Minimizing eigenvalues of the fractional Laplacian

Abstract We study the minimizers of $$\lambda _k^s(A) + |A|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>λ</mml:mi> <mml:mi>k</mml:mi> <mml:mi>s</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> where $$\lambda ^s_k(A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>λ</mml:mi> <mml:mi>k</mml:mi> <mml:mi>s</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is the k -th Dirichlet eigenvalue of the fractional Laplacian on A . Unlike in the case of the Laplacian, free boundary of minimizers exhibits distinct global behaviors. Our main results include: the existence of minimizers, optimal Hölder regularity for the corresponding eigenfunctions, and in the case where $$\lambda _k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> is simple, non-degeneracy, density estimates, separation of the free boundary, and free boundary regularity. We propose a combinatorial toy problem related to the global configuration of such minimizers.