Let \(B_1\) be the unit ball in \(\mathbb R^n\) centered at the origin and \(Q_1=B_1 \times (-1, 1)\). the authors consider a free boundary problem in \(Q_1\) resembling the Stefan problem in the fact that the heat equation is satisfied on both sides of the free boundary (with different diffusivities) and the free boundary separates the negativity and the positivity set of the solution. However, the condition replacing Stefan’s heat balance is much more general, being expressed by \[ V_\nu =-G(x,t,\nu,v^+_\nu,v^-_\nu, \kappa), \] where \(V_\nu\) denotes the normal speed of the interface, \(\nu\) is the unit normal (pointing towards the positivity set), \(v^+, v^-\) are the normal derivatives of the solution of the differential equations on the positive and negative side of the interface, \(\kappa\) is the interface curvature. The function \(G\) is continuous, has the “correct” monotonicity with respect to \(v^+_\nu, v^-_\nu, \kappa\) and goes to infinity where any of these arguments goes to infinity.
Replacing equalities by suitable inequalities, the authors define super- and subsolutions. Viscosity super- and subsolutions (and viscosity solutions) are also defined: a viscosity supersolution \(u\) has the property that if \(u>v\) on \(\partial_p Q\), where \(v\) is any supersolution, \(Q\) any subcylinder of \(Q_1\), \(\partial_p Q\) its parabolic boundary, then \(u>v\) in \(Q\). The paper presents two main regularity results:
Let \(u\) be a viscosity solution in \(Q_1\). If the free boundary is Lipschitz in some space direction, then \(u\) is Lipschitz in \(Q_{1/2}\).
Under some additional assumptions on \(G\) the free boundary belongs to \(C^{1,\alpha}\) and \(u\) is \(C^{1,\beta}\) up to the free boundary on both sides.
As a consequence the boundedness of the mean curvature of the free boundary is established.