© 2015 European Mathematical Society. We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of 1pu D g.u/ in a smooth bounded domain Rn. In particular, we obtain new Lr and W1;r bounds for the extremal solution u when the domain is strictly convex. More precisely, we prove that u 2 L1./ if n p C 2 and u 2 L np np2 ./ \ W 1;p0 ./ if n p C 2.
- Geometric inequalities
- Mean curvature of level sets
- P-laplace equations
- Regularity of stable solutions
- Schwarz symmetrization