Convexity of the free boundary for an exterior free boundary problem\n involving the perimeter

2010/10/14 by Hayk Mikayelyan, Henrik Shahgholian, Mikayelyan, Hayk +1
Mathematics · #Point processes and geometric inequalities #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1010.2960

Abstract

We prove that if the given compact set K is convex then a minimizer of the\nfunctional I(v)=
int_BR |
nabla v|^p dx+
textPer(
vgt;0
),
,1lt;plt;
infty,\n over the set v\∈ H¹₀(B_R)| , , v\≡ 1 , ,\on , , K\⊂\nB_R has a convex support, and as a result all its level sets are convex as\nwell. We derive the free boundary condition for the minimizers and prove that\nthe free boundary is analytic and the minimizer is unique.\n

Related

• Two-phase free boundary problems in convex domains
• Analytic regularity of a free boundary problem
• Free-Boundary Monotonicity for Almost-Minimizers of the Relative Perimeter
• Convexity for free boundaries with singular term (nonlinear elliptic case)
• Isoperimetric inequalities in Euclidean convex bodies