Annuli And Regions With Inner Boundaries

Why an Annulus Needs Two Boundary Checks

An annulus is the region between two concentric circles. Unlike a disk, it has a hole: points inside the inner circle are not part of the region. For absolute-extrema problems, this means there are two separate circular boundaries to examine. Checking only the outer circle can miss the minimum or maximum on the inner one.

The method combines three searches: find critical points in the interior, restrict the function to the inner circle, and restrict it to the outer circle. Then compare the function values from all three searches. The boundary calculations are one-variable problems, using circular parametrizations and the chain rule.

Definition and Candidate Principle

For radii \(0<r<R\), the closed annulus centered at the origin is the set

Its interior is \(r^2<x^2+y^2<R^2\). Its inner boundary is the circle \(x^2+y^2=r^2\), and its outer boundary is the circle \(x^2+y^2=R^2\). The assumption \(r>0\) ensures there is a genuine hole; \(R>r\) ensures the region has positive width.

Suppose \(f\) is continuous on \(A\) and differentiable on its interior, with differentiable restrictions to the boundary circles. Because \(A\) is closed and bounded, the extreme value theorem guarantees that \(f\) attains an absolute maximum and minimum on \(A\). To find them, consider the following candidates:

• Interior points where \(f_x=0\) and \(f_y=0\).
• Stationary points of the restriction of \(f\) to the inner circle.
• Stationary points of the restriction of \(f\) to the outer circle.

The last two searches are separate one-variable optimization problems. A convenient parametrization of a circle of radius \(s\) is \(x=s\cos t,\;y=s\sin t\), where \(0\le t\le 2\pi\). For the inner boundary, take \(s=r\); for the outer boundary, take \(s=R\).

Why the Candidate Principle Works

Here is the justification. An absolute extremum exists by continuity on the compact annulus. Take a point where it occurs. If the point lies in the interior, Fermat’s theorem says that both first partial derivatives vanish. If it lies on a boundary circle, restrict \(f\) to that circle. The point is then an extremum of a differentiable periodic function of \(t\), so its derivative with respect to \(t\) vanishes. Thus every possible location of an absolute extremum is included in the candidate searches.

The chain rule gives a useful formula for either boundary. For \(g_s(t)=f(s\cos t,s\sin t)\),

This derivative measures how \(f\) changes as a point moves tangent to the circle. A boundary extremum must have zero change in that tangential direction. The formula applies separately with \(s=r\) and \(s=R\); it does not replace the interior critical-point search.

Worked Examples

Constant Restrictions and Common Pitfalls

A boundary restriction can be constant, as in the first example. In that case its derivative is zero at every parameter value, so the whole circle consists of stationary points of the restriction. It is still enough to record its constant function value once. More generally, boundary restrictions may have several stationary points; evaluate all their values rather than assuming a particular stationary point is a maximum or a minimum.

A frequent mistake is to treat an annulus like a disk and check only the outer circle. For example, for \(f(x,y)=x^2+y^2\) on \(1\le x^2+y^2\le 9\), checking only the outer boundary gives the value \(9\). If one then reports \(9\) as the minimum, the conclusion is wrong: the inner boundary has value \(1\), which is smaller. The correction is to optimize on both circles, in addition to checking interior critical points.