A class of integral equations and approximation of p-Laplace equations

By Hitoshi Ishii, Gou Nakamura. 2010

117 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/4edbe52ce1af5f34a9c1f6172efa075238c8e0db

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In Applied Mathematics

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Citations in the corpus

The Brezis–Nirenberg problem for the fractional p-Laplacian (2015, background): “This operator appears in some recent works, see [2,31] as well as [9] for the motivations, that led to its introduction.”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, background): “Nakamura in [36], where the authors dealt with the solvability (in the viscosity sense) of the associated Dirichlet problem and established a connection with the classical p-Laplace operator in the limit as s → 1.”
Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case (2017, background): “q . and sharp when s= 1/2, p= 2 and q= ∞. For a more detailed description of our results we refer the reader to Section 1.3. 1.2. State of the art. Equations of the type (1.1) were ﬁrst considered in [22], where viscosity solutions are studied. Existence, uniqueness, and the convergence to the p−Lap”
Equivalence of solutions to fractional p-Laplace type equations (2016, background): “ring the ellipticity. The nonlocal integro-diﬀerential equation (1.1) can be seen as a fractional nonlocal counterpart to the usual p-Laplace equation ∆pu= div(|∇u|p−2∇u) = 0. In fact, it is shown in [IN10] that the fractional viscosity solutions of (1.1) converge to the ones of p-Laplacian as s→ 1.”
Nonlocal Harnack inequalities (2014, background): “It is worth citing the higher regularity contributions in the case when s is close to 1 proven in the interesting papers [1, 17], recently extended in some extents by the authors in [7] for any s ∈ (0, 1); see, also, [4] for related existence and uniqueness results in the case when p goes to infinit”
Fractional p-Laplacian evolution equations (2016, background): “First results in this direction are obtained in [23] for a similar problem for p > 1 in the stationary case (see also [9] and [12]) and [3], [4] and [5] for nonlocal evolution problems with non degenerate kernels.”
Hitchhiker's guide to the fractional Sobolev spaces (2011, background): “In the same way as the fractional Laplacian (−∆) is related to the space W s,2 ( as its Euler-Lagrange equation or from the formula ‖u‖2W s,2 = ∫ u(−∆)u dx ) , a more general integral operator can be defined that is related to the space W s,p for any p (see the recent paper [52]).”
Local behavior of fractional $p$-minimizers (2015, background): “l results are known. Firstly, we would like to cite the higher regularity contributions for viscosity solutions in the case when s is close to 1 proven in the recent interesting paper [1]; see, also, [16]. Secondly, the analysis in the papers [3] and [18] considers the special case when p is suitabl”
A New Proof for the Equivalence of Weak and Viscosity Solutions for the p-Laplace Equation (2011, methodology): “The equivalence has turned out to be a very useful tool in certain removability questions, see [7, 13], as well as in the analysis of various approximations of the p-Laplace equation see [4, 11, 12].”