Mixed local and nonlocal elliptic operators: regularity and maximum principles

By Stefano Biagi, Serena Dipierro, Enrico Valdinoci, E. Vecchi. Communications in Partial Differential Equations, 2020

181 citations recorded by the source corpus.

Abstract We develop a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1aba97372009dd362508c7b50d8a34b10aac95d5

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Citations in the corpus

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems (2021, methodology): “Furthermore, existence, regularity, symmetry and maximum principles along various qualitative properties of solutions have been recently investigated by Biagi-DipierroValdinoci-Vecchi [12, 13], Dipierro-Proietti Lippi-Valdinoci [35, 36], Biagi-Mugnai-Vecchi [15], Dipierro-Ros-Oton-Serra-Valdinoci [3”
On the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, methodology): “16) have recently been studied by Biagi-Dipierro-Valdinoci-Vecchi [6,7], Dipierro-Proietti Lippi-Valdinoci [22,23] and Dipierro-Ros-Oton-Serra-Valdinoci [24].”
Gradient regularity in mixed local and nonlocal problems (2022, background): “Compare for instance with the discussion made in [5,8], where related functional settings are considered.”
A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators (2021, background): “This fact shows that, when Ω is sufficiently regular, X 1,p 0 (Ω) coincides with the space Xp(Ω) introduced in [6] (for p = 2) and in [9] (for a general p > 1).”
Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations (2022, background): “For the linear case p = 2, existence, local boundedness, interior Sobolev regularity and a strong maximum principle, along with other qualitative properties of solutions have been established by Biagi-Dipierro-Valdinoci-Vecchi in [6].”
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.”
A Faber-Krahn inequality for mixed local and nonlocal operators (2021, methodology): “The proof of Theorem 3.5 relies on the classical method by Stampacchia, and is essentially analogous to that of [7, Theorem 4.7] (see also [28, 55]).”
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, 8,11,12,16–18,22 The stirring motivation for the problems presented in this paper comes from the study of nonlinear problems with critical exponents, as the ones suggested…”
Combined effects in mixed local–nonlocal stationary problems (2021, background): “Some questions related to structural results like existence, maximum principle and interior Sobolev and Lipschitz regularity [1, 11, 14], symmetry results [13], Faber-Krahn type inequality [12], Neumann problems [42], Green functions estimates [26, 27] has been answered.”
A Brezis-Oswald approach for mixed local and nonlocal operator (2021, unknown): “1 cannot be deduced as a corollary of the maximum principles proved in [5] nor in [12].”
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”