# Can I use the mean value property for a function that is harmonic on only the interior of a ball?

Let $$f$$ be a continuous function on the closure of $$U$$ where $$U=\{ (x,y) \in \mathbb{R}^2 : x^2+y^2<1 \}$$ and harmonic on $$U$$, If $$f(x,y)=x^2y^2$$ on $$\partial U$$ I want to find $$f(0,0)$$.

Now I could of course solve the Laplace equation on the disk and then calculate the value at the origin, but I was wondering if I could apply the mean value property.

Now the mean value property holds if the function is harmonic on the closure of $$U$$, but here it is harmonic on only the interior, but I actually tried to compute the mean value and I got the same value that I got when I solved the Laplace equation on the disk and calculated the value at the origin, so I am guessing that the mean value property must hold.

Perhaps the fact that $$f$$ is continuous on the closure is helpful.

My question is, can I use the mean value property in this case?

The mean value property holds for any disk of radius $$1-\varepsilon$$ with $$\varepsilon$$ as close to zero as you want. Observe that all those means over the boundaries are equal, independent of $$\varepsilon$$, since all of them are $$f(0,0)$$. By (uniform) continuity, the limit of the integrals over circumferences of radius $$1-\varepsilon$$ is equal to the integral over the circumference of radius $$1$$, so the property holds.