Remarks on the regularity of the minimizers of certain degenerate functionals

Giaquinta, Mariano; Modica, Giuseppe

doi:10.1007/bf01172492

Cited by 203 publications

(126 citation statements)

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“…They proved Lipschitz regularity of minimizers to integral functionals F (v) = Ω f (Dv) dx with an integrand satisfying D 2 f (ξ) → A when |ξ| → ∞ for some elliptic bilinear form A on R N n . More general integrands were treated later by Giaquinta & Modica [18] and Raymond [23] and the case of higher order functionals has been considered by Schemm [24]. These results provide a huge class of elliptic systems, respectively integral functionals with Lipschitz solutions, respectively minimizers, which is much larger than the well known class of quasidiagonal structure.…”

Section: Introductionmentioning

confidence: 99%

Global gradient bounds for the parabolicp-Laplacian system

Bögelein

2015

Proc. London Math. Soc.

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ABSTRACT. A by now classical result due to DiBenedetto states that the spatial gradient of solutions to the parabolic p-Laplacian system is locally Hölder continuous in the interior. However, the boundary regularity is not yet well understood. In this paper we prove a boundary L ∞ -estimate for the spatial gradient Du of solutions to the parabolic p-Laplacian systemfor p ≥ 2, together with a quantitative estimate. In particular, this implies the global Lipschitz regularity of solutions. The result continues to hold for the so called asymptotically regular parabolic systems.

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Section: Introductionmentioning

confidence: 99%

Global gradient bounds for the parabolicp-Laplacian system

Bögelein

2015

Proc. London Math. Soc.

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“…The classic elliptic result of Giaquinta and Modica [8] was that there is a uniform constant c and an α ∈ (0, 1), such that Φ(θρ) ≤ cθ α Φ(ρ). It is then a standard procedure to gain the estimate of the oscillations.…”

Section: Decay For P-caloric Functionsmentioning

confidence: 99%

“…The first important step to gain BMO estimates is a decay estimate for homogeneous solutions (called p-caloric). In Theorem 3.2 we prove a decay in the spirit of Giaquinta and Modica [8] for p-caloric solutions. This decay is a distinctively stronger estimate on the Hölder behavior for the gradients of p-caloric solutions than known before.…”

Section: Introductionmentioning

confidence: 99%

Hölder–Zygmund estimates for degenerate parabolic systems

Schwarzacher

2014

Journal of Differential Equations

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We consider energy solutions of the inhomogeneous parabolic p-Laplacien system ∂ t u − div(|∇u| p−2 ∇u) = −divg). We show in the case p ≥ 2 that if the right hand side g is locally in L ∞ (BMO), then u is locally in L ∞ (C 1 ), where C 1 is the 1-Hölder-Zygmund space. This is the borderline case of the Calderón-Zygmund theorey. We provide local quantitative estimates. We also show that finer properties of g are conserved by ∇u, e.g. Hölder continuity. Moreover, we prove a new decay for gradients of p-caloric solutions for all 2n n+2 < p < ∞.

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“…Because of this lack of smoothness, the usual ellipticity condition, which is required for C 1,α regularity in all classical papers such as [18,19,21] where L is of class C 2 with respect to ξ, is replaced at first by a qualified convexity assumption called p-uniform convexity which, in the simplest case L = f (ξ), requires that for some ν > 0 the inequality…”

Section: Introductionmentioning

confidence: 99%