Abstract
Suppose \(\mathfrak {X} = \{X_1, X_2, \ldots,\,X_m\}\) is a system of real smooth vector fields on an open neighbourhood Ω of the closure of a bounded connected open set M in \(\mathbb {R}^N\) satisfying the finite rank condition of Hörmander, namely the rank of the Lie algebra generated by \(\mathfrak {X}\) under the usual bracket operation is a constant equal to N. We study the smoothness of solutions of a class of quasilinear equations of the form
where \(a_j,\,b \in C^{\infty}(\Omega \times \mathbb {R} \times \mathbb {R}^m; \mathbb {R})\). It is shown that if the matrix \(\left({\frac {\partial a_j}{\partial \xi_i}}\right)\) is positive definite on \(M \times \mathbb {R}^{m+1}\) then any weak solution \(u \in \mathcal {C}^{2,\alpha}(M, \mathfrak {X})\) belongs to C ∞(M).
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Dedicated to the memory of Sergio Companato.
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Murthy, M.K.V. A class of subelliptic quasilinear equations. J Glob Optim 40, 245–260 (2008). https://doi.org/10.1007/s10898-007-9214-5
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DOI: https://doi.org/10.1007/s10898-007-9214-5