In this paper, we investigate the symmetry properties of positive solutions $u$ to a semilinear elliptic equation under mixed Dirichlet-Neumann boundary conditions in symmetric domains. First, we establish a maximum principle tailored to mixed-boundary problems in domains of either small volume or narrow width, thereby enabling the application of the moving plane method. Secondly, in contrast to the purely Dirichlet case, a key challenge is to establish the non-vanishing of the tangential derivative of $u$ along the Neumann boundary. To address this, we employ local analysis techniques of angular derivatives, as introduced by Hartman and Wintner [Amer. J. Math., 1953]. Thirdly, we identify the signs of directional derivatives of $u$ along sections of the moving line. Using a planar sub-spherical sector as an example, we illustrate how these new innovative techniques and the moving plane method can be combined to derive symmetry and monotonicity results, particularly when the amplitude is less than or equal to $2π/3$.
where Ω is a bounded Lipschitz domain, Γ D is a portion of the boundary ∂Ω, Γ N = ∂Ω \ Γ D , and ν denotes the unit outward normal to ∂Ω.
Research on the qualitative properties of solutions is a central topic in the theory of partial differential equations. A classical and influential approach is the method of moving planes, introduced by [3,54] and further developed by Gidas, Ni, and Nirenberg [31]. In particular, [31] established radial symmetry for positive solutions of semilinear elliptic equations under purely Dirichlet boundary conditions (i.e., Γ D = ∂Ω). Later, Berestycki and Nirenberg [7] refined these results using a maximum principle in domains of small volume and introduced the sliding method to obtain monotonicity properties. These developments sparked extensive research on symmetry and monotonicity; see, for instance, [11,17,47,5,6,29,35,28]. Motivated by these developments, it is natural to ask whether, when both the domain and the boundary decomposition are symmetric, positive solutions of (1.1) inherit the same symmetry, as raised in [50]. In general, however, such a conclusion fails once Neumann boundary conditions are present; see [51,34,26,36,48] and the references therein. Establishing analogous qualitative properties becomes even more subtle for sign-changing solutions. For instance, the monotonicity of the second Neumann eigenfunction on certain planar domains has been investigated in [45,4,56,15,12].
Researchers have devoted increasing attention to qualitative properties (in particular, symmetry and monotonicity) for elliptic problems with mixed boundary conditions, where Dirichlet and Neumann conditions coexist on ∂Ω. In particular, for spherical sectors, Berestycki and Pacella [9] established radial symmetry results when the opening angle is at most π, while Zhu [62] extended these results to certain supercritical nonlinearities when the opening angle exceeds π. Building on [9], further radial symmetry results for infinite sectorial cones were obtained in [25,20,55]. The author has contributed to this area by establishing symmetry and monotonicity properties of positive solutions to elliptic equations with mixed boundary conditions in various domains; see [16,61,13,60,14,46,49]. In recent years, analogous questions have also been investigated for mixed Dirichlet-Neumann eigenfunctions; see, e.g., [2,46,39,40,42,41]. For broader discussions of qualitative properties in nonlinear mixed boundary value problems, we refer to [57,24,58,22,23,33] and the references therein.
This paper is part of a series of works [16,61,13,60,49] investigating the symmetry and monotonicity of solutions to semilinear elliptic equations with mixed boundary conditions, in bounded symmetric domains. In particular, the symmetry result for super-spherical sectors was established in [60], while a partial result for sub-spherical sectors was obtained in [13]. We now extend this investigation by studying the symmetry and monotonicity of solutions to (1.1) in a planar subspherical sector, under a broad range of geometric assumptions as described in (1.4).
1.2. Problem setting and main result. We begin by introducing the notation and the geometric setting. For α ∈ (0, π] and β ∈ (0, 2π), let Σ = Σ α,β denote a planar domain bounded by an arc Γ D and a portion of the boundary of a sector ∂C, where Γ D = {(cos θ, sin θ) ∈ R 2 : |θ| ≤ α/2} is a portion of the unit circle with opening angle α, and C = C α,β denotes the open sector with vertex V = (a, 0) and opening β, that is,
. Sub-spherical sector (left), spherical sector (middle) and super-spherical sector (right).
We consider the following equation with mixed boundary conditions:
(1.3)
where ν denotes the unit outward normal to ∂Σ. Here, we set Γ D = {(cos θ, sin θ) ∈ R2 : |θ| ≤ α/2} as the spherical part, and Γ N = ∂Σ \ Γ D as the union of two straight segments. The nonlinearity f is assumed to be locally Lipschitz continuous (f ∈ Lip loc (R)), so that the method of moving planes applies. A solution u to (1.3) is always understood in the classical sense, namely, u ∈ C 2 (Σ)∩C(Σ)∩C 1 (Σ∪Γ N ). By standard elliptic theory, it follows that u is also C 2 at the smooth boundary points. Therefore, throughout this paper, we always have u ∈ C(Σ) ∩ C 2 (Σ \ {V, P ± }).
The main result of this paper is as follows:
Theorem 1.1. Let f ∈ Lip loc (R), and let Σ, Γ D , and Γ N be as defined above. Suppose that 0 < β < α ≤ π and
Then any solution u of (1.3) satisfies the following properties:
(i) u is symmetric with respect to the line x 2 = 0;
(ii) u is monotone in x 2 in a half-domain, that is, x 2 u x 2 < 0 for x ∈ Σ with x 2 ̸ = 0; (iii) u is monotone in x 1 , that is, u x 1 < 0 in Σ.
Remark 1.2. In the previous paper [13], two results were established:
(1) Theorem 1.1 holds under the assumptions 0 < β < α ≤ π and
(1.5) α + β ≤ π and α/3 ≤ β ≤ π/3.
(2) The properties (i), (ii), and (iii) are equivalent without the condition (1.5).
1.3. Ideas of the proof. We
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