Second-order elliptic integro-differential equations: viscosity solutions' theory revisited

By Guy Barles, C. Imbert. 2007

426 citations recorded by the source corpus.

The aim of this work is to revisit viscosity solutions' theory for second-order elliptic integro- differential equations and to provide a general framework which takes into account solutions with arbitrary growth at infinity. Our main contribution is a new Jensen-Ishii's Lemma for integro-differential equations, which is stated for solutions with no restriction on their growth at infinity. The proof of this result, which is of course a key ingredient to prove comparison principles, relies on a new definition of viscosity solution for integro-differential equation (equivalent to the two classical ones) which combines the approach with test- functions and sub-superjets.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/69f557971e13c71a3294a39d423ca4bfa7b3f347

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Citations in the corpus

Regularity theory for fully nonlinear integro‐differential equations (2007, methodology): “In this paper we will only consider continuous right-hand sides f .”
Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions (2009, methodology): “Following [7] we give here a nonlocal theorem which should replace the “maximum principle”.”
On the Dirichlet Problem for Second-Order Elliptic Integro-Differential Equations (2008, methodology): “[9,14] for local equations and [19,1,4,5,13] for non-local ones) together with the comparison result of [4] provides the existence of a solution u for this integro-differential equation such that |u| ≤M.”
Generalized Dynkin games and doubly reflected BSDEs with jumps (2013, neutral): “We can thus apply a generalized Jensen-Ishii’s lemma 5 (see [3]), which leads to the desired result, by using the same arguments as in [12] (see Theorem 4.1, last case).”
Nonlocal Curvature Flows (2014, background): “See also [3,5], and [6] for other somewhat similar nonlocal evolutions, related to dislocation dynamics.”
Aleksandrov–Bakelman–Pucci Type Estimates for Integro-Differential Equations (2010, methodology): “Schwab in [2], and the comparison theory for these equations was recently improved and re-presented in [5], with the Dirichlet problem being considered in [3].”
A Brezis-Nirenberg type result for mixed local and nonlocal operators (2022, positive): “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…”
Mixed local and nonlocal elliptic operators: regularity and maximum principles (2020, background): “The mathematical study of operators with different order is not new in itself, and indeed the literature already presents results concerning, among the others, the theory of viscosity solutions (see [37, 38, 6, 11, 21, 4, 5]), the Aubry-Mather theory for sums of different fractional Laplacians (see ”
Regularity for parabolic integro-differential equations with very irregular kernels (2014, methodology): “Let α ∈ [1, 2) and suppose r ∈ (0, 1) is given.”
Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE (2012, result): “We also mention recent results on comparison principles for IPDE in [3] and references therein.”
A class of integral equations and approximation of p-Laplace equations (2010, methodology): “We refer the reader to [1,2] and the references therein for some applications of nonlocal diffusion equations like (1.1), (1.4), or (1.4) with MN in place of MD. For the viscosity solutions approach to integro-differential equations with singular kernels, we refer to the article [ 4 ].”
On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion (2009, background): “This definition is the straight forward adaptation of the corresponding definitions in [1] and [3] for elliptic problems.”
Regularity for solutions of nonlocal, nonsymmetric equations (2012, background): “Viscosity solutions provide the right framework to study fully nonlinear equations, as seen in the local case in [3], and also in the nonlocal case in [1].”
C 1, α Interior Regularity for Nonlinear Nonlocal Elliptic Equations with Rough Kernels (2013, background): “For nonlocal equations, a comparison principle was proved in [5] for translation-invariant equations, and in [1] for some classes of nonlocal operators sufficiently regular in x.”
Lipschitz regularity of solutions for mixed integro-differential equations (2011, methodology): “Thoughout this paper, we use the definition involving sub and super-jets, which was shown in [2] to be equivalent with Definition 1.”
About the Pricing Equations in Finance (2011, background): “In the case of unbounded Levy measures however, Barles and Imbert [7] (see also Jakobsen and Karlsen [63]) recently established that this Lemma (and thus the related uniqueness proofs in Barles et al.”
Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative (2016, result): “See also [5], [2] and references therein.”