Full Regularity for Convex Integral Functionals with p ⁢ ( x ) Growth in Low Dimensions

Jens Habermann

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 3, page 521-541
  • ISSN: 0392-4041

Abstract

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For Ω ⊂ 𝐑 n ; n ≥ 2 , and N ≥ 1 we consider vector valued minimizers u ∈ W l ⁢ o ⁢ c m , p ⁢ ( ⋅ ) ⁢ ( Ω , 𝐑 N ) of a uniformly convex integral functional of the type ℱ ⁢ [ u , Ω ] := ∫ Ω f ⁢ ( x , D m ⁢ u ) ⁢ 𝑑 x , where f is a Carathéorody function satisfying p ⁢ ( x ) growth conditions with respect to the second variable. We show that if the dimension n is small enough, dependent on the structure conditions of the functional, there holds D k ⁢ u ∈ C l ⁢ o ⁢ c 0 , β ⁢ ( Ω ) ⁢ for ⁢ k ∈ { 0 , ⋯ , m - 1 } , for some β , also depending on the structural data, provided that the nonlinearity exponent p is uniformly continuous with modulus of continuity ω satisfying lim sup ρ ↓ 0 ⁡ ω ⁢ ( ρ ) ⁢ log ⁡ ( 1 ρ ) = 0 .

How to cite

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Habermann, Jens. "Full Regularity for Convex Integral Functionals with $p(x)$ Growth in Low Dimensions." Bollettino dell'Unione Matematica Italiana 3.3 (2010): 521-541. <http://eudml.org/doc/290660>.

@article{Habermann2010,
abstract = {For $\Omega \subset \mathbf\{R\}^\{n\}$; $n \ge 2$, and $N \ge 1$ we consider vector valued minimizers $u \in W_\{loc\}^\{m,p(\cdot)\}(\Omega,\mathbf\{R\}^\{N\})$ of a uniformly convex integral functional of the type $$\mathcal\{F\} \left[ u,\Omega \right] := \int\_\{\Omega\} f(x,D^\{m\}u) \, dx,$$ where $f$ is a Carathéorody function satisfying $p(x)$ growth conditions with respect to the second variable. We show that if the dimension $n$ is small enough, dependent on the structure conditions of the functional, there holds $$D^\{k\}u \in C\_\{loc\}^\{0,\beta\}(\Omega) \,\, \text\{for\} \,\, k \in \\{0,\cdots,m-1\\},$$ for some $\beta$, also depending on the structural data, provided that the nonlinearity exponent $p$ is uniformly continuous with modulus of continuity $\omega$ satisfying $$\limsup\_\{\rho\downarrow 0\} \omega(\rho) \log \bigg( \frac\{1\}\{\rho\} \bigg) = 0.$$},
author = {Habermann, Jens},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {521-541},
publisher = {Unione Matematica Italiana},
title = {Full Regularity for Convex Integral Functionals with $p(x)$ Growth in Low Dimensions},
url = {http://eudml.org/doc/290660},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Habermann, Jens
TI - Full Regularity for Convex Integral Functionals with $p(x)$ Growth in Low Dimensions
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/10//
PB - Unione Matematica Italiana
VL - 3
IS - 3
SP - 521
EP - 541
AB - For $\Omega \subset \mathbf{R}^{n}$; $n \ge 2$, and $N \ge 1$ we consider vector valued minimizers $u \in W_{loc}^{m,p(\cdot)}(\Omega,\mathbf{R}^{N})$ of a uniformly convex integral functional of the type $$\mathcal{F} \left[ u,\Omega \right] := \int_{\Omega} f(x,D^{m}u) \, dx,$$ where $f$ is a Carathéorody function satisfying $p(x)$ growth conditions with respect to the second variable. We show that if the dimension $n$ is small enough, dependent on the structure conditions of the functional, there holds $$D^{k}u \in C_{loc}^{0,\beta}(\Omega) \,\, \text{for} \,\, k \in \{0,\cdots,m-1\},$$ for some $\beta$, also depending on the structural data, provided that the nonlinearity exponent $p$ is uniformly continuous with modulus of continuity $\omega$ satisfying $$\limsup_{\rho\downarrow 0} \omega(\rho) \log \bigg( \frac{1}{\rho} \bigg) = 0.$$
LA - eng
UR - http://eudml.org/doc/290660
ER -

References

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