论文标题
$ \ mathbb {r}^d $上的随机$ p $ -laplace方程
The stochastic $p$-Laplace equation on $\mathbb{R}^d$
论文作者
论文摘要
我们在$ \ mathbb {r}^d $上展示了$ p $ -laplace Evolution方程的适合性，并带有方形集成的随机初始数据，适用于任意$ 1 <p <\ intymable $ d \ in \ mathbb {n} $。方程右侧的噪声项可能是加性的或乘法的。由于在整个空间中缺乏$ p $ laplace运算符的强制性，因此在经典功能设置中应用众所周知的存在和独特定理的可能性仅限于$ 1 <p <\ infty $的某些值，还取决于空间尺寸$ d $。我们提出了一个功能空间的框架，该框架与Sobolev空间嵌入和空间维度无关。对于加性噪声，我们使用时间离散化显示存在。然后，定点参数产生了乘法噪声的结果。
We show well-posedness of the $p$-Laplace evolution equation on $\mathbb{R}^d$ with square integrable random initial data for arbitrary $1<p<\infty$ and arbitrary space dimension $d\in\mathbb{N}$. The noise term on the right-hand side of the equation may be additive or multiplicative. Due to a lack of coercivity of the $p$-Laplace operator in the whole space, the possibility to apply well-known existence and uniqueness theorems in the classical functional setting is limited to certain values of $1<p<\infty$ and also depends on the space dimension $d$. We propose a framework of functional spaces which is independent of Sobolev space embeddings and space dimension. For additive noise, we show existence using a time discretization. Then, a fixed-point argument yields the result for multiplicative noise.
