Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

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

132 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/920e7301eaf77f4a82008273e34900eb8b1e35c6

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Lp-maximal regularity of nonlocal parabolic equations and applications (2013, unknown): “It is remarked that the technique in [4] was used by Dong and Kim to derive some local Hölder estimate for nonlocal elliptic equation in order to prove their characterization.”
Lipschitz regularity of solutions for mixed integro-differential equations (2011, methodology): “Therefore we consider, as in [1] the signed measure μ = μx̄ − μȳ.”
On a new class of fractional partial differential equations (2014, methodology): “We can cite, for instance, the regularity theory of nonlocal equations developed by Fukushima [13] and Kassmann [16, 17] in the symmetric case, and Barles, Chasseigne and Imbert [3], Ca arelli and Silvestre [6], and Di Castro, Kuusi and Palatucci [10] where no such symmetry is assumed.”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, unknown): “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].”
Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result (2009, background): “The difficulty here is that Lipschitz bounds on the initial data are not propagated on uǫ and may blow up in finite time (see [12, 5, 15] for regularity results for integral Hamiltonian).”
Nonlocal elliptic equations in bounded domains: a survey (2015, unknown): “• Interior regularity of solutions: Bass–Levin [6], Kassmann [33], Caffarelli–Silvestre [13, 14, 15], Barles–Chasseigne–Imbert [4], Kassmann–Mimica [34, 35], Schwab–Silvestre [53], and Serra [54].”
Boundary regularity for fully nonlinear integro-differential equations (2014, background): “Other important references concerning interior regularity for nonlocal equations in nondivergence form are [24, 23, 17, 2, 21].”
Boundary Harnack inequality for Markov processes with jumps (2012, unknown): “Hölder continuity of harmonic functions is crucial in the theory of partial differential equations [6, 16], and the existence of limits of ratios of nonnegative harmonic functions leads to the construction of the Martin kernel and to representation of nonnegative harmonic functions ([5, 26]).”
On the strong maximum principle for second order nonlinear parabolic integro-differential equations (2010, background): “[7]) and we expect to have Strong Maximum Principle.”
Integro-Differential equations with nonlinear directional dependence (2014, result): “In [1] regularity results were obtained for kernels very similar to (1.”
Existence of densities for stable-like driven SDEʼs with Hölder continuous coefficients (2013, unknown): “Assume that α ∈ [1, 2) and that σ ∈ C1(R) for some θ1 ∈ (0, 1) and either that b is measurable (then we set θ2 = 0) or that b ∈ C 2(R) for some θ2 ∈ (0, 1).”
A class of integral equations and approximation of p-Laplace equations (2010, unknown): “We refer to [ 3 ,6] for regularity results for integro-differential equations.”
Hölder estimates for viscosity solutions of equations of fractional p-Laplace type (2014, background): “For fully nonlinear equations of fractional Laplace type, uniform estimates as s → 1 have been obtained (see for instance [3] and [8]), but they are more involved, and they follow the same strategy as the estimates for fully nonlinear (local) equations.”