Abstract
In [10], we considered a class of infinitely degenerate quasilinear equations of the form div \(A(x,w)\nabla w + \overrightarrow r (x,w) \cdot \nabla w + f(x,w) = 0\) and derived a priori bounds for high order derivatives D a w of their solutions in terms of w and ▿w. We now show that it is possible to obtain bounds in terms of just w for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness, and interior \({\varrho ^\infty }\)-regularity of solutions for the corresponding Dirichlet problem with continuous boundary data.
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Rios, C., Sawyer, E.T. & Wheeden, R.L. Hypoellipticity for infinitely degenerate quasilinear equations and the dirichlet problem. JAMA 119, 1–62 (2013). https://doi.org/10.1007/s11854-013-0001-6
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DOI: https://doi.org/10.1007/s11854-013-0001-6