Universal Moduli of Continuity for Solutions to Fully Nonlinear Elliptic Equations
By E. Teixeira. 2011
75 citations recorded by the source corpus.
This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations F(X, D2u) = 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 Ln norm of f, which corresponds to optimal regularity bounds for the critical threshold case. Optimal C1,α regularity estimates are also delivered when $${f\in L^{n+\varepsilon}}$$f∈Ln+ε . The limiting upper borderline case, $${f \in L^\infty}$$f∈L∞ , also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under the convexity assumption on F, that $${u \in C^{1,{\rm Log-Lip}}}$$u∈C1,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 $${C^{0,\frac{n-2\varepsilon}{n-\varepsilon}}}$$C0,n-2εn-ε norm of u based on the Ln-ε norm of f, where ɛ is the Escauriaza universal constant. The exponent $${\frac{n-2\varepsilon}{n-\varepsilon}}$$n-2εn-ε is optimal. When the source function f lies in Lq, n > q > n−ε, we also obtain the exact, improved sharp Hölder exponent of continuity.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/9a65cad510e731cd07e471beea9547331b785f40
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- A proof of the Cp′-regularity conjecture in the plane (2017, background): “d a C1 corrector ξ, with kξkC1 ≪ 1, such that u+ξ is p-harmonic. This will allow us to frame the Cp′ conjecture into the formalism of the so called geometric tangential analysis, e.g. [7], [2, 1] and [17, 18, 19, 20, 21, 22]. Here is the precise statement. Lemma3. Let u ∈ W1,p(B 1) be a weak solutio”
- Regularity of solutions to a class of variable–exponent fully nonlinear elliptic equations (2018, background): “When F is convex, C loc (B1) follows from [36].”