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A Constructive Approach to a Class of Overdetermined Problems

Alessandro Fortunati, Filomena Pacella

Abstract

In this paper we study an overdetermined problem which is directly related to the well known torsion problem studied by J. Serrin. A perturbed version of the latter is tackled by using asymptotic series as well as tools borrowed from the celebrated Nekhoroshev Theorem. In a similar fashion to this class of results, we establish the existence of infinitely many approximants for the perturbed problem's solution, whose approximation error is so small which can be regarded as negligible for practical applications. The approach is fully constructive and this feature is demonstrated via an example in the final section.

A Constructive Approach to a Class of Overdetermined Problems

Abstract

In this paper we study an overdetermined problem which is directly related to the well known torsion problem studied by J. Serrin. A perturbed version of the latter is tackled by using asymptotic series as well as tools borrowed from the celebrated Nekhoroshev Theorem. In a similar fashion to this class of results, we establish the existence of infinitely many approximants for the perturbed problem's solution, whose approximation error is so small which can be regarded as negligible for practical applications. The approach is fully constructive and this feature is demonstrated via an example in the final section.
Paper Structure (25 sections, 6 theorems, 184 equations, 1 figure, 1 table)

This paper contains 25 sections, 6 theorems, 184 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem set-up and main result
  3. Formal algorithm
  4. Hierarchy of equations and formal lemma
  5. Dirichlet boundary condition
  6. Neumann boundary conditions
  7. Construction of the (infinitely many) pair(s) $\left(u^{[k]},w^{[k]}\right)$.
  8. Remainders
  9. Resolvability defect $\mathcal{R}^{[k]}$
  10. Dirichlet boundary defect $\mathcal{E}_{D}^{[k]}$
  11. Neumann boundary defect $\mathcal{E}_{N}^{[k]}$
  12. Quantitative estimates
  13. Estimates on the known terms
  14. Nonlinear Forcing
  15. Dirichlet Boundary condition
  16. ...and 10 more sections

Key Result

Theorem 2.2

Suppose $g \in \mathfrak{H}_{\rho,\sigma;\mu}$ and satisfying Then there exists $\mu_0>0$ and infinitely many $w^* \in \mathfrak{H}_{\rho/2,\sigma/2;\mu}$ such that, for each $w^*$, one can determine a unique $u^* \in \mathfrak{H}_{\rho/2,\sigma/2;\mu}$ such that the pair $(u^*,w^*)$ satisfies the following bounds for all $\mu \in (0,\mu_0]$ and some $O(1)$ real constants $\mathscr{C}_j$, i.e. i

Figures (1)

  • Figure 1: In panels (a) and (b), the solution $u^{[2]}(r,\theta)$ plotted for $\mu=0.25$ and $\mu=0.5$, respectively. Panel (c) shows the plot of $\max_{\theta \in \mathbb{T}}\left| u^{[2]} (1+\mu g(\theta), \theta)\right|_{}$ as a function of $\mu$, while the behaviour of $\max_{\theta \in \mathbb{T}}\left| \partial_{\nu} u^{[2]} (1+\mu g(\theta), \theta)\right|_{}-1/2$ is shown in panel (d).

Theorems & Definitions (21)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1: Formal
  • Example 3.2
  • Remark 3.1
  • ...and 11 more