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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.Comment: Revised version containing minor clarification
On the differential structure of metric measure spaces and applications
The main goals of this paper are: i) To develop an abstract differential
calculus on metric measure spaces by investigating the duality relations
between differentials and gradients of Sobolev functions. This will be achieved
without calling into play any sort of analysis in charts, our assumptions
being: the metric space is complete and separable and the measure is Borel, non
negative and locally finite. ii) To employ these notions of calculus to
provide, via integration by parts, a general definition of distributional
Laplacian, thus giving a meaning to an expression like , where
is a function and is a measure. iii) To show that on spaces with
Ricci curvature bounded from below and dimension bounded from above, the
Laplacian of the distance function is always a measure and that this measure
has the standard sharp comparison properties. This result requires an
additional assumption on the space, which reduces to strict convexity of the
norm in the case of smooth Finsler structures and is always satisfied on spaces
with linear Laplacian, a situation which is analyzed in detail.Comment: Clarified the dependence on the Sobolev exponent of various
objects built in the paper. Updated bibliography. Corrected typo
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