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- Open Access
Reverse Poincaré-type inequalities for the difference of superharmonic functions
- Josip Pečarić^{1},
- Muhammad Shoaib Saleem^{2}Email author,
- Hamood Ur Rehman^{2},
- Abdul Majeed Nizami^{2} and
- Abid Hussain^{2}
https://doi.org/10.1186/s13660-015-0916-9
© Pečarić et al. 2015
- Received: 26 May 2015
- Accepted: 29 November 2015
- Published: 15 December 2015
The Erratum to this article has been published in Journal of Inequalities and Applications 2016 2016:59
Abstract
In this paper, we develop the weighted square integral inequalities for the difference of two smooth superharmonic functions. Then we prove the existence and integrability of the Sobolev derivative for superharmonic functions. The inequalities are generalized for the difference of two weak superharmonic functions. We also establish that the superharmonic approximation is indeed the better imitation of the exact unknown solution rather than the usual uniform approximation.
Keywords
- concave functions
- superharmonic functions
- weak derivative
- weight function
- compact support
- weak superharmonic function
- superharmonic majorants
1 Introduction and statement of the main result
The role of mathematical inequalities within the mathematical branches as well as in its enormous applications should not be underestimated. The square integral estimate for the first derivative of convex function was established in [1, 2]. Then the results were improved by Hussain et al. in [3]. Such kinds of inequalities are very useful for the hedging problems in mathematical finance.
The negative of a convex function is concave (concave down) functions. It is well known in modern calculus that the natural generalization of concave functions to a function of several independent variables is a superharmonic function, related to the famous Laplace operator. So it is also interesting to develop similar inequalities for the superharmonic functions. The latter functions are often considered as a powerful tool for the study of solvability of the classical Poisson and Dirichlet problems in the theory of partial differential equations.
Throughout the paper we will use the following notations:
\(D, D\subset R^{n}\) is bounded and having a smooth boundary, \(B=B(x_{0},r)\) is the open ball in \(R^{n}\) with center \(x_{0}\) and radius r (\(r>0\)), B̅ is its closure. \(L^{\infty}(B)\) is the space of bounded (a.e. dx) on B.
Δ is the n dimensional Laplace operator.
\(C^{2}_{0}(B)\) is the space of twice continuously differentiable functions having compact support on B.
Now we formulate our main result.
Theorem 1.1
We will organize the paper in the following way: In the second section we will establish the inequality for the smooth superharmonic functions and then by a standard mollification technique we will approximate the weak superharmonic functions by the smooth ones. In the last section we will prove the existence and integrability of weak superharmonic functions and then establish the proof of our main result. At the end we will also explain that a superharmonic approximation is better than the usual uniform approximation.
2 The case of smooth superharmonic functions and mollification of weak superharmonic functions
Our starting point will be the following theorem.
Theorem 2.1
Proof
Remark 2.2
Writing the above remark for an arbitrary ball B, \(B=B(x_{0},r)\subset R^{n}\), we get
Remark 2.3
Now we approximate the weak superharmonic function (1.2) by the smooth ones. To this aim, we will use the classical mollification technique.
Theorem 2.4
Proof
Take \(\hat{\epsilon}=\frac{r}{2(k+2)}\). By definition it is trivial that \(u_{\epsilon}(x)\), \(\epsilon>0\), is infinitely differentiable w.r.t. x. Now we will see that for arbitrary \(x\in B_{k}\) the function \(\eta_{\epsilon}(x-y)\) has compact support on B as a function of y.
3 Sobolev gradient existence and proof of the main result
Theorem 3.1
Proof
Now we will give a proof of our main result.
Proof of Theorem 1.1
Take \(u_{m,i}(x)\), \(i=1,2\), the mollification of the weak superharmonic functions \(u_{i}(x)\), \(i=1, 2\).
By the definition of mollification, we know that for a ball \(B_{k+l}\), there exists an integer \(m_{k+l}\) such that each function \(u_{m,i}\), \(i=1,2\), is a smooth superharmonic function on the ball \(B_{k+l}\) if \(m\geq m_{k+l}\).
Notes
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
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