I'm working on an unassessed course problem to show the solution $p$ to a PDE satisfies an inequality. I think I can show that of the 2 boundary conditions of the PDE, $p$ is equal to another solution $f$ in one of the BCs and greater-than-or-equal-to $f$ in the other BC. I understand that if both BCs matched then $p=f$. Given one equality and one inequality, does this mean $p\geq f$?
For those familiar with the context topic (finance), here's the problem.
Consider a European put option with strike price $X$ and maturity time $T>0$. For any time $0\leq t\leq T$, let $S$ denote the spot price of the underlying stock and let $p(S_t,t)$ be the price of a European put option at time $t$. Assume that all spot interest rates are constant and equal to $r$. Assume also that the underlying stock price follows the Ito process $$\text{d}S = rS\text{ d}t + \sigma S \text{ d}B.$$ Verify that $f(S_t,t)=Xe^{-r(T-t)}-S$ is a solution to the Black-Scholes equation, $$\frac{\partial f}{\partial t}+rS\frac{\partial f}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2f}{\partial S^2}=rf.$$ Use this to deduce that $p\geq Xe^{-rT}-S.$
I get \begin{align} & \lim_{S\downarrow 0}p(S_t,t)=Xe^{-r(T-t)}=\lim_{S\downarrow 0}f(S_t,t), \\ & \lim_{t\uparrow T}p(S_t,t)=\max(\{X-S_T,0\})\geq X-S_T=\lim_{t\uparrow T}f(S_t,t) \end{align} and then find myself asking the question in the title.
IIRC this does not work in general unless your equation satisfies the maximum principle. If your equation is linear and for boundary conditions $a \leq b$ solutions are $f$ and $g$ respectively, you get that $h = f-g$ is also a solution (due to linearity) and $h = (a - b) \leq 0$ on the boundary. Hence $h$ must be non-positive everywhere, otherwise it would have a maximum not on the boundary. That is $a\leq b$ implies $f\leq g$ everywhere. For heat equation (which you can get from BS by change of variables) you have maximum principles.
Besides that, BS equation has explicitly known fundamental solution $\phi$ (solution for boundary condition given by Dirac delta), so that for any terminal (boundary) condition (payoff) $F(x, T)$ you get that the solution is $$ f(x,t) = \int_\Bbb R F(x, T)\phi(x)\mathrm dx. \tag{1} $$ From that you can also get your inequalities.
Note that $(1)$ is the formula which states that $f(x,t) = \mathrm e^{-r(T - t)}\Bbb E_\Bbb Q[F(x, T)]$ i.e. the value of the contigent claim now is a discounted expected value of its payoff w.r.t. the martingale measure.