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3 papers
Integrability Properties of Functions with a Given Behavior of Distribution Functions and Some Applications
A. A. Kovalevskyab a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Abstract:
We establish that if the distribution function of a measurable function
$v$
defined on a bounded domain
$\Omega$ in
$\mathbb R^n$ (
$n\ge 2$) satisfies,
for sufficiently large
$k$, the estimate
${\rm meas}\{\vert v\vert>k\}\le k^{-\alpha}\varphi(k)/\psi(k)$,
where
$\alpha>0$,
$\varphi\,\colon[1,+\infty)\to\mathbb R$
is a nonnegative nonincreasing measurable function such that
the integral of the function
$s\to\varphi(s)/s$ over
$[1,+\infty)$ is finite,
and
$\psi\,\colon[0,+\infty)\to\mathbb R$ is a positive continuous function
with some additional properties,
then $\vert v\vert^\alpha\psi(\vert v\vert)\in L^1(\Omega)$.
In so doing, the function
$\psi$ can be either bounded or unbounded.
We give corollaries of the corresponding theorems for some specific ratios
of the functions
$\varphi$ and
$\psi$.
In particular, we consider the case where the distribution function
of a measurable function
$v$ satisfies, for sufficiently large
$k$,
the estimate ${\rm meas}\{\vert v\vert>k\}\le Ck^{-\alpha}(\ln k)^{-\beta}$
with
$C,\alpha>0$ and
$\beta\ge 0$.
In this case, we strengthen our previous result for
$\beta>1$ and, on the whole,
we show how the integrability properties of the function
$v$ differ depending on
which interval,
$[0,1]$ or
$(1,+\infty)$, contains
$\beta$.
We also consider the case where the distribution function
of a measurable function
$v$ satisfies, for sufficiently large
$k$,
the estimate ${\rm meas}\{\vert v\vert>k\}\le Ck^{-\alpha}(\ln\ln k)^{-\beta}$
with
$C,\alpha>0$ and
$\beta\ge 0$. We give examples showing the accuracy
of the obtained results in the corresponding scales of classes close to
$L^\alpha(\Omega)$.
Finally, we give applications of these results to entropy and weak solutions
of the Dirichlet problem for second-order nonlinear elliptic equations
with right-hand side in some classes close to
$L^1(\Omega)$
and defined by the logarithmic function or its double composition.
Keywords:
integrability, distribution function, nonlinear elliptic equations, right-hand side in classes close to $L^1$, Dirichlet problem, weak solution, entropy solution.
UDC:
517.518,
517.956
MSC: 26B35,
35J25,
35J60 Received: 16.10.2018
Revised: 01.11.2018
Accepted: 05.11.2018
DOI:
10.21538/0134-4889-2019-25-1-78-92
Fulltext:
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