Regularity Theory for Non-autonomous Partial Differential Equations Without Uhlenbeck Structure View Full Text


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Article Info

DATE

2022-07-16

AUTHORS

Peter Hästö, Jihoon Ok

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

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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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    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
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    43 equations
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    67 regularity theory
    68 restriction
    69 results
    70 solution
    71 special case
    72 special functions
    73 specific structure
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    79 schema:name Regularity Theory for Non-autonomous Partial Differential Equations Without Uhlenbeck Structure
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