A geometrical approach to monotone functions in $\mathbb R^n$

Alberti, Giovanni; Ambrosio, Luigi

doi:10.1007/pl00004691

Cited by 127 publications

(203 citation statements)

References 36 publications

Supporting

Mentioning

202

Contrasting

Order By: Relevance

“…Let Ω be a half-ball of radius one, let Ω ′ be the concentric half ball of radius 1 2 , and let u ∈ W 1,2 Ω; R 2 be a weak solution of the Neumann problem…”

Section: Theorem 37mentioning

confidence: 99%

“…As in the proof of Theorem 3.13 it is enough to consider Γ \ (Γ cusps ∪ Γ cuts ). Fix a closed subarc Γ ′ ⊂ Γ \ (Γ cusps ∪ Γ cuts ), 1 2 < σ 0 < 1, and a point z 0 = (x 0 , y 0 ) in Γ ′ . As in the proof of Theorem 3.13, for all 0 < r ≤ r 1 we may extend u to the ball B (z 0 , r) in such a way that (3.67) holds.…”

Section: Bymentioning

confidence: 99%

“…It is well known (see for instance [1]) that a rotation of π/4 transforms the extended graph of g i,n into the graph of a 1-Lipschitz function (the rotation should be clockwise for the non-decreasing function g 1,n and counterclockwise for the non-increasing function g 2,n ). Exploiting this observation and (4.7) one can see that it is possible to extend the functions u n to the rectangle R 2 in such a way that the resulting functions, still denoted by u n , are equibounded in W 1,p R 2 ; R 2 .…”

Section: Appendix: Korn's Inequalitiesmentioning

confidence: 99%

See 2 more Smart Citations

Equilibrium Configurations of Epitaxially Strained Crystalline Films: Existence and Regularity Results

Fonseca

Fusco

Leoni

et al. 2007

Arch Rational Mech Anal

150

View full text Add to dashboard Cite

Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elasticity. The total free energy of the system is assumed to be the sum of the energy of the free surface of the film and the strain energy. Because of the lattice mismatch between film and substrate, flat configurations are in general energetically unfavourable and a corrugated or islanded morphology is the preferred growth mode of the strained film. After specifying the functional setup where the existence problem can be properly framed, a study of the qualitative properties of the solutions is undertaken. New regularity results for volume constrained local minimizers of the total free energy are established, leading, as a byproduct, to a rigorous proof of the zero contact-angle condition between islands and wetting layers.

show abstract

“…Let Ω be a half-ball of radius one, let Ω ′ be the concentric half ball of radius 1 2 , and let u ∈ W 1,2 Ω; R 2 be a weak solution of the Neumann problem…”

Section: Theorem 37mentioning

confidence: 99%

Section: Bymentioning

confidence: 99%

Section: Appendix: Korn's Inequalitiesmentioning

confidence: 99%

See 1 more Smart Citation

Equilibrium Configurations of Epitaxially Strained Crystalline Films: Existence and Regularity Results

Fonseca

Fusco

Leoni

et al. 2007

Arch Rational Mech Anal

150

View full text Add to dashboard Cite

show abstract

“…Appendix E explains how to extend the results of the paper from the setting of compact measured length spaces to the setting of complete pointed locally compact measured length spaces. Appendix F has some bibliographic notes on optimal transport and displacement convexity.The results of this paper were presented at the workshop "Collapsing and metric geometry" in Münster, August [1][2][3][4][5][6][7] 2004. After the writing of the paper was essentially completed we learned of related work by 42].…”

mentioning

confidence: 99%

Ricci curvature for metric-measure spaces via optimal transport

2009

View full text Add to dashboard Cite

Abstract. We define a notion of a measured length space X having nonnegative N -Ricci curvature, for N ∈ [1, ∞), or having ∞-Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P 2 (X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdorff limits. We give geometric and analytic consequences.This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results.To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X, d) in which the distance between two points equals the infimum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having "curvature bounded below by K", with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the Gromov-Hausdorff topology on compact metric spaces (modulo isometries); they form a closed subset.In view of Alexandrov's work, it is natural to ask whether there are metric space versions of other types of Riemannian curvature, such as Ricci curvature. This question takes substance from Gromov's precompactness theorem for Riemannian manifolds with Ricci curvature bounded below by K, dimension bounded above by N and diameter bounded above by D [23, Theorem 5.3]. The precompactness indicates that there could be a notion of a length space having "Ricci curvature bounded below by K", special cases of which would be Gromov-Hausdorff limits of manifolds with lower Ricci curvature bounds. Gromov-Hausdorff limits of manifolds with Ricci curvature bounded below have been studied by various authors, notably Cheeger and Colding [15,16,17,18]. One feature of their work, along with the earlier work of Fukaya [21], is that it turns out to be useful to add an auxiliary Borel probability measure ν and consider metric-measure spaces (X, d, ν). (A compact Riemannian manifold M has a canonical measure ν given by the normalized Riemannian density.) There is a measured Gromov-Hausdorff topology on such triples (X, d, ν) (modulo isom...

show abstract

“…A mapping f : R n → R n is called monotone if F (x) − F (y), x − y ≥ 0 for all x, y ∈ R n , where ·, · is the inner product [2,27]. In other words, F is monotone if the angle formed by the vectors F (x) − F (y) and x − y is at most π/2.…”

Section: Introductionmentioning

confidence: 99%

Doubling measures, monotonicity, and quasiconformality

Kovalev

Maldonado

2007

Math. Z.

View full text Add to dashboard Cite

Abstract. We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure.

show abstract

A geometrical approach to monotone functions in $\mathbb R^n$

Cited by 127 publications

References 36 publications

Equilibrium Configurations of Epitaxially Strained Crystalline Films: Existence and Regularity Results

Equilibrium Configurations of Epitaxially Strained Crystalline Films: Existence and Regularity Results

Ricci curvature for metric-measure spaces via optimal transport

Doubling measures, monotonicity, and quasiconformality

Contact Info

Product

Resources

About