On a variational problem of nematic liquid crystal droplets
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Qinfeng Li
Abstract
Let μ > 0 be a fixed constant, and we prove that minimizers to the following energy functional: E(u, Ω) := ∫ Ω |∇u|2 + μP(Ω) exist among pairs (Ω, u) such that Ω is an M-uniform domain with finite perimeter and fixed volume, and u ∈ H1(Ω, S2) with u = vΩ, the measure-theoretical outer unit normal, almost everywhere on the reduced boundary of Ω. The uniqueness of optimal configurations in various settings is also obtained. In addition, we consider a general energy functional given by Ef (u, Ω) := ∫ Ω ∇u(x) 2 dx + ∫ ð Ω f (u(x) ⋅ vΩ(x)) dH2(x), where ð ∗Ω is the reduced boundary of Ω and f is a convex positive function on R. We prove that minimizers of Ef also exist among M-uniform outer-minimizing domains Ω with fixed volume and u ∈ H1(Ω, S2).
Abstract
Let μ > 0 be a fixed constant, and we prove that minimizers to the following energy functional: E(u, Ω) := ∫ Ω |∇u|2 + μP(Ω) exist among pairs (Ω, u) such that Ω is an M-uniform domain with finite perimeter and fixed volume, and u ∈ H1(Ω, S2) with u = vΩ, the measure-theoretical outer unit normal, almost everywhere on the reduced boundary of Ω. The uniqueness of optimal configurations in various settings is also obtained. In addition, we consider a general energy functional given by Ef (u, Ω) := ∫ Ω ∇u(x) 2 dx + ∫ ð Ω f (u(x) ⋅ vΩ(x)) dH2(x), where ð ∗Ω is the reduced boundary of Ω and f is a convex positive function on R. We prove that minimizers of Ef also exist among M-uniform outer-minimizing domains Ω with fixed volume and u ∈ H1(Ω, S2).
Chapters in this book
- Frontmatter I
- Foreword V
- Contents VII
- Yau’s conjecture on the dimension of harmonic polynomials 1
- On Carrasco Piaggio’s theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces 23
- On trace theorems for weighted mixed-norm Sobolev spaces and applications 49
- Sharp stability of the logarithmic Sobolev inequality in the critical point setting 77
- On a variational problem of nematic liquid crystal droplets 103
- Estimates for the variable order Riesz potential with applications 127
- A remark on the atomic decomposition in Hardy spaces based on the convexification of ball Banach spaces 157
- A Bliss–Adams inequality 179
- Trudinger-type inequalities in RN with radial increasing mass-weight 197
- In response to David R. Adams’ October 12, 2001, letter 215
- Some remarks on capacitary integrals and measure theory 235
- Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights 265
- Index 287
Chapters in this book
- Frontmatter I
- Foreword V
- Contents VII
- Yau’s conjecture on the dimension of harmonic polynomials 1
- On Carrasco Piaggio’s theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces 23
- On trace theorems for weighted mixed-norm Sobolev spaces and applications 49
- Sharp stability of the logarithmic Sobolev inequality in the critical point setting 77
- On a variational problem of nematic liquid crystal droplets 103
- Estimates for the variable order Riesz potential with applications 127
- A remark on the atomic decomposition in Hardy spaces based on the convexification of ball Banach spaces 157
- A Bliss–Adams inequality 179
- Trudinger-type inequalities in RN with radial increasing mass-weight 197
- In response to David R. Adams’ October 12, 2001, letter 215
- Some remarks on capacitary integrals and measure theory 235
- Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights 265
- Index 287
