Symmetry and uniqueness for a hinged plate problem in a ball

TL;DR

The paper analyzes the semilinear biharmonic Steklov problem in the ball, given by $\Delta^2u=|u|^{p-1}u$ in $B_R$ with $u=\Delta u-(1-\sigma)\kappa u_n=0$ on $\partial B_R$ and $\kappa=\frac{1}{R}$, focusing on symmetry, radial monotonicity, and uniqueness. It proves radial monotonicity for broad biharmonic models, establishes symmetry of ground states for an unbounded subset of $\sigma$ via rearrangement and Talenti's principle, and applies a Dalmasso-style ODE argument to obtain uniqueness for all $\sigma>-1$. The paper also extends positivity of ground states to a class of nonconvex domains near limaçons de Pascal by combining linear positivity results with dual-cone methods and Green-function estimates. Collectively, these results provide a detailed understanding of the structure of higher-order semilinear PDEs with Steklov-type boundaries in plate models, including monotonicity, symmetry, positivity, and uniqueness beyond convex domains.

Abstract

In this paper we address some questions about symmetry, radial monotonicity, and uniqueness for a semilinear fourth-order boundary value problem in the ball of $\mathbb R^2$ deriving from the Kirchhoff-Love model of deformations of thin plates. We first show the radial monotonicity for a wide class of biharmonic problems. The proof of uniqueness is based on ODE techniques and applies to the whole range of the boundary parameter. For an unbounded subset of this range we also prove symmetry of the ground states by means of a rearrangement argument which makes use of Talenti's comparison principle. This paper complements the analysis in [G. Romani, Anal. PDE 10 (2017), no. 4, 943-982], where existence and positivity issues have been investigated.

Figures (1)

• Figure 1: Limaçons for resp. $a = 0$, $\frac{1}{4}$, $\frac{\sqrt6}{6}$, $\frac{1}{2}\,$.

Theorems & Definitions (20)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Lemma 2.1
• proof : Proof.
• Lemma 2.2
• proof : Proof.
• proof : Proof of Proposition \ref{['rad_decr']}