# Regularity Theory for Non-autonomous Partial Differential Equations Without Uhlenbeck Structure

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### Article Info

DATE

2022-07-16

AUTHORS ABSTRACT

We establish maximal local regularity results of weak solutions or local minimizers of divA(x,Du)=0andminu∫ΩF(x,Du)dx,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} \mathrm {div}A(x, Du)=0 \quad \text {and}\quad \min _u \int _\Omega F(x,Du)\,\mathrm{d}x, \end{aligned}\end{document}providing new ellipticity and continuity assumptions on A or F with general (p, q)-growth. Optimal regularity theory for the above non-autonomous problems is a long-standing issue; the classical approach by Giaquinta and Giusti involves assuming that the nonlinearity F satisfies a structure condition. This means that the growth and ellipticity conditions depend on a given special function, such as tp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^p$$\end{document}, φ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (t)$$\end{document}, tp(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{p(x)}$$\end{document}, tp+a(x)tq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^p+a(x)t^q$$\end{document}, and not only F but also the given function is assumed to satisfy suitable continuity conditions. Hence these regularity conditions depend on given special functions. In this paper we study the problem without recourse to, special function structure and without assuming Uhlenbeck structure. We introduce a new ellipticity condition using A or F only, which entails that the function is quasi-isotropic, i.e. it may depend on the direction, but only up to a multiplicative constant. Moreover, we formulate the continuity condition on A or F without specific structure and without direct restriction on the ratio qp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{q}{p}$$\end{document} of the parameters from the (p, q)-growth condition. We establish local C1,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document}-regularity for some α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} and Cα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\alpha }$$\end{document}-regularity for any α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases. More... »

PAGES

1401-1436

### References to SciGraph publications

• 2011. Lebesgue and Sobolev Spaces with Variable Exponents in NONE
• 2018-03-14. Regularity for general functionals with double phase in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
• 2021. Partial Differential Equations in Anisotropic Musielak-Orlicz Spaces in NONE
• 2017-02-09. Hölder regularity of quasiminimizers under generalized growth conditions in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
• 2009-05-26. Everywhere regularity of functionals with φ-growth in MANUSCRIPTA MATHEMATICA
• 2006-08. Regularity of minima: An invitation to the dark side of the calculus of variations in APPLICATIONS OF MATHEMATICS
• 2019-05-02. Harnack Inequality for Quasilinear Elliptic Equations in Generalized Orlicz-Sobolev Spaces in POTENTIAL ANALYSIS
• 2021-09-14. Lipschitz Bounds and Nonautonomous Integrals in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
• 1987-06. Growth conditions and regularity, a counterexample in MANUSCRIPTA MATHEMATICA
• 2015-03-17. Bounded Minimisers of Double Phase Variational Integrals in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
• 2014-08-15. Regularity for Double Phase Variational Problems in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
• 2019. Orlicz Spaces and Generalized Orlicz Spaces in NONE
• 1983-06. Differentiability of minima of non-differentiable functionals in INVENTIONES MATHEMATICAE
• 2001-02. Regularity Results for a Class of Functionals with Non-Standard Growth in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
• 1989-09. Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS

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DOI

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DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1149525006

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21 schema:description We establish maximal local regularity results of weak solutions or local minimizers of divA(x,Du)=0andminu∫ΩF(x,Du)dx,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} \mathrm {div}A(x, Du)=0 \quad \text {and}\quad \min _u \int _\Omega F(x,Du)\,\mathrm{d}x, \end{aligned}\end{document}providing new ellipticity and continuity assumptions on A or F with general (p, q)-growth. Optimal regularity theory for the above non-autonomous problems is a long-standing issue; the classical approach by Giaquinta and Giusti involves assuming that the nonlinearity F satisfies a structure condition. This means that the growth and ellipticity conditions depend on a given special function, such as tp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^p$$\end{document}, φ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (t)$$\end{document}, tp(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{p(x)}$$\end{document}, tp+a(x)tq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^p+a(x)t^q$$\end{document}, and not only F but also the given function is assumed to satisfy suitable continuity conditions. Hence these regularity conditions depend on given special functions. In this paper we study the problem without recourse to, special function structure and without assuming Uhlenbeck structure. We introduce a new ellipticity condition using A or F only, which entails that the function is quasi-isotropic, i.e. it may depend on the direction, but only up to a multiplicative constant. Moreover, we formulate the continuity condition on A or F without specific structure and without direct restriction on the ratio qp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{q}{p}$$\end{document} of the parameters from the (p, q)-growth condition. We establish local C1,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document}-regularity for some α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} and Cα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\alpha }$$\end{document}-regularity for any α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases.
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27 schema:keywords Giaquinta
28 Giusti
29 Uhlenbeck structure
30 approach
31 assumption
32 cases
33 classical approach
34 conditions
35 constants
36 continuity assumption
37 continuity conditions
38 differential equations
39 direct restriction
40 direction
41 ellipticity
42 ellipticity condition
43 equations
44 function
45 function structure
46 growth
47 growth conditions
48 issues
49 local minimizers
50 local regularity results
51 minimizers
52 multiplicative constant
53 non-autonomous partial differential equations
54 non-autonomous problems
55 nonlinearity f
56 optimal regularity theory
57 paper
58 parameters
59 partial differential equations
60 problem
61 quasi
62 ratio
63 recourse
64 regularity
65 regularity conditions
66 regularity results
67 regularity theory
68 restriction
69 results
70 solution
71 special case
72 special functions
73 specific structure
74 structure
75 structure conditions
76 suitable continuity conditions
77 theory
78 weak solutions
79 schema:name Regularity Theory for Non-autonomous Partial Differential Equations Without Uhlenbeck Structure
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