groupe des transformations affines unimodulaires

conformal metrics with prescribed curvature functions on manifolds with boundary we study the dirichlet problem for a class of fully nonlinear elliptic equations related to conformal deformations of metrics on riemannian manifolds with boundary as a consequence we prove the existence of a conformal metric given its value on the boundary as a prescribed metric conformal to the induced background metric with a prescribed curvature function of the schouten tensor

nonsmooth solutions for a class of fully nonlinear pde s on lie groups abstract in this paper we prove the existence of nonsmooth viscosity solutions for dirichlet problems involving a class a fully non linear operators on lie groups in particular we consider the elementary symmetric functions of the eigenvalues of the hessian built with left invariant vector fields

the exterior dirichlet problem for special lagrangian equations in dimensions n 4 abstract in this paper we establish the existence theorem for the exterior dirichlet problem for special lagrangian equations with prescribed asymptotic behavior at infinity this extends the previous results on monge ampere equations and hessian equations to special lagrangian equations in dimensions n 4 which is from calibrated geometry more generally we prove that the result is also true for hessian quotient equations with 0 l k n in dimensions n 3

mean curvature flows of lagrangian submanifolds with convex potentials this article studies the mean curvature flow of lagrangian submanifolds in particular we prove the following global existence and convergence theorem if the potential function of a lagrangian graph in t 2n is convex then the flow exists for all time and converges smoothly to a flat lagrangian submanifold we also discuss various conditions on the potential function that guarantee global existence and convergence

meromorphic extensions from small families of circles and holomorphic extensions from spheres abstract let b be the open unit ball in c and let a b c be three points in c which do not lie in a complex line such that the complex line through a bmeetsb and such that if one of the points a b is inb and the other in c b then 6 1 and such that at least one of the numbers is different from 1 we prove that if a continuous function f on bb extends holomorphically into b along each complex line which meets a b c then f extends holomorphically through b this generalizes the recent result of l baracco who proved such a result in the case when the points a b c are contained in b

weingarten hypersurfaces with prescribed gradient image we prove the existence of globally smooth convex solutions u of a class of curvature equations subject to the boundary condition du omega omega where omega and omega are smooth uniformly convex domains in mathbb r n the results generalize some of our previous work on the two dimensional case and on hessian equations in all dimensions

rigidity theorems for the entire solutions of 2 hessian equation abstract in this paper we prove some rigidity theorems for the entire 2 convex solutions of 2 hessian equation in euclidean space as an application we obtain a bernstein type theorem for global special lagrangian graphs

on the exterior dirichlet problem for hessian quotient equations abstract in this paper we establish the existence and uniqueness theorem for solutions of the exterior dirichlet problem for hessian quotient equations with prescribed asymptotic behavior at infinity this extends the previous related results on the monge ampere equations and on the hessian equations and rearranges them in a systematic way based on the perron s method the main ingredient of this paper is to construct some appropriate subsolutions of the hessian quotient equation which is realized by introducing some new quantities about the elementary symmetric polynomials and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation

geometric inequalities on locally conformally flat manifolds through the study of some elliptic and parabolic fully nonlinear pdes we establish conformal versions of the quermassintegral inequality the sobolev inequality and the moser trudinger inequality for the geometric quantities associated to the schouten tensor on locally conformally flat manifolds

near to far field transformation by a time domain spherical multipole analysis the contribution deals with a new time domain near to far field nff transformation which is particularly suited for the finite difference time domain fdtd method the far field is derived from the tangential field components on a surface enclosing the scatterer by employing a time domain spherical multipole representation the necessary time domain convolution is performed as on the fly in parallel with the fdtd time stepping algorithm the efficiency of the method is improved by incorporating a temporal linear interpolation of the near field data with the once obtained multipole amplitudes an analytic series representation of the time domain far field is achieved which allows a physical interpretability of the result moreover this expansion serves as an ideal basis for a systematic and efficient post processing the proposed technique may also be useful for other numerical and for asymptotic methods

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

transverse fully nonlinear equations on sasakian manifolds and applications abstract we prove a priori estimates for a class of transverse fully nonlinear equations on sasakian manifolds and give some geometric applications such as the transversion calabi yau theorem for transverse balanced and strongly gauduchon metrics we also explain that similar results hold on compact oriented taut transverse hermitian foliated manifold of complex co dimension n and give some geometric applications such as the transverse calabi yau theorems for transverse hermitian and strongly gauduchon metrics