The Inhomogeneous Dirichlet problem for natural operators on manifolds

Abstract

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Comparison principles by monotonicity and duality for constant coefficient nonlinear potential theory and PDEs

“…Attention is restricted to the constant coefficient case. This discussion goes beyond what is already given in [33]. It is helpful to be guided by two types of examples from the pure second order case.…”

“…The compatibility of part (a) is immediate from (11.89). For the proof of tameness (b), we refer the reader to Propositon 6.11 of [33], which makes use (11.87). Tame implies topologically tame.…”

“…The properties (a) and (c) in Proposition 11.32 imply that (11.93) comparison for the operator g Γ ⇔ comparison for the subequation Γ, by Theorem 11.13. Property (b) plays a key role in comparison on domains Ω for inhomogeneous equations g(D 2 u) = ψ(x) with ψ a continuous function on Ω (see [33]).…”

“…See Proposition 6.11 of[33] for the definition of the canonical operator, its construction and many additional examples. See also subsection ??…”

Principal eigenvalues for k-Hessian operators by maximum principle methods

“…Remark 6.2. Using the language of Harvey-Lawson [17], one could also say that (S k , Σ k ) is a compatible operator-subequation pair (see Definition 2.4 of [17]) and since the continuous boundary data ϕ ≡ 0 has its values in S k (Σ k ), the result follows also from Theorem 2.7 of [17].…”