Lipschitz Bounds and Nonautonomous Integrals

By Cristiana De Filippis, Giuseppe Mingione. Archive for Rational Mechanics and Analysis, 2020

151 citations recorded by the source corpus.

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range from those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and also yield new, optimal regularity criteria in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/43b23293c1687dc004ddfc67f2f6bd895cf5ad77

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Regularity for Double Phase Problems at Nearly Linear Growth (2023, methodology): “…when considering again the vectorial case for functionals as in (7.2), we can still obtain local Lipschitz regularity of minima provided (7.1) holds together with G ( z ) ≡ ˜ G ( | z | ) that and ˜ G ( · ) is C 2 - regular outside the origin (see for instance [8,23] for precise assumptions).”
New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems (2022, background): “Moreover, recent results for nonuniformly elliptic variational problems and nonautonomous functionals can be found in the papers of Beck-Mingione [9, 10], De Filippis-Mingione [20, 21, 22, 23, 24] and Hästö-Ok [39].”
Existence of solutions for singular double phase problems via the Nehari manifold method (2021, unknown): “We mention the papers of Baroni-Colombo-Mingione [4, 5, 6], Baroni-Kuusi-Mingione [7], Byun-Oh [9, 10], Byun-Youn [11], Colombo-Mingione [14, 15, 16], De Filippis-Mingione [18, 19, 20], De Filippis-Palatucci [21], EspositoLeonetti-Mingione [23], Esposito-Leonetti-Petricca [24], Marcellini [37, 38, 3”
Gradient regularity in mixed local and nonlocal problems (2022, background): “2) connect to a large family of problems featuring anisotropic operators and integrands with so-called nonstandard growth conditions [22, 24, 26, 27, 34, 43, 52], and to some other classes of anisotropic nonlocal problems [14–17, 28, 53].”