Ellipsoidal Positivity Sets for Fractional Obstacle Problems with Quadratic Forcing

Abstract: Let (n\ge1), (0<s\<1\), \(c\>0), and let (A) be a positive definite symmetric matrix. We prove that the unique decaying viscosity solution of [ \min{u,\,(-Δ)^s u-(c-\langle Ax,x\rangle)}=0 \qquad\text{in }\mathbb R^n ] has the form [ u(x)=K\max{1-\langle Bx,x\rangle,0}^{1+s} ] for some (K>0) and some positive definite symmetric matrix (B). In particular, its positivity set is an ellipsoid. For (s=1/2) and (n\ge2), this result, combined with the classification of Fernández-Real and Yu, implies that cubic global thin obstacle solutions with nonempty bounded positivity set have ellipsoidal positivity sets. This proves the conjecture of Fernández-Real and Yu in the nonempty bounded-positivity case.