$$C^{2,\alpha }$$ C 2 , α estimates for nonlinear elliptic equations of twisted type

Let (X, α) be a Kähler manifold of dimension n, and let [ω] ∈ H 1,1 (X, R). We study the problem of specifying the Lagrangian phase of ω with respect to α, which is described by the nonlinear elliptic equationwhere λi are the eigenvalues of ω with respect to α. When h(x) is a topological constant, this equation corresponds to the deformed Hermitian-Yang-Mills (dHYM) equation, and is related by Mirror Symmetry to the existence of special Lagrangian submanifolds of the mirror. We introduce a notion of subsolution for this equation, and prove a priori C 2,β estimates when |h| > (n − 2) π 2 and a subsolution exists. Using the method of continuity we show that the dHYM equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists. Finally, we discover some stability-type cohomological obstructions to the existence of solutions to the dHYM equation and we conjecture that when these obstructions vanish the dHYM equation admits a solution. We confirm this conjecture for complex surfaces.

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“…Proof. The proof is by a standard blow-up argument; see, for instance [9]. We give the details for the convenience of the reader.…”

Section: Higher Order Estimatesmentioning

confidence: 99%

$(1,1)$ forms with specified Lagrangian phase: <i>a priori</i> estimates and algebraic obstructions

Collins

Jacob

Yau

2020

Cambridge Journal of Mathematics

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“…If the linearized operator for F satisfies a Cordes-Nirenberg condition, one can also obtain this boosting (see Section 5). Since the 1980s, it has been a challenge to find equations with the regularity boosting property that are niether convex nor concave, see for example [CY00], [Yua01], [CC03] , [Col16], [SW16], [Pin16]. Savin [Sav07] proved interior C 2,α (and higher) estimates for viscosity solutions of (1.2) that are sufficiently close to a quadratic polynomial, for F smooth.…”

Section: Introductionmentioning

confidence: 99%

$C^{2,α}$ estimates for solutions to almost linear elliptic equations

Bhattacharya,

Warren

2019

Preprint

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“…A similar problem is examined in [11]. In that paper, the author produces a priori estimates for the solutions to…”

Section: Introductionmentioning

confidence: 99%

Regularity theory for the Isaacs equation through approximation methods

Pimentel¹

2018

Preprint

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In this paper, we propose an approximation method to study the regularity of solutions to the Isaacs equation. This class of problems plays a paramount role in the regularity theory for fully nonlinear elliptic equations. First, it is a model-problem of a non-convex operator. In addition, the usual mechanisms to access regularity of solutions fall short in addressing these equations. We approximate an Isaacs equation by a Bellman one, and make assumptions on the latter to recover information for the former. Our techniques produce results in Sobolev and Hölder spaces; we also examine a few consequences of our main findings.

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$$C^{2,\alpha }$$ C 2 , α estimates for nonlinear elliptic equations of twisted type

Cited by 16 publications

References 21 publications

$(1,1)$ forms with specified Lagrangian phase: <i>a priori</i> estimates and algebraic obstructions

$(1,1)$ forms with specified Lagrangian phase: <i>a priori</i> estimates and algebraic obstructions

$C^{2,α}$ estimates for solutions to almost linear elliptic equations

Regularity theory for the Isaacs equation through approximation methods

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