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This article is cited in 3 scientific papers (total in 3 papers)
Mathematics and mechanics
The concept and criteria of the capacitive type of the non-compact Riemannian manifold based on the generalized capacity
V. M. Keselman MIREA — Russian Technological University
Abstract:
Let $M^n$ be a non-compact $n$-dimensional Riemannian manifold and let $p>1$ be a fixed real number.
We call $(F, p)$-capacity of a compact set $K\subset M^n$ a value $\inf\int_{M^n}(F(x, \nabla u))^p dv$,
where the exact lower bound is taken over all smooth functions $u$ finite in $M^n$
and such that $u \geq 1$ on $K$.
Function $F = F(x, \xi)$, $(x, \xi)\in TM^n$ is smooth, non-negative
and satisfies certain general conditions.
A special case of $(F, p)$-capacity is, e. g., the conformal capacity when $F(x, \xi) = |\xi|$ and $p = n$.
We based this notion of $(F, p)$-capacity on the work of G. Choquet, V.G. Mazya, and V.M. Miklyukov.
Let us introduce the concept of the type of a non-compact manifold $M^n$ as follows.
We say that $M^n$ is of $(F, p)$-parabolic type, if the $(F, p)$-capacity
of some non-degenerate compact $K\subset M^n$ is zero.
Otherwise, we say that manifold $M^n$ is of $(F, p)$-hyperbolic type.
Like in the classical case, this notion of $(F, p)$-type of the non-compact Riemannian manifold
is invariant with respect to the specific choice of the compact set $K$.
We prove the criteria for the manifold to be of $(F, p)$-parabolic or $(F, p)$-hyperbolic type.
Special cases of these are the well-known criteria of conformal type of a Riemannian manifold
expressed in terms of growth of the volume $V(r)$ of geodesic balls
or of area $S(r)$ of their boundary spheres of radius $r$.
In the general case of criteria of $(F, p)$-type of manifold $M^n$
the role of the class of complete metrics conformal to the initial metric of the manifold
takes on the class of exhaustion functions $h$ of manifold $M^n$,
and the roles of $V(r)$ and $S(r)$ are taken by functions
$V_{F, p, h}(r)=\int_{h\leq r}(F(x,\nabla h))^p dv$ and
$ S_{F, p, h}(r) = \int_{h = r}(F(x, \nabla h))^p(d\sigma /|\nabla h|)$, respectively.
The criteria themselves are expressed in terms of the growth of these functions.
For instance, the following conditions
$$\int^{+\infty}\left(\frac{r}{V_{F, p, h}(r)}\right)^{\frac{1}{p-1}}dr =
\infty, \; \int^{+\infty}\left(\frac{1}{S_{F, p, h}(r)}\right)^{\frac{1}{p-1}}dr = \infty $$
characterize the $(F, p)$-parabolic type of the non-compact Riemannian manifold.
Keywords:
Riemannian manifold, capacity, conformal type, p-parabolictype, p-hyperbolic type, volume of a geodesic ball, area of the geodesic sphere, exhaustion function.
Received: 19.03.2019
Citation:
V. M. Keselman, “The concept and criteria of the capacitive type of the non-compact Riemannian manifold based on the generalized capacity”, Mathematical Physics and Computer Simulation, 22:2 (2019), 21–32
Linking options:
https://www.mathnet.ru/eng/vvgum252 https://www.mathnet.ru/eng/vvgum/v22/i2/p21
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