Regularity for scalar integrals without structure conditions

By Michela Eleuteri, Paolo Marcellini, Elvira Mascolo. 2018

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Abstract Integrals of the Calculus of Variations with p,q{p,q}-growth may have not smooth minimizers, not even bounded, for general p,q{p,q} exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand f⁢(x,ξ){f(x,\xi)} with dependence on the modulus of the gradient, i.e. f⁢(x,ξ)=g⁢(x,|ξ|){f(x,\xi)=g(x,|\xi|)}. Without imposing structure conditions, we prove that if qp{\frac{q}{p}} is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous.

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

View Paper: https://www.semanticscholar.org/paper/3fc241f85acb5c1b3f82686d1f97b9e52819d487

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Citations in the corpus

Growth conditions and regularity for weak solutions to nonlinear elliptic pdes (2020, background): “From different points of view Eleuteri-Marcellini-Mascolo [41], [42], [43], [44] and DeFilippis [33], DeFilippis-Ho [35].”
Recent developments in problems with nonstandard growth and nonuniform ellipticity (2021, unknown): “Similar structures have been considered in [101].”
Maximal regularity for local minimizers of non-autonomous functionals (2019, background): “|2) q 2 |λ|2, |F(x,z)−F(y,z)| 6 ω(|x−y|)(1+|z|q). This case was introduced and systematically studied by Marcellini [64, 65, 66, 67, 68]. Several other researchers also contributed to the theory, cf. [11, 38, 71]. Forinstance, Marcellini [65] started by showing that that every minimizer in W1,q loc ”
A variational approach to parabolic equations under general and p,q-growth conditions (2020, background): “Among them there are variational integrands fulfilling a standard growth condition from below and above, functionals of non-standard p, q−growth and double phase (as in [4–6,23,24] and [21,29,34,35]), functionals with exponential growth (see [7,44,46]), variable exponents [21,26,33,36,37] and Orlicz”