Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributions in spaces with thick submanifolds

Jiajia Ding, Jasson Vindas, Yunyun Yang

TL;DR

This work extends the theory of thick distributions from point and curve singularities to general thick submanifolds $\Sigma\subset\mathbb{R}^n$ by introducing test-function spaces $\mathcal{D}_{\ast,\Sigma}(\Omega)$ and $\mathcal{E}_{\ast,\Sigma}(\Omega)$ based on strong tubular expansions. It defines thick distributions $\mathcal{D}'_{\ast,\Sigma}(\Omega)$ as the dual of $\mathcal{D}_{\ast,\Sigma}(\Omega)$ and develops a robust calculus with thick derivatives, a natural projection $\Pi$ to Schwartz distributions, and a Hadamard-type finite-part regularization $\operatorname{Pf}$ to embed $\mathcal{E}_{\ast,\Sigma}(\Omega)$ into thick distributions. The paper constructs explicit thick delta distributions $\delta_{\ast,\Sigma}^{[j]}$ and multilayers, and derives derivative formulas for these objects, all in tubular coordinates with normal frames. This general framework enables rigorous modeling of singularities along submanifolds in PDEs and physics, providing new tools for analytic techniques where classical distributions are inadequate, including well-defined normal-type derivatives and curvature terms.

Abstract

We present the construction of a theory of distributions (generalized functions) with a ``thick submanifold'', that is, a new theory of thick distributions on $\mathbb{R}^n$ whose domain contains a smooth submanifold on which the test functions may be singular. We define several operations, including ``thick partial derivatives'', and clarify their connection with their classical counterparts in Schwartz distribution theory. We also introduce and study a number of special thick distributions, including new thick delta functions, or more generally thick multilayer distributions along a submanifold.

Distributions in spaces with thick submanifolds

TL;DR

This work extends the theory of thick distributions from point and curve singularities to general thick submanifolds by introducing test-function spaces and based on strong tubular expansions. It defines thick distributions as the dual of and develops a robust calculus with thick derivatives, a natural projection to Schwartz distributions, and a Hadamard-type finite-part regularization to embed into thick distributions. The paper constructs explicit thick delta distributions and multilayers, and derives derivative formulas for these objects, all in tubular coordinates with normal frames. This general framework enables rigorous modeling of singularities along submanifolds in PDEs and physics, providing new tools for analytic techniques where classical distributions are inadequate, including well-defined normal-type derivatives and curvature terms.

Abstract

We present the construction of a theory of distributions (generalized functions) with a ``thick submanifold'', that is, a new theory of thick distributions on whose domain contains a smooth submanifold on which the test functions may be singular. We define several operations, including ``thick partial derivatives'', and clarify their connection with their classical counterparts in Schwartz distribution theory. We also introduce and study a number of special thick distributions, including new thick delta functions, or more generally thick multilayer distributions along a submanifold.
Paper Structure (12 sections, 16 theorems, 79 equations, 1 figure)

This paper contains 12 sections, 16 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The tubular neighborhood theorem
  4. $\delta$-derivatives
  5. Strong asymptotic expansions
  6. The space of test functions
  7. Distributions with a thick submanifold
  8. Examples of thick distributions along a submanifold
  9. Finite part $\Sigma$-thick distributions
  10. Thick deltas and multilayers along $\Sigma$
  11. Partial derivatives of some $\Sigma$-thick distributions
  12. Conclusion

Key Result

Lemma 2.1

There is a continuous function $\epsilon:\Sigma\to (0,\infty)$ such that every point $x\in \Sigma^{\epsilon}$ possesses a unique closest pointThat is, a unique point $\pi(x)\in \Sigma$ such that $\operatorname*{dist}(x,\Sigma)=\|x-\pi(x)\|$.$\pi(x)$ to the submanifold $\Sigma$ and the retraction $\p

Figures (1)

  • Figure 1: Local tube coordinates $x=\xi+ \rho \omega_1 \mathbf{n}_1 +\rho \omega_2 \mathbf{n}_2$ for a curve $\Sigma$ in $\mathbb{R}^3$

Theorems & Definitions (36)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • Theorem 3.5
  • ...and 26 more