\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 125, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/125\hfil On Pacard's regularity]
{On Pacard's regularity for the equation $-\Delta u = u^p$}
\author[D. R. Adams\hfil EJDE-2012/125\hfilneg]
{David R. Adams}
\address{David R. Adams \newline
Department of Mathematics,
University of kentucky, POT 714,
Lexington, KY 40506, USA}
\email{dave@ms.uky.edu}
\thanks{Submitted June 4, 2012. Published August 2, 2012.}
\subjclass[2000]{35D10, 35J60}
\keywords{Weak solutions; singular set; regularity}
\begin{abstract}
It is shown that the singular set for a positive solution of
the PDE $-\Delta u = u^p$ has Hausdorff dimension
less than or equal to $n - 2p'$, as conjectured by Pacard \cite{P} in
1993.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks
\def\Xint#1{\mathchoice
{\XXint\displaystyle\textstyle{#1}}%
{\XXint\textstyle\scriptstyle{#1}}%
{\XXint\scriptstyle\scriptscriptstyle{#1}}%
{\XXint\scriptscriptstyle\scriptscriptstyle{#1}}%
\!\int}
\def\XXint#1#2#3{{\setbox0=\hbox{$#1{#2#3}{\int}$}
\vcenter{\hbox{$#2#3$}}\kern-.5\wd0}}
\def\dashint{\Xint-}
\section{Results}
This note concerns the open question mentioned by Pacard in\cite{P},
especially its regularity criterion for positive weak solutions to
$-\Delta u = u^p$ in a domain $\Omega \subset \mathbb{R}^n$, $p \geq n/(n-2)$, $n \geq 3$.
By this we shall mean: $u \in L_{\rm loc}^p(\Omega)$ and
\begin{equation} \label{eq1}
-\int \Delta\phi \cdot u \, dx = \int u^p \phi \, dx
\end{equation}
for all $\phi \in C^\infty_0 (\Omega)$. The main question here is to describe
the size of the set $\operatorname{Sing}(u) \subset \Omega$ where a solution $u$ becomes
$+\infty$ and such that $u \in C^\infty(\Omega \setminus \operatorname{Sing}(u))$.
Examples where such a set exists includes the simple case
$u(x) = c_0 |\bar{x}|^{-2/(p-1)}$, $x = (\bar{x}, \hat{x})$,
$\bar{x} \in \mathbb{R}^{n-d}$, $\hat{x} \in \mathbb{R}^d$, a solution in the ball $B(0,R)$,
centered at zero of radius $R$, and some constant $c_0$.
Here $\operatorname{Sing}(u) = \mathbb{R}^d \cap B(0,R)$ and necessarily
$d < n - 2p'$, $p' = p/(p-1)$. Note that when $p = n/(n-2)$, it is well
known that \eqref{eq1} can have isolated singularities (here $d = 0$; see \cite{GS}).
Furthermore, $n - 2p' = 0$ when $p = n/(n-2)$, because then $p' = n/2$.
The case $p = (n+2)/(n-2)$, the ``Yamabe case," has been also well studied
in the literature; see \cite{SY}. And several authors have constructed
solutions to \eqref{eq1} with a prescribed singular set $\operatorname{Sing}(u)$; e.g.~\cite{R}, \cite{F}, \cite{MP}. But in all cases, it appears that solutions $u$ to \eqref{eq1} behave like
\begin{equation} \label{eq2}
u(y) \sim \operatorname{dist}(y,\operatorname{Sing}(u))^{-2/(p-1)}
\end{equation}
as $y \to \operatorname{Sing}(u)$ in $\Omega$.
The Pacard conjecture is that the Hausdorff dimension of
$\operatorname{Sing}(u)$ is always $\leq n - 2p'$, which certainly appears
to be the case in all the examples considered. Pacard proves this,
in \cite{P}, under an additional hypothesis, his hypothesis ``H".
However, it soon becomes clear that hypothesis H is much too strong,
for it precludes isolated singularities when $p = n/(n-2)$, and for that
matter any singularities when $n/(n-2) \leq p < (n/(n-2)) + \varepsilon$,
for some $\varepsilon > 0$.
Thus the purpose of this note is to prove:
\begin{theorem} \label{thm1}
Let $u$ be a positive weak solution of \eqref{eq1}, then there exists an open
set $\Omega' \subset \Omega$ such that $u \in C^\infty(\Omega')$ and
$C_{2, p'}(\Omega \setminus \Omega') = 0$.
\end{theorem}
The presentation of this note follows closely that of \cite{P}, so it is
recommended that the reader have a copy of \cite{P} at hand while reading
the present note.
Here $C_{\alpha, p}(\cdot)$ is the capacity set function associated with
the Sobolev space $W^{\alpha, p}(\Omega)$, $\alpha =$ positive integer.
Also, one recalls from \cite{AH} that any set of $C_{2, p'}$-capacity zero
has Hausdorff dimension $\leq n - 2p'$. Furthermore, it is not surprising
that $\operatorname{Sing}(u) = \Omega \setminus \Omega'$ is of
$C_{2, p'}$-capacity zero, given that this condition characterizes removable
sets for equation \eqref{eq1}; see \cite{AP}.
For $p' < n/2$, we can use the standard definition of $C_{2, p'}$ using Riesz
potentials on $\mathbb{R}^n$ especially when $\partial \Omega = $ boundary of
$\Omega$ is smooth. For any compact $K \subset \mathbb{R}^n$
\[
C_{2,p'}(K) = \inf\{\|f\|^{p'}_{L^{p'}} \colon f \geq 0, I_2f \geq 1 \text{ on } K\}.
\]
Here
\[
I_2f(x) = \int_{R^n} |x-y|^{2-n} f(y) \, dy.
\]
Notice this definition easily implies
\begin{equation} \label{eq3}
C_{2,p'}(\{x \colon I_2f \geq \lambda\}) \leq \frac{1}{\lambda^p}
\cdot \|f\|_{L^{p'}}^{p'}.
\end{equation}
The proof of our Theorem constitutes the main body of this note, 1-6.
In 7, 8 and 9, we include further speculations.
1. If $u = u(x)$ is a positive weak solution to \eqref{eq1}, then $u$ belongs
to the Morrey space $L^{p,2p'}(\Omega)$.
\begin{proof}
(This result is due to Pacard \cite{Pa}, and it has also been observed by Brezis.)
The Morrey space in question --- here we extend functions outside $\Omega$ by zero ---
is those $f \in L_{\rm loc}^p(\mathbb{R}^n)$ such that
\[
\Big(\sup_{x \in \mathbb{R}^n,\; r > 0} r^{\lambda - n} \int_{B(x,r)} |f(y)|^p
\, dy\Big)^{1/p} \equiv \|f\|_{L^{p,\lambda}} < \infty,
\]
for $1 \leq p < \infty$, $0 < \lambda \leq n$. Again, recall that we will only
be dealing with the case $p' < n/2$. The case $p' = n/2$ can be handled using
the usual modifications; see \cite{AH}.
So now set $\phi(x) = \eta\left(\frac{x - x_0}{r}\right)^\sigma$, $\eta \in C_0^\infty(B(0,1))$
for $\sigma > 2p'$. Then
\begin{equation} \label{eq4}
\int u^p \eta^{\sigma}
\leq \frac{C}{r^2} \Big( \int u^p \eta^{\sigma} \Big)^{1/p} \cdot r^{n/p'}
\end{equation}
by H\"older's inequality. The result follows.
\end{proof}
2. A modified Pacard Lemma \cite{P}:
\begin{lemma} \label{lem1}
Let $u$ be a positive weak solution of \eqref{eq1}, then there are constants
$c_p$ such that for $x \in \Omega$ and $r$ small
\begin{equation} \label{eq5}
\dashint_{B(x,r)} u^p \leq c_p\Big\{ \Big(\dashint_{B(x,2r)} u^{p-1}\Big)^{p'}
+ \dashint_{B(x,2r)} u(y)^p \Big(\int_{B(y,2r)} |y-z|^{2-n} u(z)^{p-1} \, dz\Big)
dy\Big\}
\end{equation}
for $p \geq 2$, and
\begin{equation} \label{eq6}
\dashint_{B(x,r)} u^p \leq c_p \Big\{ \Big(\dashint_{B(x,2r)} u \Big)^p
+ \dashint_{B(x,2r)} u(y)^p \Big( \int_{B(y,2r)} |y-z|^{2-n} u(z)^{p-1} \, dz
\Big) dy\Big\}
\end{equation}
for $1 < p < 2$.
\end{lemma}
Here, the integrals with a bar denote integral averages.
\begin{proof}
(Outline from \cite{P}.)
Inequalities \eqref{eq5} and \eqref{eq6} follow from the following inequality
for positive weak solutions to \eqref{eq1}; see \cite{P} or \cite{HK}:
\begin{equation} \label{eq7}
u(y) \leq
\dashint_{B(y,r)} u + \frac{r^n}{n(n-2)} \,
\dashint_{B(y,r)} |y-z|^{2-n} u(z)^p \, dz.
\end{equation}
To get our result, simply multiply \eqref{eq7} through by $u^{p-1}$ and
integrate over a ball centered at $x$ of radius $r$.
\end{proof}
This Lemma is important for at least two reasons:
(a) If the quantity
\begin{equation} \label{eq8}
\int_{B(y,R)} |y-z|^{2-n} u(z)^{p-1} \, dz
\end{equation}
can be made uniformly small for $R$ small and all $y$ in some neighborhood
of $x \in \Omega$, then \eqref{eq5} or \eqref{eq6} can be used to
engage the theory of reverse H\"older inequalities; see \cite{G} or \cite{BF}.
In each case, one can then deduce that $u \in L^q$ in that neighborhood of $x$,
where $q > p$. This, it turns out, is the crucial step in proving
$C^\infty$-regularity in that neighborhood. We return to this below in section 6.
(b) It is less than intuitive that the potential $I_2 u^{p-1}$ (or some part of it)
should play a significant role here in describing the pointwise behavior of $u$
near $\operatorname{Sing}(u)$ in $\Omega$. One expects $u = I_2 u^p$ to be of
some service here but not $I_2 u^{p-1}$. Notice that the section 1 result plus
the embeddings of Morrey spaces under the Riesz potential operator $I_2$
imply that $I_2 u^{p-1} \in BMO$, the John-Nirenberg space of functions of
bounded mean oscillations; see \cite{AH} or \cite{A}. This fact alone suggests
that $\exp(c \cdot I_2 u^{p-1})$ might be of interest here. We speculate further
on this in section 8.
Notice that $u(x) = c I_2 u^p(x)$ in $\Omega$ for some constant $c$,
hence
$$
I_2 u^{p-1} = c I_2 (I_2 u^p)^{p-1}.
$$
This is precisely the classical non-linear potential from \cite{AH}; i.e.,
for $(\alpha, p)$:\\ $I_\alpha(I_\alpha \mu)^{p'-1}$, when $\alpha = 2$,
and $p'$ is replaced by $p$,
and the measure $d\mu = u^p \, dx$.
3. $I_2 u^{p-1}(x) < \infty$ implies
\begin{equation} \label{eq9}
\lim_{r \to 0} r^{2p'-n} \int_{B(x,r)} u(y)^p \, dy = 0.
\end{equation}
\begin{proof}
This follows from a fundamental estimate from Nonlinear Potential Theory; see \cite{AH} or \cite{AM}. The estimate is for the so-called ``nonlinear potentials" associated with the capacities $C_{2, p'}$:
\begin{equation} \label{eq10}
I_2 (I_2 u^p)^{p-1}(x) \geq c \cdot W^{u^p\, dy}_{2,p'} (x),
\end{equation}
where the $W$-potential here is the associated Wolff potential
\[
W_{\alpha,p}^\mu (x) \equiv \int_0^\infty [ r^{\alpha p - n} \mu(B(x,r))]^{p'-1}
\, \frac{dr}{r},
\]
for $0 < \alpha < n$, $1 < p < n/\alpha$, and $\mu =$ non-negative Borel measure
on $\mathbb{R}^n$. In \eqref{eq10}, $d\mu = u^p \, dy$. Our result follows
since both $r^{2p' - n}$ and $\int_{B(x,r)} u^p$ are monotone functions of $r$.
It should perhaps be added here that the reverse inequality to \eqref{eq10} may
fail for $p > 2(n-1)/(n-2)$; see \cite{AM}.
\end{proof}
4. $\xi_u(x) =$ the jump discontinuity of $I_2 u^{p-1}$ at $x$ when
$I_2 u^{p-1}(x) < \infty$.
\begin{proof}
Here we compute
\[
\overline{\lim_{y \to x}} I_2 u^{p-1}(y) = \xi_u(x) + I_2 u^{p-1}(x)
\]
where
\begin{equation} \label{eq11}
\xi_u(x) \equiv \overline{\lim_{y \to x}} (n-2) \int_0^{|x-y|} r^{2-n}
\Big(\int_{B(y,r)} u^{p-1}\Big) \, \frac{dr}{r}.
\end{equation}
Notice that $\xi_u(x) = 0$, when $u$ is continuous at $x$.
In fact, Fubini's theorem gives
\begin{equation} \label{eq12}
I_2 u^{p-1}(y) = (n-2)\int_0^\infty r^{2-n} \Big(\int_{B(y,r)} u^{p-1}\Big) \,
\frac{dr}{r}.
\end{equation}
And writing \eqref{eq12} as
$\big(\int_0^{|x-y|} \cdots + \int_{|x-y|}^\infty\big) (n-2)$,
we easily see that the last integral tends to $I_2 u^{p-1}(x)$ as $y \to x$
since $B(y,r) \subset B(x,2r)$ and $I_2 u^{p-1}(x) < \infty$ allows us to use
dominated convergence. Hence the result follows. Note that we also have
\begin{equation} \label{eq13}
\xi_u(x) = \overline{\lim_{y \to x}} \int_{|y-z| < |x-y|} |y-z|^{2-n} u(z)^{p-1} \, dz
\end{equation}
since
\[
\lim_{r \to 0} r^{2-n} \int_{B(x,r)} u(y)^{p-1} \, dy = 0
\]
follows from $I_2 u^{p-1}(x) < \infty$.
\end{proof}
Thus the jump discontinuity $\xi_u(x)$ is generally $\geq 0$
for $x \in \operatorname{Sing}(u)$. But notice that $\xi_\varphi(x) = 0$
for any $\varphi \in C_0^\infty (\mathbb{R}^n)$.
5. $C_{2,p'} (\operatorname{Sing}(u)) = 0$.
\begin{proof}
Here we set
\begin{equation} \label{eq14}
\operatorname{Sing}{}_\lambda(u) = \{x \in \Omega \colon \xi_u(x) \geq \lambda\}.
\end{equation}
And for $\operatorname{Sing}(u)$ needed in our Theorem, we take $\lambda$
in \eqref{eq14} to be $1/(4c_p)$,
$c_p$ the constant in the Pacard Lemma (section 2).
Now if $x \in \operatorname{Sing}(u)$, then for any
$y \in N(x) \cap \operatorname{Sing}(u)$, $N(x) =$ some neighborhood of $x$,
\[
\lambda \leq \xi_u(x) \leq c I_2(|u^{p-1} - \varphi|)(y) + \lambda/2,
\]
hence
\[
C_{2,p'}(N(x) \cap [I_2(|u^{p-1} - \varphi|) > \lambda/2])
\leq \big(\frac{2}{\lambda}\big)^{p'} \|u^{p-1} - \varphi\|^{p'}_{L^{p'}(\Omega)}.
\]
So taking $\varphi$ to be an $L^{p'}$ smooth approximation to $u^{p-1}$
yields $C_{2,p'}(N(x)\cap\operatorname{Sing}(u)) = 0$
and the final result follows due to the countable subadditivity of $C_{2,p'}$;
see \cite{AH}.
\end{proof}
6. Deducing $u \in C^\infty(\Omega \setminus \operatorname{Sing}(u))$.
(Here we follow the path forged by Pacard \cite{P}.)
\begin{proof}
The reason for our choice of $\lambda = 1/(4c_p)$ above now becomes clear:
for $x \in \Omega - \operatorname{Sing}_\lambda(u)$, \eqref{eq8} then does not exceed
$1/(2c_p)$ for some $R > 0$ and all $y$ in a neighborhood of $x$.
This together with the modified Pacard Lemma yields that $u \in L^q$ in
that neighborhood of $x$, for some $q > p$ by the reverse H\"older inequality
theory mentioned earlier. We are now in position to use
Lemmas 4 and 5 from \cite{P}. Using \eqref{eq9}, we have:
there exists constant $\theta \in (0,1)$ such that
\begin{equation} \label{eq15}
\frac{1}{(\theta R)^{n-2p'}} \int_{B(x,\theta R)} u^p
\leq \frac{1}{2} \, \frac{1}{R^{n-2p'}} \int_{B(x,R)} u^p.
\end{equation}
Iterating \eqref{eq15} yields: for such $x$ as above
\begin{equation} \label{eq16}
\frac{1}{(\theta^k R_1)^{n-2p'}} \int_{B(x,\theta^k R_1)} u^p
\leq 2^{-k} \frac{1}{R_1^{n-2p'}} \int_{B(x,R_1)} u^p
\end{equation}
for all $k \in \mathbb{Z}^+$. Now one can choose a $\mu < 2p'$ such that
$\theta^{2p'-\mu} > 1/2$ and derive that in fact in this neighborhood of
$x$ that $u \in L^{p,\mu}$ (note that the notation here differs from
that in \cite{P}, a fact we prefer). And now, as in \cite{P},
we can easily get $u \in C^\infty$ in this neighborhood since $\mu < 2p'$.
\end{proof}
7. We mention a simple regularity criterion that can be used, for example,
to get $u \in C^\infty$ in all of $\Omega$: if $u \in L^{n(p-1)/2,\lambda}(\Omega)$
for some $\lambda < n$, then, in fact, $u \in C^\infty(\Omega)$.
This might be stated as a corollary to the main theorem, for one immediately
sees that this condition implies that $\xi_u(x) = 0$ for all $x \in \Omega$;
i.e., $I_2 u^{p-1}$ is continuous on $\Omega$ and our theory implies then
that $u \in C^\infty (\Omega)$. Notice that this condition also implies that
there are no bounded point discontinuities for $u$ in $\Omega$ (a fact well known),
but this then confirms that indeed $\operatorname{Sing}(u)$ is made up of points
where $u(y) \to +\infty$ as $y \to \operatorname{Sing}(u)$.
And that agrees, of course, with \eqref{eq2}.
8. A conjecture seems to now be in order: there is a function $\beta(x) > 0$
such that for all $x \in [I_2 u^{p-1} = +\infty]$
\begin{equation} \label{eq17}
u(y) \sim \exp\big(\beta(x) I_2 u^{p-1} (y) \big)
\end{equation}
as $y \to x \in \operatorname{Sing}(u)$. Since $I_2 u^{p-1} = I_2 (I_2 u^p)^{p-1}$
and the equivalence of this nonlinear potential with the Wolff potential,
at least for $p < \frac{1}{2}(\frac{n-1}{n-2})$, we expect $\beta(x)$
to be something like
\begin{equation} \label{eq18}
\frac{2}{p-1} \, \frac{1}{D(x)^{p-1}}
\end{equation}
where $D(x) = \underline{\lim}_{r \to 0} r^{2p'-n} \int_{B(x,r)} u^p$,
$x \in [I_2 u^{p-1} = +\infty]$, by comparing this with the examples where
\eqref{eq2} holds.
9. A further conjecture is that one can prove our Theorem for $-\Delta$
replaced by the differential operator $L = -\sum_{i,j}(a_{ij}u_{x_i})_{x_j} + cu$
studied in \cite{AP}.
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\end{document}