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Regularity properties of viscosity solutions of a non-Hörmander degenerate equation. (English) Zbl 1026.35024

In the paper the authors obtain regularity results of viscosity solutions of the following nonlinear degenerate equation \[ {\L}u=f \quad \text{in } S_T \equiv \mathbb{R}^2 \times ]0,T[ \] where \(T\) is sufficiently small and the nonlinear operator \({\L}\) is defined by \[ {\L}u= \partial_{xx}u+ u \partial_y u-\partial_t u \] where \(z=(x,y,t)\) is the generic point belongs to \(\mathbb{R}^3.\) The authors, using geometric properties of some Hörmander operator naturally associated to \({\L},\) prove that the viscosity solutions of \({\L}u=f\) are classical solutions.
Also, there are established some properties of \(u_x\) that, with a propagation principle, allows the authors to obtain a sufficient condition to have different from zero the partial derivative \(\partial_x u(z)\), for all \(z \in \Omega\), where \(\Omega\) is an open bounded set in \(\mathbb{R}^3.\)

MSC:

35D10 Regularity of generalized solutions of PDE (MSC2000)
35K55 Nonlinear parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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