Ellipsoidal Positivity Sets for Fractional Obstacle Problems with Quadratic Forcing

Let \(n\ge1\), \(00\), 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 \