Abstract
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–Ampère type operators in optimal transportation and geometric optics, the general theory here embraces Neumann problems arising from prescribed mean curvature problems in conformal geometry as well as general oblique boundary value problems for augmented k-Hessian, Hessian quotient equations and certain degenerate equations.
Highlights
In this paper we develop the essentials of a general theory of classical solutions of oblique boundary value problems for certain types of fully nonlinear elliptic partial differential equations, which we describe as augmented Hessian equations
Such problems arise in various applications, notably to optimal transportation, geometric optics and conformal geometry and our critical domain and augmenting matrix convexity notions are adapted from those introduced in [31,40,45] for regularity in optimal transportation
Our theory embraces a wide class of examples which we present as well as a key application to semilinear Neumann problems arising in conformal geometry, where remarkably our adaptation of optimal transportation domain convexity from [40,45] enables us to replace the rather strong umbilic boundary condition for second derivative bounds, assumed in previous work [3,19], by more general natural convexity conditions
Summary
In this paper we develop the essentials of a general theory of classical solutions of oblique boundary value problems for certain types of fully nonlinear elliptic partial differential equations, which we describe as augmented Hessian equations. Remark 1.1 A stronger condition than regularity of the matrix function A is necessary in the above hypotheses as it is known from the Monge–Ampère case that one cannot expect second derivative estimates for general oblique boundary value problems for A ≡ 0, which is a special case of regular A but not strictly regular, see [47,52]. In its formulation we will assume the existence of subsolutions and supersolutions to provide the necessary solution estimates and an appropriate interval I in our boundary convexity conditions For this purpose we will say that functions u and u, in C2( ) ∩ C1( ̄ ), are respectively subsolution and supersolution of the boundary value problem (1.1)–(1.2) if. Since our main concern here is a priori estimates for classical solutions the reader may assume that all functions and domains are C∞ smooth
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