Symmetry and monotonicity of positive solutions of elliptic equations with mixed boundary conditions in a super-spherical cone

Symmetry of positive solutions of elliptic equations with mixed boundary conditions in a sub-spherical sector

“…The maximum principle is an important tool in elliptic and parabolic partial differential equations; see [18,30]. In the process of MMP, several versions of maximum principle are established, see [3,4,34]. Now we are in the position to prove a new maximum principle for mixed boundary problems, which is helpful to prove (2.12).…”

“…When the domain is a spherical sector, Berestycki and Pacella [5] proved that the radial symmetry results for spherical sector domains with mixed boundary conditions provided the amplitude of spherical sector is less or equal to π, and Zhu [35] proved a similar result for singular solutions when the amplitude may be greater than π and f satisfies some supercritical growth conditions. The first and third authors [7,34] proved some symmetry properties of positive elliptic solutions in a standard spherical cone and in a super-spherical cone. Researchers are also concerned about the symmetry results for nonlinear boundary problems, see [10,11,32,33,36].…”

“…The proof of this theorem is based on the ideas developed in [5,16,34], and its main difficulty is to obtain the direction of ∇u along the Neumann boundary Γ N . So the crucial step of this proof is to show the monotonicity of u along the direction of upper and lower Neumann boundary in the whole domain, i.e., ∇u • (cos(β/2), ± sin(β/2)) < 0 in Σ.…”

A shape variation result via the geometry of eigenfunctions

“…Perdo and Tobis [15] have studied the symmetry and monotonicity of the solutions of (1.1) with Neumann boundary condition on balls with a particular non-linearity, f (x, u) = λ p |u| p−1 u + µ p with appropriate constants λ p and µ p . The symmetry and monotonicity of the positive solutions of the Zaremba problem of the semi-linear equation (1.1), i.e., the mixed boundary problem of (1.1), are studied only in the spherical cones, see [6,8,39,41]. To the best of our knowledge, there are no results available on the geometry of the eigenfunctions of the Laplace operator for annular domains.…”