In my setting I'm working on an open subset $Y$ of a Riemann surface, but I don't think that matters too much. I have a smooth function $f: Y \to \mathbb{R}$ that is strictly subharmonic. If another smooth function $g: Y \to \mathbb{R}$ is sufficiently close (in the Whitney $C^\infty$ topology), then will it also be strictly subharmonic? I'm finding this very difficult to show with my definition of strictly subharmonic (if this result is true at all...which I suspect it is, because this topology is that strong).

Here is my definition of strictly subharmonic: Let $A$ denote the set of all relatively compact, open subsets of $Y$ on which a solution to the Dirichlet problem exists for any arbitrary continuous boundary condition. If $D \in A$ and $u: Y \to \mathbb{R}$ is continuous, denote by $P_D(u): \overline{D} \to \mathbb{R}$ the function which solves the Dirichlet problem on $\overline{D}$ (with boundary values $u|_{\partial D}$), and coincides with $u$ on $Y \setminus D$. A continuous function $u: Y \to \mathbb{R}$ is strictly subharmonic if $P_D(u) > u$ on $Y$ for all $D \in A$.

(By the Dirichlet problem I mean the following. Let $q: \partial D \to \mathbb{R}$ be a continuous function. We wish to find a continuous function $u: \overline{D} \to \mathbb{R}$ which coincides with $q$ on $\partial Y$ and is harmonic in $Y$.)


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