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Neil S. Trudinger

Publications and source records attributed to Neil S. Trudinger.

14 recordsLinked to original sources
arXiv2026

On a class of Cauchy problems with applications in nonlinear partial differential equations

In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of nonlinear partial differential equations in n-dimensional Euclidean space. As applications of the general framework, we present such results for two series of nonlinear equations: a series of k-Hessian type equations and a new series of P-k-Hessian type equations. These results are also proved for the more general p-Hessian matrices, with p > 1, and degenerate non homogeneous terms.

math.AP↗
arXiv2024

On the maximum principle for general linear elliptic equations

We consider maximum principles and related estimates for linear second order elliptic partial differential operators in n-dimensional Euclidean space, which improve previous results, with H-J Kuo, through sharp Lp dependence on the drift coefficient b. As in our previous work, the ellipticity is determined through the principal coefficient matrix A lying in sub-cones of the positive cone, which are dual cones of the Garding k-cones. Our main results are maximum principles for bounded domains, which extend those of Aleksandrov in the case k = n, together with extensions to unbounded domains, depending on appropriate integral norms of A, and corresponding local maximum principles. We also consider applications to local estimates in the uniformly elliptic case, including extensions of the Krylov-Safonov Holder and Harnack estimates.

math.AP↗
arXiv2019

On the Dirichlet problem for general augmented Hessian equations

In this paper we apply various first and second derivative estimates and barrier constructions from our treatment of oblique boundary value problems for augmented Hessian equations, to the case of Dirichlet boundary conditions. As a result we extend our previous results on the Monge-Ampere and k-Hessian cases to general classes of augmented Hessian equations in Euclidean space

math.AP↗
arXiv2018

On the second boundary value problem for Monge-Ampere type equations and geometric optics

In this paper, we prove the existence of classical solutions to second boundary value prob- lems for generated prescribed Jacobian equations, as recently developed by the second author, thereby obtaining extensions of classical solvability of optimal transportation problems to problems arising in near field geometric optics. Our results depend in particular on a priori second derivative estimates recently established by the authors under weak co-dimension one convexity hypotheses on the associated matrix functions with respect to the gradient variables, (A3w). We also avoid domain deformations by using the convexity theory of generating functions to construct unique initial solutions for our homotopy family, thereby enabling application of the degree theory for nonlinear oblique boundary value problems.

math.AP↗
arXiv2017

Oblique boundary value problems for augmented Hessian equations I

In this paper, we study global regularity for oblique boundary value problems of augmented Hessian equations for a class of general operators. By assuming a natural convexity condition of the domain together with appropriate convexity conditions on the matrix function in the augmented Hessian, we develop a global theory for classical elliptic solutions by establishing global a priori derivative estimates up to second order. Besides the known applications for Monge-Amp`ere type operators in optimal transportation and geometric optics, the general theory here embraces prescribed mean curvature problems in conformal geometry as well as oblique boundary value problems for augmented k-Hessian, Hessian quotient equations and certain degenerate equations.

math.AP↗
arXiv2016

Oblique boundary value problems for augmneted Hessian equations II

In this paper, we continue our investigations into the global theory of oblique boundary value problems for augmented Hessian equations. We construct a global barrier function in terms of an admissible function in a uniform way when the matrix function in the augmented Hessian is only assumed regular. This enables us to derive global second derivative estimates in terms of boundary estimates which are then obtained by strengthening the concavity or monotonicity conditions in our previous work on the strictly regular case. Finally we give some applications to existence theorems which embrace standard Hessian equations as special cases.

math.AP↗
arXiv2015

On the Neumann problem for Monge-Ampère type equations

In this paper, we study the global regularity for regular Monge-Ampère type equations associated with semilinear Neumann boundary conditions. By establishing a priori estimates for second order derivatives, the classical solvability of the Neumann boundary value problem is proved under natural conditions. The techniques build upon the delicate and intricate treatment of the standard Monge-Ampère case by Lions, Trudinger and Urbas in 1986 and the recent barrier constructions and second derivative bounds by Jiang, Trudinger and Yang for the Dirichlet problem. We also consider more general oblique boundary value problems in the strictly regular case.

math.AP↗
arXiv2014

On the Dirichlet problem for a class of augmented Hessian equations

In this paper, we consider the Dirichlet problem for a new class of augmented Hessian equations. Under sharp assumptions that the matrix function in the augmented Hessian is regular and there exists a smooth subsolution, we establish global second order derivative estimates for the solutions to the Dirichlet problem in bounded domains. The results extend the corresponding results in the previous paper [11] from the Monge-Ampere type equations to the more general Hessian type equations.

math.AP↗
arXiv2006

New maximum principles for linear elliptic equations

We prove extensions of the estimates of Aleksandrov and Bakel$'$man for linear elliptic operators in Euclidean space $\Bbb{R}^{\it n}$ to inhomogeneous terms in $L^q$ spaces for $q < n$. Our estimates depend on restrictions on the ellipticity of the operators determined by certain subcones of the positive cone. We also consider some applications to local pointwise and $L^2$ estimates.

math.AP↗
arXiv2005

Boundary regularity for the Monge-Ampere and affine maximal surface equations

In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomogeneous term is only assumed to be Holder continuous. As a consequence of our approach, we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the affine maximal surface equation and affine mean curvature equation.

math.DG↗
arXiv2004

The affine Plateau problem

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.

math.DG↗
arXiv1999

Hessian measures II

In our previous paper on this topic, we introduced the notion of k-Hessian measure associated with a continuous k-convex function in a domain \Om in Euclidean n-space, k=1,...,n, and proved a weak continuity result with respect to local uniform convergence. In this paper, we consider k-convex functions, not necessarily continuous, and prove the weak continuity of the associated k-Hessian measure with respect to convergence in measure. The proof depends upon local integral estimates for the gradients of k-convex functions.

math.FA↗