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\markboth{\hfil Average value of eigenfunctions \hfil EJDE/Conf/10}
{EJDE/Conf/10 \hfil Stephen B. Robinson \hfil}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Fifth Mississippi State Conference on Differential Equations and
Computational Simulations, \newline
Electronic Journal of Differential Equations,
Conference 10, 2003, pp 251--256. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\vspace{\bigskipamount} \\
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On the average value for nonconstant eigenfunctions of the
$p$-Laplacian assuming Neumann boundary data
%
\thanks{ {\em Mathematics Subject Classifications:} 35P30, 35J20, 35J65.
\hfil\break\indent
{\em Key words:} Nonlinear eigenvalue problem, $p$-Laplacian, variational methods.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Published February 28, 2003. } }
\date{}
\author{Stephen B. Robinson}
\maketitle
\begin{abstract}
We show that nonconstant eigenfunctions of the $p$-Laplacian do not
necessarily have an average value of 0, as they must when $p=2$.
This fact has implications for deriving a sharp variational
characterization of the second eigenvalue for a general class of
nonlinear eigenvalue problems.
\end{abstract}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\section{Introduction}
In this paper we show that the nonconstant solutions of
\begin{equation} \label{evp}
\begin{gathered}
-\Delta_p u -\lambda |u|^{p-2}u= 0 \quad \mbox{a.e. in }\Omega, \\
\frac{\partial u}{\partial \nu}=0 \quad \mbox{on }\partial\Omega,
\end{gathered}
\end{equation}
do not necessarily satisfy $\int_{\Omega}u=0$. This fact has
implications for deriving a sharp variational characterization of
the second eigenvalue for a broad class of nonlinear eigenvalue
problems including (\ref{evp}). We assume that
$\Omega\subset\mathbb{R}^N$ is a smooth bounded domain, $\lambda$
is a real number, and $\Delta_p $ is the $p$-Laplacian, i.e.
$\Delta_p u:=\nabla\cdot (|\nabla u|^{p-2}\nabla u)$, for some
$p\in (1,\infty)$.
In some respects (\ref{evp}) is already well understood. Since Neumann
boundary conditions are assumed, it is straightforward to see that the
principle eigenvalue is $\lambda_1=0$ with simple eigenspace $W:=span\{1\}$.
Recent work in \cite{AT}, \cite{CDG}, \cite{CDG2}, and \cite{DR} has provided
a detailed description of the second eigenvalue, $\lambda_2$, which is defined
as the smallest real number greater than $\lambda_1$ such that (\ref{evp})
has a nontrivial solution. In particular, it is known that $\lambda_2>0$,
and that eigenfunctions associated with $\lambda_2$ are sign-changing with
exactly two nodal domains and are in the set
$V_p:=\{v\in W^{1,p}(\Omega):\int_{\Omega}|v|^{p-2}v=0\}$. Also, $\lambda_2$ satisfies
variational characterizations that generalize from the linear case in a
natural way. We should point out that the references above impose Dirichlet
boundary conditions, but provide a framework that works just as well for
(\ref{evp}). In section 2 we will provide a sketch of how some of these
facts can be proved.
There are several situations where it is straightforward to see
that second eigenfunctions have an average value of $0$. Of
course, if $p=2$, then (\ref{evp}) reduces to the standard
eigenvalue problem for the Laplace operator with Neumann boundary
data, and it is clear that every nonconstant eigenfunction lies in
$V_2=W^{\perp}=\{u\in W^{1,2}(\Omega):\int_{\Omega}u=0\}$. For
arbitrary $p$, if we examine the ODE case, then it is possible to
exploit the symmetry of $\Omega=(a,b)$ to prove that nonconstant
eigenfunctions once again satisfy $\int_a^bu=0$. This ODE argument
can be extended to eigenfunctions on ``boxes" in $\mathbb{R}^N$
with $N>1$, i.e. $\Omega=(a_1,b_1)\times\cdots\times (a_N,b_N)$.
But what of the average value for second eigenfunctions over more
general domains?
This question arose while studying eigenvalue problems for a class
of quasilinear operators that generalize the $p$-Laplacian, i.e.
\[
Q(u):=\sum_{1\leq |\alpha|\leq m} (-1)^{|\alpha |}
D^\alpha A_\alpha (x,\xi'_m (u)),
\]
where $Q$ is a $2m$-th order quasilinear operator satisfying
general growth, ellipticity and monotonicity conditions. For
boundary value problems associated with such operators some
interesting existence theorems have been proved by Shapiro,
et.al., where a {\it second eigenvalue} is defined and used as an
upper bound in certain key growth estimates. This second
eigenvalue is obtained via the minimization of an appropriate
functional, essentially a Raleigh quotient, over the space $V_2=\{u\in
W^{1,p}(\Omega):\int_{\Omega}u=0\}$. (More details are provided in
section 2 and in the references \cite{RS} and \cite{S2}.) This
allows something like an orthogonal splitting of the
Banach Space $W^{1,p}(\Omega)$ so that saddle point theorems can be applied in a standard way. An open question that arose as a result
of these papers was whether or not this orthogonal splitting leads
to a sharp characterization of the second eigenvalue. Our main
result in this paper shows that it does not. It follows that an improved
characterization should lead to an improvement of the existence
results in the papers listed above. These improved existence
theorems are described in subsequent work.
\section{ Preliminaries }
We begin with a standard variational formulation of the problem, and
briefly present some straightforward properties and definitions.
Details can be checked in the references. Let $W^{1,p}(\Omega)$
be defined in the usual way, as in \cite{A}. Let
\[
E(u):=\int_{\Omega}|\nabla u|^p, \mbox{ for } u\in W^{1,p}(\Omega).
\]
It is well known that $E$ is a $C^1$ functional with
\[
E'(u)\cdot v=p\int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla v.
\]
Moreover, if we consider $E$ constrained to the surface
$\mathcal{S} :=\{u\in W^{1,p}(\Omega):\int_{\Omega}|u|^p=1\}$, then any critical point, $\phi$,
satisfies
\begin{equation}
\int_{\Omega}|\nabla \phi|^{p-2}\nabla \phi\cdot\nabla v
=\lambda\int_{\Omega}|\phi|^{p-2}\phi v
\label{lagrange}
\end{equation}
for some $\lambda\in\mathbb{R}$ and all $v\in W^{1,p}(\Omega)$. Hence, the critical points
of the constrained functional correspond to eigenfunctions, and the associated
Lagrange multipliers correspond to eigenvalues. (Substitute $v=\phi$
into (\ref{lagrange}) to see that $\lambda=E(\phi)$.) Notice that by
constraining the functional to the $L^p$ unit sphere we are simply recognizing
that all nontrivial eigenfunctions can be rescaled so that they are elements
of $\mathcal{S}$.
$E$ clearly attains a global minimum of $0$ at
$\pm\phi_1=\pm(\frac{1}{|\Omega|})^{\frac{1}{p}}$. Also, it is clear that
$E(u)>0$ for any nonconstant $u$. Thus $\lambda_1=0$ is a simple eigenvalue
with eigenspace $W:=span\{1\}$.
If $\lambda>0$ is an eigenvalue with associated eigenfunction $\phi$, then we
can substitute $v=1$ into (\ref{lagrange}) to see that $\phi\in V_p$.
Hence our search for critical points can be restricted to the set
$V_p\bigcap\mathcal{S}$. Members of this set are clearly sign-changing. Using the
fact that $V_p\bigcap\mathcal{S}$ is weakly closed, and that $E$ is bounded below
and weakly lower semicontinuous, we see that $E$ attains its positive
infimum on $\mathcal{S}$. Hence, one variational characterization of $\lambda_2$ is
\begin{equation}
\lambda_2:=\inf_{\mathcal{S}\cap V_p}E .\label{lambda21}
\end{equation}
Let $\phi_2$ represent an associated eigenfunction and consider the curve
\[
h:\mathbb{R}\to\mathcal{S}: h(t)=\frac{\phi_2+t}{||\phi_2+t||_{L^p}}.
\]
Then
\[ E(h(t))=\frac{\int_{\Omega}|\nabla\phi_2|^p}{\int_{\Omega}|\phi_2 +t|^p},
\mbox{ and }
\frac{d}{dt}E(h(t))=\frac{-p\int_{\Omega}|\nabla\phi_2 |^p\int_{\Omega}|\phi_2
+t|^{p-2}(\phi_2+t)}{\left ( \int_{\Omega}|\phi_2 +t|^{p}\right )^2}.
\]
Thus $E(h(t))$ reaches a global maximum of $\lambda_2$ only at $t=0$.
Moreover, $\lim_{t\to\pm\infty}h(t)=\pm\phi_1$ and
$\lim_{t\to\pm\infty}E(h(t))=0$. Thus $h(t)$ can be modified to create
a continuous curve $\gamma :[-1,1]\to\mathcal{S}$ such that $\gamma(\pm 1)=\pm\phi_1$,
$\gamma (0)=\phi_2$, and such that $E(\gamma (t))$ achieves a maximum value
of $\lambda_2$ precisely when $t=0$. Conversely, any continuous curve on $\mathcal{S}$
connecting $\pm\phi_1$ must cross $V_p$ and hence must contain a point,
$\gamma(t)$, where $E(\gamma(t))\geq\lambda_2$. Thus we deduce a second,
equivalent, variational characterization of $\lambda_2$ which is
\begin{equation}
\lambda_2:=\inf_{\gamma\in\Gamma}\sup_{-1\leq t\leq 1}E(\gamma (t)),
\label{lambda22}
\end{equation}
where $\Gamma:=\{\gamma :[-1,1]\to\mathcal{S} : \gamma \mbox{ is continuous, }
\gamma(\pm 1)=\pm\phi_1\}$. The proof that $\phi_2$ has exactly $2$ nodal
domains relies on the fact that if $\phi_2$ has more than $2$ nodal domains
then a curve can be constructed that contradicts (\ref{lambda22}) .
Details can be found in \cite{CDG2} or \cite{DR}.
Let $\mu_2$ represent the parameter characterized in \cite{S2} and
\cite{RS}. For homogeneous problems, such as (\ref{evp}), this reduces to
\begin{equation}
\mu_2:=\inf_{\mathcal{S}\cap V_2}E .\label{mu2}
\end{equation}
If we compare the characterization (\ref{lambda22}) with (\ref{mu2}),
we observe that every curve in $\Gamma$ must cross at least one point
in $V_2$, and thus the maximum value of $E$ over any such curve is at least
as large as $\mu_2$. It follows that $\mu_2\leq\lambda_2$. Now suppose
that we can show that $\phi_2\not\in V_2$. If we examine the special
curve $\gamma$, constructed above, we see that $\gamma$ crosses $V_2$
at a point $\gamma(t)\neq\phi_2$, so $E(\gamma (t))<\lambda_2$, and thus
$\mu_2<\lambda_2$. This would show that (\ref{mu2}) is not a sharp
characterization of $\lambda_2$. In section 3 we will prove that
$\phi_2\not\in V_2$ for certain asymmetric domains. An interesting
open question might be to classify the domains where $\mu_2=\lambda_2$,
and it is reasonable to conjecture that this depends upon a symmetry condition.
It is important to note that the quasilinear operators in \cite{RS} and
\cite{S2} are not assumed to be homogeneous, so the associated eigenvalue
problems could not be restricted to $\mathcal{S}$. Hence, the more general
characterization had to consider the infimum of
$\frac{E(u)}{\int_{\Omega}|u|^p}$ over $V_2\bigcap r\mathcal{S}$ and then compute a
$\liminf$ as $r\to\infty$.
\section {Comparing $\lambda_2$ and $\mu_2$}
\begin{theorem} \label{thm1}
There is at least one domain $\Omega\subset\mathbb{R}^N$ such that
the associated second eigenvalue, $\lambda_2$, has an associated
eigenfunction, $\phi_2$, that does not lie in $V_2$.
\end{theorem}
\paragraph{Proof}
Consider the problem
\begin{equation}
\begin{gathered}
-\Delta_p u-\lambda_2^{\epsilon}|u|^{p-2}u=0 \quad\mbox{in } \Omega_{\epsilon},\\
\frac{\partial u}{\partial\nu}=0\quad \mbox{on }\partial\Omega_{\epsilon},
\end{gathered} \label{bvpe}
\end{equation}
where $\Omega_{\epsilon}:=\left ((0,2)\times (0,2))\bigcup ((2,3)
\times (0,\epsilon)\right )\bigcup \left ((3,4)\times (0,1)\right )$
for $0\leq\epsilon\leq 1$, and where $\lambda_2^{\epsilon}$ is characterized
by (\ref{lambda21}) and (\ref{lambda22}). $\Omega_0$ will refer to the
limiting case which is simply the union of the two disjoint rectangles.
Let $\phi_{2,\epsilon}\in V_p\bigcap\mathcal{S}$ represent an associated second
eigenfunction. When $\epsilon=0$ this will simply indicate a function
that is a positive constant over one rectangle and a negative constant over
the other, where the constants are balanced to fit the constraints.
First, we find an upper bound for $\lambda_2^{\epsilon}$. Let
\[
u_2:= \begin{cases}
1 & \mbox{for } (x,y)\in [0,2]\times[0,2],\\
-2x+5 & \mbox{for } (x,y)\in [2,3]\times [0,\epsilon],\\
-1 & \mbox{for } (x,y)\in [3,4]\times [0,1]
\end{cases}
\]
Also, let $\gamma (\alpha,\beta)=\alpha u_2^+ -\beta u_2^-$, where
$u_2^+:=\max \{u_2,0\}$, $u_2^-:=\max \{-u_2,0\}$, and $\alpha$ and
$\beta$ are nonnegative scalars such that
$\alpha^p||u_2^+||_{L^p}^p+\beta^p||u_2^-||_{L^p}^p=1$. Notice that $\gamma$
is a curve on $\mathcal{S}$ connecting the points $\frac{u_2^+}{||u_2^+||_{L^p}}$
and $-\frac{u_2^-}{||u_2^-||_{L^p}}$. By the Intermediate Value Theorem
$\gamma$ crosses the surface $V_p$. Hence the maximum of
$E(\gamma (\alpha,\beta))$ must be greater than $\lambda_2^{\epsilon}$.
However,
\[ \nabla \gamma (\alpha,\beta)=
\begin{cases}
(0,0) & \mbox{for } (x,y)\in [0,2]\times[0,2],\\
(-2\alpha,0) & \mbox{for } (x,y)\in [2,\frac{5}{2}]\times[0,\epsilon],\\
(-2\beta,0) & \mbox{for } (x,y)\in (\frac{5}{2},3]\times[0,\epsilon],\\
(0,0) & \mbox{for } (x,y)\in (3,4]\times[0,1]
\end{cases}
\]
Thus $\int_{\Omega_{\epsilon}}|\nabla\gamma (\alpha,\beta)|^p\leq 2^p
\max\{\alpha^p,\beta^p\}\epsilon$. But $||u_2^+||^p_{L^p}\geq 4$ and
$||u_2^-||^p_{L^p}\geq 1$, so $\alpha^p\leq\frac{1}{4}$ and $\beta^p\leq 1$.
Therefore
$\int_{\Omega_{\epsilon}}|\nabla\gamma (\alpha,\beta)|^p\leq 2^p\epsilon$.
It follows that
$\lambda_2^{\epsilon}\leq \max_{(\alpha,\beta)}E(\gamma (\alpha,\beta))
\leq 2^p\epsilon$, so $\lim_{\epsilon\to 0}\lambda_2^{\epsilon}=0$.
We will now show that $\int_{\Omega_{\epsilon}}\phi_{2,\epsilon}\neq 0$
for some $\epsilon$. Since $\lambda_2^{\epsilon}\to 0$, straightforward
estimates now show that $\phi_{2,\epsilon}\to\phi_{2,0}$ in
$W^{1,p}(\Omega_0)$, where $\nabla\phi_{2,0}\equiv 0$ and
$\int_{\Omega_0}|\phi_{2,0}|^p=1$. Moreover,
$\int_{\Omega_0}|\phi_{2,0}|^{p-2}\phi_{2,0}=\lim_{\epsilon\to 0}
\int_{\Omega_{\epsilon}}|\phi_{2,\epsilon}|^{p-2}\phi_{2,\epsilon}=0$.
It must be that there are constants $a,b\in\mathbb{R}$ such that
$\phi_{2,0}\equiv a$ in $[0,2]\times [0,2]$ and $\phi_{2,0}\equiv b$
in $[3,4]\times [0,1]$. Moreover, it follows that $4|a|^p+|b|^p=1$, $a$
and $b$ have opposite signs, and $4|a|^{p-1}-|b|^{p-1}=0$.
Thus $|b|=4^{\frac{1}{p-1}}|a|$. It can now be checked that
$\int_{\Omega_0}\phi_{2,0}=\pm (4-4^{\frac{1}{p-1}})|a|\neq 0$ for $p\neq 2$.
Hence $\int_{\Omega_{\epsilon}}\phi_{2,\epsilon}\neq 0$ for some $\epsilon>0$.
\hfill$\square$\smallskip
As an immediate consequence we have the following statement.
\begin{corollary} \label{coro1}
If $\Omega$ is the domain given in Theorem \ref{thm1}, then $\mu_2<\lambda_2$.
\end{corollary}
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Journal of Differential Equations, vol. 181 (2002), pp. 58-71
\bibitem{RS} A.J. Rumbos and V.L. Shapiro,
{\em One-sided resonance for a quasilinear variational problem},
Cont. Math., Vol. 208 (1997), pp. 277-299.
\bibitem {S1} V.L. Shapiro,
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Comm in PDEs, 16 (1991), 1819-1855.
\bibitem{S2} V.L. Shapiro,
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\end{thebibliography}
\noindent\textsc{Stephen B. Robinson}\\
Department of Mathematics, Wake Forest University,\\
Winston-Salem, NC 27109, USA \\
e-mail: sbr@wfu.edu
\end{document}