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Symmetry and Approximate Symmetry of a Nonlinear Elliptic Problem over a Ring

Alaa Haj Ali, Dongsheng Li, Peiyong Wang

Abstract

A singularly perturbed free boundary problem arising from a real problem associated with a Radiographic Integrated Test Stand concerns a solution of the equation $Δu = f(u)$ in a domain $Ω$ subject to constant boundary data, where the function $f$ in general is not monotone. When the domain $Ω$ is a perfect ring, we incorporate a new idea of radial correction into the classical moving plane method to prove the radial symmetry of a solution. When the domain is slightly shifted from a ring, we establish the stability of the solution by showing the approximate radial symmetry of the free boundary and the solution. For this purpose, we complete the proof via an evolutionary point of view, as an elliptic comparison principle is false, nevertheless a parabolic one holds.

Symmetry and Approximate Symmetry of a Nonlinear Elliptic Problem over a Ring

Abstract

A singularly perturbed free boundary problem arising from a real problem associated with a Radiographic Integrated Test Stand concerns a solution of the equation in a domain subject to constant boundary data, where the function in general is not monotone. When the domain is a perfect ring, we incorporate a new idea of radial correction into the classical moving plane method to prove the radial symmetry of a solution. When the domain is slightly shifted from a ring, we establish the stability of the solution by showing the approximate radial symmetry of the free boundary and the solution. For this purpose, we complete the proof via an evolutionary point of view, as an elliptic comparison principle is false, nevertheless a parabolic one holds.

Paper Structure

This paper contains 11 sections, 23 theorems, 162 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Symmetry over a Ring
  3. Stability of the Free Boundary
  4. Construction of a solution of our problem on the perfect ring $\Omega_1$
  5. Construction of a strict subharmonic function in $\Omega$ satisfying the boundary conditions associated with our problem
  6. Construction of a strict subsolution of $\Delta u=f(u)$ on $\Omega$ satisfying the boundary conditions associated with our problem and the condition $u-\varepsilon \leq v_1 \leq u$ on $\Omega$
  7. Construction of a function $v_{01}$ strict subsolution of $\Delta u=f(u)$ on $\Omega_1$ satisfying the boundary conditions associated with our problem and the condition $u-\varepsilon \leq v_{01} \leq u$ on $\Omega_1$
  8. Construction of $w_1(x,t)$ a solution of the parabolic version of our problem on $\Omega_1 \times (0,\infty)$
  9. Convergence of the evolution to a steady state
  10. Construction of a solution of our problem on the perfect rings $\mathcal{R}$ and $\Omega_2$ respectively
  11. Comparison and Stability

Key Result

Theorem 1.1

Let $R > 1$ and $\Omega = B_R\backslash \bar{B}_1$ be the domain of a ring or shell. Suppose $f\colon \mathbb{R}_+\rightarrow\mathbb{R}$ is a $C^1$ function such that $f(s)\leq 0$ and $\inf_{\mathbb{R}_+}f'(s) > -\frac{4(n+2)}{R^2}$. Then a solution $u\in C^2(\overline{B_R\backslash B_1})$ of (bvp)

Figures (2)

  • Figure 1: $\Pi(\lambda)$ for $R = 2$, $\lambda = 1.5, 1.25, 1, 0.75, 0.5, 0.25, 0$, respectively
  • Figure 2: The spheres $B_1$, $B_{R_1}$, $B_R(Z)$, $B_R$, and $B_{R_2}$ for $\delta = 0.5$ and $R = 4$

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Definition 3.1
  • ...and 16 more