| Abstract |
"We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form inhomogeneous equations, and also continuity of weak solutions to homogeneous equations. For example, we consider the family {f[sigma]}[sigma]>0 with f[sigma] (x) = e -( 1 [pipe]x[pipe] ) [sigma] , -[infinity] <x< [infinity], of infinitely degenerate functions at the origin, and show that all weak solutions to the associated infinitely degenerate quasilinear equations of the form divA (x, u) gradu = [phi] (x), A (x, z) ̃̃ In-1 0 0 f (x1) 2 , with rough data A and [phi], are locally bounded for admissible [phi] provided 0 <[sigma]< 1. We also show that these conditions are necessary for local boundedness in dimension n [greater than or equal to] 3, thus paralleling the known theory for the smooth Kusuoka-Strook operators [partial derivative]2 [partial derivative]x2 1 + [partial derivative]2 [partial derivative]x2 2 +f[sigma] (x) 2 [partial derivative]2 [partial derivative]x2 3 . We also show that subsolutions satisfy a maximum principle under the same restriction on the degeneracy. Finally, continuity of solutions is derived in the homogeneous case [phi] [triple bar] 0 under a more stringent assumption on the degeneracy, namely that f [greater than or equal to] f3,[sigma] for 0 <[sigma]< 1 where f3,[sigma] (x) = [pipe]x[pipe] (ln ln ln 1 [pipe]x[pipe] ) [sigma] , -[infinity] <x< [infinity]. As an application we obtain weak hypoellipticity (i.e. smoothness of all weak solutions) of certain infinitely degenerate quasilinear equations in the plane [partial derivative]u [partial derivative]x2 + f (x, u (x, y))2 [partial derivative]u [partial derivative]y2 = 0, with smooth data f (x, z) f3,[sigma] (x) where f (x, z) has a sufficiently mild nonlinearity and degeneracy. In order to prove these theorems, we first establish abstract results in which certain Poincarè and Orlicz Sobolev inequalities are assumed to hold. We then develop subrepresentation inequalities for control geometries in order to obtain the needed Poincarè and Orlicz Sobolev inequalities"- Provided by publisher. |
