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We show how to apply harmonic spaces potential theory in the study of the Dirichlet problem for a general class of evolution hypoelliptic partial differential equations of second order. We construct Perron-Wiener solution and we provide a sufficient condition for the regularity of the boundary points. Our criterion extends and generalizes the classical parabolic-cone criterion for the Heat equation due to Effros and Kazdan.
By an easy trick taken from caloric polynomial theory we construct a family $mathscr{B}$ of $almost regular$ domains for the caloric Dirichlet problem. $mathscr{B}$ is a basis of the Euclidean topology. This allows to build, with a basically elementa
We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Garie
A short account of recent existence and multiplicity theorems on the Dirichlet problem for an elliptic equation with $(p,q)$-Laplacian in a bounded domain is performed. Both eigenvalue problems and different types of perturbation terms are briefly di
We consider the linear second order PDOs $$ mathscr{L} = mathscr{L}_0 - partial_t : = sum_{i,j =1}^N partial_{x_i}(a_{i,j} partial_{x_j} ) - sum_{j=i}^N b_j partial_{x_j} - partial _t,$$and assume that $mathscr{L}_0$ has nonnegative characteristic fo
In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equation $partial_t^2 u-t^m Delta u=|u|^p$ with initial data $(u(0,cdot), partial_t u(0,cdot))= (u_0, u_1)$, where $tgeq 0$, $xin{mathbb R}^n$ ($nge