Reverse Poincaré-type inequalities for the weak solution of system of partial differential inequalities
- Muhammad Shoaib Saleem^{1}Email author,
- Josip Pečarić^{2},
- Hamood Ur Rehman^{1},
- Abdul Rauf Nizami^{3} and
- Abid Hussain^{3}
https://doi.org/10.1186/s13660-016-1204-z
© Shoaib Saleem et al. 2016
Received: 29 January 2016
Accepted: 12 October 2016
Published: 24 October 2016
Abstract
In this work we develop the square integral estimates for the functions of n variables which are subharmonic with respect to some variables and for other remaining variables are superharmonic. It is in a sense a generalization of reverse Poincaré type inequalities for the difference of superharmonic functions developed in (J. Inequal. Appl. 2015: doi:10.1186/s13660-015-0916-9, 2015).
Keywords
1 Introduction
The second order partial differential equations represent a large number of practical problems. One of the most important classes of linear second order partial differential equations is elliptic equations. A second order partial differential equation is uniformly elliptic if the matrix of higher order coefficients is positive definite. The particular and important case of a second order uniformly elliptic equation is the Laplace equation.
The Laplace equation not only emerges in a variety of physical problems but also arises in the study of analytic functions and probabilistic investigations of Brownian motion.
Let Δ be the second order Laplace operator of n variables and \(B(x_{o},r) \) is a ball in \(\mathbb{R}^{n} \), with center \(x_{o} \), and radius r. A function \(u(x)\in C^{2}(B(x_{o},r))\cap C(\overline{B}) \) is subharmonic if \(\Delta u(x) \geq0 \), and it is said to be superharmonic if \(\Delta u(x)\leq0\). The subharmonic functions attain their maximum and superharmonic functions attain their minimum on the boundary (see e.g. Evans Section 6.4 [2]).
Subharmonic and superharmonic functions play a key role in classical as well as in modern potential theory. These functions are most familiar in partial differential equations in the construction of solutions to the Dirichlet problem [3, 4].
There is a lot of information on subharmonic and superharmonic functions and also on their properties in [3–6].
The weighted square integral inequality for convex functions of one variable was developed by Hussain, Pečarić, and Shashiashvili [7]. Such kinds of inequalities are widely used in finance and physical problems. The function of n variables is convex if its Hessian matrix is positive definite. The natural generalization of convex functions for n variables is a subharmonic function and similarly of concave functions it is a superharmonic function. It is also clear that a function which is convex (concave) w.r.t. each of its variables may not be convex (concave) as a whole but such kinds of functions are a subclass of subharmonic functions (superharmonic functions).
The weighted square integral inequalities for superharmonic functions are developed in [1]. There is also another important class of functions which are convex w.r.t. some variables but concave w.r.t. other remaining variables. The generalization of such functions is subharmonic for some variables and superharmonic for the other variables. In this research our notations are standard.
We will organize the paper in following way. In the second section we prove the inequality for the smooth solution of system (1.1) and also approximate the weak solution of the system (1.1) by smooth ones. In the last section we prove that the continuous weak solutions possess first order weak derivatives and also we will prove the inequality for a weak solution of system (1.1).
2 The reverse Poincaré inequalities for smooth subsolution and approximation of weak subsolution by smooth ones
The following lemmas for superharmonic functions and subharmonic functions are proved in [1].
Lemma 2.1
([1]) Consider two arbitrary smooth superharmonic functions \(u_{i}(x)\), \(i=1,2\) over Domain D, \(D\subset\mathbb{R}^{n}\) (the domain is bounded and has a smooth boundary) i.e. \(u_{i}(x)\in {C^{2}}(\overline{D})\), \(i=1,2\), and \(\Delta u_{i}(x)\leq0\) if \(x\in D\), \(i=1,2\).
Lemma 2.2
We will start by the following theorem.
Theorem 2.3
Proof
Taking the infinite norm on (2.7) we get the result (2.2). □
Remark 2.4
Theorem 2.5
Let \(u(x)\) be the weak solution of system (1.1) on the ball B, \(B=B(x_{o},r)\). Then, for any \(k=1,2,\ldots\) , there exists \(\widehat{h} > 0\), such that if \(0< h<\widehat{h}\), each \(u_{h}(x) \) is a smooth solution of the system (1.1) over the ball \(B_{k} \).
Proof
Now we check that, for arbitrary \(x\in B_{k}\), \(\varphi_{h}(x-y) \) has compact support in the ball \(B(x_{o},r)\).
3 The existence and integrability of weak partial derivative and weighted square inequalities for the difference of weak subsolutions
The following theorem tells that a continuous weak subsolution of system (1.1) possesses all first order weak partial derivatives and also they are square integrable.
Theorem 3.1
Proof
The proof of the theorem can be done along similar lines to the proof of Theorem 3.1 of [1], using inequality (2.8) of the present paper instead of (3.5) of [1]. □
The next theorem will give us reverse Poincaré type inequalities for a weak subsolution of system (1.1).
Theorem 3.2
Proof
For the continuous weak sub solutions \(u_{i}(x)\), \(i=1,2 \), for system (1.1), take a smooth approximation \(u_{m},_{i}(x)\) \(i=1,2 \). In the ball \(B_{k+l} \) there exists an integer \(m_{k+l} \) s.t. the requirement that \(u_{m},_{i}(x) \) is smooth in the ball \(B_{k+l} \) and \(u_{m},_{i}(x) \) converges uniformly to \(u_{i}(x)\) \(i=1,2 \) for \(m \geq m_{k+l} \).
4 Conclusion
From our results we conclude that if a weak solution of system (1.1) is closed in a supremum norm then their weak derivatives are also closed in a weighted \(L^{2}\) norm.
Declarations
Acknowledgements
We are grateful to the reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper.
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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