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Universal Moduli of Continuity for Solutions to Fully Nonlinear Elliptic Equations

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Abstract

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations F(X, D 2 u) =  f(X), based on the weakest and borderline integrability properties of the source function f in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on u based on the L n norm of f, which corresponds to optimal regularity bounds for the critical threshold case. Optimal C 1,α regularity estimates are also delivered when \({f\in L^{n+\varepsilon}}\) . The limiting upper borderline case, \({f \in L^\infty}\) , also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under the convexity assumption on F, that \({u \in C^{1,{\rm Log-Lip}}}\) , provided f has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior a priori estimates on the \({C^{0,\frac{n-2\varepsilon}{n-\varepsilon}}}\) norm of u based on the L n-ε norm of f, where ɛ is the Escauriaza universal constant. The exponent \({\frac{n-2\varepsilon}{n-\varepsilon}}\) is optimal. When the source function f lies in L qn > q > nε, we also obtain the exact, improved sharp Hölder exponent of continuity.

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Correspondence to Eduardo V. Teixeira.

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Communicated by G. Dal Maso

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Teixeira, E.V. Universal Moduli of Continuity for Solutions to Fully Nonlinear Elliptic Equations. Arch Rational Mech Anal 211, 911–927 (2014). https://doi.org/10.1007/s00205-013-0688-7

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