数学

优雅的好题

求最小的正实数$d$，使得对任意正整数$n$及满足$a_{i,0}=a_{i,n}=a_{0,j}=a_{n,j}=0(0~\leq~i,j~\leq~n)$的任意$(n+1)^2$个整数$a_{i,j}(0~\leq~i,j~\leq~n)$,都有

$\large{\sum_{i=1}^{n}~\sum_{j=0}^{n}|a_{i,j}-a_{i-1,j}|~+~\sum_{i=0}^{n}~\sum_{j=1}^{n}|a_{i,j}-a_{i,j-1}|~\geq~\left(\sum_{i=0}^{n}~\sum_{j=0}^{n}|a_{i,j}|^d~\right)^{1/d}}$