Constraint Maps with Free Boundaries: the Obstacle Case

Abstract This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove various properties for each component. As has already been pointed out in the literature, the distance part falls under the classical obstacle problem, which is well-studied by classical methods. A perplexing issue, untouched in the literature, concerns the properties of the projected image and its higher regularity, which we show to be at most of class $$C^{2,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . In arbitrary dimensions, we prove that the image map is globally of class $$W^{3,BMO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mi>M</mml:mi> <mml:mi>O</mml:mi> </mml:mrow> </mml:msup> </mml:math> , and locally of class $$C^{2,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> around the regular part of the free boundary. The issue becomes more delicate around singular points, and we resolve it in two dimensions. In the appendix, we extend some of our results to what we call leaky maps.