论文标题
在非同伴条件下具有广义ORLICZ生长的非均匀椭圆功能的局部最小化局部最小化器的连续性和HARNACK不平等现象
Continuity and Harnack inequalities for local minimizers of non uniformly elliptic functionals with generalized Orlicz growth under the non-logarithmic conditions
论文作者
论文摘要
我们研究属于相应de giorgi类的函数的定性属性\ begin {equination*} \ int \ limits _ {b_ {r（1-σ）}（x_ {0}）} \，\ varphi（x_ {0}） γ\，\ int \ limits_ {b_ {r}（x_ {0}）} \，\ varphi \ bigG（x，x，\ frac {（u-k）_ {\ pm}}}}}} {σr} {σr} {σr} \ bigG）\ big） $ k \ in \ mathbb {r} $和函数$ \ varphi $满足非cordaritmic条件\ begin {equation*} \ big（r^{ - n} \ int \ limits_ {b_ {r}（x_ {0}）} [\ varphi \ big（x，x，\ frac {v} {r} {r} \ big）] （r^{ - n} \ int \ limits_ {b_ {r}（x_ {0}）} [\ varphi \ big（x，x，\ frac {v} {r} {r} {r} \ big）]^{ - t} { - t} \，dx \ big） c（k）λ（x_ {0}，r），\ quad r \ leqslant v \ leqslant k \，λ（r），\ end \ end {equation*}根据功能上的某些假设$λ（r）$和$λ（x_ {0}，r）$λ（x_ {0}，r）$和$ s $ $ s $ s $ s $ s $，这些条件概括了已知的对数，非同伴和非椭圆形条件。特别是，我们的结果涵盖了新的椭圆双相，脱名的双相功能和具有可变指数的功能的新案例。
We study the qualitative properties of functions belonging to the corresponding De Giorgi classes \begin{equation*} \int\limits_{B_{r(1-σ)}(x_{0})}\,\varPhi(x, |\nabla(u-k)_{\pm}|)\,dx \leqslant γ\,\int\limits_{B_{r}(x_{0})}\,\varPhi\bigg(x, \frac{(u-k)_{\pm}}{σr}\bigg)\,dx, \end{equation*} where $σ$, $r \in (0,1)$, $k\in \mathbb{R}$ and the function $\varPhi$ satisfies the non-logarithmic condition \begin{equation*} \bigg(r^{-n}\int\limits_{B_{r}(x_{0})}[\varPhi\big(x,\frac{v}{r}\big)]^{s}\,dx\bigg)^{\frac{1}{s}}\bigg(r^{-n}\int\limits_{B_{r}(x_{0})}[\varPhi\big(x,\frac{v}{r}\big)]^{-t}\,dx\bigg)^{\frac{1}{t}}\leqslant c(K) Λ(x_{0},r),\quad r\leqslant v\leqslant K\,λ(r), \end{equation*} under some assumptions on the functions $λ(r)$ and $Λ(x_{0}, r)$ and the numbers $s$, $t >1$. These conditions generalize the known logarithmic, non-logarithmic and non uniformly elliptic conditions. In particular, our results cover new cases of non uniformly elliptic double-phase, degenerate double-phase functionals and functionals with variable exponents.
