# First eigenfunction of $p=3$-Laplacian of a square domain in $\Bbb R^2$ : reference for any work on this?

In the last few decades, lots of work on first eigenfunction of $p$-Laplace with Dirichlet and other boundary conditions. But I couldn't find much on periodic boundary conditions. I have computed the first eigenfunction for $p=3$, square domain in $\Bbb R^2$. I have shown it as a grey scale image below. One period I tiled in the plane ($4\times 4$ tiles) to illustrate its meeting periodic boundary conditions.

I wonder such a simple structure not have a closed form expression. Can we guess anything on its closed form expression? Any reference to work in this direction (especially in dimension greater than $1$.)

Color picture :

Let $g_p(x)$ be the first eigenfunction of $p$-Laplacian for the domain $(0,1)$ under periodic boundry conitions. Then, for all $p\ge(d+1)$ it can be shown that $$u(\vec{x}) = g_p( \vec{x}\cdot\vec{z}), z \in \mathbb{Z}^d,$$ is an eigenfunction for the $p$-Laplacian for the domain $(0,1)^d$ under periodic boundary conditions.
• It is mentioned in loc.cit. that $\sin_3$ is the eigenfunction of the 3-Laplacian, so maybe the non-trivial part of the question is to show that this is the eigenfunction with the smallest eigenvalue. – Fan Zheng Sep 2 '17 at 8:34
• which brings me to this question: does it work if I just lift the 1-dimensional eigenfunction to the torus, say $\sin_3(x)$ regarded as a function of two variables? – Fan Zheng Sep 2 '17 at 9:15
• @FanZheng : I am at a loss. I really don't know what you are saying, and also I don't know differential geometry. So I cannot get any interpretation of what you are saying and its intent. If its simple enough, you may explain. From my point of view, I am interested only square domains in the Euclidean space $\mathbb{R}^d$. So probably it may help me if you can explain a bit. – Rajesh Dachiraju Sep 2 '17 at 12:02
• OK. Maybe I have used a little fancy terminology, but I'm really trying to say: Way don't you try $u(x,y)=\sin_3x$ (with appropriate scaling)? BTW, it seems that you missed a factor of $\sqrt2$ or something in the above computation. If that is the case, then $\sin_3x$ would have a smaller eigenvalue. – Fan Zheng Sep 2 '17 at 12:07