\documentclass[reqno]{amsart} \usepackage{hyperref} \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 $$\label{eq1} -\int \Delta\phi \cdot u \, dx = \int u^p \phi \, dx$$ 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 $$\label{eq2} u(y) \sim \operatorname{dist}(y,\operatorname{Sing}(u))^{-2/(p-1)}$$ 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 $$\label{eq3} C_{2,p'}(\{x \colon I_2f \geq \lambda\}) \leq \frac{1}{\lambda^p} \cdot \|f\|_{L^{p'}}^{p'}.$$ 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 $$\label{eq4} \int u^p \eta^{\sigma} \leq \frac{C}{r^2} \Big( \int u^p \eta^{\sigma} \Big)^{1/p} \cdot r^{n/p'}$$ 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 $$\label{eq5} \dashint_{B(x,r)} u^p \leq c_p\Big\{ \Big(\dashint_{B(x,2r)} u^{p-1}\Big)^{p'} + \dashint_{B(x,2r)} u(y)^p \Big(\int_{B(y,2r)} |y-z|^{2-n} u(z)^{p-1} \, dz\Big) dy\Big\}$$ for $p \geq 2$, and $$\label{eq6} \dashint_{B(x,r)} u^p \leq c_p \Big\{ \Big(\dashint_{B(x,2r)} u \Big)^p + \dashint_{B(x,2r)} u(y)^p \Big( \int_{B(y,2r)} |y-z|^{2-n} u(z)^{p-1} \, dz \Big) dy\Big\}$$ 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}: $$\label{eq7} u(y) \leq \dashint_{B(y,r)} u + \frac{r^n}{n(n-2)} \, \dashint_{B(y,r)} |y-z|^{2-n} u(z)^p \, dz.$$ 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 $$\label{eq8} \int_{B(y,R)} |y-z|^{2-n} u(z)^{p-1} \, dz$$ 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 $$\label{eq9} \lim_{r \to 0} r^{2p'-n} \int_{B(x,r)} u(y)^p \, dy = 0.$$ \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 $C_{2, p'}$: $$\label{eq10} I_2 (I_2 u^p)^{p-1}(x) \geq c \cdot W^{u^p\, dy}_{2,p'} (x),$$ 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 $$\label{eq11} \xi_u(x) \equiv \overline{\lim_{y \to x}} (n-2) \int_0^{|x-y|} r^{2-n} \Big(\int_{B(y,r)} u^{p-1}\Big) \, \frac{dr}{r}.$$ Notice that $\xi_u(x) = 0$, when $u$ is continuous at $x$. In fact, Fubini's theorem gives $$\label{eq12} I_2 u^{p-1}(y) = (n-2)\int_0^\infty r^{2-n} \Big(\int_{B(y,r)} u^{p-1}\Big) \, \frac{dr}{r}.$$ 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 $$\label{eq13} \xi_u(x) = \overline{\lim_{y \to x}} \int_{|y-z| < |x-y|} |y-z|^{2-n} u(z)^{p-1} \, dz$$ 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 $$\label{eq14} \operatorname{Sing}{}_\lambda(u) = \{x \in \Omega \colon \xi_u(x) \geq \lambda\}.$$ 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 $$\label{eq15} \frac{1}{(\theta R)^{n-2p'}} \int_{B(x,\theta R)} u^p \leq \frac{1}{2} \, \frac{1}{R^{n-2p'}} \int_{B(x,R)} u^p.$$ Iterating \eqref{eq15} yields: for such $x$ as above $$\label{eq16} \frac{1}{(\theta^k R_1)^{n-2p'}} \int_{B(x,\theta^k R_1)} u^p \leq 2^{-k} \frac{1}{R_1^{n-2p'}} \int_{B(x,R_1)} u^p$$ 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 $x \in [I_2 u^{p-1} = +\infty]$ $$\label{eq17} u(y) \sim \exp\big(\beta(x) I_2 u^{p-1} (y) \big)$$ 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 $$\label{eq18} \frac{2}{p-1} \, \frac{1}{D(x)^{p-1}}$$ 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}. \begin{thebibliography}{10} \bibitem{A} D. R. Adams. \newblock A note on {R}iesz potentials. \newblock {\em Duke Math. J.}, 42(4):765--778, 1975. \bibitem{AH} D. R. 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