
Contents
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11.1 Compatible Operator-Subequation Pairs and Topological Tameness 11.1 Compatible Operator-Subequation Pairs and Topological Tameness
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11.2 The Correspondence Principle for Compatible Pairs 11.2 The Correspondence Principle for Compatible Pairs
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11.3 A Structure Theorem Derived from Subequation Monotonicity 11.3 A Structure Theorem Derived from Subequation Monotonicity
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11.4 Canonical Operators for Subequations With Monotonicity 11.4 Canonical Operators for Subequations With Monotonicity
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11.5 Lipschitz Regularity of Subequation Boundaries 11.5 Lipschitz Regularity of Subequation Boundaries
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11.6 Gårding–Dirichlet Operators 11.6 Gårding–Dirichlet Operators
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11.7 Subequation Branches 11.7 Subequation Branches
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Eleven The Correspondence Principle for Compatible Operoperatoator-Subequation Pairs
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Published:October 2023
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Abstract
This chapter discusses some key issues concerning applications of the potential-theoretic comparison principles to comparison principles for constant-coefficient nonlinear operators. Attention is restricted to the constant-coefficient case. The chapter provides two types of examples from the pure second-order case: one where the operator F is defined and “elliptic” on the full jet space 𝒥2, and one where F must be restricted (constrained) to a proper subset of 𝒥2 in order to be “elliptic.” Classes of examples and illustrations are provided, including unconstrained case examples like canonical operators and constrained case examples like Garding–Dirichlet polynomials. These classes are representative but, of course, not exhaustive for the dichotomy explored in the chapter. In particular, the minimal eigenvalue operator is a canonical operator for the pure second-order subequation ℘⊂𝒮(n), and the Monge–Ampère operator is one of the most basic and important Garding–Dirichlet polynomials.
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