论文标题
关于先验假设的非自主问题的规律性理论
Regularity theory for non-autonomous problems with a priori assumptions
论文作者
论文摘要
我们研究了非自主问题的弱解决方案和最小化$ u $ $ \ operatotorname {div} a（x，du）= 0 $和$ \ min_v \int_Ωf（x，dv）\，dx $，with quasi-iSotropic $（p，q）$ - 增长。我们认为$ u $有限，Hölder连续或躺在Lebesgue空间中，并在$ a $ a或$ f $上的假设与$ u $的相应规范之间建立牢固的联系。我们证明了sobolev-poincaré不平等，更高的集成性以及$ u $和$ du $的Hölder连续性。我们的证明是优化和简化的早期研究版本，可以更容易地将其进一步扩展到其他设置。 $ a $或$ f $上的假设与$ u $上的假设之间的连接以双相能量$ f（x，ξ）= |ξ|^p + a（x）|ξ|^q $而闻名。即使在这种特殊情况下，我们也会获得更好的结果。此外，我们还涵盖了扰动的可变指数，Orlicz变量指数，退化双相，Orlicz Double相，三相，双相，双变量指数以及可变指数双相能，并且在大多数这些特殊情况下结果都是新的。
We study weak solutions and minimizers $u$ of the non-autonomous problems $\operatorname{div} A(x, Du)=0$ and $\min_v \int_ΩF(x,Dv)\,dx$ with quasi-isotropic $(p, q)$-growth. We consider the case that $u$ is bounded, Hölder continuous or lies in a Lebesgue space and establish a sharp connection between assumptions on $A$ or $F$ and the corresponding norm of $u$. We prove a Sobolev--Poincaré inequality, higher integrability and the Hölder continuity of $u$ and $Du$. Our proofs are optimized and streamlined versions of earlier research that can more readily be further extended to other settings. Connections between assumptions on $A$ or $F$ and assumptions on $u$ are known for the double phase energy $F(x, ξ)=|ξ|^p + a(x)|ξ|^q$. We obtain slightly better results even in this special case. Furthermore, we also cover perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, triple phase, double variable exponent as well as variable exponent double phase energies and the results are new in most of these special cases.
