Scalar Minimizers with Fractal Singular Sets

By Irene Fonseca, Jan Maly, Giuseppe Mingione. 2004

146 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/aa550047d55545d26f74305d2f07c52a9ae1c06f

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Citations in the corpus

C1,α-solutions to non-autonomous anisotropic variational problems (2005, background): “(ii) Due to the counterexamples of [12, 13 ] (Mingione, “personal communication”.) we cannot expect to weaken the conditions (1.4 )a nd (1.7), respectively.”
Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure (2009, background): “3 We refer the reader to [9] for another Cantor type construction of a minimizer with a large singular set.”
Lipschitz estimates for systems with ellipticity conditions at infinity (2016, background): “3) is considered sharp in view of some examples given in [15,17].”
Regularity for Double Phase Variational Problems (2015, background): “1 (Lavrentiev phenomenon [15], [17], [51]).”
On the Regularity of Minima of Non-autonomous Functionals (2019, methodology): “every x ∈ Ω. Under such assumptions, local minimizers are locally Lipschitz regular provided the condition (6.2) q p < 1+ 1 n is in force together with Lq ≡ 0. Condition (6.2) is sharp as shown in [32,34]. The bound in (6.2) corresponds to (1.27) when r → ∞ and 2−µ = p. In fact (6.1) are a partic”
Regularity for Double Phase Problems at Nearly Linear Growth (2023, background): “This was ﬁrst shown by the counterexamples in [28,30].”
Regularity of minima: An invitation to the dark side of the calculus of variations (2006, background): “Indeed, as demonstrated in the papers [111] [126], when dealing with functionals of the type (6.14), the regularity of minima is ruled by a subtle interaction between the the regularity of the function x → F (x, z), and the size of the gap q/p.”
Lipschitz Bounds and Nonautonomous Integrals (2020, methodology): “oblems one typically recovers classical theories as Schauder’s [54] and Calderón-Zygmund’s [12]. This is in general not the case for the double phase integral in (1.10), due to the counterexamples in [37,39]. Nevertheless the integrand H(·) turns out to be uniformly elliptic when using the quantity ”
New Examples on Lavrentiev Gap Using Fractals (2019, methodology): “Using out method we hope to overcome the dimensional threshold p < d < d+ α < q from [9].”
Non-autonomous functionals, borderline cases and related function classes (2016, background): “1) is sharp in the sense that the counterexamples constructed in [28, 33] show that as soon the condition (2.”
Regularity for general functionals with double phase (2017, background): “3) fail, and this eventually allows to give examples of irregular minima; see also [27].”