A 2-dimensional shape optimization problem for tree branches

Figure 1.  A stem $ \gamma_1 $ perpendicular to the sun rays is optimally shaped to collect the most light. For the stem $ \gamma_2 $ bending toward the light source, the upper leaves put the lower ones in shade

Figure 2.  When the light rays impinge from a fixed direction $ {{\bf n}} $, the optimal distribution of leaves is supported on the two rays $ \Gamma_0 $ and $ \Gamma_1 $

Figure 3.  Density profile $ u(s) $ for $ s \in [0, \ell_1] $ along the ray $ \Gamma_1 $ for $ c = 1 $ and $ \alpha = 2/3, 1/3 $

Figure 4.  According to the definition (31), the set $ \chi^-(x) $ is a curve joining the origin to the point $ x $. The set $ \chi^+(x) $ is a subtree, containing all paths that start from $ x $

Figure 5.  If the set $ \chi^+(x) $ is not contained in the slab $ \Gamma_x $ (the shaded region), by taking the perpendicular projections $ \pi^\sharp $ and $ \pi^\flat $ we obtain another irrigation plan with strictly lower cost, which irrigates a new measure $\tilde{\mu}$ gathering exactly the same amount of sunlight. Notice that here $ P $ is the point in the closed set $ \overline{\chi^+(x)}\cap{{\mathbb R}}{{\bf e}}_1 $ which has the largest inner product with $ {{\bf n}} $.

Figure 6.  After a rotation of coordinates, the sunlight comes from the vertical direction. Here the blue lines correspond to the set $ {{\mathcal B}}^* $ in (36).

Figure 7.  The construction used in the proof of Lemma 3.3.

Figure 8.  The thick portions of the curve $ \gamma $ are the only points where a left bifurcation can occur. If a horizontal branch $ \sigma $ bifurcates from $ C_j $, all the mass on this branch can be shifted downward to another branch $ \sigma^* $ bifurcating from $ C_j^* $. Furthermore, if some portion of the path $ \gamma $ between $ P^* $ and $ Q $ lies above the segment $ \gamma^* $ joining these two points, we can take a projection of $ \gamma $ on $ \gamma^* $. In both cases, the transportation cost is strictly reduced.

Figure 9.  Left: an irrigation plan for a measure $ \mu $ with two masses at $ Q $ and at $ P_1 $. Right: an irrigation plan for a modified measure $ \tilde{\mu} $ with two masses at $ \tilde{Q} $ and at $ P_1 $. The lengths of the segments $ PP^* $ and $ P^* P_1 $ will be denoted by $ \ell_a, \ell_b $, respectively.

Figure 10.  A more general configuration, considered in Lemma 4.2.

Figure 11.  Left: in the shaded region $ \Delta $ above the curve $ \gamma $, the measure $ \mu $ cannot concentrate any mass. Otherwise, by shifting this mass downward until it hits a point on $ \gamma $, we would obtain a second measure $ \tilde{\mu} $ which gathers the same amount of sunlight, but has a lower irrigation cost. As a consequence, by the interior regularity property, the flow out of $ P^* $ is locally supported on a finite number of line segments. Right: the construction used in steps 4–6 of the proof of Theorem 2.5.

Figure 12.  Changing the transport plan, when $ \theta_0>0 $ is very small. If in the irrigation problem the bifurcation angle satisfies $ \theta>\theta_0 +{\pi\over 2} $, then the original configuration, where all the mass is supported along $ \Gamma_0\cup\Gamma_1 $, would not be optimal. The analysis at (102) shows that this situation never happens.