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A primer on semiconvex functions in general potential theories

Kevin R. Payne, Davide Francesco Redaelli

TL;DR

This primer develops semiconvex analysis as a bridge between classical and viscosity theories for generalized subharmonics defined by a subequation $ ext{F}$ on the $2$-jet bundle. It systematizes four core tools in Part I—differentiability (Alexandrov), upper contact jets, the Jensen–Słodkowski machinery, and semiconvex approximation via sup-convolution—then places them in the broader context of general potential theories with duality and monotonicity in Part II. Central contributions include the HL reformulations and equivalences of Jensen–Słodkowski lemmas, paraboloidal machinery for contact points, and a robust semiconvex framework enabling comparison principles and Perron's method across constant and variable coefficient settings. The work emphasizes a deep interplay between potential theory and fully nonlinear elliptic PDEs, offering a self-contained, coordinate-friendly foundation that extends to manifolds through pullbacks and yields practical tools for both operator theory embedding and nonlinear Dirichlet problems.

Abstract

This work is dedicated to foundational aspects of general (nonlinear second order) potential theories and fully nonlinear elliptic PDEs. In particular, we systematically develop the fundamental role played by semiconvex functions as a bridge between the classical and viscosity theory of generalized subharmonics determined by a given subequation constraint set in the bundle of $2$-jets over open subsets of Euclidian spaces. The first part is dedicated to four fundamental and complementary aspects of semiconvex analysis for the viscosity theory of subharmonic functions in general potential theories: differentiability properties of first and second order for locally semiconvex functions; a detailed analysis of their upper contact jets and upper contact points; the deep analytical results which ensure the existence of sets of contact points with positive measure; the well-known device of semiconvex approximation of upper semicontinuous functions. The second part is dedicated to general potential theories. The fundamental roles of duality and monotonicity are discussed, along with the main tools in the viscosity theory of subharmonics and the essential role that semiconvex functions play in that theory. The validity of the comparison principle is examined in two distinct regimes of sufficient monotonicity and minimal monotonicity.

A primer on semiconvex functions in general potential theories

TL;DR

This primer develops semiconvex analysis as a bridge between classical and viscosity theories for generalized subharmonics defined by a subequation on the -jet bundle. It systematizes four core tools in Part I—differentiability (Alexandrov), upper contact jets, the Jensen–Słodkowski machinery, and semiconvex approximation via sup-convolution—then places them in the broader context of general potential theories with duality and monotonicity in Part II. Central contributions include the HL reformulations and equivalences of Jensen–Słodkowski lemmas, paraboloidal machinery for contact points, and a robust semiconvex framework enabling comparison principles and Perron's method across constant and variable coefficient settings. The work emphasizes a deep interplay between potential theory and fully nonlinear elliptic PDEs, offering a self-contained, coordinate-friendly foundation that extends to manifolds through pullbacks and yields practical tools for both operator theory embedding and nonlinear Dirichlet problems.

Abstract

This work is dedicated to foundational aspects of general (nonlinear second order) potential theories and fully nonlinear elliptic PDEs. In particular, we systematically develop the fundamental role played by semiconvex functions as a bridge between the classical and viscosity theory of generalized subharmonics determined by a given subequation constraint set in the bundle of -jets over open subsets of Euclidian spaces. The first part is dedicated to four fundamental and complementary aspects of semiconvex analysis for the viscosity theory of subharmonic functions in general potential theories: differentiability properties of first and second order for locally semiconvex functions; a detailed analysis of their upper contact jets and upper contact points; the deep analytical results which ensure the existence of sets of contact points with positive measure; the well-known device of semiconvex approximation of upper semicontinuous functions. The second part is dedicated to general potential theories. The fundamental roles of duality and monotonicity are discussed, along with the main tools in the viscosity theory of subharmonics and the essential role that semiconvex functions play in that theory. The validity of the comparison principle is examined in two distinct regimes of sufficient monotonicity and minimal monotonicity.
Paper Structure (42 sections, 82 theorems, 445 equations, 2 figures)

This paper contains 42 sections, 82 theorems, 445 equations, 2 figures.

Table of Contents

  1. Semiconvex apparatus
  2. Differentiability of convex functions
  3. First order theory
  4. Second order theory: Alexandrov's theorem
  5. Semiconvex functions and upper contact jets
  6. Semiconvex functions
  7. A semiconvexity characterization of $C^{1,1}$
  8. Upper contact jets
  9. The theorems on upper contact jets and on summands
  10. The lemmas of Jensen and Słodkowski
  11. Classical statements: a review
  12. Upper contact quadratic functions and the vertex map
  13. A paraboloidal proof of Słodkowski's density estimate
  14. The Jensen--Słodkowski theorem
  15. Consequences of the Jensen--Słodkowski Theorem
  16. ...and 27 more sections

Key Result

Lemma 1.3

Jensen's inequality; finite form] Let $u: X \to {\mathbb{R}}$ be convex on $X \subset {\mathbb{R}}^n$ convex. Then for each collection of points $\{x_j\}_{j=1}^N \subset X$ with $N \geqslant 2$

Figures (2)

  • Figure 3.1: The three situations described in \ref{['parab:cases']}.
  • Figure 3.2: The slab constructed in the proof of \ref{['shp']}.

Theorems & Definitions (247)

  • Definition 1.1
  • Remark 1.2: Reduction to the one-dimensional notion
  • Lemma 1.3: Jensen's inequalitysubject
  • Definition 1.4
  • Lemma 1.5: Localization
  • Remark 1.6: Examples of convex functions
  • Definition 1.7
  • Remark 1.8
  • Remark 1.9
  • Proposition 1.10: Subdifferentiability of convex funtions
  • ...and 237 more