The hot spots conjecture can be false: some numerical examples

The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10− 3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.

On the lower bound of the principal eigenvalue of a nonlinear operator

We prove sharp lower bound estimates for the first nonzero eigenvalue of the non-linear elliptic diffusion operator $$L_p$$ L p on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition, satisfying $$BE(\kappa ,N)$$ B E ( κ , N ) for $$\kappa \ne 0$$ κ ≠ 0 . Our results extends the work of Koerber Valtorta (Calc Vari Partial Differ Equ. 57(2), 49 2018) for case $$\kappa =0$$ κ = 0 and Naber–Valtorta (Math Z 277(3–4):867–891, 2014) for the p-Laplacian.

Global Stabilization of a Single-Species Ecosystem with Markovian Jumping under Neumann Boundary Value via Laplacian Semigroup

By applying impulsive control, this work investigated the global stabilization of a single-species ecosystem with Markovian jumping, a time delay and a Neumann boundary condition. Variational methods, a fixed-point theorem, and Laplacian semigroup theory were employed to derive the unique existence of the global stable equilibrium point, which is a positive number. Numerical examples illuminate the feasibility of the proposed methods.

Dynamics of Spatial Epidemic Model with Infected Population Repulsion

In this paper, we are concerned with the spatial epidemic model with infected-taxis in which the susceptible individuals could avoid the infected ones. The spatial pattern for the resulted model is investigated under homogeneous Neumann boundary condition. We gain the condition for spatial pattern induced by diffusion term and infected-taxis term. Moreover, we obtain the condition for the occurrence of pattern formations induced by infected-taxis, in which the diffusion-driven Turing instability case is excluded. We give numerical examples to support the theoretical scheme.