Weighted asymptotic Korn and interpolation Korn inequalities with singular weights

<p>In this work we derive asymptotically sharp weighted Korn and Korn-like interpolation (or first and a half) inequalities in thin domains with singular weights. The constants <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (Korn’s constant) in the inequalities depend on the domain thickness <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> according to a power rule <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper K equals upper C h Superscript alpha Baseline comma"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>h</mml:mi> <mml:mi>α<! -- α --></mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K=Ch^\alpha ,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper C greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">C>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="alpha element-of upper R"> <mml:semantics> <mml:mrow> <mml:mi>α<! -- α --></mml:mi> <mml:mo>∈<! -- ∈ --></mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \in R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are constants independent of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the displacement field. The sharpness of the estimates is understood in the sense that the asymptotics <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="h Superscript alpha"> <mml:semantics> <mml:msup> <mml:mi>h</mml:mi> <mml:mi>α<! -- α --></mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">h^\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is optimal as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="h right-arrow 0 period"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo stretchy="false">→<! -- → --></mml:mo> <mml:mn>0. </mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">h\to 0. </mml:annotation> </mml:semantics> </mml:math> </inline-formula> The choice of the weights is motivated by several factors; in particular a spatial case occurs when making Cartesian to polar change of variables in two dimensions. </p>