A First Course in Sobolev Spaces

By Giovanni Leoni. 2009

1,078 citations recorded by the source corpus.

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In Applied Mathematics

Citations in the corpus

Quantitative stability of optimal transport maps under variations of the target measure (2021, positive): “Morrey’s inequality (Theorem 11.34 and Theorem 12.15 in [25]) ensures that φ is (1 )-Hölder and that there exists a constant depending only on d, p and X such that Remark 3.2 (Morrey’s inequality for convex functions) .”
Basic Complex Analysis: A Comprehensive Course in Analysis, Part 2A (2015, background): “169], or Leoni [354] While our desire to avoid mainly topological arguments has restricted us to the smooth case, there are costs in limitations.”
Strong Solutions of Mean-Field Stochastic Differential Equations with irregular drift (2018, background): “Hence, {bn} is bounded in W (U) and thus weakly relatively compact by [27][Theorem 10.”
Plugin estimation of smooth optimal transport maps (2021, background): “The following is the classical Poincaré inequality for domains with bounded width (see for instance Leoni (2017), Theorem 12.17), stated in the special case of the unit cube.”
Noiseless regularisation by noise (2020, background): “3 from [37]) or reasoning as follows: by definition, any φ ∈ B p,q can be identified with a sequence {∆jφ}j ⊂ L(R) with suitable summability, namely such that it belongs to `(N, μ;L(R)), where μ({j}) = 2−sqj ; in the case p, q ∈ [2,∞) by the previous examples it has martingale type 2.”
Valuations on Sobolev spaces (2012, background): “Piecewise affine functions lie dense in W 1,1(Rn) (see, for example, [31]).”
Interior regularity results for inhomogeneous anisotropic quasilinear equations (2021, methodology): “In this case we observe that, for any θ > 2, it holds Here we have used that η a k ε ( ∇ u ε ) ∈ H 1 c ((cid:4)) and its support is contained in B 2 R ⊂ (see for instance [16, Theorem 12.33]).”
Distribution dependent SDEs driven by additive fractional Brownian motion (2021, background): “either interpolation or Gagliardo–Nirenberg type integral seminorms, see for instance [33].”
Hitchhiker's guide to the fractional Sobolev spaces (2011, background): “When s is an integer, we cite [58] for a complete characterization in the special case s = 1, p = 2 and n = 2, and we refer the interested reader to the recent book by Leoni [60], in which this problem is very well discussed (see, in particular, Chapter 11 and Chapter 12 there).”
On a class of degenerate parabolic equations with dynamic boundary conditions (2011, neutral): “, [33]), that”
The Takagi function and its properties (Functions in Number Theory and Their Probabilistic Aspects) (2011, methodology): “By an application of the co-area formula of geometric measure theory for functions of bounded variation (in the version in Leoni [58, Theorem 7.2 and Theorem 13.25]) to a relative of the Takagi singular function, the author with Maddock [56] determined the average number of local level sets on a…”