p$ and that $\varphi$ is a piecewise $C^1$ function such that $0 \leq \varphi \leq d({\bf x})^{-1/c}$. Then for any $\zeta \in W_0^{1,p}(\Omega)$: $$\int_\Omega {\varphi}^{p^2} |\zeta|^p d^{N}x \leq (c_p)^{p^2/c} \left(\int_\Omega |\nabla \zeta|^p d^{N}x \right)^{p/c} \left(\int |\zeta|^p d^{N}x \right)^{1-p/c}.$$ \noindent {\bf Proof:} Because $\varphi({\bf x})\leq d({\bf x})^{-1/c}$, $$\int_\Omega {\varphi}^{p^2} |\zeta|^p \le \int_\Omega d^{-p^2/c}|\zeta|^{p^2c^{-1}+p(c-p)c^{-1}},$$ which by H\"older's inequality is bounded by $$\left(\int_\Omega{|\zeta|^p\over d^p}\right)^{p/c} \left(\int_\Omega |\zeta|^p\right)^{1-p/c}.$$ With the Hardy inequality (3.3), we therefore obtain: $$\int_\Omega {\varphi}^{p^2}\zeta^p\le (c_p)^{p^2/c} \left(\int_\Omega |\nabla \zeta|^p\right)^{p/c} \left(\int_\Omega |\zeta|^p\right)^{1-p/c}.$$ \hfill $\diamondsuit$ \proclaim{Lemma III.3}. Let $\hat m$ and $\hat k$ be such that (A) holds, and let $\varphi$ be any piecewise $C^1$ function such that $0 \leq \varphi \leq d({\bf x})^{-1/c}$. %and suppose that $c \geq p$. Then for any $\zeta \in W_0^{1,p}(\Omega)$ such that $\Delta_p \zeta \in L^{p'}(\Omega)$: $$\eqalign{ \int_\Omega |\nabla (\varphi \zeta)|^p d^{N}x &\le \hat m^p\int_\Omega |\zeta\nabla \omega|^p d^{N}x + \hat k(c_p)^{p/c} \left(\int_\Omega |\nabla \zeta|^p d^{N}x \right)^{1/c}\cr &\qquad\times\left(\int_\Omega |\zeta|^p d^{N}x \right)^{p^{-1}-c^{-1}}\left(\int_\Omega \left|-\Delta_p \zeta\right|^{p'} d^{N}x \right)^{1/p'}. }$$ \noindent {\bf Proof:} From Lemma III.1 we know that $$\int_\Omega |\nabla (\varphi \zeta)|^p\le \hat m^p\int_\Omega |\zeta\nabla \varphi|^p+ \hat k\int_\Omega \zeta \varphi^p(-\Delta_p \zeta).$$ Recall that $c \geq p$. If $c>p$, then by H\"older's inequality and Lemma III.2, $$\eqalignno{ \left| \int_\Omega \zeta \varphi^p(-\Delta_p \zeta)\right|&\le \left(\int_\Omega \zeta^p \varphi^{p^2} \right)^{1/p}\left(\int_\Omega \left|-\Delta_p \zeta\right|^{p'}\right)^{1/p'} &(3.4)\cr &\le (c_p)^{p/c}\left(\int_\Omega |\nabla \zeta|^p\right)^{1/c} \left(\int_\Omega |\zeta|^p\right)^{1/p-1/c} \left(\int_\Omega \left|-\Delta_p \zeta\right|^{p'}\right)^{1/p'},&\cr }$$ yielding the claim. \noindent For $c=p$, since $\varphi^{p^2}\le d^{-p^2/c}=d^{-p}$ we have $$ \left| \int_\Omega( \zeta \varphi^p)(-\Delta_p\zeta) \right| \le \left(\int_\Omega |\zeta|^p \varphi^{p^2}\right)^{1/p} \left(\int_\Omega \left|-\Delta_p \zeta\right|^{p'}\right)^{1/p'},$$ which by Lemma III.2 is bounded by $$c_p\left(\int_\Omega |\nabla \zeta|^p\right)^{1/p} \times \left(\int_\Omega \left|-\Delta_p \zeta\right|^{p'}\right)^{1/p'}.$$ Hence the same inequality holds in this case. \hfill $\diamondsuit$ \smallskip Our next result, Theorem III.4, shows that integrals involving $\zeta$ on an $\epsilon$-neighborhood of the boundary are bounded by expressions of the form $F\cdot \epsilon^s$, where $F$ depends only on $\Omega$, $\Vert \zeta \Vert_p$, $\Vert \nabla\zeta \Vert_p$, and $\Vert \Delta_p\zeta \Vert_{p'}$. When $p=2$, and $\partial \Omega$ is smooth, our exponents $s$ reduce to the sharp values as remarked in [D2]. We adopt some notation and other conventions of [D2]; in particular, for a given $\varepsilon > 0$, we define $$\omega ({\bf x}) = {\left({\max\{d({\bf x}),\varepsilon \}}\right)} ^{-1/c} \eqno(3.5) $$ and $$\tau \left({ \bf x}\right) = \cases{ \varepsilon^{-1/c} & if $0 < d(\bf x ) \le \varepsilon$ \cr {c}^{ -1}{\varepsilon }^{ -1-1/c}\left({\left({ 1+c}\right)\varepsilon -d({\bf x} )}\right) & if $\varepsilon < d({\bf x}) \le \left({1+c}\right)\varepsilon$ \cr 0 & otherwise.\cr}\eqno(3.6)$$ \noindent (Recall that $c = \widehat{m} c_p$ with $\widehat{m}$ appearing in (A) and $c_p$ in (3.3). We remark that both functions $\omega$ and $\tau$ satisfy the conditions of the functions $\varphi$ appearing in Lemma III.1--Lemma III.3.) \proclaim{Theorem III.4}. There are (identifiable) constants $K_{1,2}$ such that given any $\zeta \in W_0^{1,p}(\Omega)$ such that $\Delta_p \zeta \in L^{p'}(\Omega)$: $$ \int_{\{ d({\bf x})<\epsilon\}\cap \Omega} {|\zeta|^p\over d^p} d^{N}x \leq \leqno(i)$$ $$ K_{1}\epsilon^{p/c} \left(\int_\Omega |\nabla \zeta|^p d^{N}x \right)^{1/c} \left(\int_\Omega |\zeta|^p d^{N}x \right)^{p^{-1} - c^{-1}} \left(\int_\Omega (-\Delta_p \zeta)^{p'} d^{N}x \right)^{1/p'}$$ for all $\epsilon > 0$. Hence also, $$ \int_{\{ d({\bf x})<\epsilon\}\cap \Omega} {|\zeta|^p} d^{N}x \leq \leqno(ii)$$ $$ K_{1} \epsilon^{p+p/c} \left(\int_\Omega |\nabla \zeta|^p d^{N}x \right)^{1/c} \left(\int_\Omega |\zeta|^p d^{N}x \right)^{p^{-1} - c^{-1}} \left(\int_\Omega (-\Delta_p \zeta)^{p'} d^{N}x \right)^{1/p'}$$ for all $\epsilon > 0$. In addition, $$ \int_{\left\{{d\left({\bf x}\right)\bf \le \varepsilon }\right\}}{\left|{\nabla \zeta }\right|}^{p}{d}^{N}x \le {K}_{2} F {\varepsilon }^{p/c}, \leqno(iii) $$ where $F$ depends only on $\Omega$, $\Vert \zeta \Vert_p$, $\Vert \nabla\zeta \Vert_p$, and $\Vert \Delta_p\zeta \Vert_{p'}$ (and is implicitly specified by the last few lines of the proof). Recall that $c = {\hat m}c_{p}$. \smallskip \noindent {\bf Proof:} We deduce from Lemmas III.2 and III.3 that $$ \int_\Omega {|\omega \zeta|^p\over d^p} \le (\hat m c_p)^p\int_\Omega | \zeta \nabla \omega|^p + I, \eqno(3.7) $$ where $$I =\hat k(pc_p)^{p+p/c} \left(\int_\Omega |\nabla \zeta|^p\right)^{1/c}\left(\int_\Omega |\zeta|^p \right)^{p^{-1} - c^{-1}}\left(\int_\Omega |-\Delta_p \zeta|^{p'}\right)^{1/p'} .$$ Let $Y({\bf x})={\omega^p\over d^p}-c^p|\nabla \omega|^p$. For $d({\bf x})\ge \epsilon$, $|\nabla \omega| = {1\over c} {\omega\over d};$ hence $Y({\bf x})\ge 0$, and for $d({\bf x})<\epsilon$, $\nabla \omega({\bf x})=0$, so $Y({\bf x})\ge {1\over \epsilon^{p/c}d^p}$. Rewriting (3.7) as $$\int_\Omega |\zeta|^p Y \le I$$ we deduce that $$\int_{\{d({\bf x})<\epsilon\}\cap \Omega} {|\zeta|^p\over d^p}\le \hat k(c_p)^{p+p/c} \epsilon^{p/c}I,$$ and hence we have part (i), from which (ii) is immediate. For part (iii), we first note that $$ \int_{\left\{{d\left({\bf x}\right)\bf < \varepsilon }\right\}}{\left|{\nabla \zeta }\right|}^{p}{d}^{N}x \le {\varepsilon }^{p/c}\int_{\Omega }{\left|{\nabla (\tau \zeta) }\right|}^{p} {d}^{N}x, \eqno(3.8) $$ and then apply Lemma III.1 to conclude that $$ \eqalign{ \int_{\Omega }{\left|{\nabla (\tau \zeta) }\right|}^{p}{d}^{N}x \le& {\widehat{m}}^{p}\int_{\left\{{d\left({\bf x}\right)<(1+c)\varepsilon }\right\}}{\left|{\zeta \nabla \tau }\right|}^{p}{d}^{N}x + \widehat{k}{c}_{p}^{p/c}{\left({\int_{\Omega }{\left|{\nabla \zeta }\right|}^{p} {d}^{N}x}\right)}^{1/c} \times \cr &{\left({\int_{\Omega }{\left|{\zeta }\right|}^{p} {d}^{N}x}\right)}^{1/p-1/c}{\left({\int_{\Omega }{\left|{-{\Delta }_{p}\zeta }\right|}^{p\prime}{d}^{N}x}\right)}^{1/p\prime}.\cr} $$ Now, $$\int_{\left\{{d\left({\bf x}\right)<(1+c)\varepsilon }\right\}}{\left|{\zeta \nabla \tau }\right|}^{p}{d}^{N}x \le {\left({{1 \over c{\varepsilon }^{1+1/c}}}\right)}^{p}\int_{\left\{{d\left({\bf x}\right)<(1+c)\varepsilon }\right\}}{\left|{\zeta }\right|}^{p}{d}^{N}x,$$ which is bounded by quantities independent of $\epsilon$ according to part (ii). Together with (3.8), this yields (iii). \hfill $\diamondsuit$ \smallskip Next we obtain a similar estimate for (3.1) for nonzero $V({\bf x})$, for which the coefficient of $\epsilon^s$ is given in terms of $\|\zeta\|_p, R(\zeta)$, and $\| - \Delta_p \zeta + V({\bf x}) |\zeta|^{p-2} \zeta \|_{p'}$. We shall assume that $V({\bf x}) = V_{1}({\bf x}) + V_{2}({\bf x})$, where $V_{1}({\bf x}) \ge 0$ and there exist finite constants $A,B,\alpha,\beta$, with $\alpha < 1$, such that $|V_2|$ satisfies $$ \int_{\Omega} |V_2|^{p'} |\zeta|^p d^Nx \leq A \int_{\Omega} |\nabla \zeta|^p d^Nx + B \int_{\Omega} | \zeta|^p d^Nx \leqno(i) $$ and $$ \displaylines{ \rlap{(ii)}\hfill \int_{\Omega} |V_2| |\zeta|^p d^Nx \leq \alpha \int_{\Omega} |\nabla \zeta|^p d^Nx + \beta \int_{\Omega} | \zeta|^p d^Nx \hfill\llap{(3.9)} \cr} $$ for all $\zeta \in C_{c}^{\infty}(\Omega)$. We remark that using the results of Section II, (3.9) will hold, for example, provided that $|V_2|^{p'} < {C_1\over d^p} +\hbox{bounded function} \Leftrightarrow |V_2|< C_2 d^{-(p-1)}+ \hbox{bounded function}$ for some constants $C_{1,2}$, since this implies that $|V_2|<{1\over c^p_p} \, {1\over d^p} + \hbox{bounded function}$. \proclaim{Theorem III.5}. Given Hardy's inequality $(3.3)$ with $c=\hat m c_p>p$, assume that $V$ satisfies (3.9) and that $\zeta \in W_{0}^{1,p}$ with $-\Delta _p\zeta + V |\zeta|^{p-2} \zeta \in W_{0}^{1,p} \cap L^{p'}(\Omega)$. Then there are quantities $F_{1,2}$ depending only on $\Omega$, $\|\zeta\|_p, R(\zeta),$ and $\| - \Delta_p \zeta + V({\bf x}) |\zeta|^{p-2} \zeta \|_{p'}$ such that $$\int_{\{ d({\bf x})<\epsilon\}\cap \Omega} {|\zeta|^p\over d^p} d^{N}x \leq F_{1} \epsilon^{p/\hat mc_p} \leqno(i)$$ for all $\epsilon > 0$. Hence also, $$\int_{\{ d({\bf x})<\epsilon \}\cap \Omega} {|\zeta|^p} d^{N}x \leq F_{1} \epsilon^{p+p/\hat mc_p} \leqno(ii) $$ for all $\epsilon > 0$. In addition, $$ \int_{\left\{{d\left({\bf x}\right)\bf \le \varepsilon }\right\}}{\left|{\nabla \zeta }\right|}^{p}{d}^{N}x\le {K}_{2} F_{2} {\varepsilon }^{p/c}. \leqno(iii)$$ \noindent {\bf Proof:} We proceed as in the proof of Theorem III.4 until the stage where we call on Lemma III.3. Instead of dominating $\int\zeta \omega^p(-\Delta_p\zeta)$ as in $(3.4)$, we bound it above by $$ \displaylines{ \int\zeta \omega^p(-\Delta _p\zeta + V_1 |\zeta|^{p-2}\zeta) \cr \le \left(\int |\zeta|^p \omega^{p^c}\right)^{1/p} \left(\|-\Delta_p\zeta + V|\zeta|^{p-2}\zeta\|_{p'} + \|V_2|\zeta|^{p-2}\zeta\|_{p'}\right). } $$ The claim requires that we control the final term, which to the $p'$ power is $$ \eqalign{ \int |V_2|^{p'}|\zeta|^p \le& A\int|\nabla\zeta|^p + B\int|\zeta|^p \cr \le& A\left(\int|\nabla\zeta|^p+V|\zeta|^p+|V_2\zeta^p|\right)+B\|\zeta\|^p_p \cr \le& (A R(\zeta) + B)\|\zeta\|^p_p+A\int|V_2\zeta^p|, \cr} $$ so it remains to control $\int|V_2\zeta^p|$. This we do using part (ii) of (3.7) as follows. $$\int|V_2\zeta^p|\le\alpha \left(\int |\nabla\zeta|^p + V|\zeta|^p+|V_2\zeta^p|\right) +\beta\int| \zeta|^p,$$ so $$\int |V_2\zeta^p|\le {1\over 1-\alpha} \, (\alpha R(\zeta)+ \beta) \|\zeta\|^p_p.$$ \hfill $\diamondsuit$ \bigbreak \centerline{\bf IV. Some inequalities} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip \noindent In this section we establish a family of elementary but refined algebraic inequalities, needed to apply the estimates of Section III to the $p$-Laplacian for various values of $p$. First we establish some algebraic inequalities for a binomial in a scalar real variable $x$, taken to the power $p$. Then we use them to derive vectorial inequalities which imply the basic algebraic bound (A) of Section III. \proclaim{Lemma IV.1}. For $p \ge 2$ and $x \in {\Bbb R}$, $${\left|{x-1}\right|}^{p}\le { \left( {p-1} \right) }^{p}+{p}^{2-p} {\left({p-1}\right)}^{p-1}\left({{\left|{x}\right|}^{p}-p {\left|{x}\right|}^{p -2}x}\right).\eqno(4.1)$$ \smallskip \noindent {\bf Remark:} Essentially we dominate the left side by a constant plus two terms from its expansion for large $|x|$. The inequality is sharp in the sense that the constant $(p-1)^p$ on the right is minimal. \smallskip \noindent {\bf Proof:} Because of the absolute values, we need to consider separately three cases, $1 < x$, $0 \leq x \leq 1$, and $x < 0$. \noindent Case 1. For $0 < x < 1$, we let $$f_{2}(x) = (1-x)^{-p} [(p-1)^p + p^{p-2}(p-1)^{p-1}(x^p-px^{p-1})],$$ and calculate the derivative $$f_{2}'(x) = p^{3-p} (p-1)^p (1-x)^{-p-1} [p^{p-2}-x^{p-2}] > 0,$$ so the minimal value of $f_{2}$ on this interval is $f_{2}(0) = (p-1)^p \ge 1$. \noindent Case 2. For $1 < x $, we claim that $$f_{1}(x) = (x-1)^{-p} [(p-1)^p + p^{p-2}(p-1)^{p-1}(x^p-px^{p-1})]$$ achieves its unique minimum for $x=p$. This is because a calculation reveals that $$f_{1}'(x) = p^{3-p} (p-1)^p (x-1)^{-p-1} [x^{p-2}-p^{p-2}],$$ which is zero uniquely for $x=p$ and otherwise has the same sign as $x-p$. \noindent Case 3. For convenience, for the case when $x < 0$, we replace $x$ by $-x$. Thus we need to show that for $x > 0$, $$(1+x)^{p}\le { (p-1)}^{ p} + {p}^{2-p}{(p-1)}^{p-1}\left({x^ {p}+px^{p-1}}\right),\eqno(4.2)$$ or in other words that $$f_{3}(x) := {{ (p-1)^p + p^{2-p} (p-1)^{p-1}(x^p+px^{p-1})}\over{(1+x)^p}} \geq 1. \eqno(4.3)$$ Again we differentiate, finding $$f_{3}'(x)=p^{3-p}(p-1)^p(1+x)^{-p-1}(x^{p-2}-p^{p-2}),$$ which reveals that $f_{3}'$ vanishes uniquely at p and elsewhere has the same sign as $x-p$. Hence $f_{3} (x) \geq f_{3} (p)= (2p-1) \left({{p-1}\over{p+1}}\right)^{p-1}$. It remains to show that $f_3(p)\geq 1$, or equivalently that $f_4(y)\geq 1$ for $y\geq 2$ where $$f_4(y) = (2y-1)\left({{y-1}\over{y+1}}\right)^{y-1}.$$ We note that $f_4(2)=1$. We prove now that $f_4'>0$: $$f_4'(y)= f_4(y) B(y),$$ where $B(y)={{2}\over{2 y - 1}} + {{2}\over{y+1}} + Ln\left({{y-1}\over{y+1}}\right)$. Hence we wish to prove that $B(y) > 0$, which is true for $y=2$. Now, $$B'(y)=4 {{N(y)}\over{D(y}},$$ with $D(y)=(2y-1)^2(y+1)^2(y-1)>0$ and $N(y)=-y^3+3y^2-3y+2$. Since $N'(y)=-3(y-1)^2<0$, $N\leq 0$ and thus $B'(y)<0$, i.e., B is a decreasing function. As $y$ tends to $\infty$, $B(y)\rightarrow 0$. Hence $B > 0$ and $f_4'> 0$ for $y > 2$. Therefore $f_4(y)\geq 1$ for all $y \geq 2$. \hfill $\diamondsuit$ \proclaim{Lemma IV.2}. For $p \le 2$ and $x \in {\Bbb R}$, $${\left|{x-1}\right|}^{p} \le m_{p}^{p}+ \left({{\left|{x}\right|}^{p} - p{\left|{x}\right|}^{p -2}x}\right), \eqno(4.5)$$ where $ m_{p}^{p}$ is defined by $$m^p_p=\max_{0\le x\le 1}((p-x)x^{p-1}+(1-x)^p). \eqno(4.6)$$ {\bf Remarks:} In comparison with Lemma IV.1, for $p \ge 2$, the second constant on the right has been simplified to $1$, while the first one has a different form. Both sharp inequalities trivialize to the same identity for $(x-1)^2$ when $p$ becomes $2$. \noindent Observe that $m_2^2=\max(1)=1$, and that if ${h}_{p}\left({x}\right):=(p-x){x}^{p-1}+{(1-x)}^{p},$ then $m_{p}^{p} \ge \max (h(0), h(1)) = \max (1, p-1)$. \smallskip \noindent {\bf Proof:} We need to show $|x-1|^p\le m^p_p+ (|x|^p-p|x|^{p-2}x)$ for $x\in {\Bbb R}$. As before, we consider three cases. \noindent Case 1. $0\le x\le 1$. The desired bound holds by the definition of $m^p_p$. \noindent Case 2, $x \ge 1$. Let $$\phi=(x-1)^p, \qquad\psi=m^p_p+x^p-px^{p-1}.$$ We see that $\phi (1)=0 < \psi (1)$ and define $$r:={\psi'\over \phi'}={(x-(p-1))x^{p-2}\over (x-1)^{p-1}}.$$ It is easy to see that $\lim\limits_{x\downarrow 1}r(x)= +\infty$ and $\lim\limits_{x\rightarrow \infty }r(x)= 1$, and to calculate that $r'(x) =\hbox{(positive)} \times (p-2)<0$ on this interval. Thus $r>1$, which implies the bound in this case. \noindent Case 3. $x < 0.$ As before, it is convenient to redefine $x\leftrightarrow -x$ and compare the functions $$\phi=(1+x)^p \hbox{ and }\psi=m^p+x^p+px^{p-1}$$ for $x > 0$. We define $r = \psi'/\phi'$, and calculate as for case 2 that $r'=\hbox{positive} \times (p-2) < 0$. By examining the limits $\lim\limits_{x\downarrow 0}r(x)= +\infty$ and $\lim\limits_{x\rightarrow \infty }r(x)= 1$, we conclude that $r(x) > 1$ on this interval, implying the desired bound. \hfill $\diamondsuit$ We now proceed to deduce vectorial inequalities from the scalar inequalities of Lemma IV.1 and Lemma IV.2. \proclaim{Lemma IV.3}. For $p>q>1$, the following inequalities hold $\forall$ ${\bf Y}\in{\Bbb R}^N$, $$\|{\bf Y}\|_p \underbrace{\le}_{(1)} \|{\bf Y}\|_q\underbrace{\le}_{(2)} N^{(p-q)/pq}\|{\bf Y}\|_p.$$ where $\|{\bf Y}\|_p = \left\{ \sum\limits^N_{i=1} |y_i|^p\right\}^{1/p}$. \noindent {\bf Proof:} (1) By a homothety, it is sufficient to consider the case $$\displaylines{ \sum\limits^N_{i=1} |y_i|^q\ge 1\quad\hbox{with}\quad |y_i|\le 1,\forall \, i=1,\dots, N \cr \sum^N_{i=1} |y_i|^p\le \sum^N_{i=1} |y_i|^q \cr} $$ so that $$\left(\sum^N_{i=1} |y_i|^p\right)^q\le \left(\sum^N_{i=1}|y_i|^q\right)^q \le \left(\sum^N_{i=1} |y_i|^q\right)^p\Rightarrow \|{\bf Y}\|_p\le\|{\bf Y}\|_q$$ \noindent (2) Letting $x_i=|y_i|^q$, by convexity we have $$ \displaylines{ {x_1+\cdots + x_N\overwithdelims ()N}^{p/q}\le {1\over N}\left( x_1^{p/q} + \cdots + x_N^{p/q}\right) \cr {1\over N^{p/q}}\, (|y_1|^q+\cdots + |y_N|^q)^{p/q}\le {1\over N} (|y_1|^p+\cdots + |y_N|^p) \cr \| {\bf Y}\|_q \le (N^{p/q-1})^{1/p} \| {\bf Y}\|_p=N^{(p-q)/pq}\| {\bf Y}\|_p. \cr}$$ \hfill $\diamondsuit$ \noindent {\bf Remarks:} The constant $1$ in (1) is optimal: take $y_2=\cdots = y_N=0$. The constant $N^{(p-q)/pq}$ in (2) is likewise optimal: take $y_1=y_2=\cdots =y_N=1$; in that case, (2) becomes $N^{1/q}\le N^{(p-q)/pq}N^{1/p}$. \proclaim{Lemma IV.4}. Suppose that for $m\ge 1$ and $k>0$ it has been established that $$\forall y,z\in{\Bbb R}: |y-z|^p\le m^p|z|^p+k|y|^p-kp|y|^{p-2}yz. \eqno(4.7)$$ Then the following inequalities hold for any ${\bf Y}$ and ${\bf Z}\in{\Bbb R}^n$: \item{(i)} For $p\ge 2$, $\| {\bf Y}-{\bf Z}\|^p_2\le 2^{(p/2)-1} \left\{ m^p\| {\bf Z}\|^p_2+k\|{\bf Y}\|^p_2-kp\|{\bf Y}\|^{p-2}_2{\bf Y} \cdot {\bf Z}\right\}$ \item{(ii)} For $1

p$, from Lemma IV.3 we find $$\eqalign{ \| {\bf Y}-{\bf Z}\|^p_2&=\left\{ (z_1-1)^2+z^2_2\right\}^{p/2}\le |z_1-1|^p+|z_2|^p\cr &\le m^p(|z_1|^p+|z_2|^p)+k-kpz_1\qquad\hbox{from (4.7)}\cr &\le m^p2^{1-(p/2)} (z^2_1+z^2_2)^{p/2} + k-kpz_1.}$$ From the second relation of Lemma IV.3, we obtain here: $$(|z_1|^p+|z_2|^p)^{1/p}\le 2^{(1/p)-(1/2)}(z^2_1+z^2_2)^{1/2},$$ and hence $$\|{\bf Y}-{\bf Z}\|^p_2\le m^p2^{1-(p/2)} \| {\bf Z}\|^p_2+k\|{\bf Y}\|^p_2-kp\|{\bf Y}\|^{p-2} {\bf Y} \cdot {\bf Z}.$$ \hfill $\diamondsuit$ By combining the lemmas of this section, we obtain the estimates needed for Section III. \proclaim{Proposition IV.5}. For any ${\bf X}$ and ${\bf Z}\in{\Bbb R}^n$, \noindent (i) For $p\ge 2$: $$\| {\bf X}+{\bf Z}\|^p_2\le 2^{(p/2)-1} \left\{ (p-1)^p\| {\bf X}\|^p_2+p^{2-p}(p-1)^{p-1} \left({\|{\bf Z}\|^p_2 + p \|{\bf Z}\|^{p-2}_2{\bf Z} \cdot {\bf X}}\right) \right\}.$$ \noindent (ii) For $1

\varepsilon\}$. We shall find it convenient to define $\Gamma_{\varepsilon} = \{{\bf x} \in \Omega/ d({\bf x})<\varepsilon\}$ and $S_{\varepsilon} = \Omega_{\varepsilon} \cap \Gamma_{2\varepsilon}.$ \par \noindent We denote by $\lambda_1(\Omega) $ the first eigenvalue of the Dirichlet $p$-Laplacian on $\Omega$. By the variational principle, we have $$ \lambda_1(\Omega) \leq \lambda_1(\Omega_{\varepsilon}) .$$ Our main result in this section is the following \proclaim{Theorem V.1}. There exists a positive constant $k$ depending only on $p$, $N$, and $\Omega$, such that for $\varepsilon$ sufficiently small, $$\lambda_1(\Omega_{\varepsilon}) \leq \lambda_1(\Omega) + k \varepsilon^{{p\over{\hat m c_p}}}.$$ \noindent {\bf Proof:} We introduce $\mu: \Omega \longrightarrow [0; +\infty)$ defined by $$ \mu({\bf x}) = \cases{ 0 & if ${\bf x} \in \Gamma_{\varepsilon}$, \cr \varepsilon^{-1} (d({\bf x})-\varepsilon) & if ${\bf x} \in S_{\varepsilon}$, \cr 1 & if ${\bf x} \in \Omega_{2\varepsilon}$. \cr} $$ Let $\phi_1$ be the first eigenfunction of the Dirichlet $p$-Laplacian on $\Omega$ such that $\Vert \phi_1\Vert_{L^p}=1$. We have $$\eqalign{ \int_{\Omega} \left(|\nabla (\mu\phi_1)|^p - |\nabla \phi_1|^p\right) =& \int_{\Gamma_{2\varepsilon}} \left(|\nabla (\mu\phi_1)|^p - |\nabla \phi_1|^p\right) \cr \leq& \int_{S_{\varepsilon}}\left(|\nabla (\mu\phi_1)|^p - |\nabla \phi_1|^p \right) \cr \leq& \int_{S_{\varepsilon}}\left[\left(|\nabla \phi_1| + |{\phi_1\over \varepsilon} | \right)^p - |\nabla \phi_1|^p \right] \cr \leq & p \int_{S_{\varepsilon}} |{\phi_1\over \varepsilon} | \left(|\nabla \phi_1| + |{\phi_1\over \varepsilon} | \right)^{p-1} \cr \leq & K\int_{S_{\varepsilon}} |{\phi_1\over \varepsilon} |^p + K \left(\int_{S_{\varepsilon}} |{\phi_1\over \varepsilon} |^p\right)^{{1\over p}} \left(\int_{S_{\varepsilon}}|\nabla \phi_1|^p\right)^{{1\over p'}}.\cr }$$ From Theorem III.4, we deduce that $$\int_{\Omega} \left(|\nabla (\mu\phi_1)|^p - |\nabla \phi_1|^p \right) \leq K' \varepsilon^{{p\over {\hat m c_p}}} + K''\varepsilon^{{p\over {\hat m c_p}}\left({1\over p}+ {1\over p'}\right)} \leq K \varepsilon^{{p\over {\hat m c_p}}}.$$ Hence $$\int_{\Omega} |\nabla (\mu\phi_1)|^p \leq \lambda_1(\Omega) + K\varepsilon^{{p\over {\hat m c_p}}}.$$ From the variational principle we conclude that $$\int_{\Omega} |\nabla (\mu\phi_1)|^p \geq \lambda_1(\Omega_{\varepsilon}) \int_{\Omega}|\mu\phi_1|^p . $$ Now, $$\eqalign{ \int_{\Omega}|\phi_1|^p =& \int_{\Omega}|\mu\phi_1+(1-\mu)\phi_1|^p \cr \leq& \int_{\Omega}|\mu\phi_1|^p + \int_{\Omega}(1-\mu)^p|\phi_1|^p \cr \leq& \int_{\Gamma_{2\varepsilon}}|\phi_1|^p + \int_{\Omega}|\mu\phi_1|^p \cr \leq& K \varepsilon^{{p\over {\hat m c_p}}+p}+ \int_{\Omega}|\mu\phi_1|^p .\cr} $$ Thus $$ \int_{\Omega} |\nabla (\mu\phi_1)|^p \geq \lambda_1(\Omega_{\varepsilon}) \left[1-K\varepsilon^{{p\over {\hat m c_p}}+p}\right],$$ and hence for $\varepsilon$ sufficiently small $$\eqalign{ \lambda_1(\Omega_{\varepsilon}) \leq & {{\lambda_1(\Omega)+K\varepsilon^{{p\over {\hat m c_p}}}} \over {1 - K\varepsilon^{p+{p\over {\hat m c_p}}}}} \cr \leq & \lambda_1(\Omega)+K(1+2\lambda_1(\Omega)) \varepsilon^{{p\over {\hat m c_p}}} \cr \leq & \lambda_1(\Omega) + k \varepsilon^{{p\over{\hat m c_p}}}. } $$ \hfill $\diamondsuit$ Estimates of this type apply, with the same power of $\varepsilon$ under conditions as in Section III, to the $p$-Laplacian with a potential. \bigbreak \centerline{\bf VI. $L^s(\Omega)$ estimates for solutions of $|u|^{p-2}{u}_{t}= {\Delta }_{p}u-V({\bf x})|u|^{p-2} u$} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip \noindent In this section we turn our attention to the Cauchy problem for evolution equations of the form $$|u|^{p-2}{u}_{t}= {\Delta }_{p}u-V({\bf x})|u|^{p-2} u. \eqno(6.1) $$ The reason for the factor $|u|^{p-2}$ on the left side is that it guarantees that the equation is homogeneous (see the definition $(1.3)$ of the $p$-Laplacian). In this section, we assume that $V({\bf x}) = V_{1}({\bf x}) + V_{2}({\bf x})$, where $V_{1}({\bf x}) \ge 0$ and $|V_2|$ satisfies a bound of the form $$\int_{\Omega }\left|{V_2}\right|{\left|{\zeta }\right|}^{p}\ {d}^{N}x \le \alpha \int_{\Omega }{\left|{\nabla \zeta }\right|}^{p} {d}^{N}x+\beta \int_{\Omega }{\left|{\zeta }\right|}^{p}\ {d}^{N}x, \eqno(6.2) $$ with $\alpha < \infty$. We recall that in Section II we provided some criteria for this bound; for instance, by Corollary II.4, if $N>p$, then the negative part of $V({\bf x})$ may be bounded in magnitude by a sufficiently small constant, proportional to $\alpha$, times a sum of terms with local divergences of the form ${1 \over {\left|{\bf x-{\bf x}_{0}}\right|}^{p}}$. Belyi and Semenov [B2] and Liskevich [L1] have shown that for certain linear differential operators the growth in time $t$ of $\|u(t,x)\|_{L^p(\Omega)}$ can be estimated when the negative part of $V$ is relatively form bounded. In this section we show that similar estimates are valid for solutions of $(6.1)$. We consider only classical solutions of $(6.1)$ on regular domains, with vanishing Dirichlet boundary conditions, and content ourselves with two theorems, which sufficiently well illustrate the idea. \proclaim{Theorem VI.1}. Assume that u is a classical solution of equation (6.1), $u$ belongs to $W_{0}^{1,p}(\Omega) \cap L^s(\Omega)$, $s \geq p$, and $-\Delta_p u \in L^{\infty}(\Omega)$. Assume moreover that the potential $V({\bf x})$ satisfies (6.2) with $\alpha \le \left({s+1-p}\right){\left({{p \over s}}\right)}^{p}$. Let ${f}_{s,u}\left({t}\right):={\| u\left({t;\bf x}\right)\| }_{{L}^s\left({\Omega }\right)}$. Then $${f}_{s,u}\left({t}\right)\le {f}_{s,u}\left({0}\right)\exp \left({\beta t}\right).$$ \noindent {\bf Proof:} We write $r = s - p$ and multiply (6.1) by $|u|^{r}u$ and integrate. We find $$\eqalign{ {1\over p+r} \, {d\over dt}\int_\Omega|u|^{p+r} &= \int\left\{ |u|^ru\nabla\cdot (|\nabla u|^{p-2}\nabla u)-V|u|^{p+r}\right\}\cr &\le -\int\left\{\nabla (|u|^ru)\cdot|\nabla u|^{p-2}\nabla u\right\} +\int|V_2| |u|^{p+r}\cr &= -(r+1)\int\left\{ |u|^r|\nabla u|^p\right\} + \int |V_2| |u|^{p+r}\cr &\le -(r+1)\int |u|^r|\nabla u|^p + \alpha\int\left| \nabla \left( u^{(p+r)/p}\right)\right|^p + \beta \int |u|^{(p+r)} \cr &=\left(\alpha {p+r\overwithdelims () p}^p-(r+1)\right)\int |u|^r|\nabla u|^p + \beta \int |u|^{(p+r)}}. $$ The assumption on $\alpha$ makes the first term in the final line $\le 0$, so we drop it, obtaining $$ {d \over dt}{\| u\| }_s^s\le \beta s{\| u\| }_s^s,$$ which implies the claim. \hfill $\diamondsuit$ \proclaim{Theorem VI.2}. Assume that u is a positive solution of a differential equation for which the differential inequality $$|u|^{p-2}{u}_{t}\leq {\Delta }_{p}u-V({\bf x})|u|^{p-2} u. \eqno(6.3)$$ holds, that $u \in W_{0}^{1,p}(\Omega) \cap L^s(\Omega)$, $s \geq p$, and $-\Delta_p u \in L^{\infty}(\Omega)$. Assume moreover that the potential $V({\bf x})$ satisfies (6.2) with $\alpha \le \left({s+1-p}\right){\left({{p \over s}}\right)}^{p}$. Let $${f}_{s,u}\left({t}\right):={\| u\left({t;\bf x}\right)\| }_{{L}^s\left({\Omega }\right)}.$$ Then $${f}_{s,u}\left({t}\right)\le {f}_{s,u}\left({0}\right) \exp \left({\beta t}\right).$$ \noindent {\bf Proof:} Exactly as for Theorem VI.1; positivity matters because the proof requires the inequality to be multiplied by a power of $u$. \hfill $\diamondsuit$ \medskip \noindent{\bf Acknowledgments.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The authors wish to thank W. D. Evans, D. J. Harris, V. Liskevich, and P. Tak\'a\v{c} for their useful conversations and references. \bigbreak \centerline{\bf References} \item{[B1]} J. Barta, Sur la vibration fondamentale d'une membrane, {\it C.R. Acad. Sci. Paris} 204(1937), 472-473. \item{[B2]} A.G. Belyi and Yu.A. Semenov, On the $L^p$-theory of Schr\"odinger semigroups, {\it Sibirsk. Mat. J.} {\bf 31}(1990), 16-26; English translation in {\it Siberian. Math. J.} {\bf 31}(1991), 540-549. \item{[B3]} T. Boggio, Sull'equazione del moto vibratorio delle membrane elastiche, {\it Accad. Lincei, sci. fis.}, ser. 5a{\bf 16}(1907), 386-393. \item{[D1]} E.B. Davies, {\it Heat kernels and spectral theory}, Cambridge, University Press, 1989. \item{[D2]} E.B. Davies, Sharp boundary estimates for elliptic operators, preprint 1998. \item{[D3]} P. Dr\'abek, P. Krej\v{c}\'\i , and P. Tak\'a\v{c}, {\it Nonlinear Differential Equations}, Boca Raton, FL, CRC Press, to appear. \item{[D4]} R.J. Duffin, Lower bounds for eigenvalues, {\it Phys. Rev.} {\bf 71}(1947), 827-828. \item{[E1]} D.E. Edmunds and W.D. Evans, {\it Spectral theory and differential operators}, Oxford, Clarendon Press, 1987. \item{[E2]} W.D. Evans, D.J. Harris, and R. Kauffman, Boundary behaviour of Dirichlet eigenfunctions of second order elliptic equations, {\it Math. Z.} {\bf 204}(1990), 85-115. \item{[H1]} G.H. Hardy, J.E. Littlewood, and G. P\'olya, {\it Inequalities}. Cambridge, University Press, 1959. \item{[L1]} V. Liskevich, On $C_0$-semigroups generated by elliptic second order differential expressions on $L^p$-spaces, {\it Diff. and Int. Eqns.} {\bf 9}(1996), 811-826. \item{[M1]} M. Marcus, V. J. Mizel, and Y. Pinchover, On the best constant for Hardy's inequality in ${\Bbb R}^n$, {\it Trans. Amer. Math. Soc.} 350(1998), 3237-3255. \item{[M2]} E. Mitidieri, A simple approach to Hardy's inequalities, preprint. \item{[P1]} M.M.H. Pang, Approximation of ground state eigenvalues of eigenfunctions of Dirichlet Laplacians. {\it Bull. London Math. Soc.} {\bf 29}(1997), 720-730. \item{[S1]} B. Simon, Schr\"odinger semigroups, {\it Bull. Amer. Math. Soc.} 7(1982), 447-526. \bigskip % Erratum for Vol. 1999 No. 38 \centerline{{\bf ERRATUM}: Submitted on April 28, 2003.} \smallskip \noindent In Corollary II.4, the formula $$ \Big({p \over N-p}\Big)^{p-1} \int_{\Omega }\big|\nabla \zeta|^p d^N x\ge \int_\Omega \Big|{\zeta \over |{\bf x}|} \Big|^p d^N x. $$ should be replaced by $$ \Big({p \over N-p}\Big)^p \int_{\Omega }\big|\nabla \zeta|^p d^N x\ge \int_\Omega \Big|{\zeta \over |{\bf x}|} \Big|^p d^N x. $$ \medskip \noindent Jacqueline Fleckinger \hfill\break CEREMATH \& UMR MIP, Universit\'e Toulouse-1 \hfill\break 21 all\'ees de Brienne \hfill\break 31000 Toulouse, France \hfill\break e-mail address: jfleck@univ-tlse1.fr \noindent Evans M. Harrell II \hfill\break School of Mathematics, Georgia Tech \hfill\break Atlanta, GA 30332-0160, USA, and \hfill\break UMR MIP, Universit\'e Paul Sabatier \hfill\break 31062 Toulouse, France \hfill\break e-mail address: harrell@math.gatech.edu \noindent Fran\c cois de Th\'elin \hfill\break UMR MIP, Universit\'e Paul Sabatier\hfill\break 31062 Toulouse, France \hfill\break e-mail address: dethelin@mip.ups-tlse.fr \bye