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Hypoellipticity for infinitely degenerate quasilinear equations and the Dirichlet problem. (English) Zbl 1310.35081

Summary: In [C. Rios et al., Differ. Integral Equ. 21, No. 1–2, 131–200 (2008; Zbl 1224.35149)], we considered a class of infinitely degenerate quasilinear equations of the form div \(A(x,\omega)\nabla\omega+\overrightarrow\gamma(x,\omega)\cdot\nabla\omega+f(x,\omega)=0\) and derived a priori bounds for high order derivatives \(D^\alpha\omega\) of their solutions in terms of \(\omega\) and \(\omega\). We now show that it is possible to obtain bounds in terms of just \(\omega\) 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.

MSC:

35H10 Hypoelliptic equations
35J62 Quasilinear elliptic equations
35J70 Degenerate elliptic equations
35D35 Strong solutions to PDEs

Citations:

Zbl 1224.35149
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References:

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