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arXiv · 2401.17912

Monotonicity of positive solutions to semilinear elliptic equations with mixed boundary conditions in triangles

Rui Li·Ruofei Yao
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Abstract

We study positive solutions of semilinear elliptic equations in a planar triangular domain under mixed boundary conditions, consisting of homogeneous Dirichlet boundary conditions on one side and homogeneous Neumann boundary conditions on the remaining two sides. Using the method of moving planes, we prove that if the Neumann vertex is non-obtuse, then every positive solution is strictly increasing in the direction of the inward unit normal to the Dirichlet side. If the Neumann vertex is obtuse, we show that monotonicity instead holds in the direction of the outward normal direction to the longer Neumann side, under certain technical conditions. Furthermore, by applying the maximum principle, we demonstrate that these monotonicity properties extend to the first mixed eigenfunction of the Laplacian: its unique global maximum lies on the longer Neumann side and coincides with the Neumann vertex precisely when that vertex is non-obtuse or when the Neumann sides have equal length. This answers a question raised in the Polymath7 research thread 1 regarding the location of extrema for mixed boundary eigenfunctions in triangles.

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Rui Li, Ruofei Yao. 2024-01-31. Monotonicity of positive solutions to semilinear elliptic equations with mixed boundary conditions in triangles. https://arxiv.org/abs/2401.17912

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