论文标题
$ p $ laplace方程的非古典解决方案
Non-classical solutions of the $p$-Laplace equation
论文作者
论文摘要
在本文中，我们回答了Iwaniec和Sbordone的猜想\ cite {ib94}，涉及$ p $ -laplace方程的非常弱的解决方案。也就是说，一方面，我们表明$ w^{1，r} $ in $ p \ neq 2 $和$ p \ neq 2 $和$ r> \ max \ {1，p-1 \} $的分配解决方案是经典的弱解决方案，如果它们的弱衍生物属于某些锥体。另一方面，如果未满足这种锥体条件，我们通过凸集成构建非能量分布解决方案，从而对iWaniec和Sbordone的猜想进行负面反应。
In this paper we answer Iwaniec and Sbordone's conjecture \cite{IB94} concerning very weak solutions to the $p$-Laplace equation. Namely, on one hand we show that distributional solutions of the $p$-Laplace equation in $W^{1,r}$ for $p \neq 2$ and $r>\max\{ 1,p-1\}$ are classical weak solutions if their weak derivatives belong to certain cones. On the other hand, we construct via convex integration non-energetic distributional solutions if this cone condition is not met, thus answering negatively Iwaniec and Sbordone's conjecture in general.
