论文标题
椭圆形PDE的有界弱解决方案，以及在Orlicz空间中的数据
Bounded weak solutions to elliptic PDE with data in Orlicz spaces
论文作者
论文摘要
一个经典的规律性结果是，在有界域$ω$中对dirichlet问题的非负解决方案$ΔU= f $，其中$ f \ in l^q（ω）$，$ q>> \ frac {n} 2 $，满足$ \ \ \ \ \ \ \ \ | _ {l^\ iffty（l^\ infty（ω） c \ | f \ | _ {l^q（ω）} $。我们以三种方式扩展了这一结果：我们用退化的椭圆运算符代替Laplacian；我们表明，我们可以在$ l^{\ frac {n} {2}}}（ω）$和$ l^q（ω）$，$ q>> \ frac {n} 2 $;中严格介于$ l^{\ frac {n} {2}}} $介于$ l^{\ frac {n} {2}}}之间的数据$ f $我们表明，我们可以通过较小的表达方式来替换右侧的$ l^a $规范，该表达式涉及“熵凸起的对数” $ \ | f \ | _ {l^a（ω）}/\ | | f \ | _ {l^{l^{l^{\ frac {\ frac {n} {2} {2}} {
A classical regularity result is that non-negative solutions to the Dirichlet problem $Δu =f$ in a bounded domain $Ω$, where $f\in L^q(Ω)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(Ω)} \leq C\|f\|_{L^q(Ω)}$. We extend this result in three ways: we replace the Laplacian with a degenerate elliptic operator; we show that we can take the data $f$ in an Orlicz space $L^A(Ω)$ that lies strictly between $L^{\frac{n}{2}}(Ω)$ and $L^q(Ω)$, $q>\frac{n}2$; and we show that that we can replace the $L^A$ norm in the right-hand side by a smaller expression involving the logarithm of the "entropy bump" $\|f\|_{L^A(Ω)}/\|f\|_{L^{\frac{n}{2}}(Ω)}$, generalizing a result due to Xu.
