On Some Uniform Estimates of Gauge Functions with Respect to Domains
Abstract
<jats:p> We establish uniform estimates and properties of gauge functions for domains <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Omega_\varepsilon</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varepsilon\in[0,1]</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , defined by the Minkowski sum <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Omega_\varepsilon=\Omega_0+\varepsilon\Omega</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>ε</mml:mi> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> where <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Omega_0</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Omega</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> are convex and bounded subsets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{R}^n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . These estimates are in fact needed when one deals with shape derivatives in PDE-constrained shape optimization problems using this Minkowski sum as a deformation as it is done in a recent paper of A. Boulkhemair and A. Chakib [On a shape derivative formula with respect to convex domains, J. Convex Analysis 21/1 (2014) 67–87] for example. We first show that this class of domains <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Omega_\varepsilon</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> satisfies the so-called uniform ball property which is equivalent to the positiveness of its reach. Then, we establish the said uniform estimates on the gauge function of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Omega_\varepsilon</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and its gradient as well as its hessian, with respect to the parameter <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varepsilon</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . </jats:p>