Lipschitz Bounds and Nonuniform Ellipticity

By Lisa Beck, Giuseppe Mingione. Communications on Pure and Applied Mathematics, 2019

191 citations recorded by the source corpus.

We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve natural limiting function spaces and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. Moreover, we provide new and natural growth a priori estimates whose validity was an open problem. Finally, we find new results also in the classical uniformly elliptic case. Beyond the specific results, the paper proposes a new approach to nonuniform ellipticity that, in a sense, allows us to reduce nonuniform elliptic problems to uniformly elliptic ones via potential theoretic arguments that are for the first time applied in this setting. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/6333f6e5162aa55f20e2d70d801cce4eb45095fb

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Citations in the corpus

Lipschitz Bounds and Nonautonomous Integrals (2020, background): “ Br0(x0) (v−κ0)2 + dx !1/2 +cM max{δ,n−2 2 } 1 h M2P f1 1 (x0,2r0)+M3P f2 1 (x0,2r0) i (3.4) holds too, with c≡ c(n,˜c,δ ). We conclude with another, more classical iteration lemma (see references of [3]). Lemma 2. Let Z: [,ξ] → [0,∞) be a bounded function. Let ε∈ (0,1), a1,a2,b ≥ 0 be numbers. If Z”
Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold (2020, result): “We also mention the recent works of BeckMingione [8], [9] concerning nonuniformly elliptic variational problems.”
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”
Growth conditions and regularity for weak solutions to nonlinear elliptic pdes (2020, result): “In this context we quote the recent article [5] by Beck-Mingione where, among other results, the authors extend Theorem 1.”
A new class of double phase variable exponent problems: Existence and uniqueness (2021, background): “Moreover, recent results for nonuniformly elliptic variational problems and nonautonomous functionals can be found in the papers of Beck-Mingione [11, 12], De Filippis-Mingione [23] and Hästö-Ok [43].”
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].”
Regularity for Double Phase Problems at Nearly Linear Growth (2023, methodology): “In turn, this inequality follows along the lines of the estimates in [8, Lemma 5.6].”