Korn and Poincaré-Korn inequalities for functions with a small jump set
Filippo Cagnetti, Antonin Chambolle, Lucia Scardia
Abstract
Abstract In this paper we prove a regularity and rigidity result for displacements in $$GSBD^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mi>S</mml:mi><mml:mi>B</mml:mi><mml:msup><mml:mi>D</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:mrow></mml:math> , for every $$p>1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> and any dimension $$n\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> . We show that a displacement in $$GSBD^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mi>S</mml:mi><mml:mi>B</mml:mi><mml:msup><mml:mi>D</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:mrow></mml:math> with a small jump set coincides with a $$W^{1,p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>W</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math> function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in $$GSBD^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mi>S</mml:mi><mml:mi>B</mml:mi><mml:msup><mml:mi>D</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:mrow></mml:math> .