Let $Ω := \mathbb{R}^n \times [0, 1]$ and let $F: Ω \to \mathbb{R}^{n \times n}_\mathrm{sym}$ be a function on $Ω$ valued in symmetric matrices.
We have two assumptions on $F$: There exists a $R > 0$ such that
- $F|_{B_R \times \{0, 1\}} = 0$.
- $F|_{(\mathbb{R}^n ∖ B_R) \times [0, 1]} = - ε r^2 I_{n+1}$, where $r$ is the distance function to the origin in $\mathbb{R}^n$ and $I_{n+1}$ the identity matrix.
By $≥$ for symmetric matrices I mean that the difference is positive semidefinite.
I'm interested in questions such as:
How do you construct (or just show existence of) a function $u$ on $Ω$ satisfying $u|_{∂ Ω} = 0$ and $∇^2 u ≥ F$?
I would want $u$ to be compactly supported, but I don't even know how to find any such $u$.
Actually, I don't even know how to write down a strictly convex function on a square whose boundary value is zero!
Remark: Something like condition 2 on $F$ is necessary. If $F ≥ δ > 0$ uniformly then $u(x, 1/2) > 0$ for large $x$ (just integrate the ODE you get when restricting to $\mathbb{R} x \times [0, 1]$) but then $u(x, 1/2) > \frac12 (u(x, 0) + u(x, 1))$