Concavity and hot spots in elliptic problems under mixed boundary conditions

We consider the torsion function and the first Laplacian eigenfunction in a convex curvilinear sector in the plane, under homogeneous Neumann conditions on the two straight lateral sides and a homogeneous Dirichlet condition on the remaining part of the boundary. We prove that they are, respectively, strictly $(\frac 1 2)$-concave and strictly log-concave, provided the interior angles at the vertices are at most $\frac \pi 2$. Under the same assumption, we further establish a billiard-type concavity, obtained through reflections across the Neumann sides. As a consequence, we deduce that there exists a unique hot spot, located at the Neumann--Neumann vertex, and that some monotonicity properties hold along suitable segments. Finally, we prove that the associated variational energies, namely the mixed torsional rigidity and the first mixed Laplacian eigenvalue, satisfy Brunn--Minkowski type inequalities in the class of convex curvilinear sectors with a fixed opening angle.