Example of a uniformly convex function on the unit square, that is zero on the boundary.

Question

I would like to know of an example of a smooth function $u:[0,1]^2\to\mathbb R$, that is zero on the boundary, i.e., $$u(0,y)=u(1,y)=u(x,0)=u(x,1)=0,$$ that is uniformly convex. By uniformly convex, I mean that the Hessian of $u$, $D^2u$, is strictly positive definite on all of $[0,1]^2$.

The only examples that I could think of fail to be uniformly convex on the boundary, for instance $$u = x(x-1)y(y-1),$$ and $$u = -\sin(\pi x)\sin(\pi y).$$

Thanks in advance!

Answer 1

The condition has to fail on the boundary since e.g. $\partial_{xx} u(x,0)=0$ and then the Hessian can not be strictly pos def.