\input amstex \documentstyle{amsppt} \loadmsbm \magnification=\magstephalf \hcorrection{1cm} \vcorrection{-6mm} \nologo \TagsOnRight \NoBlackBoxes \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--2000/44\hfil Symmetry Theorems \hfil\folio} \def\leftheadline{\folio\hfil L. Ragoub \hfil EJDE--2000/44} \def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations, Vol.~{\eightbf 2000}(2000), No.~44, pp.~1--11.\hfil\break ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \hfill\break ftp ejde.math.swt.edu (login: ftp)\bigskip} } \topmatter \title Symmetry theorems via the continuous steiner symmetrization \endtitle \thanks {\it 2000 Mathematics Subject Classifications:} 28D10, 35B05, 35B50, 35J25, 35J60, 35J65.\hfil\break\indent {\it Key words and phrases:} Moving plane method, Steiner Symmetrization, \hfil\break\indent Overdetermined problems, Local Symmetry. \hfil\break\indent \copyright 2000 Southwest Texas State University and University of North Texas.\hfil\break\indent Submitted October 1, 1999. Published June 12, 2000. \endthanks \author L. Ragoub \endauthor \address L. Ragoub \hfill\break\indent Riyadh College of Technology, Mathematics Department \hfill\break\indent P.O.Box 42, 826 Riyadh 11 551, Saudi Arabia \endaddress \email f66m002\@ksu.edu.sa \endemail \abstract Using a new approach due to F. Brock called the Steiner symmetrization, we show first that if $u$ is a solution of an overdetermined problem in the divergence form satisfying the Neumann and non-constant Dirichlet boundary conditions, then $\Omega$ is an $N$-ball. In addition, we show that we can relax the condition on the value of the Dirichlet boundary condition in the case of superharmonicity. Finally, we give an application to positive solutions of some semilinear elliptic problems in symmetric domains for the divergence case. \endabstract \endtopmatter \document \head 1. Introduction \endhead This paper deals with an overdetermined boundary value problem and an application to positive solutions of some semilinear elliptic problems in symmetric domains. In Section 1A we describe the first eigenvalue problem concerning a free membrane. This problem was resolved recently by Henrot and Philippin [3] who applied Brock's [1] continuous Steiner symmetrization and the domain derivative due to F. Murat and J. Simon [6], J. Simon [8], and J. Sokolowski and J. P. Zolesio [9]. Assuming that $\phi > 0$, that $\psi > 0$ is an increasing function of $r$, and $\lambda$ is the first eigenvalue of the Laplacian, they showed that if the first eigenvector $u$ satisfies $$ \gather \Delta u + \lambda\phi(r) u=0 \quad \text{on }\Omega, \tag 1 \\ u=0\quad\text{on }\partial\Omega,\tag 2 \\ {\partial u \over \partial n} = \psi(r^{2}) \quad \text{on } \partial\Omega, \tag 3 \endgather $$ then $\Omega$ is an $N$-ball, where $N\,\geq \,3$ and $r:=|x|$. In this section, we generalize this problem to more general operators while using the same technique. We formulate this generalized overdetermined problem, which we will denote $(P_{\epsilon})$, as follows: $$ \gather - \operatorname{Div} (a(u,{|{\nabla u}|}) {\nabla u}) =\phi (r) F(x,u)\quad\text{in }\Omega, \tag 4 \\ u=0\quad\text{on}\,\,\partial\Omega, \tag 5 \\ {\partial u \over \partial n}=\epsilon {\psi_{{\epsilon}}}({r^{2}})\quad\text{on }\partial\Omega. \tag 6 \endgather $$ Using the Steiner symmetrization method and two fundamental theorems of F. Brock [2], we prove that the problem cited above (with appropriate conditions on $a$ and $F$) is solvable only if $\Omega$ is an $N$-ball. Next we show that the theorem holds for some other choices of the function $\psi$. To be more precise, using the lemma of Mitidieri [5] concerning a radial positive function $\psi$ which satisfies $$ \Delta \psi \leq 0\quad\text{in}\,\,\Omega, \eqno(7) $$ we prove the same result without assuming that $\psi$ is increasing. We show that some conditions for Steiner symmetrization of functions can be relaxed when the functions are subharmonic (respectively superharmonic), radial, and positive (resp.\ negative). We need to recall some properties of Steiner symmetrization of functions as well as those of the domain derivative so that we can use them for our proof. We begin with the domain derivative following [8]. \head 1a. Differentiability and integrability of functions with respect to the domain \endhead We denote by $S$ the set of all bounded, open, connected, and regular domains of ${\Bbb R}^{N}$. We will assume that $J^{*},$ $J_{*}$, $G,$ and $J$ are defined on $S$. Using variational calculus, we compute the differentiability of $A$ and $B$ (defined below) with respect to the domain $\Omega$, and find an $\Omega$ realizing $ \inf_{\Omega}\, J(\Omega) \text{ where}\, J(\Omega)\, $ is a functional domain defined by: $$ J(\Omega):=\int\limits_{\Omega} {{a(u, {|{\nabla}\,u|}^{2}) {|{\nabla}\,u|}^{2}}} - \phi(r) F(x,u)\, dx.$$ To begin, suppose that $u$ is a solution of the following boundary value problem $$\gather A(u):=0 \quad \text{in } \Omega, \tag 8 \\ B(u):=0 \quad \text{on } \Gamma:=\partial\Omega. \tag 9 \endgather$$ We start with the directional derivative of $J$ at $\Omega$. For this, we make a variation of $\Omega$ by considering a continuous one-parameter family of domains $\Omega_{t}$ defined by $$\Omega_{t}:=\{ x+tV(x), x\in\Omega, t>0 \} $$ where $V$ is any vector field in $C^{2}({\Bbb R}^{N},{\Bbb R}^{N})$ defined by $$ V(x) := \frac{\textstyle\partial H}{\textstyle\partial t} (x,0) := x'(0). \eqno(10) $$ We observe that $V$ can be understood as a field of deformation for $\Omega$. At the same time we introduce a real parameter $t$ and a map $H$ such that $ \textstyle\Omega \rightarrow \textstyle\Omega_{t},$ $ x \rightarrow x_{t}:=H(x,t) = x + t V(x) ,$ $\Omega_{0} := \Omega,\,\,\text{and}\,\,H(.,0) := I \ \text{in}\,\Omega.$ We note that the application $Id + tV$ is a perturbation of the identity which will be a $C^{2}$ diffeomorphism for sufficiently small $t$. If we think of $t$ as time, $V$ is the deformation speed at the origin of the open set $\Omega$.\newline Consequently, we obtain the new map described below: $$\textstyle\Omega_{t} \rightarrow \textstyle y_{t} := y_{\Omega_{t}} \rightarrow J(t) := J(\Omega_{t}).$$ In fact, the maps $J$ and $H$ are defined by $J: \textstyle O \rightarrow \textstyle {\Bbb R},$ $H: \textstyle \tilde{O} \rightarrow \textstyle O,$ where $O$ is the set of all domains $\Omega_{t},\,\,t>0$, and $\tilde{O}$ is the set of all domains $\bar{\Omega}$ for which $\Omega$ is a bounded domain of ${\Bbb R}^{N}$. In this way, the directional derivative of $J$ at $\Omega$ in the $V$ direction is equivalent to the derivative of $J$ evaluated at $t=0$: $$dJ(\Omega;V) := J '(0).\eqno(11) $$ Our first problem is equivalent to finding a $t$ which realizes $\inf_{t}\,J(t).$ In practice, $\Omega$ will usually depend on one parameter $u$. In this case, we assume that $\Omega = \Omega_{u}$ is the image of a fixed domain $\bar\Omega$ under a map which depends on $u$: $H: \bar\Omega \rightarrow \Omega_{u} := H(\bar\Omega,u),$ $\bar{x} \rightarrow x_{u} := H(\bar{x},u),$ and we define $\bar J$ from ${\Bbb R}$ to ${\Bbb R}$ by $\bar{J}(x) := J(\Omega_{x}).$ By putting $\Omega_{t} := \Omega_{u + tv} \text{, so that }$ $\Omega_{0} := \Omega_{u} ,$ we can formulate the derivatives of $\bar J$ in terms of the function $J$ as $d \bar{J} (u;V) := dJ(\Omega_{u}; V).$ Returning to our goal, we can now deduce the necessary properties of the domain derivative: $$ \gather \frac{\textstyle\partial A} {\textstyle\partial u} \,\,\frac{\textstyle \partial u} {\textstyle\partial V}=0\quad\text{in }\Omega, \tag 12 \\ \frac{\textstyle\partial B} {\textstyle\partial u}\,\,\frac{\textstyle \partial u} {\textstyle\partial V} + V\cdot n \frac{\textstyle\partial B} { \textstyle\partial n}=0\quad\text{on }\Gamma. \tag 13 \endgather$$ Here $n$ denotes the outward normal on $\Gamma$ and $$ \frac{\textstyle\partial A}{\textstyle\partial u} V := \frac{\textstyle \partial A (u+tV) }{\textstyle\partial t} |_{t=0}.$$ We close this subsection with some integral derivatives with respect to the domain $\Omega$. Let $J^{*}$ and $J_{*}$ be given domain functionals defined by $$ \gather J^{*} := \int_{\Omega} f(\Omega)\, d \text{\bf {x}}, \tag 14\\ J_{*} := \int_{\partial\Omega} g(\Omega)\, d \text{\bf{s}}, \tag 15 \endgather$$ where $f$ and $g$ are positive $C^{2}$ functions on $\bar{\Omega}$. We can compute their integral domain derivatives: $$\gather d{J^{*}}(\Omega;V) := \int\limits_{\Omega} f'(\Omega)\, d\text{\bf {x}} + \int\limits_{\partial\Omega} f(\Omega){{V\cdot n}}\, d\text{\bf s} \tag 16 \\ d{J_{*}}(\Omega;V) := \int\limits_{\partial\Omega} g'(\Omega)\, d\text{\bf s} + \int\limits_{\partial \Omega}(N-1)Kg (\Omega){ {V\cdot n}}\, d \text{\bf s} + \displaystyle \int\limits_{\partial\Omega} \frac{\textstyle\partial g (\Omega)}{\textstyle\partial n} { {V\cdot n}} \, d\text{\bf s},\tag 17 \endgather$$ where $K$ denotes the mean curvature of the boundary of $\Omega$. \head 1b. Steiner symmetrization for functions in ${\Bbb F}({\Bbb R} ^{N})$ \endhead As defined in [1] and [2], the set ${\Bbb F}({\Bbb R}^{N})$ is the set of real symmetrizable functions. We say that $u \in {\Bbb F}({\Bbb R} ^{N})$ if and only if $u$ is measurable on ${\Bbb R}^{N}$ and for every $c\,>\,\inf \, {u}$ the level sets $\left \{x\in{\Bbb R}^{N}:\,\,u(x)\,>\,c \right \}$ have finite Lebesgue measure. We denote by $\left\{ u\,>\,c \right\}^{t}$ the continuous symmetrization of the set $\left\{ u\,>\,c \right\}$. For more details concerning the continuous symmetrization and its properties, we refer to the work of F. Brock [2]. (1) \quad Let $u \in {\Bbb F}(\Bbb R^{N})$ and let $m_{u} (c)$ be the corresponding distribution function defined by $$m_{u} (c) := |\left\{ x\in \Bbb R^{N} \,:\,\,u(x)\,>\,c \,\right\} |.$$ The inverse function is denoted by $u^{*}\in {\Bbb F}(\Bbb R)$ and is called the {\it symmetrization of $u$}. The function $u^{*}$ satisfies the relations $$c=u^{*}(x)=u^{*}(-x),\,\,c\,>\, \inf \, {u}.$$ (2)\quad Let $u \in {\Bbb F}(\Bbb R^{N})$, $N \geq 2$ and $y:=x_{N}$ where $x := (x',x_{N}),$ $x' := (x_{1},...,x_{N-1})$. For almost every $x'\in\Bbb R^{N-1}$ there exists a distribution function $$m_{u} (x',c) := |\left\{ y\in\Bbb R \,:\,\,u(x',y)\,>\,c \,\right\}|,\,\,c\,>\, \inf \, {u}.$$ $u^{*}$ is called the {\it symmetrization of $u$ with respect to $y$}. \head 1c. Continuous Steiner symmetrization for functions in ${\Bbb F}({\Bbb R}^{N})$ \endhead Let $u \in {\Bbb F}(\Bbb R^{N})$. The set of functions $u^{t}$, $t\in\Bbb R^{+}$, defined by the relations $$\gathered \left\{ u^{t}\,>\,c \right\}\,\,=\,\,\left\{ u\,>\,c \right\}^{t},\,\,c\,>\,\inf \, {u},\\ \left\{ u^{t}\,=\, \inf \, {u} \right\}\,\,=\,\,\Bbb R^{N}\setminus {\bigcup}_{c\,>\, \inf \,{u}}\left\{ u\,>\,c \right\}^{t}, \text{ and }\\ \left\{ u^{t}\,=\,\infty \right\}\,\,=\,\,\Bbb R^{N}\setminus {\bigcap}_{c\,>\,\inf \, {u}}\left\{ u\,>\,c \right\}^{t}, \endgathered$$ is called the {\it continuous Steiner symmetrization of $u$ with respect to $y$} in the case $N \geq 2$ and the {\it continuous symmetrization} in the case $N=1$. The set $\left\{ u^{t}\,>\,c \right\}$ is the set of all $x\in\Omega$ such that $u^{t}\,>\,c$, where $c$ is a constant. For $u\in {\Bbb F}(\Bbb R^{N})$ take $u :=u^{0}$ and $u^{\infty} := u^{*}$ (the Steiner symmetrization of $u$ with respect to $y$). Before citing some properties of the continuous Steiner symmetrization defined above, it is necessary for us to recall some definitions for Steiner symmetrization of sets. (For more details see [1]). \vskip .1 in {\bf {Hardy-Littlewood Inequality}:} Let $u, v \in {\Bbb F}({\Bbb R} ^{N})$ and let $t$ be a real positive parameter. Then, $$\int\limits_{{\Bbb R}^{N}} u(\text{\bf x}) v(\text{\bf x}) \, d\text{\bf x} \leq \int\limits_{{\Bbb R}^{N}} u^{t}(\text{\bf x})v^{t}(\text{\bf x}) \, d\text{\bf x}.$$ \vskip .1 in \head 2. The Main Theorem \endhead We assume the following smoothness conditions to ensure uniqueness of the solution of the problem (4), (5), and (6) in divergence form and convergence of the integrals (in particular convergence of the convex functional $J$). Note that the uniform ellipticity condition corresponds to these inequalities in the special case $p=2$. For more details concerning the uniqueness theorem, see [4]. Let $u$ be a positive solution of the overdetermined boundary value problem $(4)$-$(6)$ and let $a$ be a real valued function defined on $\Bbb R\times{\Bbb R}^{+}$ which satisfies the following conditions: $$\gather a(u,s)\,\geq\,k_{1} s^{p-2}, \tag 18 \\ a(u,s) + s a_{s}(u,s)\,\geq\,k_{2} s^{p-2}, \text{ and } \tag 19 \\ |a(u,s)| +| a_{u}(u,s)| + s a_{s}(u,s)\,\leq\,k_{3}(s^{p-2} + 1) \tag 20 \endgather$$ for every $s\in{\Bbb R}^{+}$, every $x\in\Omega$, and for some appropriate positive constants $k_{1}, k_{2}, k_{3}$. We denote the partial derivative of $a$ with respect to $u$ and $s$ by $a_{u}$ and $a_{s}$ respectively. The number $p$ appearing in (18), (19), and (20) ranges over the interval $(1, {+\infty})$. \vskip .1 in We assume that the function $\phi$ in $(4)$ is continuous, positive, and satisfies the following condition: $$\int\limits_{\Omega} {\phi}^{\frac{N}{2}}(\text{\bf x}) \, d\text{\bf x} \,<\,\infty. \eqno(21) $$ \proclaim {Main Theorem} Let $\Omega$ be a convex domain in $\Bbb R^{N}$ and $u=u(${\bf x}$)$ be a nonnegative solution of $(4)$-$(6)$. We assume that $F(\text{\bf x},u)$ satisfies the following conditions: (1)\quad $F$ is measurable on $\Bbb R^{N} \times ({\Bbb R}^{+} \cup \{0\}) $; (2)\quad $F$ is differentiable with respect to $u$, and $\frac{\textstyle\partial F}{\textstyle\partial u}$ is even and nonincreasing in $x_{N}$; and (3)\quad $\psi_{\epsilon} :=\psi$ is a given positive continuous nondecreasing function on $\Bbb R^{N}$ for $\epsilon \neq +1$ and $\epsilon \neq -1.$ \vskip .1 in Also we suppose that $\psi_{\epsilon}$ satisfies one of the two following conditions: (3a)\quad $\psi_{\epsilon}$ is a given positive (resp.\ negative) continuous function on $\Bbb R^{N}$ which is superharmonic in $\Bbb R^{N},$ for $N=2$ and $\epsilon=1$ (respect. $\epsilon=-1$). (3b)\quad ${|\nabla\,u|}^{2} := \ln {r^{2(N-2)}\psi}$ on the boundary $\partial\Omega$ of $\Omega$ where $\psi$ is a given positive continuous function, which is superharmonic in $\Bbb R^{N}$ for $N \geq 3$. \vskip .1 in Then we conclude that $\Omega$ must be an $N$-ball. \endproclaim For the proof of this theorem, we recall a lemma due to Mitidieri [5] and two important theorems of F. Brock [2]. \proclaim{Lemma 1} (Mitidieri) If $\psi \in C^{2} (\Bbb R^{N}),$ $N \geq 3,$ is positive, radial, and superharmonic (i.e. $\Delta\,\psi \leq 0$ in $\Bbb R^{N})$, then for every $r\in (0, \infty)$ we have (in the obvious notation) $$r {\psi'}(r) + (N-2) \psi (r) \,\geq\,0. \eqno(22) $$ \endproclaim The first theorem of F. Brock [2] concerns some properties of continuous Steiner symmetrization of functions. \proclaim {Theorem 1} (F. Brock) Let $u \in {\Bbb F}^{+} (\Bbb R^{N})$ and let $F := F(\text{\bf x},u)$ be measurable on $\Bbb R^{N} \times ( {\Bbb R}^{+} \cup \{0\}) .$ Further, assume that $F$ is differentiable with respect to $u$ and that the function $F_{u} (\text{\bf x},u)$ (the first derivative of F with respect to $u$) is even and nonincreasing in $y$. Then it follows that for every $t \in [0, + \infty]$, $$\int\limits_{\Bbb R^{N}} F(\text{\bf x},u) \, {d \text{\bf x}} \leq \int\limits_{\Bbb R^{N}} F(\text{\bf x},u^{t}) \, {d \text{\bf x}}. \eqno(23)$$ \endproclaim The second theorem of F. Brock [2] uses one more important condition which is the {\it local symmetry} of the positive solution $u$ as defined in [2]. \proclaim{ Theorem 2} (F. Brock) Let $\Omega$ be a bounded convex domain and let $u$ be a positive function in ${H_{0}^{1}} (\Omega) \cap {C_{loc}^{1}}(\Omega)$ that is locally symmetric in every direction - i.e. such that $$\lim_{t \rightarrow 0} \frac{1}{t} \left\{ \int\limits_{\Omega_{t}}{{{ { { {{|\nabla}} \,u^{t} \,|^{2} }}}}\, d \text{\bf x} - \int\limits_{\Omega_{t}}{{ { {{|\nabla}} \,u \,|^{2}}}}} \, d\text{\bf x} \right\}<0 \eqno(24) $$ holds for every hyperplane $H$ through the origin. Then $u$ has the following form: $$u(\text{\bf x})=f_{k} (|\text{\bf {x}}-\text{\bf {x}}_{k}|) \eqno(25) $$ $\text{in } C_{k}:= \left\{ \text{\bf x} \in \Bbb R^{N}\, |\,\, r_{k}\,<\,|\text{\bf {x}}-\text{\bf{x}}_{k}|\,<\,R_{k} \right\},$ $k=1,...,m,$ and is piecewise constant in $G,$ where $G$ and $C_k$ are disjoint subsets of $\Omega$ such that $\Omega=\bigcup _{k=1}^{m} {C_{k} \cup G}.$ Here $C_{k} := C_{k}(\text{\bf x}_{k}, r_{k}, R_{k})$ are $m$ $( \leq \infty)$ disjoint ring-shaped regions centered at {\bf x}$_{k}$ with interior and exterior radii $0 \leq r_{k} \,<\, R_{k}$ and $G$ is the subset of critical points of $u$. \endproclaim Now we will prove our main theorem. \newline \proclaim {Proof of Main Theorem} \endproclaim To begin, we suppose that the conditions $(1)$-$(3)$ of the theorem are realized and we show that $\Omega$ is an $N$-ball.\newline We argue by contradiction, assuming that $\Omega$ is not a ball and constructing a deformation field $v$ such that $dF(\Omega;v) \leq 0$, where $F(\Omega)$ is the domain functional defined by: $$\Omega_{t} := \left\{ (\text{\bf x}',y) \in {\Bbb R} ^{N} \,|\, \text{\bf x}'\in \Omega',\, y_{1} (\text{\bf x}') - t \bar{y} (\text{\bf x}')0$ the equation $$u(x_{1},...,x_{N-1},y)=c \eqno(36) $$ has only a finite (even) number of solutions $y=y_{k},$ $(k=1,...,2m)$ and $$\inf \{|\frac{\partial u(x)}{\partial y}|: \frac{\partial u(x)}{\partial y} \quad \text{exists and is non-zero}\}\,>\,0.$$ Inequality (35), based on the continuous Steiner symmetrization, is essential for our overdetermined problem in view of the following theorem. We consider a positive solution $u$ in $W^{1,p}(\Bbb R^{N})$, $1\leq p<{+\infty},$ and Remark 2 of [2]. The following lemma, which is a key step in the proof, summarizes the cited remark of [2]. \proclaim {Lemma 2} Good functions are dense in $W_{+}^{1,p}(\Bbb R^{N})$ in the $W^{1,p}(\Bbb R^{N})$ norm. \endproclaim For the sake of completeness, we cite the important theorem due to F. Brock which allows us to establish our main inequality - Lemma 3. \proclaim {Theorem 3} Let $u$ be a Good function. We assume: the functions $F(\text{\bf x}',u,z)$, $a(\text{\bf x}',y,z),$ $a_{i}(\text{\bf x}',u), i=1,...,n-1$, $(\text{\bf x}' \in \Bbb R^{N-1}, u,z \in (\Bbb R^{+} \cup \{ 0 \}) ),$ are nonnegative and continuous in all arguments, $a(\text{\bf x}',y,z)$ is even and convex in $y$ and $F(\text{\bf x}',u,z)$ is monotone, nondecreasing, and convex in $z$. Then, $$ \gather \int_{\Bbb R^{N}}(F(\text{\bf x}',u, \{a^{2}(\frac{\partial u}{\partial n})^{2} + \Sigma_{i=1}^{N-1} a_{i}^{2}(\frac{\partial u} {\partial x_{i}})^{2}\}^{\frac{1}{2}})\, d \text{\bf x} \\ \geq \int_{\Bbb R^{N}} (F(\text{\bf x}',u,\{\tilde{a}^{2}(\frac{\partial u^{t}}{\partial n})^{2} + \Sigma_{i=1}^{N-1} \tilde{a}_{i}^{2} (\frac{\partial u^{t}} {\partial x_{i}})^{2} \}^{\frac{1}{2}}) \, d \text{\bf x}, \endgather$$ for every $t \in [0, + \infty]$, where for simplicity we wrote\newline $u=u(\text{\bf x}), u^{t}= u^{t}(\text{\bf x}),\,a=a(\text{\bf x},u(\text{\bf x})), \,\tilde{a}=a(\text{\bf x}',u^{t}(\text{\bf x}))$ $a_{i}=a_{i}(\text{\bf x}',u(\text{\bf x}))$,\newline and $\tilde{a}_{i}=a_{i}(\text{\bf x}',u^{t}(\text{\bf x})) $, $i=1,...,N-1$. \endproclaim We are now able to prove our inequality by assuming that $a(u,|\nabla\, u|)$ satisfies the following conditions:\newline \quad\quad (i) \quad $a$ is nonnegative and continuous in all the arguments, and \newline \quad\quad (ii) \quad $a$ is monotone, nondecreasing, and convex in $|\nabla u|$. \proclaim {Lemma 3} We suppose that $(i)$ and $(ii)$ are satisfied by the function $a$. Then: $$ \int\limits_{\Bbb R^{N}} a(u,\{(\frac{\partial u}{\partial n})^{2} + \Sigma_{i=1}^{N-1} (\frac{\partial u}{\partial x_{i}})^{2} \}^{\frac{1}{2}}) \, d\text{\bf x} \geq \int\limits_{\Bbb R^{N}} a(u,\{(\frac{\partial u^{t}}{\partial n})^{2} + \Sigma_{i=1}^{N-1} (\frac{\partial u^{t}}{\partial x_{i}})^{2} \}^{\frac{1}{2}}) \, d\text{\bf x}.$$ \endproclaim \noindent{\bf Proof.} By a suitable choice of $a_{i}$, $a$, $\tilde{a}_{i}$, $\tilde{a}$ we see that this is an immediate consequence of Theorem 3 above. To complete the argument, it is necessary to apply the well known theorem of Ladyzhenskaya and Uralt'sceva $[4]$ on $W_{0}^{1,p}(\Omega)$ which says that if the function $a$ satisfies the smoothness conditions $(18)$, $(19),$ and $(20)$ then the following elliptic problem, $$-\operatorname{Div} (a(u,|\nabla \, u|) \nabla \, u)=\phi (r) F(x,u)\quad\text{in } \Omega,$$ $$u=0\quad\text{on}\,\,\partial\Omega,$$ has as a unique solution $u$ which is characterized as the minimum of $J(\Omega)$ where $$ J(\Omega):=\int\limits_{\Omega} a(u, |\nabla\,u|^{2}) |\nabla\,u|^{2} - \phi(r)F(x,u))\, d \text{\bf x}.$$ Let us define a new functional $G$ as follows: $$G(\Omega):=\int\limits_{\Omega} \psi(r^{2})\, \text{d\text{\bf{x}}}. \eqno(37)$$ Since $\psi$ does not depend on $\Omega$, the derivative of $G(\Omega)$ with respect to the deformation field $v$ is given by the classical Hadamard formula $[7], [8], [9]$: $$dG(\Omega, v):=\int\limits_{\partial\Omega} \psi(r^{2}) v \cdot n \, d\text{\bf x}. \eqno(38)$$ As mentioned in $[3]$, the derivative of $G(\Omega)$ with respect to $v$ is given by: $$d\,G(\Omega; v^{*})=\int\limits_{P_{r}(\Omega)}\bar{\text{\bf {y}}} (\text{\bf {x}}')[\psi(|\text{\bf {x}}'|^{2}+{y_{1}}^{2} (\text{\bf {x}}'))-\psi(|\text{\bf {x}}'|^{2}+{y_{2}}^{2} (\text{\bf {x}}'))] \,\text{{d\text{\bf{x}}}}'.$$ It is clear that $d\,G \leq 0$ since $\psi$ is nondecreasing. Finally, we claim the positive solution $u$ is locally symmetric in the sense of Brock [2]. Using (31) and (37) we conclude: \proclaim{Lemma 4} -$d\,G=d\,F$ where $G$ and $F$ are defined by $(37)$ and $(4)$. \endproclaim By Theorem $(2)$ and Lemma $(2)$, $$\left\{\int\limits_{\Omega} a(u, |\nabla\,u|^{2}) \, d \text{\bf x} - \int\limits_{\Omega_{t}} a(u^{t}, |\nabla\,u^{t}\,|^{2}) \, d \text{\bf x} \right\}\leq \int\limits_{{\Bbb R}^{N}} {\ {{\ F(\text{\bf x},u)}}}\, d \text{\bf x} - \int\limits_{{\Bbb R} ^{N}} F(\text{\bf x},u^{t}) \, d \text{\bf x}. \eqno(39) $$ Multiplying both sides of (39) by $\frac{1}{t}$ and letting $t$ tend to zero, we obtain, $$d\,J(\Omega,v):=\left\{\int\limits_{\Omega} a(u, |\nabla\,u|^{2}) \, d \text{\bf x} - \int\limits_{\Omega_{t}} a(u^{t}, |\nabla\,u^{t}\,|^{2})\, d \text{\bf x} \right\} \leq 0.$$ Consequently, Lemma $(3)$ gives $$0\leq\left\{\int\limits_{\Omega} a(u, |\nabla\,u|^{2}) \, d\text{\bf x} - \int\limits_{\Omega_{t}} a(u^{t}, |\nabla\,u^{t}\,|^{2}) \, d \text{\bf x}\right\}\leq 0.$$ Since the hyperplane is arbitrary, we have proved that $u$ is locally symmetric in every direction. We complete the proof of the Main Theorem using the famous Theorem $2$ of F. Brock. We observe that $G$ does not reach the boundary $\partial\Omega$ as $ |\nabla\,u|^{2}:=\psi(r^{2})$ is positive on this boundary by $(3)$ (third condition of the Theorem 1) . Moreover, the ring-shaped subsets $C_{k}$ are locally finite near the boundary. Indeed, let $x \in\partial\Omega$ and assume that there exists a sequence of disjoint subsets $C_{x_{k},r_{k},R_{k}}$ such that $x_{k} \rightarrow x$, $ r_{k} \rightarrow 0$, and $R_{k} \rightarrow 0$ as $k \rightarrow \infty$. Select two points $\xi_{k},\,\eta_{k}\,\in\,C_{k}$ such that $x_{k}=\frac{1}{ 2}(\xi_{k} +\eta_{k})$. From $(28)$ we have $\nabla\,u_{k}=-\nabla\,u_{k}$. This implies $$\nabla\,u(x)=\lim_{k}\,\nabla\,u(\xi_{k})=-\lim_{k}\,\nabla\,u(\eta_{k})=- \nabla\,u(x)=0,$$ in contradiction to $|\nabla\,u|^{2}=\psi(r^{2})$. Since $\Omega$ is convex it is clear that $\partial\Omega$ coincides with the exterior boundary of a single ring-shaped subset $C_{k}$. The proof is as above for both of the possible conditions (a) and (b). \head {An application to positive solutions of some semilinear elliptic problems in symmetric domains} \endhead In this section, we give an application to this new alternative approach to symmetric domains. This application generalizes Theorem 14 of F. Brock [2]. In fact, we will prove the following theorem. \proclaim{Theorem} Let $u\in {W_{0}}^{1,p}(\Omega)$ $(10$, then (ii)\quad $u$ is symmetric and decreasing in $y$. Finally, in the case of an $N$-ball $(\Omega=B_{R})$ with a positive radius $R$ and with $f=f(|x|,u)$ and $F=\phi(r)f(|x|,u)$ monotonously nonincreasing in $|x|$, then (iii)\quad $u$ is locally symmetric in every direction. \endproclaim \noindent{\bf {Proof}}:\newline (i)\quad\quad Since $u$ is an element of ${W_{0}}^{1,p}(\Omega),$ then $u^{t}$ is an element of ${W_{0}}^{1,p}(\Omega)$ and by (4), we obtain, $$\int\limits_{\Omega} a(u, |\nabla\,u|) |\nabla\,(u^{t}-u)|\, d\text{\bf x} \geq \int\limits_{\Omega}{\ \phi(r) f(x,u)}(u^{t}-u)\, d\text{\bf x},$$ for all $t \in [0,+ \infty]$. Using the inequality $$\int\limits_{\Omega}{\ \phi(r) f(x,u)} \, d \text{\bf x} \leq \int\limits_{\Omega^{t}}{\ \phi(r) f(x,u^{t})}\, d \text{\bf x} ,$$ we conclude that $$\lim_{t \rightarrow 0} \frac{1}{t} \int\limits_{\Omega} a(u^{t}, |\nabla\,u^{t}|) |\nabla\,(u^{t})|^{2} \,d \text{\bf x} - \int\limits_{\Omega} a(u, |\nabla\,u|) |\nabla\,(u)|^{2} \, d\text{\bf x}=0.$$ Consequently, $u$ is locally symmetric, which is the desired result.\newline (ii)\quad\quad If $f$ is nondecreasing in the positive variable $y$, we can find $$x^{1}=(x_{0}',y_{1}),\,\, x^{2}=(x_{0}',y_{2})\,\text{in} \,\,\Omega,$$ with $$y_{1}+y_{2} \ne 0. \eqno(40)$$ By the hypothesis on $u$, $\frac{\partial u}{\partial y} > 0$ at $x^{1}$. \newline Let $U_{1}$ denote the (maximal) connected component of $\Omega \cap \left\{ x: u_{y}(x) > 0 \right\}$ containing $x^{1}$, where $x^{2}=(x_{0}',y_{2}) \in \Omega$, $y_{1} < y_{2}$,\newline and $u(x^{1})=u(x^{2}) < u(x_{0}',y)$ for all $y$ in $(y_{1},y_{2}).$ Then, for all $(x^{1},y)\in U_{1}$, $u(x^{1},y)=(x^{1},y_{1} + y_{2} - y) < u(x^{1},z)$ for all $z$ in $(y,y_{1}+ y_{2} - y).$ \newline We put $v(x^{1},y)=u(x^{1},y_{1}+ y_{2} - y)$ and apply Theorem 13 of Brock to conclude that $v(x)=u(x)$ for all $x \in U \subset V(x^{1})$. Using the equation in the statement of the problem, $w:=u-v$ which is zero. Then, $$-\text{Div} (a(w,{|{\nabla w}|}) {|\nabla w|})=\phi(r)[f(x^{1},y,u)-f(x^{1},y_{1} + y_{2} - y,u)]\quad\text{in}\quad U^{1},$$ which contradicts (40).\newline (iii)\quad\quad Let $\Omega$ be the ball $B_{R}$ and $f=f(|x|,u)$. If we associate to $x$ the value $\xi$ defined by $\xi:=({\xi}^{1},\eta)$ for an arbitrary rotation of the coordinate system about the origin, we see that f is not even and nonincreasing in $\eta$. By the above considerations, this yields the last assertion of the theorem. \vskip 0.3 cm {\bf Acknowledgement:} $\ $ I thank F. Brock from Leipzig, for helpful discussions. Also my particular thanks go to the referees for their remarks and suggestions. \vskip 0.5 cm \Refs \ref \no 1 \by F. Brock \pages 25--48 \paper Continuous Steiner symmetrization \yr 1995 \vol 172 \jour Math Nachr \endref \ref \no 2 \by F. 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