On the Regularity of Minima of Non-autonomous Functionals

By Cristiana De Filippis, Giuseppe Mingione. Journal of Geometric Analysis, 2019

120 citations recorded by the source corpus.

We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of functionals with nearly linear growth. The analysis is carried out provided certain necessary approximation-in-energy conditions are satisfied. These are related to the occurrence of the so-called Lavrentiev phenomenon that non-autonomous functionals might exhibit, and which is a natural obstruction to regularity. In the case of vector valued problems, we concentrate on higher gradient integrability of minima. Instead, in the scalar case, we prove local Lipschitz estimates. We also present an approach via a variant of Moser’s iteration technique that allows to reduce the analysis of several non-uniformly elliptic problems to that for uniformly elliptic ones.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5ddffc638c29113fc94d669f80f6f1017d2bc718

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Citations in the corpus

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].”
Fractional double-phase patterns: concentration and multiplicity of solutions (2020, methodology): “For various types of double-phase problems, we refer to the recent papers [32,55,56,14,62,63,72].”
Recent developments in problems with nonstandard growth and nonuniform ellipticity (2021, methodology): “Theorem 10 (De Filippis & Mingione [89]).”
Growth conditions and regularity for weak solutions to nonlinear elliptic pdes (2020, background): “A special mention to [34] by DeFilippis-Mingione with some interesting considerations about the so-called Lavrentiev phenomenon.”
Regularity for Double Phase Problems at Nearly Linear Growth (2023, background): “A characterization of fractional Sobolev spaces via finite difference is contained in the following lemma, that can be retrieved for instance from [22].”