Lipschitz regularity of solutions for mixed integro-differential equations

By Guy Barles, Emmanuel Chasseigne, Adina Ciomaga, C. Imbert. 2011

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We establish new Holder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii–Lionsʼs method. We thus extend the Holder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local–nonlocal interaction, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c24dd11e2267349723c8771a29f39290af30dfb3

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Citations in the corpus

Lp-maximal regularity of nonlocal parabolic equations and applications (2013, unknown): “On the other hand, Hölder regularity theory for the viscosity solutions of fully non-linear and nonlocal elliptic equations was also developed by Caffarelli and Silvestre [8], and Barles, Chasseigne and Imbert [4] (see also [3] and the series of works of Silverstre [30,31, 33,32], etc. .”
Mixed local and nonlocal elliptic operators: regularity and maximum principles (2020, background): “For regularity results involving mixed order diffusive operators in different directions see also [4, 34, 35].”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, background): “There are also many important results in this direction including [17], [60], [16], [15], [57], [22], [58], [13], [12], [19], [23], [25], [47], [26], [42] and [56].”
Linear theory for a mixed operator with Neumann conditions (2020, unknown): “In terms of theory and applications, we recall that operators with mixed classical and fractional orders have been studied under different points of views, see for instance [1,2,4,6–10,12,13,18,19,21,25–27, 31–33,40,47,48] and the references therein.”
Regularity results for solutions of mixed local and nonlocal elliptic equations (2022, background): “Moreover, mixed operators have recently received a great attention from different points of view, including regularity theory [2, 3, 8, 10, 12, 14, 15, 17, 25], existence and non-existence results [1, 6, 28], eigenvalue problems [13, 16, 22], shape optimization and calculus of variations [7, 11], sy”
The Bernstein Technique for Integro-Differential Equations (2020, methodology): “We point out that the results in both [17] and [4] apply to viscosity solutions.”
A Faber-Krahn inequality for mixed local and nonlocal operators (2021, background): “Operators of this type present interesting mathematical questions, especially due to the lack of scale invariance and in view of the combination of local and nonlocal behaviors, see [41, 42, 5, 22, 10, 18, 3, 17, 4, 53, 16, 23, 32, 24, 8, 7, 29, 15, 1, 14, 21, 45].”
A Brezis-Nirenberg type result for mixed local and nonlocal operators (2022, unknown): “The contemporary investigation has speciﬁcally focused on several problems in the existence and regularity theory (see [1,4,5,8,11,12,16–18, The stirring motivation for the problems presented in this paper comes from the study of nonlinear problems with critical exponents, as the ones suggested by…”