914 research outputs found
Isoperimetric inequality in the Grushin plane
No full textIn this article, we prove a sharp isoperimetric inequality in the generalized Grushin plane depending on a parameter . For each we compute the corresponding isoperimetric sets. We also discuss the connection of the problem with the Heisenberg isoperimetric problem
John domains for the control distance of diagonal vector fields
No full textWe study John domains in the metric space associated with a system of diagonal vector fields
Levi umbilical surfaces in complex space
No full textWe define a complex connection on a real hypersurface of Cnþ1 which is
naturally inherited from the ambient space. Using a system of Codazzi-type equations, we
classify connected real hypersurfaces in Cnþ1, nf2, which are Levi umbilical and have non
zero constant Levi curvature. It turns out that such surfaces are contained either in a sphere
or in the boundary of a complex tube domain with spherical section
Pseudohermitian invariants and classification of CR mappings in generalized ellipsoids
No full textGiven a strictly pseudoconvex hypersurface M 1⁄2 Cn+1, we
discuss the problem of classifying all local CR diffeomorphisms between open
subsets N;N0 1⁄2 M. Our method exploits the Tanaka–Webster pseudohermitian
invariants of a contact form # on M, their transformation formulae,
and the Chern–Moser invariants. Our main application concerns a class of
generalized ellipsoids where we classify all local CR mappings
Kelvin transform for Grushin operators and semilinear critical equations
No full textWe study positive entire solutions u = u(x, y) of the critical equation
xu + (α + 1)2|x|2αyu = −u(Q+2)/(Q−2) in Rn = Rm × Rk, (1)
where (x, y) ∈ Rm ×Rk,α > 0, and Q = m+k(α+1). In the first part of the article,
exploiting the invariance of the equationwith respect to a suitable conformal inversion,
we prove a “spherical symmetry” result for solutions. In the second part, we show
how to reduce the dimension of the problem using a hyperbolic symmetry argument.
Given any positive solution u of (1), after a suitable scaling and a translation in the
variable y, the function v(x) = u(x, 0) satisfies the equation
divx (p∇xv) − qv = −pv(Q+2)/(Q−2), |x| < 1, (2)
with a mixed boundary condition. Here, p and q are appropriate radial functions. In
the last part, we prove that if m = k = 1, the solution of (2) is unique and that for
m ≥ 3 and k = 1, problem (2) has a unique solution in the class of x-radial functions
Regular domains in homogeneous groups
No full textWe study John, uniform and non-tangentially accessible domains
in homogeneous groups of steps 2 and 3. We show that C1,1 domains in groups
of step 2 are non-tangentially accessible and we give an explicit condition which
ensures the John property in groups of step 3
Positive solutions of anisotropic Yamabe-type equations in
No full textWe study entire positive solutions to the partial differential equa-
tion in Rn , n+2
∆x u + (α + 1)2 |x|2α ∆y u = −|x|2α u n−2 ,
where x ∈ R 2 , y ∈ Rn−2 , n ≥ 3 and α > 0. We classify positive solutions with
second order derivatives satisfying a suitable growth near the set x = 0
Trace theorems for vector fields
No full textIn the setting of Carnot-Carathéodory spaces we prove some trace theorems for Sobolev functions. We consider the trace on a non characteristic surface for Hörmander vector fields of step r ≥ 1 and the trace on the boundary of a class of domains in the Grushin plane
Multiexponential maps in Carnot groups with applications to convexity and differentiability
No full textWe analyze some properties of a class of multiexponential maps appearing naturally in the geometric analysis of Carnot groups. We will see that such maps can be useful in at least two interesting problems: first, in relation to the analysis of some regularity properties of horizontally convex sets. Then, we will show that our multiexponential maps can be used to prove the Pansu differentiability of the subRiemannian distance from a fixed point
John and Uniform Domains in Generalized Siegel Boundaries
No full textGiven the pair of vector fields X = ∂x + |z|2my∂t and Y = ∂y −|z|2mx∂t,where (x,y,t) = [InlineMediaObject not available: see fulltext.], we give a condition on a bounded domain [InlineMediaObject not available: see fulltext.] which ensures that Ω is an (ε,δ)-domain for the Carnot-Carathéodory metric. We also analyze the Ahlfors regularity of the natural surface measure induced on ∂Ω by the vector fields
- …
