Regularity Results for Nonlocal Equations by Approximation

By Luis Caffarelli, Luis Silvestre. 2009

289 citations recorded by the source corpus.

We obtain C1,α regularity estimates for nonlocal elliptic equations that are not necessarily translation-invariant using compactness and perturbative methods and our previous regularity results for the translation-invariant case.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/59c8146b161360b6f6e8762f8a2efe9a8d7ce48c

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Citations in the corpus

On the spectrum of two different fractional operators (2014, unknown): “In particular, two notions of fractional operators were considered in the literature, namely the integral one (which reduces to the classical fractional Laplacian, see, for instance, [7, 8, 9, 10, 14, 15, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] and references therein) and the”
Integro-Differential equations with nonlinear directional dependence (2014, background): “The first robust results appeared in [9], [10], [11], where a whole program was developed involving C1,β and then classical regularity (the Evans-Krylov Theorem for the integro-differential setting) for fully nonlinear equations which captures all of the existing second order theory as a limit α → 2”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, result): “When p = 2, the literature is richer: similar results have been obtained in [5, 37, 39, 40] and [59, 19, 20].”
Regularity for solutions of nonlocal, nonsymmetric equations (2012, methodology): “Let φ(x) = min(1,C(|x| − 1)+), where C and α have been chosen as in [5].”
WEAK AND VISCOSITY SOLUTIONS OF THE FRACTIONAL LAPLACE EQUATION (2014, background): “As a technical remark, we also notice that the barrier used in [5] does not belong to a good energy space like H(Ω) and therefore it cannot be used to prove directly the boundary continuity of weak solutions.”
Aleksandrov–Bakelman–Pucci Type Estimates for Integro-Differential Equations (2010, methodology): “− becomes much harder to tackle. In conclusion, all these issues underline the lack of geometric equations for integro-diﬀerential operators. 8. Important Limits As σ→ 2 In the series of works, [16], [17], and [18], all of the results were obtained in a fashion in which they are preserved as σ→ 2 an”
Global H\"older regularity for the fractional $p$-Laplacian (2014, background): “For non-local, fully non-linear, uniformly elliptic equation, higher interior regularity (up to C) is proved in [5], [6], [21] when the kernel satisfies additional structural and regularity assumption, but no such result is known for the s-fractional p-Laplacian.”
A logistic equation with nonlocal interactions (2016, background): “h τ= 0, (−∆)su m ≤ kσ mk 2 L∞(Ω) ≤ kσ0kL∞(Ω) +1 2 . Notice that the latter quantity does not depend on m. Thus, by fractional elliptic regularity (see e.g. Proposition 1.1 in [ROS14] and Lemma 4.3 in [CS11]) we have that um converges uniformly in Ω to some u0 as m→ 0, and u0 solves (−∆)su 0 = (σ0 − ”
Ground state solutions of scalar field fractional Schrödinger equations (2015, background): “In addition, nonlocal fractional equations appear inmany fields and a lot of interest has been devoted to this kind of problems and to their concrete applications; see, for instance the seminal papers [12–14] and [3,9,15,22,40,57], as well as the references therein.”
On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (2011, background): “11) holds with infB3/4 u replaced by ‖u‖L∞(Rn), which is proved in [22], [23] and [24] in a much more general setting of fully nonlinear nonlocal equations.”
Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case (2017, methodology): “However, in order not to lose the Hölder exponent in this transfer, we need to employ a blow-up argument similar to that used by Caffarelli and Silvestre in [12], see Proposition 6.”