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$μ$-ellipticity and nonautonomous integrals

Cristiana De Filippis

TL;DR

This work addresses gradient regularity for minimizers of μ-elliptic, nonautonomous functionals with nonuniform ellipticity and possible degeneracy. It develops an anisotropic μ-ellipticity framework, establishes gradient boundedness and Hölder continuity in autonomous and nonautonomous settings, and introduces intrinsic Bernstein functions together with nonlinear potential theory to handle genuine nonuniform ellipticity, including nearly linear growth. The results yield sharp Schauder-type estimates and reveal both regular and singular regimes, supported by fractal-cone constructions and malicious-competitor examples that delineate the limits of regularity. The analysis advances the understanding of when gradient regularity can be expected under nonstandard growth, with implications for variational problems in nonlinear elasticity, fluids with shear-thinning behavior, and related PDE models.

Abstract

$μ$-ellipticity is a form of nonuniform ellipticity arising in various contexts from the calculus of variations. Understanding regularity properties of minimizers in the nonautonomous setting is a challenging task fostering the development of delicate techniques and the discovery of new irregularity phenomena.

$μ$-ellipticity and nonautonomous integrals

TL;DR

This work addresses gradient regularity for minimizers of μ-elliptic, nonautonomous functionals with nonuniform ellipticity and possible degeneracy. It develops an anisotropic μ-ellipticity framework, establishes gradient boundedness and Hölder continuity in autonomous and nonautonomous settings, and introduces intrinsic Bernstein functions together with nonlinear potential theory to handle genuine nonuniform ellipticity, including nearly linear growth. The results yield sharp Schauder-type estimates and reveal both regular and singular regimes, supported by fractal-cone constructions and malicious-competitor examples that delineate the limits of regularity. The analysis advances the understanding of when gradient regularity can be expected under nonstandard growth, with implications for variational problems in nonlinear elasticity, fluids with shear-thinning behavior, and related PDE models.

Abstract

-ellipticity is a form of nonuniform ellipticity arising in various contexts from the calculus of variations. Understanding regularity properties of minimizers in the nonautonomous setting is a challenging task fostering the development of delicate techniques and the discovery of new irregularity phenomena.

Paper Structure

This paper contains 12 sections, 10 theorems, 60 equations, 1 figure.

Table of Contents

  1. Anisotropic $\mu$-ellipticity
  2. Nonuniform ellipticity and degeneracy
  3. The autonomous case
  4. Superlinear nonuniform ellipticity
  5. Uniformly elliptic Schauder estimates
  6. Classical Schauder (a model case)
  7. More uniformly elliptic Schauder
  8. Soft nonuniform ellipticity and hard irregularity
  9. Fractal cones and malicious competitors
  10. Schauder estimates and $\mu$-ellipticity
  11. Nearly linear Schauder and intrinsic Bernstein functions
  12. Back to the beginning

Key Result

Theorem 3.1

Let $u\in W^{1,1}_{\textnormal{loc}}(\Omega)$ be a minimizer of functional fun with $\textnormal{F}(x,z)\equiv \textnormal{F}(z)$ satisfying $\textnormal{F}(z)\lesssim \lvert z\rvert^q+1$, superl and 0.5 with $g(\cdot)\equiv 1$, $q \in (1,2)$, $1\le \mu<2$ such that Then $u\in W^{1,\infty}_{\textnormal{loc}}(\Omega)$.

Figures (1)

  • Figure 1: Competitor $u_{*}$ vs coefficient $a(\cdot)$. Figure \ref{['fig']} is a modification of the one in bds20.

Theorems & Definitions (12)

  • Theorem 3.1: Fuchs and Mingione fm00
  • Remark : Vectorial problems
  • Theorem 3.2: Giaquinta gia87; Marcellini mar87mar89
  • Theorem 3.3: Hirsch and Schäffner hs21
  • Theorem 3.4: dkk24
  • Theorem 4.1: Baroni, Colombo and Mingione bcm18cm15
  • Theorem 4.2: Fonseca, Malý and Mingione fmm04
  • Theorem 4.3: Balci, Diening and Surnachev bds20bds23
  • Theorem 5.1: dm23b
  • Theorem 5.2: dm23bddp24
  • ...and 2 more