论文标题
弱和粘度解决方案与均质$ \ texttt {p}（\ cdot）$ - laplacian in $ \ mathbb {r}^n $中的弱和粘度解决方案等效的新证明
A New Proof of the Equivalence of Weak and Viscosity Solutions to the Homogeneous $\texttt{p}(\cdot)$-Laplacian in $\mathbb{R}^n$
论文作者
论文摘要
我们提供了一个新的证明，以等效于$ \ Mathbb {r}^n $中的$ \ texttt {p}（\ cdot）$ laplace方程的潜在理论弱解决方案和粘度解决方案。 juutinen，Lukkari和Parviainen（2010）首先给出了欧几里得空间中可变指数案例中等价性的证明，并扩大了潜在的理论弱解决方案和粘度解决方案与$ \ fexttt {p} $ - laplace-laplace方程的等效性，并延长了$ \ nly的laplace方程。 Manfredi（2001）。在固定指数案例和变量指数案例中，主要参数基于半连续函数的最大原理，几个近似值，并且还应用了粘度解决方案理论的完整独特性机制。本文扩展了Julin和Juutinen（2012）对于固定指数案例的方法，因此我们采用虚拟卷积来提供直接的，新的证据，以等效于$ \ texttt {p}（\ cdot）$ laplace方程的潜在理论弱解决方案和粘度解决方案。
We present a new proof for the equivalence of potential theoretic weak solutions and viscosity solutions to the $\texttt{p}(\cdot)$-Laplace equation in $\mathbb{R}^n$. The proof of the equivalence in the variable exponent case in Euclidean space was first given by Juutinen, Lukkari, and Parviainen (2010) and extended the equivalence of potential theoretic weak solutions and viscosity solutions to the $\texttt{p}$-Laplace equation in $\mathbb{R}^n$, given by Juutinen, Lindqvist, and Manfredi (2001). In both the fixed exponent case and the variable exponent case, the main argument is based on the maximum principle for semicontinuous functions, several approximations, and also applied the full uniqueness machinery of the theory of viscosity solutions. This paper extends the approach of Julin and Juutinen (2012) for the fixed exponent case, and so we employ infimal convolutions to present a direct, new proof for the equivalence of potential theoretic weak solutions and viscosity solutions to the $\texttt{p}(\cdot)$-Laplace equation in $\mathbb{R}^n$.
