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Green functions of mixed boundary value problems for stationary Stokes systems in two dimensions

Jongkeun Choi, Minsuk Yang

Abstract

We establish the existence, uniqueness, and various estimates for Green functions of mixed Dirichlet-conormal derivative problems for the stationary Stokes system with measurable coefficients in a two-dimensional Reifenberg flat domain with a rough separation.

Green functions of mixed boundary value problems for stationary Stokes systems in two dimensions

Abstract

We establish the existence, uniqueness, and various estimates for Green functions of mixed Dirichlet-conormal derivative problems for the stationary Stokes system with measurable coefficients in a two-dimensional Reifenberg flat domain with a rough separation.
Paper Structure (8 sections, 13 theorems, 116 equations)

This paper contains 8 sections, 13 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Mixed boundary value problem and Green function
  4. Main result
  5. Auxiliary results
  6. Embeddings
  7. Solvability
  8. Proof of Theorem \ref{['M1']}

Key Result

Theorem 2.4

Let $\Omega$ be a bounded Reifenberg flat domain in $\mathbb{R}^2$ satisfying Assumptions A1 and A2. Then there exist Green functions $(G, \Pi)$ and $(G^*, \Pi^*)$ for $\mathcal{L}$ and $\mathcal{L}^*$, respectively, such that and that for all $x,y\in \Omega$ with $x\neq y$. Moreover, the following estimates hold. In the above, $(C, q_0)=(C, q_0)(\lambda, R_0, \kappa, \operatorname{diam}(\Omega

Theorems & Definitions (26)

  • Definition 2.1
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 16 more