A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group

1999 ◽  
Vol 6 (2) ◽  
pp. 191-206 ◽  
Author(s):  
Francesco Uguzzoni
Keyword(s):  
Dirichlet Problem ◽  
Critical Exponent ◽  
Heisenberg Group ◽  
Existence Theorem

scholarly journals The Dirichlet Problem for elliptic equations in unbounded domains of the plane

2008 ◽  
Vol 6 (1) ◽  
pp. 47-58 ◽  
Author(s):  
Paola Cavaliere ◽  
Maria Transirico
Keyword(s):  
Dirichlet Problem ◽  
Existence Theorem ◽  
Elliptic Equations ◽  
Unbounded Domains ◽  
Suitable Condition ◽  
Second Order ◽  
Order Linear ◽  
Uniqueness And Existence

In this paper we prove a uniqueness and existence theorem for the Dirichlet problem inW2,pfor second order linear elliptic equations in unbounded domains of the plane. Here the leading coefficients are locally of classVMOand satisfy a suitable condition at infinity.

The polynomial growth solutions to some sub-elliptic equations on the Heisenberg group

2019 ◽  
Vol 21 (01) ◽  
pp. 1750069 ◽  
Author(s):  
Hairong Liu ◽  
Tian Long ◽  
Xiaoping Yang
Keyword(s):  
Dirichlet Problem ◽  
Heisenberg Group ◽  
Elliptic Equations ◽  
Polynomial Growth ◽  
Bounded Solution ◽  
Type Theorem ◽  
Divergence Form ◽  
Liouville Type ◽  
Periodic Coefficients ◽  
Liouville Type Theorem

We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [Formula: see text]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [Formula: see text]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be constant.

Polyharmonic Dirichlet problem on the Heisenberg group

2008 ◽  
Vol 53 (12) ◽  
pp. 1103-1110 ◽  
Author(s):  
Ajay Kumar ◽  
Mukund Madhav Mishra
Keyword(s):  
Dirichlet Problem ◽  
Heisenberg Group

The Dirichlet Problem for the Subelliptic Laplacian on the Heisenberg Group II

1985 ◽  
Vol 37 (4) ◽  
pp. 760-766 ◽  
Author(s):  
Bernard Gaveau ◽  
Jacques Vauthier
Keyword(s):  
Dirichlet Problem ◽  
Continuous Function ◽  
Heisenberg Group ◽  
Open Subset ◽  
Martingale Problem ◽  
Three Dimensions ◽  
True Solution ◽  
Group Ii ◽  
Image Position

Let H3 be the Heisenberg group in three dimensions, Δ the fundamental subelliptic laplacian on H3 (see Section 1 for notations and definitions) and U be an open subset of H3 If φ is a continuous function on the boundary ∂U of U, the Dirichlet problem is thus,(1)In [3], p. 104, it was asserted by the first author that, when dU is regular (see Section 1 for this definition), the problem (1) has a solution continuous on D and a probabilistic formula was given. In [3], we prove that our probabilistic formula gives a solution of the so called “martingale problem” associated to Δ on U (see [5] for this notion). But it appears that the connection between the solution in the martingale problem sense and the true solution is not at all clear in the subelliptic case; in particular it is not obvious at all that the probabilistic formula is a C2 function.

scholarly journals Nonexistence results of solutions to systems of semilinear differential inequalities on the Heisenberg group

2004 ◽  
Vol 2004 (2) ◽  
pp. 155-164 ◽  
Author(s):  
Abdallah El Hamidi ◽  
Mokhtar Kirane
Keyword(s):  
Critical Exponent ◽  
Heisenberg Group ◽  
Critical Exponents ◽  
Differential Inequalities

We establish nonexistence results to systems of differential inequalities on the(2N+1)-Heisenberg group. The systems considered here are of the type(ESm). These nonexistence results hold forNless than critical exponents which depend onpiandγi,1≤i≤m. Our results improve the known estimates of the critical exponent.

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