论文标题
De Giorgi类和应用的扩展
An Extension of De Giorgi Class and Applications
论文作者
论文摘要
我们提出了经典的De Giorgi类的扩展，然后我们表明该新类中的功能是局部界限的，并且在局部局部连续。给出了一些应用程序。作为第一个应用程序，我们给出了本地最小值$ u：ω\ subset \ mathbb r^4 \ rightArrow \ Mathbb r^4 $的定期结果，其中一类特殊类PolyConvex函数的分配形式在四维欧几里得空间中。在能量密度的某些结构条件下，我们证明了局部最小化$ u $的每个组件$ u^α$属于广义de giorgi类，然后可以得出它是局部界限的，并且局部在本地界限。我们的结果可以应用于polyConvex积分，其原型为$ \int_Ω\ big（\ sum_ {α= 1}^4 | du^α|^p + sum_ {β= 1}^6 |（{\ rm adj） ）^γ|。作为第二个应用程序，我们考虑了$$ - \ mbox {div}（a（x）\ nabla u）= - \ mbox {div} f，$ 0 <a（x）\leβ<+β<+bbox f，$ box {a（x）\leβ<+\β<+\ iffty $。我们凭借广义的De Giorgi类证明，任何弱解决方案都是本地界限的，并且只要$ \ frac 1 {a（x）} $和$ f（x）$属于某些合适的本地集成功能空间，则在本地界限。作为第三个应用程序，我们表明我们的定理可以应用于处理具有非标准生长条件的椭圆方程的规律性问题。作为第四次应用，我们使用准线性椭圆系统处理。在适当的关于系数的假设下，我们表明其任何弱溶液都是局部界限的，并且局部hölder连续。
We present an extension of the classical De Giorgi class, and then we show that functions in this new class are locally bounded and locally Hölder continuous. Some applications are given. As a first application, we give a regularity result for local minimizers $u:Ω\subset \mathbb R^4 \rightarrow \mathbb R^4$ of a special class of polyconvex functionals with splitting form in four dimensional Euclidean spaces. Under some structural conditions on the energy density, we prove that each component $u^α$ of the local minimizer $u$ belongs to the generalized De Giorgi class, then one can derive that it is locally bounded and locally Hölder continuous. Our result can be applied to polyconvex integrals whose prototype is $$ \int_Ω\Big(\sum_{α=1}^4 |Du^α|^p + \sum_{β=1}^6 |({\rm adj}_2 Du )^β| ^q +\sum_{γ=1}^4 |({\rm adj}_3 Du )^γ| ^r +|\det Du|^s \Big ) \mathrm {d}x $$ with suitable $p,q,r,s\ge 1$. As a second application, we consider a degenerate linear elliptic equation of the form $$ -\mbox {div} (a(x)\nabla u)=-\mbox {div}F, $$ with $0<a(x) \le β<+\infty$. We prove, by virtue of the generalized De Giorgi class, that any weak solution is locally bounded and locally Hölder continuous provided that $\frac 1 {a(x)}$ and $F(x)$ belong to some suitable locally integrable function spaces. As a third application, we show that our theorem can be applied in dealing with regularity issues of elliptic equations with non-standard grow conditions. As a fourth application we treat with quasilinear elliptic systems. Under suitable assumptions on the coefficients, we show that any of its weak solutions is locally bounded and locally Hölder continuous.
