子域中指数无界的p（x）-调和函数
p(x)-Harmonic functions with unbounded exponent in a subdomain
摘要：
我们研究了$Ω$中的Dirichlet问题$-÷(|\nabla u|^{p(x)-2} \nabla u) =0 $，其中$u=f$在$\partial Ω$上，$p(x) = \infty$在$D$中，$Ω$是参考域$Ω$的一个子域.主要的问题是给一个正确的意义是什么解决方案。为此，我们认为作为$n \to \infty$的解决方案$u_n$相应的问题时，$p_n(x) =p(x) \wedge n$，特别是与$p_n = n$在$D$的限制。在适当的假设下的数据，我们发现，这样的限制存在，它可以被描述为唯一的解决方案的变分最小化问题，这是，此外，$\infty$ -调和内$D$。此外，我们检查这个极限的粘性意义，并找到它满足整个$Ω$的边值问题。
英文摘要：
We study the Dirichlet problem $-÷(|\nabla u|^{p(x)-2} \nabla u) =0 $ in $Ω$, with $u=f$ on $\partial Ω$ and $p(x) = \infty$ in $D$, a subdomain of the reference domain $Ω$. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as $n \to \infty$ of the solutions $u_n$ to the corresponding problem when $p_n(x) =p(x) \wedge n$, in particular, with $p_n = n$ in $D$. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, $\infty$-harmonic within $D$. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of $Ω$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
