1. **State the problem:** Simplify or analyze the expression $uxx + uxxyy + uyy$ where subscripts denote partial derivatives. 2. **Identify notation:** Here, $uxx$ means $\frac{\partial^2 u}{\partial x^2}$, $uyy$ means $\frac{\partial^2 u}{\partial y^2}$, and $uxxyy$ means $\frac{\partial^4 u}{\partial x^2 \partial y^2}$. 3. **Rewrite the expression:** The expression is $$uxx + uxxyy + uyy = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^4 u}{\partial x^2 \partial y^2} + \frac{\partial^2 u}{\partial y^2}.$$ 4. **Interpretation:** This is a sum of second and fourth order mixed partial derivatives of $u$. 5. **No further simplification** is possible without additional information about $u$. 6. **Summary:** The expression combines second derivatives in $x$ and $y$ plus a mixed fourth derivative term. Final answer: $$uxx + uxxyy + uyy.$$