\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 135, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/135\hfil Some inequalities for Sobolev integrals] {Some inequalities for Sobolev integrals} \author[N. Dimitrov, S. Tersian\hfil EJDE-2009/135\hfilneg] {Nikolaj Dimitrov, Stepan Tersian} % in alphabetical order \dedicatory{Dedicated to Professor Nedyu Popivanov on his 60-th birthday} \address{Nikolaj Dimitrov \newline Department of Mathematical Analysis, University of Rousse,\newline 8, Studentska, 7017 Rousse, Bulgaria} \email{acm189@abv.bg} \address{Stepan Tersian \newline Department of Mathematical Analysis, University of Rousse,\newline 8, Studentska, 7017 Rousse, Bulgaria} \email{sterzian@ru.acad.bg} \thanks{Submitted February 8, 2009. Published October 21, 2009.} \subjclass[2000]{26D10, 34A40} \keywords{Sobolev integrals; Dirichlet principle; divided differences; \hfill\break\indent Hermite polynomials} \begin{abstract} We present some inequalities for Sobolev integrals for functions of one variable which are generalization of Dirichlet principle for harmonic functions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} In this note we present some inequalities for Sobolev integrals which are generalizations of Dirichlet principle for functions of one variable. The Dirichlet principle for harmonic functions, also known as Thomson's principle, states that there exists a function $u$ that minimizes the functional $E(u)=\int_{\Omega }|\nabla u(x)|^{2}dx,$ called the Dirichlet integral for $\Omega \subset \mathbb{R}^{n}$, $n\geq 2$, among all the functions $u\in C^2(\Omega)\cap C(\bar{\Omega})$ which take given on values $\varphi$ on the boundary $\partial\Omega$ of $\Omega$. The minimizer $u$ satisfies the Dirichlet problem for Laplace equation \begin{gather*} \Delta u(x)=0, \quad x\in \Omega, \\ u(x)=\varphi(x), \quad x\in \partial\Omega, \end{gather*} which is the Euler-Lagrange equation associated to the Dirichlet integral. The term Dirichlet principle" is identified by Bernhard Riemann in Theory of Abelian Functions", published 1857. It is one of main steps in the history of potential theory and calculus of variations (see \cite{M}). The direct method of the calculus of variations was developed near the middle of nineteenth century. The existence of a minimum of $E(u)$ was considered, in heuristic way, as a trivial consequence of its positivity. Weierstrass, gave in 1870, a counterexample that such an evidence is not valid for the one dimensional case, by showing that for $C^{2}$ functions $u:[-1,1] \to \mathbb{R}$ such that $u(-1)=a$, $u(1)=b$, $a\neq b$, the integral $\int_{-1}^1|xu'(x)|^{2}dx,$ has no minimum. Let $f$ be $n$ times continuously differentiable functions of one variable $x$. We assume that the independent variable $x\in I:=[ 0,1]$ and all derivatives of function $f$, except one, are zero at the end points $0$ and $1$ of the interval $I$. Denote by $N$ the set $\{0,1,\dots ,n-1\}$. Our main result is as follows. \begin{theorem} \label{thm1} (a) Let $f\in C^{n}(I)$ be a real-valued differentiable function such that $f(0)=f(1)=\dots =f^{(n-2)}(0)=f^{(n-2)}(1)=0$ and $f^{(n-1)}(0)=A$, $f^{(n-1)}(1)=B$. Then $$\int_0^1|f^{(n)}(x)|^{2}dx\geq n^{2}A^{2}+(-1)^{n}2nAB+n^{2}B^{2}. \label{f1}$$ (b) Let $f\in C^{n}(I)$ be a real-valued differentiable function such that $f(0)=a$, $f(1)=b$ and $f'(0)=f'(1)=\dots =f^{(n-1)}(0)=f^{(n-1)}(1)=0.$ Then $$\int_0^1|f^{(n)}(x)|^{2}dx\geq (2n-1)!\binom{2n-2}{n-1}(a-b)^{2}. \label{f2}$$ (c) Let $f\in C^{n}(I)$ be a real-valued differentiable function such that $f^{(k)}(0)=A,\quad f^{(k)}(1)=B,\quad f^{(j)}(0)=f^{(j)}(1)=0,\quad k\in N, \; j\in N\backslash \{k\}.$ Then $$\int_0^1|f^{(n)}(x)|^{2}dx\geq \alpha _{n,k}A^{2}-\gamma _{n,k}AB+\alpha _{n,k}B^{2}, \label{f3}$$ where \begin{gather*} \alpha _{n,k}=\frac{(2n-k-1)!}{k!}\binom{2n-2k-2}{n-k-1}\binom{2n-k-1}{k}, \\ \begin{aligned} \gamma _{n,k}&= (-1)^k \frac{(2n-k-1)!}{k!} \Big\{ \binom{2n-2k-2}{n-k-1}\binom{2n-k-1}{k} \\ &\quad+\sum_{t=0}^{k}\binom{n-k-1+t}{t}\binom {2n-2k-2+t}{n-2k-1+t}\Big\} . \end{aligned} \end{gather*} \end{theorem} Theorem \ref{thm1} is proved in Section 2. Direct calculations are used in the proof of (a) and (b). It is mentioned for which functions the equality holds. We prove Corollary \ref{coro2.1} as a consequence of (\ref{f1}) and (\ref{f2}), which is a generalization of an inequality proved in \cite[Lemma 3]{B}. The method of divided differences (cf. \cite{G}) in interpolation theory is used in the proof of general statement (c). We simplify some coefficients in (\ref{f3}) by reflection method. As a result, we have Corollary \ref{coro2.2}, which is a combinatorial identity, proved by variational'' tools. The present paper is a continuation of a problem, formulated by second author in \cite{T}. \section{Proof of the main result} \subsection*{Proof of Theorem \ref{thm1}} \textbf{(a)} We divide the proof into the following steps. \noindent\textbf{Claim 1.} The polynomial of order $2n-1$ $P(x)=\frac{(x-x^{2})^{n-1}}{(n-1)!} \big(A+x((-1)^{n-1}B-A)\big)$ satisfies the boundary conditions of the problem. \noindent\textit{Proof:} We have $(x^{n})^{(k)}=n(n-1)\dots (n-k+1)x^{n-k}$ and $((1-x)^{n})^{(k)}=(-1)^{k}n(n-1)\dots (n-k+1)(1-x)^{n-k}.$ Then, by Leibnitz formula, $P^{(k)}(0) =P^{(k)}(1)=0$ for $0\leq k\leq n-2$. Further \begin{gather*} P^{(n-1)}(0)=(1-x)^{n-1}(A+x((-1)^{n-1}B-A))|_{x=0}=A, \\ P^{(n-1)}(1)=(-1)^{n-1}x^{n}(A+x((-1)^{n-1}B-A))|_{x=1}=B. \end{gather*} \noindent\textbf{Claim 2.} We have $\int_0^1|P^{(n)}(x)|^{2}dx=n^{2}A^{2}+(-1)^{n}2nAB+n^{2}B^{2}.$ \textit{Proof:} By the boundary conditions for $P$ and integration by parts we obtain \begin{align*} I_{n} &= \int_0^1|P^{(n)}(x) |^{2}dx=\int_0^1P^{(n)}(x)dP^{(n-1)}(x) \\ &= P^{(n)}(1)B-P^{(n)}(0) A-\int_0^1P^{(n+1)}(x)P^{(n-1)}(x)dx \\ &= P^{(n)}(1)B-P^{(n)}(0)A-\int_0^1P^{(n+2)}(x)P^{(n-2)}(x)dx \\ &= \dots \\ &= P^{(n)}(1)B-P^{(n)}(0)A. \end{align*} Let us calculate $P^{(n)}(1)$ and $P^{(n)}(0)$. By the Leibnitz formula we have \begin{align*} &P^{(n)}(0)\\ &= \frac{1}{(n-1)!} \sum_{j=0}^{n}\binom{n}{j}((x-x^{2})^{n-1}) ^{(j)}(A+x((-1)^{n-1}B-A))^{(n-j)}|_{x=0} \\ &= \frac{1}{(n-1)!}\big[((x-x^{2}) ^{n-1})^{(n)}|_{x=0}A+n((x-x^{2})^{n-1}) ^{(n-1)}|_{x=0}((-1)^{n-1}B-A)\big] \\ &= -(n^{2}-n)A+n((-1)^{n-1}B-A)\\ &= -n^{2}A+(-1)^{n-1}n B, \end{align*} and by the same argument, $P^{(n)}(1)=n^{2}B+(-1)^{n}nA.$ Then \begin{align*} I_{n} &= (n^{2}B+(-1)^{n}nA)B-(-n^{2}A+(-1)^{n-1}nB)A \\ &= n^{2}A^{2}+(-1)^{n}2nAB+n^{2}B^{2}. \end{align*} \noindent\textbf{Claim 3 (Dirichlet principle).} Suppose that $f$ satisfies the boundary conditions of the problem. Then $\int_0^1|f^{(n)}(x)|^{2}dx\geq \int_0^1|P^{(n)}(x)|^{2}dx.$ \textit{Proof.} Denote $h(x)=f(x)-P(x)$. The function $h$ satisfies homogeneous boundary conditions $h^{(j)}(0)=h^{(j)}(1)=0,\quad j\in N.$ Then \begin{align*} \int_0^1|f^{(n)}(x)|^{2}dx &= \int_0^1|P^{(n)}(x)|^{2}dx+2\int_0^1P^{(n)}(x)h^{(n)}(x) dx+\int_0^1|h^{(n)}(x)|^{2}dx \\ &\geq \int_0^1|P^{(n)}(x) |^{2}dx+\int_0^1|h^{(n)}(x)|^{2}dx \\ &\geq \int_0^1|P^{(n)}(x)|^{2}dx, \end{align*} because $\int_0^1P^{(n)}(x)h^{(n)}(x)dx=0$. It follows by boundary conditions for $h$ and integration by parts as follows: \begin{align*} \int_0^1P^{(n)}(x)h^{(n)}(x)dx &= \int_0^1P^{(n)}(x)dh^{(n-1)}(x)\\ &=-\int_0^1P^{(n+1)}(x)h^{(n-1)}(x)dx \\ &= +\int_0^1P^{(n+2)}(x)h^{(n-2)}( x)dx \\ &= \dots \\ &= (-1)^{n}\int_0^1P^{(2n)}(x)h(x)dx=0, \end{align*} because $P$ is a polynomial of order $2n-1$. This completes the proof of (a). (b) Suppose that $f$ satisfies the boundary conditions and $Q(x)=cx^{2n-1}+a_{1}x^{2n-2}+\dots +a_{2n-1}$ be the unique polynomial of order $2n-1$ satisfying boundary conditions $Q(0)=a$, $Q(1)=b$ and $Q'(0)=Q'(1)=\dots =Q^{(n-1)}(0) =Q^{(n-1)}(1)=0$. Then as in Claim 3 we can prove that $$\int_0^1|f^{(n)}(x)|^{2}dx\geq \int_0^1|Q^{(n)}(x)|^{2}dx. \label{e}$$ To compute the right hand side of this equation, we use Hermite interpolation formula and divided differences. Recall some notions and facts on divided differences (cf. \cite[pp.\ 96--104]{G}). Let $g$ be a sufficiently smooth function defined in $m+1$ points $x_0\leq x_{1}\leq \dots \leq x_{m}$ points and \$x_0=\dots =x_{\nu _{i}-1}=t_{1}