1,720,968 research outputs found
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
Full text linkWe 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
Full text linkGiven 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
A trace theorem for Martinet-type vector fields
Full text linkIn R3 we consider the vector fields X_1 and X_2 in R^3. We prove a trace theorem for Sobolev functions on a half space. The trace is estimated by means of a suitable Besov space that is defined using the Carnot–Carathéodory metric associated with the vector fields and the related perimeter measure
Anisotropic estimates of subelliptic type
Full text linkWe discuss some estimates of subelliptic type related with vector fields satisfying the Hormander condition. Our approach makes use of a class of approximate exponentials studied in our previous papers. Such kind of estimates arises naturally in the study of regularity theory of weak solutions of degenerate elliptic equations
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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
Full text linkWe 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
Levi umbilical surfaces in complex space
Full text linkWe 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
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