Circular symmetrization and extremal Robin conditions (Q1284451)

The authors consider the problem of finding the maximum and minimum average temperatures in a homogeneous disk subject to uniform heating in the interior, and Newton's law of cooling on its boundary. More specifically, let \(D\) be the unit disk, \(h\) the conductivity on the boundary, \(v_h\) the temperature inside \(D\), subject to Newton's law \(hv+\partial v/\partial n=0\), and with \(\Delta v=-1\) in \(D\). The problem is to find the minimum and maximum average temperature \(\mathcal T(h)=\int_D v_h dx\) over all \(h\) subject to the constraints \(h_1\leq h(x)\leq h_2\), \( \int_{\partial D} h ds= 2\pi(h_1\gamma+h_2(1-\gamma))\), for given constants \(h_1,h_2\) and \(\gamma\in (0,1)\). The estimates derived by the authors are the following: \[ {\pi\over 2}\Big({1\over h_1\gamma+h_2(1-\gamma)}+ {1\over 4}\Big)\leq \mathcal T(h)\leq {\pi\over 2} \Big({\gamma\over h_1}+{{1-\gamma}\over h_2}+{1\over 4}\Big), \] for all \(h\) satisfying the above constraints. Moreover, equality on the left is achieved when \(h=h_1\gamma+h_2(1-\gamma)\), and equality on the right when \(h\) is the function taking the value \(h_1\) on a circular arc of length \(2\pi\gamma\), and the value \(h_2\) on the complementary arc. The first step in the proof is to write the variational characterization of this problem, which almost immediatley yields the left inequality. The main tool used for the right-hand side inequality is the fact that the Dirichlet integral \(\int_D | \nabla v| ^2 dx\) decreases when \(v\) is replaced by its circular symmetrization, for any \(v\in H^1(D)\). The authors provide a self-contained proof of this fact, which was well known under some additional smoothness assumptions.