Example: Given the following PDE: $$ \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} =0 $$ , subjected to the Dirichlet boundary conditions: $$ \left\{ \begin{align*} V(0,y)&=f_1(y)\\ V(m,y)&=f_2(y)\\ V(x,0)&=g_1(x)\\ V(x,n)&=g_2(x) \end{align*} \right. $$
Let $V_1$ be the solution of the PDE above.
Let $V_2$ be a function that also satisfies the PDE and that: $$ \left\{ \begin{align*} V_2(0,y)&=f_1(y)+\epsilon_1(y)\\ V_2(m,y)&=f_2(y)+\epsilon_2(y)\\ V_2(x,0)&=g_1(x)+\delta_1(x)\\ V_2(x,n)&=g_2(x)+\delta_2(x) \end{align*} \right. $$ where $\epsilon ,\delta \approx 0$.
My question is: Does $V_2$ "look" similar to $V_1$? By this I mean: $$ V_2(x,y)-V_1(x,y) \approx 0 \text{ for any } x,y:0\leq x \leq m,0\leq y\leq n $$ I would also appreciate if you can refer me to any theory that can answer my question.
Yes: since $W=V_2-V_1$ is harmonic and satisfies the boundary conditions $$ \left\{ \begin{align*} W(0,y)&=\epsilon_1(y)\\ W(m,y)&=\epsilon_2(y)\\ W(x,0)&=\delta_1(x)\\ W(x,n)&=\delta_2(x) \end{align*} \right., $$ by the Maximum Principle it is bounded by the largest value of $\epsilon,\delta$ on the boundary.