Article Contents
Article Contents

# A 2-dimensional shape optimization problem for tree branches

• * Corresponding author: Alberto Bressan
• The paper is concerned with a shape optimization problem, where the functional to be maximized describes the total sunlight collected by a distribution of tree leaves, minus the cost for transporting water and nutrient from the base of trunk to all the leaves. In a 2-dimensional setting, the solution is proved to be unique and explicitly determined.

Mathematics Subject Classification: 49Q10, 49Q20.

 Citation:

• 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.

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