# zbMATH — the first resource for mathematics

The Dirichlet problem for sublaplacians on nilpotent Lie groups - geometric criteria for regularity. (English) Zbl 0601.31007
Given a sublaplacian L on a stratified Lie algebra $${\mathcal N}$$ of type I, geometric criteria for the regularity (irregularity resp.) of open subsets are given. Assuming that $${\mathcal N}$$ is of step r and that $${\mathcal N}$$ is identified with the underlying space $${\mathbb{R}}^ m$$ such an operator is of the form $$L=X^ 2_ 1+...+X^ 2_ n$$ with smooth vectorfields $$X_ i$$ which together with their commutators up to order r span the tangent space at each point of $${\mathbb{R}}^ m$$. In terms of Euclidean geometry the following results are obtained: If $$n=2$$ and $$m>2$$ then an exterior (r/2-$$\epsilon)$$-Hölder condition is not sufficient for U to be regular. Whenever $$n\geq 3$$ and $$m>3$$ even an exterior (r- $$\epsilon)$$-Hölder condition is not sufficient. The only cases where all bounded domains with smooth boundary are regular are the cases $$r\leq 2$$ and the few cases where $$n=2$$ and $$r\leq 4$$. The results generalize the well known cone conditions for the classical Laplacian.

##### MSC:
 31D05 Axiomatic potential theory 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 22E25 Nilpotent and solvable Lie groups
##### Keywords:
sublaplacian; stratified Lie algebra; regularity
Full Text:
##### References:
 [1] Bauer, H.: Harmonic spaces and associated Markov processes. In: Potential theory (CIME, 1{$$\deg$$} Ciclo, Stresa 1969), Ed. Cremonese, 23-67 (1970) [2] Bliedtner, J., Hansen, W.: Potential theory ? An analytic and probabilistic approach to Balayage. Universitext. Berlin, Heidelberg, New York: Springer 1986 · Zbl 0706.31001 [3] Bony, J.M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier19, 277-304 (1969) · Zbl 0176.09703 [4] Friedman, A., Pinsky, M.A.: Dirichlet problem for degenerate elliptic equations. Trans. AMS186, 359-383 (1973) · Zbl 0274.35026 [5] Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents. Acta math.139, 96-153 (1977) · Zbl 0366.22010 [6] Gaveau, B., Vauthier, J.: The Dirichlet problem for the subelliptic laplacian on the Heisenberg group. II. Can. J. Math.37, 760-766 (1985) · Zbl 0578.22009 [7] Hervé, R.-M., Hervé, M.: Les fonctions surharmoniques dans l’axiomatique de M. Brelot associées à un opérateur elliptique dégénéré. Ann. Inst. Fourier22, 131-145 (1972) · Zbl 0224.31014 [8] Hueber, H.: Wiener criterion in potential theory with applications to nilpotent Lie groups. Math. Z.190, 527-542 (1985) · Zbl 0585.31004 [9] Hueber, H.: Examples of irregular domains for some hypoelliptic differential operators. BIBOS Preprint Nr. 107, Univ. Bielefeld (1985) · Zbl 0597.58036 [10] Hueber, H.: Further examples of some irregular domains for some hypoelliptic differential operators. BIBOS Preprint Nr. 114, Univ. Bielefeld (1985) · Zbl 0597.58036 [11] Landkof, N.S.: Foundations of Modern Potential Theory. Grundl. d. math. Wiss. 180. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0253.31001 [12] Maeda, F.Y.: Energy of functions on a self-adjoint harmonic space. I. Hiroshima Math. J.2, 313-337 (1972) · Zbl 0273.31015 [13] Maeda, F.Y.: Energy of functions on a self-adjoint harmonic space. II. Hiroshima Math. J.3, 37-60 (1973) · Zbl 0273.31016 [14] Manankiandrianana, D.: Comportement du noyau de la chaleur d’un opérateur hypoelliptique dégénéré pour des temps petits. C.R. Acad. Sc. Paris288, Sér. A, 1061-1063 (1979) · Zbl 0424.35029 [15] Negrini, P., Scornazzani, V.: Wiener criterion for a class of degenerate elliptic operators. Preprint · Zbl 0633.35018 [16] Rothschild, L.P., Stein, E.M.: Hypoelliptic differential operators and nilpotent Lie groups. Acta math.137, 247-320 (1977) · Zbl 0346.35030 [17] Sanchez-Calle, A.: Fundamental solutions and geometry of the sum of squares of vector fields. Invent. Math.78, 143-160 (1984) · Zbl 0582.58004 [18] Strook, D., Taniguchi, S.: Regular points for the first boundary value problem associated with degenerate elliptic operators. Preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.