This is a rather vague question. Also, I will restrict my description to a bivariate function $u(x,y)$ only for simplicity.
Roughly speaking, the standard regularity result theory of Laplace equation shows that if the derivatives $\partial_{xx}^2u$ and $\partial_{yy}^2u$ exist, then $\partial_{xy}^2$ exists as well. (For example, this holds if we consider locally integrable functions on $\mathbb{R}^2$ and say that "a derivative exists" if the weak derivative is in $L^2(\mathbb{R}^2)$.)
Now, if we think of derivatives as elements in $\mathbb{N}^2$, then we see that $\partial_{xy}^2=(1,1)$ is simply the midpoint of $\partial_{xx}^2=(2,0)$ and $\partial_{yy}^2=(0,2)$, and the set of available derivatives is thus convex.
Does this observation hold in general? For example, if $\partial^4_{xxxx}$ and $\partial_{yyyy}^4$ exist, does $\partial_{xxxy}^4$ exist? Or if $\partial^2_{xx}$ and $\partial_{yyyy}^4$ exist, does $\partial_{xyy}^3$ exist? (I.e. are there specific settings/meanings/spaces in which this holds?)