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Register a new userOn the Dirichlet Problem for hypoelliptic evolution equations: Perron-Wiener solution and a cone-type criterion
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Authors
Alessia E. Kogoj
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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.
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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 elementary procedure, the Perron solution to the caloric Dirichlet problem on every bounded domain.
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-Gariepys Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials.
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 discussed. Special attention is paid to possibly coercive, resonant, subcritical, critical, or asymmetric right-hand sides.
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 form and satisfies the Olev{i}nik--Radkeviv{c} rank hypoellipticity condition. These hypotheses allow the construction of Perron-Wiener solutions of the Dirichlet problems for $mathscr{L}$ and $mathscr{L}_0$ on bounded open subsets of $mathbb R^{N+1}$ and of $mathbb R^{N}$, respectively. Our main result is the following Tikhonov-type theorem: Let $mathcal{O}:= Omega times ]0, T[$ be a bounded cylindrical domain of $mathbb R^{N+1}$, $Omega subset mathbb R^{N},$ $x_0 in partial Omega$ and $0 < t_0 < T.$ Then $z_0 = (x_0, t_0) in partial mathcal{O}$ is $mathscr{L}$-regular for $mathcal{O}$ if and only if $x_0$ is $mathscr{L}_0$-regular for $Omega$. As an application, we derive a boundary regularity criterion for degenerate Ornstein--Uhlenbeck operators.
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 $tgeq 0$, $xin{mathbb R}^n$ ($nge 3$), $minmathbb N$, $p>1$, and $u_iin C_0^{infty}({mathbb R}^n)$ ($i=0,1$). We show that there exists a critical exponent $p_{text{crit}}(m,n)>1$ such that the solution $u$, in general, blows up in finite time when $1<p<p_{text{crit}}(m,n)$. We further show that there exists a conformal exponent $p_{text{conf}}(m,n)> p_{text{crit}}(m,n)$ such that the solution $u$ exists globally when $p>p_{text{conf}}(m,n)$ provided that the initial data is small enough. In case $p_{text{crit}}(m,n)<pleq p_{text{conf}}(m,n)$, we will establish global existence of small data solutions $u$ in a subsequent paper.
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