On Convexity and Starshapedness of Level Sets for Some Nonlinear Elliptic and Parabolic Problems on Convex Rings

We consider some degenerate parabolic problems on a convex (or starshaped) ring. We prove that if the initial data have convex (or starshaped) level sets, then the solution u(t, •) has the same property for any positive t. Similar results are shown for the corresponding stationary problems. Our results imply in particular the convexity (or starshapedness) of certain free boundaries. Other nonlinear parabolic problems are also discussed.