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The Minkowski problem for the $k$-torsional rigidity

Xia Zhao, Peibiao Zhao

Abstract

P. Salani [Adv. Math., 229 (2012)] introduced the $k$-torsional rigidity associated with a $k$-Hessian equation and obtained the Brunn-Minkowski inequalities $w.r.t.$ the torsional rigidity in $\mathbb{R}^3$. Following this work, we first construct, in the present paper, a Hadamard variational formula for the $k$-torsional rigidity with $1\leq k\leq n-1$, then we can deduce a $k$-torsional measure from the Hadamard variational formula. Based on the $k$-torsional measure, we propose the Minkowski problem for the $k$-torsional rigidity and confirm the existence of its smooth non-even solutions by the method of a curvature flow. Specially, a new proof method for the uniform lower bound estimation in the $C^0$ estimation for the solution to the curvature flow is presented with the help of invariant functional $Φ(Ω_t)$.

The Minkowski problem for the $k$-torsional rigidity

Abstract

P. Salani [Adv. Math., 229 (2012)] introduced the -torsional rigidity associated with a -Hessian equation and obtained the Brunn-Minkowski inequalities the torsional rigidity in . Following this work, we first construct, in the present paper, a Hadamard variational formula for the -torsional rigidity with , then we can deduce a -torsional measure from the Hadamard variational formula. Based on the -torsional measure, we propose the Minkowski problem for the -torsional rigidity and confirm the existence of its smooth non-even solutions by the method of a curvature flow. Specially, a new proof method for the uniform lower bound estimation in the estimation for the solution to the curvature flow is presented with the help of invariant functional .

Paper Structure

This paper contains 9 sections, 12 theorems, 183 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Convex hypersurface
  4. Symmetric functions and Hessian operators
  5. Variational formula and associated functionals of the geometric flow
  6. A priori estimates
  7. $C^0$, $C^1$ estimates
  8. $C^2$ estimate
  9. The convergence of the flow

Key Result

Theorem 1.2

Let $1\leq k\leq n-1$, $u(x,t)$ be a smooth admissible solution of (eq101) in $\Omega_t$ and $\partial\Omega_0$ be a smooth, closed and strictly convex hypersurface in $\mathbb{R}^n$ containing the origin in its interior, $f$ is a positive smooth function on $S^{n-1}$. Then the flow (eq110) has an u

Theorems & Definitions (23)

  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • ...and 13 more