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An optimal boundary control approach to the Cherrier-Escobar problem

Cheikh Birahim Ndiaye, Abdul-Malik Saiid

TL;DR

The paper studies an optimal boundary control problem for the boundary obstacle problem associated with the conformal Laplacian and conformal Robin operator on compact manifolds with boundary, under a positive Cherrier-Escobar invariant. It constructs and analyzes the state map $T_h$, proves its idempotence and positive homogeneity, and establishes monotonicity formulas for the Cherrier-Escobar energy and its boundary control variant, leading to rigidity that minimizers lie in the fixed-point set of $T_h$. By comparing ground-state energies, it shows equivalence between the variational problems $\mathcal{E}^h_p$ and $\mathcal{I}^h_p$ and proves existence of an optimal boundary control under an Aubin-type assumption; on the unit ball, it obtains a sharp Sobolev trace inequality and proves that standard bubbles are the only optimizers and equal to their associated optimal states. Collectively, these results connect optimal control of the boundary obstacle problem to the Cherrier-Escobar framework, yielding conformal metrics with zero scalar curvature and constant mean curvature, and provide sharp characterizations in the symmetric Ball setting.

Abstract

We study an optimal boundary control problem associated to the boundary obstacle problem for the couple conformal Laplacian and conformal Robin operator on n-dimensional compact Riemannian manifolds with boundary and with n\geq 3. When the Cherrier-Escobar invariant of the compact Riemannian manifold with boundary is positive, we show that the optimal controls are equal to their associated optimal states. Moreover, we show that the optimal controls are minimizers of the Cherrier-Escobar functional, and hence induce conformal metrics with zero scalar curvature and constant mean curvature. Furthermore, we show the existence of an optimal control under an Aubin type assumption. For the standard unit ball, we derive a sharp Sobolev trace type inequality and prove that the standard bubbles-namely conformal factor of metrics conformal to the standard one with zero scalar curvature and constant mean curvature -- are the only optimal controls and hence equal to their associated optimal states.

An optimal boundary control approach to the Cherrier-Escobar problem

TL;DR

The paper studies an optimal boundary control problem for the boundary obstacle problem associated with the conformal Laplacian and conformal Robin operator on compact manifolds with boundary, under a positive Cherrier-Escobar invariant. It constructs and analyzes the state map , proves its idempotence and positive homogeneity, and establishes monotonicity formulas for the Cherrier-Escobar energy and its boundary control variant, leading to rigidity that minimizers lie in the fixed-point set of . By comparing ground-state energies, it shows equivalence between the variational problems and and proves existence of an optimal boundary control under an Aubin-type assumption; on the unit ball, it obtains a sharp Sobolev trace inequality and proves that standard bubbles are the only optimizers and equal to their associated optimal states. Collectively, these results connect optimal control of the boundary obstacle problem to the Cherrier-Escobar framework, yielding conformal metrics with zero scalar curvature and constant mean curvature, and provide sharp characterizations in the symmetric Ball setting.

Abstract

We study an optimal boundary control problem associated to the boundary obstacle problem for the couple conformal Laplacian and conformal Robin operator on n-dimensional compact Riemannian manifolds with boundary and with n\geq 3. When the Cherrier-Escobar invariant of the compact Riemannian manifold with boundary is positive, we show that the optimal controls are equal to their associated optimal states. Moreover, we show that the optimal controls are minimizers of the Cherrier-Escobar functional, and hence induce conformal metrics with zero scalar curvature and constant mean curvature. Furthermore, we show the existence of an optimal control under an Aubin type assumption. For the standard unit ball, we derive a sharp Sobolev trace type inequality and prove that the standard bubbles-namely conformal factor of metrics conformal to the standard one with zero scalar curvature and constant mean curvature -- are the only optimal controls and hence equal to their associated optimal states.
Paper Structure (8 sections, 24 theorems, 166 equations)

This paper contains 8 sections, 24 theorems, 166 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Notations and Preliminaries
  3. Boundary obstacle problem for the couple $(L_h, B_h)$.
  4. Conformal transformation rules of $T_{h}$ and $\mathcal{I}^{h}$.
  5. Monotonicity formula for $\mathcal{E}^h_p$, $h\in [g]$, and $1\leq p\leq 2^\#-1$.
  6. Monotonicity formula for $\mathcal{I}^h_p$, $h\in [g]$, $1\leq p\leq 2^\#-1$.
  7. Comparing $\mathcal{\mu}^p(M, \partial{M}, h)$ and $\mathcal{\mu}_{oc}^p(M,\partial{M}, h)$, $h\in[g]$, and $1\leq p\leq 2^\#-1$
  8. Obstacle problem and Sobolev trace type inequality

Key Result

Theorem 1.1

Assuming that $\mathcal{\mu}(M, \partial M, \;[g])>0$, then 1) For $u\in H^1_+(\overline{M}, g)$, with $g_u=u^{\frac{4}{n-2}}g$. 2) If there exists $u_{min} \in C^{\infty}_+(\overline{M}, g)$ such that $\mathcal{I}^g(u_{min})=\mathcal{\mu}_{oc}(M, \partial M, \;[g])$, then there exists also $u^{min}\in C^{\infty}_+(\overline{M}, g)$ such that with $g_{u^{min}}=(u^{min})^{\frac{4}{n-2}}g$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • ...and 22 more