# An Unstable Two-Phase Membrane Problem and Maximum Flux Exchange Flow

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## Abstract

*U*be a bounded open connected set in \(\mathbb {R}^n\) (\(n\ge 1\)). We refer to the unique weak solution of the Poisson problem \(-\Delta u = \chi _A\) on

*U*with Dirichlet boundary conditions as \(u_A\) for any measurable set

*A*in

*U*. The function \(\psi :=u_U\) is the torsion function of

*U*. Let

*V*be the measure \(V:=\psi \,\mathscr {L}^n\) on

*U*where \(\mathscr {L}^n\) stands for

*n*-dimensional Lebesgue measure. We study the variational problem

### Keywords

Two-phase membrane problem Isoperimetric inequality Pólya–Szegö inequality Spherical cap symmetrisation### Mathematics Subject Classification

26D10 35J20 35J60## 1 Introduction and Motivation

*U*be a bounded open connected set in \(\mathbb {R}^n\) (\(n\ge 1\)). We refer to the unique weak solution of the Poisson problem

*A*in

*U*. The function \(\psi :=u_U\) is the torsion function of

*U*. Let

*V*be the measure \(V:=\psi \,\mathscr {L}^n\) on

*U*where \(\mathscr {L}^n\) stands for the

*n*-dimensional Lebesgue measure. For \(p\in (0,1)\) consider the variational problem

*E*in (1.2) will be called an

*optimal configuration*for the data (

*U*,

*p*). If

*E*is an optimal configuration and \(u=u_E\), then (

*u*,

*E*) will be called an

*optimal pair*.

*u*,

*E*) for the data (

*U*,

*p*). In Proposition 3.3 we characterise the optimal configuration

*E*as a super level set of \(u/\psi \); that is, \(E=\{u>c\psi \}\) for some \(c\in (0,1)\) up to \(\mathscr {L}^n\)-a.e. equivalence. The derivation assumes that

*U*is a \(C^{1,1}\) domain. Under this last assumption, we show in Corollary 3.4 that

*u*satisfies the following semi-linear elliptic partial differential equation with discontinuous nonlinearity. Put \(v:=u-c\psi \) with

*c*as above. Then

*v*is a strong solution of the problem

*U*with the unit ball

*B*in \(\mathbb {R}^n\) (\(n\ge 2\)). For \(p\in (0,1)\) we show that any optimal configuration

*E*for the data (

*B*,

*p*) possesses spherical cap symmetry \(\mathscr {L}^n\)-a.e. (see Theorem 4.1).

In the remainder of the article, we study the problem (1.2) in the one-dimensional case \(n=1\) and take \(B=(-1,1)\). In Theorem 9.5 we show that any optimal configuration *E* with data (*B*, *p*) is \(\mathscr {L}^1\)-a.e. equivalent to an open interval abutting a boundary point of *B*. A first step in obtaining this result is to transform the problem using an analog of the ground-state transformation (with the torsion function in place of the ground-state) (see Proposition 9.2). We then obtain an isoperimetric inequality on *B* with volume density \(\psi \) and perimeter density \(\psi ^{3/2}\) (Theorem 6.3) and a corresponding Hardy-Littlewood type inequality (Theorem 6.6) and a Pólya–Szegö inequality (Theorem 7.10). We also study the case of equality in the isoperimetric and Pólya–Szegö inequalities (Theorem 6.4 and Corollary 8.7 respectively). We have been guided by [2] in obtaining these results, though our setting and proofs are slightly different.

We have not obtained an analog of Theorem 9.5 in the case \(n\ge 2\). At least part of our method transfers to higher dimensions. There is a counterpart of the isoperimetric inequality Theorem 6.3 (though its derivation is more involved with the usual difficulties around regularity and stability) and the Hardy–Littlewood inequality is a ready consequence. A potential stumbling block is the validity of a corresponding Pólya–Szegö inequality. We note that the sufficient conditons given in [22] are stringent.

*U*in a state of steady flow. The densities of the fluids are labelled \(\rho \), \(\rho ^\prime \) with \(\rho > \rho ^\prime \) and each fluid has unit viscosity. The pressure

*p*has constant gradient \(\partial p/\partial z = -G\) on

*U*. Suppose the fluid with density \(\rho \) occupies a region

*A*in

*U*. By the Navier–Stokes equations, the vertical component of velocity

*u*satisfies

*U*and it is assumed that

*u*and its gradient are continuous on the interface between the two regions

*A*and \(U\setminus A\).

*G*lies in the interval \((\rho ^\prime g,\rho g)\) which allows the possibility of bidirectional flow. On rescaling (and relabelling the velocities) we obtain the system

*A*which satisfy the flux balance condition \((u,1) = 0\) with constant \(\lambda \); the other in which we optimize also over \(\lambda \). In detail,

*E*for the problem (1.5) with data \((B,\lambda )\) is \(\mathscr {L}^1\)-a.e. equivalent to an open interval abutting a boundary point of

*B*in Theorem 9.8. Moreover, any optimal configuration

*E*for the problem (1.4) is \(\mathscr {L}^1\)-a.e. equivalent to either \((-1,0)\) or (0, 1).

## 2 Existence of Optimal Configurations

*f*. The first main result runs as follows.

**Theorem 2.1**

- (i)
there exists \(f\in \mathscr {V}_t\) such that \(\beta (U,t) = J(f)\);

- (ii)
\((\psi ,f)=t\);

- (iii)
*f*has the form \(f=\chi _E\) for some measurable set*E*in*U*.

**Corollary 2.2**

For each \(p\in (0,1)\) the problem (1.2) admits an optimal pair (*u*, *E*) for the data (*U*, *p*).

*Proof*

Let \(p\in (0,1)\) and put \(t:=p V(U)\). Let *E* be as in Theorem 2.1 (iii). Then \(V(E)=(\psi ,\chi _E)=t=pV(U)\). Let \(A\subset U\) be a measurable set with \(V(A)=pV(U)\). Then \(f=\chi _A\in \mathscr {V}_t\) so \(J(E)\ge J(A)\). \(\square \)

We prepare a few lemmas before proving Theorem 2.1.

**Lemma 2.3**

Let *X*, *Y* be (real) Banach spaces and suppose that \(X\subset Y\) with continuous embedding. Let \((x_h)\) be a sequence in *X* which converges weakly in *X* to \(x\in X\). Then \((x_h)\) converges weakly to *x* in *Y*.

*Proof*

Note that for any \(g\in Y^\prime \), \(g\vert _X\in X^\prime \). \(\square \)

*G*stand for the corresponding Green operator.

**Lemma 2.4**

- (i)
the functional \(J:\,\mathscr {V}_t\rightarrow \mathbb {R}\) is continuous in the topology of weak sequential convergence;

- (ii)
\(J:\,\mathscr {V}_t\rightarrow \mathbb {R}\) is convex.

*Proof*

*h*and \(\varphi \in L^2(U)\),

*G*so that \((u_h,\,\varphi )\rightarrow (f,\,G\varphi )= (u,\,\varphi )\) as \(h\rightarrow \infty \) where \(u:=Gf\). We also have that

*u*in \(\mathscr {F}\).

Note that \(W^{1,2}_0(U)\subset W^{1,1}_0(U)\) and \(\Vert u\Vert _{W^{1,1}_0(U)}\le \sqrt{2\,|U|}\,\Vert u\Vert _{W^{1,2}_0(U)}\) for each \(u\in W^{1,2}_0(U)\). By Lemma 2.3, \((u_h)\) converges weakly to *u* in \(W^{1,1}_0(U)\).

*u*in \(L^1(U)\). Now

In the case \(n=1\) we use the fact that \(W^{1,2}_0(U)\) is compactly embedded in \(C^0(\overline{U})\) (see [13, Theorem 7.22]) and hence in \(L^1(U)\).

*E*is concave so that

*J*is convex. \(\square \)

**Lemma 2.5**

Let \(t\in (0,\,V(U))\). A function *f* in the convex set \(\mathscr {V}_t\subset L^2(U)\) is extremal only if \(f=\chi _A\) \(\mathscr {L}^n\)-a.e. on *U* for some \(A\subset U\) measurable with \((\psi ,\chi _A)\le t\).

*Proof*

*f*on

*U*is \(\mathscr {L}^n\)-a.e. equivalent to \(\chi _A\) for some \(A\subset U\) measurable if and only if \(f(1-f)=0\) \(\mathscr {L}^n\)-a.e. on

*U*. Suppose that \(f\in \mathscr {V}_t\) is an extremal element and assume that \(\vert \{f(1-f)\ne 0\}\vert >0\) for a contradiction. Then there exists \(\varepsilon >0\) and a measurable set

*E*in

*U*with positive \(\mathscr {L}^n\)-measure such that \(\varepsilon \le f\le 1-\varepsilon \text { on }E\). Decompose

*E*into two disjoint sets \(E_1,\,E_2\) each with positive \(\mathscr {L}^n\)-measure. Choose \(\alpha =(\alpha _1,\,\alpha _2)\in \mathbb {R}^2\setminus \{0\}\) such that \(\alpha _1(\psi ,\,\chi _{E_1})+\alpha _2(\psi ,\,\chi _{E_2})=0\) and define

*f*is not extremal as \(f=(1/2)\left\{ f_\tau +f_{-\tau }\right\} \) for such \(\tau \). \(\square \)

*Proof of Theorem 2.1*

We now argue as in [8, Corollary 6.2]. By [5, Chapitre II §7 Proposition 1.1 (EVT II.58)], *J* attains its supremum on \(\mathscr {V}_t\) at an extremal point *f*. We then invoke Lemma 2.5 to conclude that *f* has the form \(f=\chi _A\) \(\mathscr {L}^n\)-a.e. on *U* for some measurable set *A* in *U* and hence (iii). \(\square \)

## 3 Some Partial Regularity Results

**Proposition 3.1**

Suppose that *U* is a \(C^{1,1}\) domain. Let \(E\subset U\) be a measurable set and \(u=u_E\). Then \(V(\{u=t\psi \})=0\) for each \(t\in (0,1)\).

*Proof*

By [13, Theorem 9.15], \(u\in W^{2,p}(U)\) for any \(1<p<\infty \). Put \(v:=u-t\psi \), \(N_t:=\{v=0\}\) and assume that \(|N_t|>0\). By [13, Lemma 7.7], we derive that \(D^\alpha v=0\) \(\mathscr {L}^n\)-a.e. and hence *V*-a.e. on \(N_t\) for any multi-index \(\alpha \) with \(|\alpha |=1\). Observe that \(D^\alpha v\) belongs to \(W^{1,p}(U)\) for \(|\alpha |\le 1\). Applying the last-mentioned lemma once more, we see \(D^\alpha v=0\) *V*-a.e. on \(N_t\) for any multi-index \(\alpha \) with \(|\alpha |\le 2\). So \(-\Delta v=0\) *V*-a.e. on \(N_t\). But \(-\Delta v=\chi _E-t\,\chi _U\) *V*-a.e. on *U*. This leads to a contradiction. \(\square \)

*X*. Given \(0<v<\mu (X)\), consider the variational problem

*A*,

*B*in

*X*are equivalent \(\mu \)-a.e. and write \(A=B\) if and only if \(\mu (A\Delta B)=0\).

**Theorem 3.2**

*Proof*

*A*in

*X*with \(\mu (A)=v\),

*E*is an optimiser for (3.1).

*A*is a measurable set in

*X*with \(\mu (A)=v\) which is not \(\mu \)-a.e. equivalent to

*E*. Then

*A*is \(\mu \)-a.e. equivalent to

*E*. By countable additivity,

Let *U* be a \(C^{1,1}\) domain and \(p\in (0,1)\). Let (*u*, *E*) be an optimal pair for (1.2) with data (*U*, *p*). By [13, Corollary 9.18] we may assume that \(u\in C^0(\overline{U})\).

**Proposition 3.3**

*U*is a \(C^{1,1}\) domain. Let \(p\in (0,1)\) and suppose that (

*u*,

*E*) is an optimal pair for (1.2) with data (

*U*,

*p*). Then \(V(E\Delta \{u>c\psi \})=V(E\Delta \{u\ge c\psi \})=0\) where \(c\in (0,1)\) is uniquely determined by the condition

*Proof*

*c*as in (3.4). Assume for a contradiction that \(V(E\Delta F)>0\). We consider a version of Problem (3.1) on

*U*with \(\rho \) replaced by \(w:=u_E/\psi \) and \(\mu \) replaced by

*V*. By Proposition 3.1, \(V(\{w=t\})=0\) for each \(t>0\); thus condition (3.2) holds. By uniqueness of the optimiser in Theorem 3.2 and the Cauchy–Schwarz inequality,

*E*is an optimal configuration. The identity \(V(E\Delta \{u\ge c\psi \})=0\) follows from Proposition 3.1. \(\square \)

**Corollary 3.4**

*U*is a \(C^{1,1}\) domain. Let \(p\in (0,1)\) and suppose that (

*u*,

*E*) is an optimal pair for (1.2) with data (

*U*,

*p*). Put \(v:=u-c\psi \) where

*c*is given by (3.4). Then

*v*is a strong solution of the problem

*Proof*

By [13, Theorem 9.15], \(u\in W^{2,p}(U)\) for any \(1<p<\infty \) and *u* is a strong solution of \(-\Delta u=\chi _E\). By Proposition 3.3, *u* is a strong solution of \(-\Delta u=\chi _{\{u>c\psi \}}\). The result follows from the fact that \(-\Delta (c\psi )=c\chi _U\) and Proposition 3.1.\(\square \)

**Lemma 3.5**

Let \(p\in (0,1)\) and (*u*, *E*) be an optimal pair for the data (*U*, *p*). Then \((\psi -u,U\setminus E)\) is an optimal pair for the data \((U,1-p)\).

*Proof*

Put \(\Gamma _\pm (v)=\partial \Omega _\pm (v)\cap U\).

**Lemma 3.6**

Suppose that *U* is a \(C^{1,1}\) domain. Suppose that (*u*, *E*) is an optimal pair for the data \((U,\,p)\) and let *v* be as in Corollary 3.4. Then \(\Gamma _+(v)=\Gamma _-(v)\).

*Proof*

*B*(

*x*,

*r*) and \(u(x)=c\psi (x)\). By Proposition 3.3, \(V(B(x,r)\setminus E)\le V(\{u\ge c\psi \}\setminus E)=0\) and \(B(x,r)\setminus E\) is a Lebesgue null set. Let \(\Phi \) stand for the fundamental solution of Laplace’s equation in \(\mathbb {R}^n\). By the mean-value formula (see [10, 2.5 Problem 3] for example), for any \(0<\tau <r\),

Now suppose that \(x\in \Gamma _-(v)\setminus \Gamma _+(v)\). As before, there exists \(r>0\) such that \(u\le c\,\psi \) on *B*(*x*, *r*) and \(u(x)=c\psi (x)\); alternatively, \(\psi -u\ge (1-c)\,\psi \) on *B*(*x*, *r*) and \((\psi -u)(x)=(1-c)\psi (x)\). By Lemma 3.5, \((\psi -u,U\setminus E)\) is an optimal pair for the data \((U,1-p)\). We then get a contradiction as above. \(\square \)

Put \(\Gamma (v):=\Gamma _+(v)=\Gamma _-(v)\) and \(\Gamma ^*(v):=\Gamma (v)\cap \{|\nabla v|\ne 0\}\). The next theorem follows as in [20, Theorem 4.24].

**Theorem 3.7**

Suppose that *U* is a \(C^{1,1}\) domain. Suppose that \((u,\,E)\) is an optimal pair for the data \((U,\,p)\) and that \(x_0\in \Gamma ^*(v)\). Then there exists \(r>0\) such that \(\Gamma (v)\cap B(x,r)\) is a real-analytic hypersurface in *B*(*x*, *r*).

## 4 Spherical Cap Symmetry

In this section, we replace *U* by the open unit ball *B* in \(\mathbb {R}^n\) (\(n\ge 2\)) centred at the origin. We prove the following symmetry result. The notion of spherical cap symmetry is defined below.

**Theorem 4.1**

Let \(p\in (0,1)\). Suppose that (*u*, *E*) is an optimal pair for the data (*B*, *p*). Then *E* possesses spherical cap symmetry \(\mathscr {L}^n\)-a.e.

*B*(see [4] and references therein). For \(\nu \in \mathbb {S}^{n-1}\) the closed half-space \(H=H_\nu \) is defined by

*H*by \(\mathscr {H}\). The polarisation \(f_H\) of \(f\in L^1_+(B)\) with respect to \(H\in \mathscr {H}\) is defined as follows. Choose an \(\mathscr {L}^n\)-version \(\widetilde{f}\) of

*f*. Set

*f*. The definition is well-defined due to the fact that if \(\widetilde{f}=\widetilde{g}\) \(\mathscr {L}^n\)-a.e. on

*B*then \(\widetilde{f}=\widetilde{g}\) \(\mathcal {H}^{n-1}\)-a.e. on \(\mathbb {S}^{n-1}_\tau \) for \(\mathscr {L}^1\)-a.e. \(0<\tau <1\), and vice-versa.

*G*(

*x*,

*y*) for

*B*is given by

**Theorem 4.2**

Let \(f\in L^1_+(B)\) and \(H\in \mathscr {H}\). Then \(J(f)\le J(f_H)\) with equality if and only if either \(f=f_H\) or \(f\circ \tau _H=f_H\) \(\mathscr {L}^n\)-a.e. on *B*.

*Proof*

*f*. Define

*J*(

*f*) but without composition with reflection. We may then write

In the case of equality, it holds that either \(|A^+|=0\) or \(|S^+|=0\). In the former case, \(f=f_H\) while in the latter, \(f\circ \tau _H = f_H\) \(\mathscr {L}^n\)-a.e. on *B*. \(\square \)

*E*in

*B*put

*L*is Borel measurable The spherical cap symmetrisation of

*E*is the set

*B*(use Fubini’s Theorem [1, 1.74] for example) and \(|C_\omega E|=|E|\). We say that the Borel set \(E\subset B\) possesses spherical cap symmetry \(\mathscr {L}^n\)-a.e. if \(C_\omega E=E\) up to \(\mathscr {L}^n\)-a.e. equivalence for some \(\omega \in \mathbb {S}^{n-1}\).

*f*. Put \(m_{\widetilde{f}}(\tau ,t):=\mathcal {H}^{n-1}(\{\widetilde{f}>t\}\cap \mathbb {S}^{n-1}_\tau )\) for \(t\in \mathbb {R}\) and \(0\le \tau <1\). The function \(m_{\widetilde{f}}(\tau ,\cdot )\) is non-increasing and right continuous. Define its right continuous inverse by

*f*.

*Q*with the \(\ell ^1\)-norm \(\Vert x\Vert _1:=|x_1|+|x_2|\) and define a mapping \(\varphi :Q\rightarrow Q\) via \((x_1,\,x_2)\mapsto (x_1\vee x_2,\,x_1\wedge x_2)\); \(\varphi \) folds

*S*onto

*R*. A geometric argument establishes the following lemma.

**Lemma 4.3**

For any \(x,\,y\in Q\), \(\Vert \varphi x - \varphi y\Vert _1\le \Vert x-y\Vert _1\) with strict inequality if and only if \(x\in R\) and \(y\in \overline{S}\) or \(x\in \overline{R}\) and \(y\in S\) or the same with the rôles of *x* and *y* interchanged.

For \(\omega \in \mathbb {S}^{n-1}\) introduce the collection of closed half-spaces \( \mathscr {H}_\omega :=\Big \{H_\nu :\,\nu \in \mathbb {S}^{n-1}\text { and }\nu \cdot \omega \ge 0\Big \}\).

**Lemma 4.4**

- (i)
for any \(f,g\in L^1_+(B)\), \( \Vert f_H - g_H\Vert _{L^1(B)}\le \Vert f - g\Vert _{L^1(B)}; \)

- (ii)
for any \(f\in L^1_+(B)\), \((C_\omega f)_H=C_\omega f\) \(\mathscr {L}^n\)-a.e. on

*B*; - (iii)for any \(f\in L^1_+(B)\),$$\begin{aligned} \Vert f_H - C_\omega f\Vert _{L^1(B)}\le \Vert f - C_\omega f\Vert _{L^1(B)} \end{aligned}$$(4.4)
with strict inequality if \(|\left\{ f\circ \tau _H > f\right\} \cap H|>0\).

*Proof*

*f*. For \(x\in B\cap H\), \(x\cdot \omega \ge (\tau _H x)\cdot \omega \) so \((C_\omega \widetilde{f})_H=C_\omega \widetilde{f}\) on

*B*. Therefore,

**Lemma 4.5**

*Proof*

Note that \(|\tau _h x-\tau x|\le 4|\nu _h-\nu |\) for each \(x\in \mathbb {S}^{n-1}\) and *h*. Now use the density of \(C(\mathbb {S}^{n-1})\) in \(L^1(\mathbb {S}^{n-1},\mathcal {H}^{n-1})\) and the fact that each \(\tau ,\tau _h\) is an isometry on \(L^1(\mathbb {S}^{n-1},\mathcal {H}^{n-1})\).\(\square \)

The next lemma is a spherical cap symmetrisation counterpart to [4, Lemma 6.3] and extends [23, Lemma 3.9].

**Lemma 4.6**

*Proof*

*f*,

*g*on

*B*, \(f=g\) if and only if \(|\{f>t\}\Delta \{g>t\}|=0\) for any \(t>0\). As \(f\ne C_\omega f\) there exists \(t>0\) such that

We claim there exists \(H\in \mathscr {H}_\omega \) such that \(|A\cap \tau _H A^\prime |>0\). Taking this as read, on \(A\cap \tau _H A^\prime \) we have that \(C_\omega f>t\ge C_\omega f\circ \tau _H\) so that \(A\cap \tau _H A^\prime \subset H\). Also, \(f\le t<f\circ \tau _H\) there. In short, \(A\cap \tau _H A^\prime \subset \left\{ f\circ \tau _H>f\right\} \cap H\). So \(|\left\{ f\circ \tau _H>f\right\} \cap H|>0\) and strict inequality holds by Lemma 4.4 (iii).

*F*be a countable dense subset in \(\mathbb {S}^{n-1}\cap H_\omega \). Then

*r*-section of

*A*and likewise for \(A^\prime \). Let \(\nu \in \mathbb {S}^{n-1}\cap H_\omega \) with corresponding reflection \(\tau =\tau _{H_\nu }\). Select a sequence \((\nu _h)\) in

*F*which converges to \(\nu \) in \(\mathbb {S}^{n-1}\) and write \(\tau _h\) for the reflection associated to the closed half-space \(H_{\nu _h}\). For \(r\in (0,\,1)\setminus N\),

*x*for \(A_r\) lying in \(A_r\) using [1, Corollary 2.23] for example; that is,

*x*in the sense that

*y*in \(A^\prime _r\) similarly so that \(A^\prime _r\) has density 1 at

*y*. Then \(C_\omega f(x)>t\ge C_\omega f(y)\). So there exists \(\nu \in \mathbb {S}^{n-1}\cap H_\omega \) such that with \(\tau =\tau _{H_\nu }\) we have that \(\tau y = x\). But then

*Proof of Theorem 4.1*

*E*be an optimal configuration for the data (

*U*,

*p*). Assume for a contradiction that \(E\ne C_\omega E\) \(\mathscr {L}^n\)-a.e. for any \(\omega \in \mathbb {S}^{n-1}\). Then there exists \(\omega \in \mathbb {S}^{n-1}\) such that

*E*is an optimal configuration for the data (

*U*,

*p*). The result now follows. \(\square \)

## 5 Preliminaries on Weighted Dirichlet Forms

*w*with the property

- (A)
*w*is a positive function in \(C_0(U)\).

*E*in

*U*is given by \(m(E):=\int _E w\,dx\). We introduce the further assumption

- (B)
\(w\in C^1(U)\) and \(w^\prime /w\in L^2(U,m)\).

**Lemma 5.1**

- (i)
\((\mathscr {D},\mathscr {E})\) is closable in \(L^2(U,m)\) with closure denoted \((D(\mathscr {E}),\mathscr {E})\);

- (ii)
\((D(\mathscr {E}),\mathscr {E})\) is a symmetric Dirichlet form in \(L^2(U,m)\).

*Proof*

Given a real-valued function *u* on \(\mathbb {R}_+\) (or \(\mathbb {R})\) define the function \(\theta _t u\) on \(\mathbb {R}_+\) for each \(t>0\) by \((\theta _t u)(x):=u(x+t)\) for \(x\in \mathbb {R}_+\) (or \(\mathbb {R})\).

**Lemma 5.2**

- (i)
\(\theta _t\in B(L^2(\mathbb {R}_+,\lambda \mathscr {L}^1))\) for each \(t>0\);

- (ii)
\(\Vert u-\theta _t u\Vert _{L^2(\mathbb {R}_+,\lambda \mathscr {L}^1)}\rightarrow 0\) as \(t\downarrow 0\) for each \(u\in L^2(\mathbb {R}_+,\lambda \mathscr {L}^1)\).

*Proof*

*(i)*lead to the result using a \(3\varepsilon \)-argument. \(\square \)

- (C)
*w*is unimodal on*U*.

**Lemma 5.3**

*Proof*

*u*in \(L^2(U,m)\). Then \((u_h^\prime )\) is a Cauchy sequence in \(L^2(U,m)\) with limit \(v\in L^2(U,m)\) (say). For \(\phi \in C^\infty _c(U)\),

*u*is weakly differentiable on

*U*with weak derivative \(u^\prime =v\in L^2(U,m)\).

*U*such that \(u^\prime \in L^2(U,m)\). Multiplying by a partition of unity we may assume that \(u=0\) near

*b*. Denote by \(\overline{u}\) the extension of

*u*to \(\mathbb {R}\) by zero. For \(t>0\) put \(v_t:=(\theta _t\overline{u})\vert _U\). Note that \(v_t\) is weakly differentiable and \(v_t^\prime =\theta _t(\overline{u^\prime })\vert _U\). For \(t>0\), \(v_t, v_t^\prime \in L^2(U,m)\). Let \((\rho _\varepsilon )_{\varepsilon >0}\) be a family of mollifiers on \(\mathbb {R}\) (cf. [1, 2.1]). For \(t>0\) and \(\varepsilon >0\) small, \((\rho _\varepsilon \star (\theta _t\overline{u}))^\prime =\rho _\varepsilon \star (\theta _t\overline{u^\prime })\) on

*U*. The operation \(\star \) stands for convolution. For \(t>0\) and \(\varepsilon >0\) small put \(w_{t,\varepsilon }:=\rho _\varepsilon \star (\theta _t\overline{u})\vert _U\in C^\infty (\overline{U})\). Now

*m*under \(\Phi \); thus \(\hat{m}=\hat{w}\mathscr {L}^1\) on \(\hat{U}\) where \(\hat{w}:=\varphi (w\circ \Phi )\). Define a coercive bilinear formin \(L^2(\hat{U},\hat{m})\) with domain

**Lemma 5.4**

- (i)
\((D(\hat{\mathscr {E}}),\hat{\mathscr {E}})\) is a symmetric Dirichlet form in \(L^2(\hat{U},\hat{m})\);

- (ii)
the mapping \(D(\hat{\mathscr {E}})\rightarrow D(\mathscr {E});u\mapsto \overline{u}:=u\circ \Phi ^{-1}\) is a Hilbert space isomorphism.

*Proof*

*u*is weakly differentiable and \(\varphi ^{-1}u^\prime =v\); that is, \(u\in D(\hat{\mathscr {E}})\). It then follows that \((u_h)\) converges to

*u*in \(\hat{\mathscr {E}}^{1/2}\)-norm. So \((D(\hat{\mathscr {E}}),\hat{\mathscr {E}})\) is a symmetric closed form in \(L^2(\hat{U},\hat{m})\).

(ii) Let \(u\in D(\hat{\mathscr {E}})\). Note that \((u^\prime /\varphi )\circ \Phi ^{-1}\in L^2(U,m)\) (use [1, (2.47)]) and \(\overline{u}\) is weakly differentiable on *U* with \(\overline{u}^\prime =(u^\prime /\varphi )\circ \Phi ^{-1}\in L^2(U,m)\). Thus the mapping is well-defined. For \(u,v\in D(\hat{\mathscr {E}})\), \(\hat{\mathscr {E}}(u,v)=\mathscr {E}(\overline{u},\overline{v})\) again using [1, (2.47)]. In particular, the mapping \(u\mapsto \overline{u}\) is injective. Now let \(u\in D(\mathscr {E})\) and put \(\hat{u}:=u\circ \Phi \). Then \(\hat{u}\in L^2(\hat{U},\hat{m})\), \(\hat{u}\) is weakly differentiable on \(\hat{U}\) with weak derivative \(\hat{u}^\prime =\varphi (u^\prime \circ \Phi )\) and \(\varphi ^{-1}\hat{u}^\prime \in L^2(\hat{U},\hat{m})\); in other words, \(\hat{u}\in D(\hat{\mathscr {E}})\). This shows that the mapping in (ii) is surjective.\(\square \)

*f*,

*g*with the properties

- (A.1)
*f*is a positive function in*C*(*B*); - (A.2)
*g*is a positive unimodal function in \(C_0(B)\).

*E*in

*B*is given by \(V(E):=\int _E f\,dx\). Put \(\rho :=f/g\). We introduce the further assumption

- (A.3)
\(\rho \in L^1(B,\mathscr {L}^1)\), \(g\in C^1(B)\) and \(g^\prime /f\in L^2(B,V)\).

*B*under

*R*; \(\check{B}\) is an open bounded interval in \(\mathbb {R}\). Then \(R:B\rightarrow \check{B}\) is a \(C^1\) bijection. Define \(\check{g}:=g\circ R^{-1}\) on \(\check{B}\). Define the measure \(\check{V}:=\check{g}\mathscr {L}^1\) on \(\check{B}\).

*U*, \(w:=\check{g}\) satisfies properties (A)–(C) above in light of the assumptions (A.1)–(A.3). We derive

**Lemma 5.5**

- (i)
\((D(\check{\mathscr {E}}),\check{\mathscr {E}})\) is a symmetric Dirichlet form in \(L^2(\check{B},\check{V})\);

- (ii)
\(C^\infty (\overline{\check{B}})\) is dense in \(D(\check{\mathscr {E}})\) with respect to the \(\check{\mathscr {E}}^{1/2}\)-norm;

- (iii)
\((D(\hat{\mathscr {E}}),\hat{\mathscr {E}})\) is a symmetric Dirichlet form in \(L^2(B,V)\);

- (iv)
the mapping \(D(\hat{\mathscr {E}})\rightarrow D(\check{\mathscr {E}});u\mapsto u\circ R^{-1}\) is a Hilbert space isomorphism.

## 6 An (*f*, *g*)-Isoperimetric Inequality

*B*,

*B*such that

**Theorem 6.1**

Suppose that *E* is a Caccioppoli set in *B* with \(|E|>0\). Then there exist \(N\in \mathbb {N}\cup \{\infty \}\) and closed intervals \(E_h=[a_{2h-1},a_{2h}]\subset \mathbb {R}\) \((h=1,\ldots ,N)\) with non-empty interior and separated by open neighbourhoods in \(\mathbb {R}\) such that *E* is \(\mathscr {L}^1\)-a.e. equivalent to the union of the \(E_h\).

The statement that the collection of intervals \((E_h)\) is separated by open intervals means that \(\inf _{k\ne h}d(E_h,E_k)>0\) for each *h*. Here, *d* denotes the standard metric on \(\mathbb {R}\).

*Proof*

*B*[1, 1.40]. Define

*w*is left-continuous on

*B*by the Hahn decomposition and inner/outer regularity [1, 1.43]. By Fubini’s theorem, \(w\in \mathrm {BV}_{\mathrm {loc}}(B)\) and \(Dw=\mu \). By [1, Proposition 3.2], \(u=c+w\) \(\mathscr {L}^1\)-a.e. on

*B*for some \(c\in \mathbb {R}\). Let

*A*be the set of atoms of \(\mu \) in

*B*. Note that

*w*is continuous on \(B\setminus A\) and \(w(t+)-w(t)=\mu (\{t\})\) at each \(t\in A\). As \(c+w\in \{0,1\}\) \(\mathscr {L}^1\)-a.e. on

*B*, \(\mu (\{t\})\in \{-1,1\}\) for each \(t\in A\). Let \(\Omega \) be a relatively compact open set in

*B*. Then \(\mathrm {Card}(A\cap \Omega )=|\mu |(A\cap \Omega )\le |\mu |(\Omega )<\infty \). Thus the set of atoms

*A*accumulates at \(\partial B\) (if at all). By the observations above, the function \(c+w\) is constant on each connected component of \(B\setminus A\) with values in the set \(\{0,1\}\). Let the sets \(E_h\) be the closure of the open intervals in \(B\setminus A\) where \(c+w\) takes the value 1.\(\square \)

*g*be a positive lower semicontinuous function on

*B*. Let

*E*be a Caccioppoli set in

*B*. The

*g*-perimeter of

*E*relative to

*B*is defined by

**Lemma 6.2**

*g*be a positive lower semicontinuous function on

*B*and

*E*a Caccioppoli set in

*B*. Then

*A*stands for the set of atoms of \(D\chi _E\) in

*B*.

*Proof*

A direct computation gives \(D\chi _E=\sum _{a\in A}D\chi _E(\{a\})\delta _a\) and \(|D\chi _E|=\sum _{a\in A}\delta _a\) which gives the result. We use \(\delta _a\) to stand for the Dirac measure at *a*.\(\square \)

*f*,

*g*be densities on

*B*satisfying conditions (A.1)–(A.3). The weighted volume of an \(\mathscr {L}^1\)-measurable set

*E*in

*B*is the measure given by \(V(E):=\int _E f\,dx\). Define \(F:[-1,1]\rightarrow [0,V(B)]\) by \(F(x):=V((-1,x))\) and

- (A.4)
\(J(p)=J(V(B)-p)\) for \(0<p<V(B)\);

- (A.5)
for all \(p,q>0\) with \(p+q<V(B)\), \(J(p+q)<J(p)+J(q)\).

*E*in

*B*the \(\bigstar \)-rearrangement of

*E*is defined by \(E^\bigstar :=(-1,F^{-1}(V(E)))\). We then have that the following (

*f*,

*g*)-isoperimetric inequality is valid.

**Theorem 6.3**

Assume (A.1)–(A.5). Suppose that *E* is a Caccioppoli set in *B*. Then \(P_g(E,B)\ge P_g(E^\bigstar ,B)\).

*Proof*

*p*,

*q*are extremal.

*E*has the form \(E=\bigcup _{h=1}^N[a_{2h-1},a_{2h}]\) with \(-1\le a_1<a_2<\cdots <a_{2N-1}<a_{2N}\le 1\) for some \(N\in \mathbb {N}\). Put \(p_j:=F(a_j)\) so that \(0\le p_1<\cdots < p_{2N}\le V(B)\) and \(\sum _{h=1}^N(p_{2h}-p_{2h-1})=p\). By Lemma 6.2, (6.2) and (A.5),

*E*as in the statement follows by Theorem 6.1, the monotone convergence theorem and continuity of

*J*. \(\square \)

We now investigate the equality case in the isoperimetric inequality.

**Theorem 6.4**

Assume (A.1)–(A.5). Suppose that *E* is a Caccioppoli set in *B*. Assume that \(P_g(E,B) = P_g(E^\bigstar ,B)\). Then either \(E=E^\bigstar \) or \(B\setminus E=(B\setminus E)^\bigstar \) *V*-a.e.

*Proof*

*E*is the union of closed intervals \(E_h\subset \mathbb {R}\) \((h=1,\ldots ,N, N\in \mathbb {N}\cup \{\infty \})\) with non-empty interior and separated by open neighbourhoods in \(\mathbb {R}\) as in Theorem 6.1. In virtue of (6.1) we may take \(N>1\). We can then find \(x\in B\setminus E\) such that \(V(E\cap (x,1))>0\) and \(V(E\cap (-1,x))>0\). Put \(E_-:=E\cap (-1,x)\), \(E_+:=E\cap (x,1)\), \(F_-:=E_-^\bigstar \) and \(F_+:=B\setminus (B\setminus E_+)^\bigstar \). Note that \(V(E_-)=V(F_-)=p_-\) and \(V(E_+)=V(F_+)=p_+\) for some \(p_\pm \in (0,V(B))\). We have

**Proposition 6.5**

Assume (A.1)–(A.3). If \(g^\prime /f\) is strictly decreasing on *B* then (A.5) holds.

*Proof*

*J*is differentiable on (0,

*V*(

*B*)). Moreover, \((F^{-1})^\prime (p)=1/(f\circ F^{-1})(p)\) and

*J*is strictly concave on (0,

*V*(

*B*)) as \((0,V(B))\mapsto (-1,1):\,p\mapsto (F^{-1})(p)\) is increasing. For \(0<p<q<V(B)\) we have that

*u*be a real-valued \(\mathscr {L}^1\)-measurable function on

*B*. Put \(\mu _u(t):=V(\{|u|>t\})\) for \(t\ge 0\). The function \(\mu _u:[0,\infty )\rightarrow [0,V(B)]\) is non-increasing, right-continuous and \(\mu _u(t)\rightarrow 0\) as \(t\rightarrow \infty \). Define its right-continuous inverse \(u^\sharp :[0,V(B)]\rightarrow [0,\infty ]\) by

*B*. Note that \(\mu _u(t)>s\) if and only if \(u^\sharp (s)>t\) (see Lemma 10.1). It follows that

The following result is a Hardy–Littlewood type inequality and can be proved as in [14, 13.10] (see also [9, Theorem 3]).

**Theorem 6.6**

*u*,

*v*be real-valued \(\mathscr {L}^1\)-measurable functions on

*B*. Then

The next non-expansivity result can be found in [9, Corollary 1].

**Theorem 6.7**

*u*,

*v*be real-valued \(\mathscr {L}^1\)-measurable functions on

*B*. Then

*B*as follows. The length of a piecewise \(C^1\) parametrised curve \(\gamma :[\alpha ,\beta ]\rightarrow B\) in \((B,\hat{d})\) is

*B*connecting

*x*to

*y*; \(\hat{d}(\cdot ,\cdot )\) is a metric on

*B*.

**Lemma 6.8**

For \(x,y\in B\), \(\hat{d}(x,y)=d(R(x),R(y))\).

*Proof*

Let \(\gamma :[\alpha ,\beta ]\rightarrow B\) be a piecewise \(C^1\) parametrised curve in *B* connecting *x* to *y*. Then \(\hat{L}[\gamma ]=L[R\circ \gamma ]\) in an obvious notation. So \(\hat{d}(x,y)\le d(R(x),R(y))\). A similar argument gives the reverse inequality.\(\square \)

Note that for each \(\mathscr {L}^1\)-measurable set *E* in *B*, \(\check{V}(R(E))=V(E)\).

**Lemma 6.9**

*B*resp. \(\check{B}\). Moreover, let

*E*be a Caccioppoli set in

*B*. Then

- (i)
\(|D\chi _{R(E)}|=R_\sharp |D\chi _{E}|\);

- (ii)
\(P_{\check{g}}(R(E),\check{B})=P_g(E,B)\).

*Proof*

The function \(\check{F}:\mathbb {R}\rightarrow [0,V(B)]\) defined by \(\check{F}(x):=\check{V}(\check{B}\cap (-\infty ,x))\) is the cumulative distribution function of \(\check{g}\). Let *u* be a real-valued \(\mathscr {L}^1\)-measurable function on \(\check{B}\). Put \(\check{\mu }_{u}(t):=\check{V}(\{|u|>t\})\) for \(t\ge 0\) and denote by \(\check{u}^{\sharp }:[0,\check{V}(\check{B})]\rightarrow [0,\infty ]\) its right-continuous inverse (as in the Appendix). Define \(u^\star :=\check{u}^{\sharp }\circ \check{F}\) on \(\check{B}\).

**Proposition 6.10**

Let *u* be a real-valued \(\mathscr {L}^1\)-measurable function on *B* and put \(v:=u\circ R^{-1}\). Then \(u^\bigstar =v^\star \circ R\). In particular, for any \(\mathscr {L}^1\)-measurable set \(E\subset B\), \(R(E)^\star =R(E^\bigstar )\).

*Proof*

We have that \(\mu _u(t)=\check{\mu }_{v}(t)\) for each \(t\ge 0\); hence \(u^\sharp =\check{v}^{\sharp }\) on [0, *V*(*B*)]. Now , \(\check{F}\circ R=F\) on \([-1,1]\). This leads to the first claim. The second then follows straightforwardly.\(\square \)

**Corollary 6.11**

Assume (A.1)–(A.5). Suppose that *E* is a Caccioppoli set in \(\check{B}\). Then \(P_{\check{g}}(E,\check{B})\ge P_{\check{g}}(E^\star ,\check{B})\).

*Proof*

**Corollary 6.12**

Assume (A.1)–(A.5). Suppose that *E* is a Caccioppoli set in \(\check{B}\). Assume that \(P_{\check{g}}(E,\check{B}) = P_{\check{g}}(E^\star ,\check{B})\). Then either \(E=E^\star \) or \(\check{B}\setminus E=(\check{B}\setminus E)^\star \) \(\check{V}\)-a.e.

Finally, we state a counterpart of Theorem 6.7.

**Theorem 6.13**

*u*,

*v*be real-valued \(\mathscr {L}^1\)-measurable functions on \(\check{B}\). Then

## 7 A Pólya–Szegö Inequality

*r*-neighbourhood of an \(\mathscr {L}^1\)-measurable set

*E*in \((\check{B},d)\); by convention, \(\emptyset _r=\emptyset \). The Minkowski content of

*E*is the quantity

**Lemma 7.1**

*E*be a finite union of open intervals in \((\check{B},d)\). Then

- (i)
*E*is a Caccioppoli set in \(\check{B}\); - (ii)
\(\check{V}^+(E)=P_{\check{g}}(E,\check{B})\).

*Proof*

(i) The set *E* is a finite union of disjoint open intervals, \(\overline{E}\) is a finite union of closed intervals in \(\mathbb {R}\) with non-empty interior and separated by open sets in the sense of Theorem 6.1 and \(\overline{E}=E\cup I\) for a finite set \(I\subset \mathbb {R}\). So *E* is \(\mathscr {L}^1\)-a.e. equivalent to \(F:=\overline{E}\cap \check{B}\); in particular, *E* is a Caccioppoli set in \(\check{B}\) by Theorem 6.1.

*E*as in the statement follows from the property that the closed intervals in \(\overline{E}\) are separated by open sets.\(\square \)

*B*.

**Lemma 7.2**

Let *E* be an \(\mathscr {L}^1\)-measurable set in \(\check{B}\). Then \(\check{V}(E_r)\ge \check{V}((E^\star )_r)\) for each \(r>0\). In particular, the rearrangement \(\cdot ^\star \) is smoothing in the sense that \((E^\star )_r\subset (E_r)^\star \) for each \(\mathscr {L}^1\)-measurable set *E* in \(\check{B}\) and \(r>0\).

*Proof*

*r*-neighbourhood (\(r>0\)) of any open ball in \(\check{B}\) is an open ball in \(\check{B}\). Let

*E*be a finite union of open intervals in \((\check{B},d)\). By Lemma 7.1 and Corollary 6.11,

*E*in \(\check{B}\) with \(0<\check{V}(E)<V(B)\) and \(r>0\). The result then extends to \(\mathscr {L}^1\)-measurable sets in \(\check{B}\).\(\square \)

**Lemma 7.3**

Let *A*, *E* be \(\mathscr {L}^1\)-measurable sets in \(\check{B}\) with \(A\subset E\). Then \(d(A^\star ,\check{B}\setminus E^\star )\ge d(A,\check{B}\setminus E)\).

Here, \(d(A,E):=\inf \{d(x,y):x\in A,y\in E\}\) with the understanding that \(\inf \emptyset =+\infty \).

*Proof*

We use the criterion that for \(r>0\), \(A_r\subset E\) if and only if \(d(A,\check{B}\setminus E)\ge r\). Put \(r:=d(A,\check{B}\setminus E)\); we may assume that \(r>0\). By the criterion, \(A_r\subset E\) and hence \((A_r)^\star \subset E^\star \). By Lemma 7.2, \(\check{V}((A_r)^\star )=\check{V}(A_r)\ge \check{V}((A^\star )_r)\) meaning \((A^\star )_r\subset (A_r)^\star \subset E^\star \) which entails that \(d(A^\star ,\check{B}\setminus E^\star )\ge r\) by the criterion. \(\square \)

*u*on \(\check{B}\) is defined by

*u*is uniformly continuous on \(\check{B}\) if and only if \(\lim _{t\downarrow 0}\omega _u(t)=0\). We state the following criterion without proof.

**Lemma 7.4**

Let *u* be a real-valued function on \(\check{B}\) and \(t,\tau >0\). Then \(\omega _u(t)>\tau \) if and only if there exist \(s,s^\prime \in \mathbb {R}\) with \(s>s^\prime +\tau \) such that \(d(\{u>s\},\check{B}\setminus \{u>s^\prime \})<t\).

**Proposition 7.5**

- (i)
Let

*u*be a real-valued \(\mathscr {L}^1\)-measurable function on \(\check{B}\). Then \(\omega _u(t)\ge \omega _{u^\star }(t)\) for each \(t>0\). - (ii)
If

*u*is uniformly continuous on \(\check{B}\) then so is \(u^\star \). - (iii)
If

*u*is Lipschitz continuous on \(\check{B}\) then so is \(u^\star \) and \(\mathrm {Lip}(u^\star ,\check{B})\le \mathrm {Lip}(u,\check{B})\).

*Proof*

*u*be a Lipschitz continuous function on \((\check{B},d)\). By Rademacher’s theorem (cf. [1, Theorem 2.14])

*u*is differentiable \(\mathscr {L}^1\)-a.e. on \(\check{B}\) and its derivative coincides with the weak derivative on a set of full measure. Put

**Lemma 7.6**

*u*be a nonnegative Lipschitz continuous function on \((\check{B},d)\). Then

- (i)
\(\check{\mu }_{u}\in \mathrm {BV}(\mathbb {R})\);

- (ii)
\(D\check{\mu }_{u}=-u_\sharp \check{V}\);

- (iii)
- (iv)
- (v)
\(A:=\Big \{t\in \mathbb {R}:\mathscr {L}^1(Z\cap \{u=t\})>0\Big \}\) is the set of atoms of \(D\check{\mu }_{u}\) and Open image in new window ;

- (vi)\(\check{\mu }_u\) is differentiable \(\mathscr {L}^1\)-a.e. on \(\mathbb {R}\) with derivative given byfor \(\mathscr {L}^1\)-a.e. \(t\in \mathbb {R}\);$$\begin{aligned} \check{\mu }_{u}^\prime (t)=-\int _{(\check{B}\setminus Z)\cap \{u=t\}}\frac{\check{g}}{|u^\prime |}\,d\mathcal {H}^{0} \end{aligned}$$
- (vii)
\(\mathrm {Ran}(u)=\mathrm {supp}(D\check{\mu }_u)\).

The notation above \(D\check{\mu }_{u}^a\), \(D\check{\mu }_{u}^s\), \(D\check{\mu }_{u}^j\) stands for the absolutely continuous resp. singular resp. jump part of the measure \(D\check{\mu }_{u}\) (see [1, 3.2] for example).

*Proof*

*u*, \(D\check{\mu }_{u}=-u_\sharp \check{V}\) (cf. [1, 1.70]). By (7.1),for any \(\mathscr {L}^1\)-measurable set

*A*in \(\mathbb {R}\). In light of the above, we may identify Open image in new window and Open image in new window . The set of atoms of \(D\check{\mu }_{u}\) is defined by \(A:=\{t\in \mathbb {R}:D\check{\mu }_{u}(\{t\})\ne 0\}\). By [13, Lemma 7.7], we may write

*A*as in (v). The monotone function \(\check{\mu }_u\) is a good representative within its equivalence class and is differentiable \(\mathscr {L}^1\)-a.e. on \(\mathbb {R}\) with derivative given by the density of \(D\check{\mu }_{u}\) with respect to \(\mathscr {L}^1\) by [1, Theorem 3.28]. Item (vii) follows from (ii).\(\square \)

**Lemma 7.7**

Let *u* be a nonnegative Lipschitz continuous function on \((\check{B},d)\). Then \(\int _{\check{B}\cap Z}u^2\,d\check{V}=\int _{\check{B}\cap Z_\star }(u^\star )^2\,d\check{V}\).

*Proof*

*Z*has finite \(\mathscr {L}^1\)-measure, \(A\subset \mathbb {R}\) is a countable set. Thus

**Theorem 7.8**

Assume (A.1)–(A.5). Let *u* be a Lipschitz continuous function on \((\check{B},d)\). Then \(u,u^\star \in D(\check{\mathscr {E}})\) and \(\check{\mathscr {E}}(u,u)\ge \check{\mathscr {E}}(u^\star ,u^\star )\).

*Proof*

Given a Lipschitz continuous function *u* on \((\check{B},d)\), \(u\in W^{1,\infty }(\check{B})\) and \(\Vert u^\prime \Vert _{L^\infty (\check{B})}=\mathrm {Lip}(u,\check{B})\) (see [1, Proposition 2.13]) so \(u\in D(\check{\mathscr {E}})\). The same is true for \(u^\star \) by Proposition 7.5. Replacing *u* by |*u*| and using the contraction property of the Dirichlet form \((D(\check{\mathscr {E}}),\check{\mathscr {E}})\) we may assume that *u* is nonnegative.

*t*comprise a set of full measure in the range of

*u*. Then \(\partial \{u>t\}=\check{B}\cap \{u=t\}\cup \partial \check{B}\cap \partial \{u>t\}\) and \(\{u>t\}\) is a Caccioppoli set in \(\check{B}\) with finite \(\check{g}\)-perimeter; and likewise for \(u^\star \). From Corollary 6.11,

**Corollary 7.9**

Let \(u\in D(\check{\mathscr {E}})\). Then \(u^\star \in D(\check{\mathscr {E}})\) and \(\check{\mathscr {E}}(u,u)\ge \check{\mathscr {E}}(u^\star ,u^\star )\).

*Proof*

*u*in \((D(\check{\mathscr {E}}),\check{\mathscr {E}})\); each \(u_h\) is Lipschitz continuous on \(\check{B}\). By Theorem 6.13, \((u^\star _h)\) converges to \(u^\star \) in \(L^2(\check{B},\check{V})\). By Theorem 7.8, each \(u^\star _h\in D(\check{\mathscr {E}})\) and \(\check{\mathscr {E}}(u^\star _h,u^\star _h)\le \check{\mathscr {E}}(u_h,u_h)\), so the sequence \((\check{\mathscr {E}}(u^\star _h,u^\star _h))_h\) is uniformly bounded in \(\mathbb {R}\). By the Banach-Alaoglu theorem (cf. [19, A2 Theorem 2.1]) we may assume that \(u^\star _h\rightarrow v\) weakly as \(h\rightarrow \infty \) in \((D(\check{\mathscr {E}}),\check{\mathscr {E}})\) for some \(v\in D(\check{\mathscr {E}})\) by selecting a subsequence if necessary. We may identify

*v*with \(u^\star \) thanks to the \(L^2(\check{B},\check{V})\) convergence and the Banach-Saks theorem (cf. [19, A2 Theorem 2.2]); hence \(u^\star \in D(\check{\mathscr {E}})\). By [16, Theorem 10.1.5],

**Corollary 7.10**

Let \(u\in D(\hat{\mathscr {E}})\). Then \(u^\bigstar \in D(\hat{\mathscr {E}})\) and \(\hat{\mathscr {E}}(u,u)\ge \hat{\mathscr {E}}(u^\bigstar ,u^\bigstar )\).

*Proof*

## 8 Equality Case in the Pólya–Szegö Inequality

We now investigate the equality case in the Pólya–Szegö inequality.

**Lemma 8.1**

- (i)
Put \(v:=u\wedge t\). Then \(\int _{\check{B}}|v^\prime |^2\,d\check{V}=\int _{\check{B}\cap \{u>t\}}|u^\prime |^2\,d\check{V}=\int _{\check{B}\cap \{u\ge t\}}|u^\prime |^2\,d\check{V}\).

- (ii)
Put \(v:=u\vee t\). Then \(\int _{\check{B}}|v^\prime |^2\,d\check{V}=\int _{\check{B}\cap \{u<t\}}|u^\prime |^2\,d\check{V}=\int _{\check{B}\cap \{u\le t\}}|u^\prime |^2\,d\check{V}\).

*Proof*

*u*is continuous so \(v=u\) and \(v^\prime =u^\prime \) there. By [13, Lemma 7.7], \(v^\prime =0\) \(\mathscr {L}^1\)-a.e. on \(\{u\ge t\}\).\(\square \)

Suppose that \(u\in W^{1,2}_{\mathrm {loc}}(\check{B})\) is precisely represented in the sense of [18, (2.5)]. Then the set \(\{u=t\}\) is finite or countably infinite for \(\mathscr {L}^1\)-a.e. \(t\in \mathbb {R}\) and the coarea formula (7.1) holds for *u* by [18, Theorem 1.1]. With *Z* as before it follows that \(Z\cap \{u=t\}=\emptyset \) for \(\mathscr {L}^1\)-a.e. \(t\in \mathbb {R}\) and hence \(N:=u(Z)\subset \mathbb {R}\) is \(\mathscr {L}^1\)-negligible.

**Lemma 8.2**

Let *u* be a nonnegative function in \(W^{1,2}_{\mathrm {loc}}(\check{B})\) precisely represented in the sense of [18, (2.5)]. Then statements (i)–(vii) of Lemma 7.6 hold.

*Proof*

This runs as in Lemma 7.6. \(\square \)

**Lemma 8.3**

Let \(u\in D(\check{\mathscr {E}})\) be nonnegative. Then \(\int _{\check{B}\cap Z}u^2\,d\check{V}=\int _{\check{B}\cap Z_\star }(u^\star )^2\,d\check{V}\).

*Proof*

The proof proceeds as in Lemma 7.7. \(\square \)

**Lemma 8.4**

- (i)for \(t^\prime ,t^{\prime \prime }\in \mathbb {R}\) with \(0\le t^\prime <t^{\prime \prime }\),$$\begin{aligned} \int _{\check{B}\cap \{t^\prime <u\le t^{\prime \prime }\}}\Big (u^2+|u^\prime |^2\Big )\,d\check{V} \ge \int _{\check{B}\cap \{t^\prime <u\le t^{\prime \prime }\}}\Big ((u^\star )^2+|(u^\star )^\prime |^2\Big )\,d\check{V}; \end{aligned}$$
- (ii)for \(\mathscr {L}^1\)-a.e. \(t\in \mathbb {R}\),where \(\tilde{u}\) is the unique continuous representative of$$\begin{aligned} \int _{(\check{B}\setminus Z)\cap \{\tilde{u}=t\}}\frac{\tilde{u}^2+|\tilde{u}^\prime |^2}{|\tilde{u}^\prime |}\check{g}\,d\mathcal {H}^0 \ge \int _{(\check{B}\setminus Z_\star )\cap \{u^\star =t\}}\frac{(u^\star )^2+|(u^\star )^\prime |^2}{|(u^\star )^\prime |}\check{g}\,d\mathcal {H}^0 \end{aligned}$$
*u*(cf. [1, Definition 3.31 and after]).

*Proof*

*u*,

*v*. The statement then follows from Corollary 7.9.

*w*is absolutely continuous [1, Definition 3.31]. By [1, Theorem 3.28],

*w*is differentiable \(\mathscr {L}^1\)-a.e. on \(\mathbb {R}\) and \(w^\prime =\rho \) \(\mathscr {L}^1\)-a.e. on \(\mathbb {R}\). The same holds for the function \(w_\star \) defined as for

*w*but with \(u^\star \) in place of

*u*. Note that \(\widetilde{u^\star }=u^\star \). The statement then follows from (i).\(\square \)

We state the following lemma without proof.

**Lemma 8.5**

Let *u* be a continuous real-valued function on \(\check{B}\). Suppose that for \(\mathscr {L}^1\)-a.e. \(t>0\) the set \(\{u>t\}\) is either an open interval in \(\check{B}\) abutting a boundary point or \(\{u>t\}=\emptyset \). Then *u* is monotone on \(\check{B}\).

**Theorem 8.6**

Let \(u\in D(\check{\mathscr {E}})\) be nonnegative and suppose that \(\check{\mathscr {E}}(u,u)=\check{\mathscr {E}}(u^\star ,u^\star )\). Then \(\tilde{u}\) is monotone on \(\check{B}\).

*Proof*

*u*and we may replace the sign \(\ge \) with the equality sign. In particular,

**Corollary 8.7**

Let \(u\in D(\hat{\mathscr {E}})\) be nonnegative and suppose that \(\hat{\mathscr {E}}(u,u)=\hat{\mathscr {E}}(u^\bigstar ,u^\bigstar )\). Then \(\tilde{u}\) is monotone on *B*.

*Proof*

*B*.\(\square \)

## 9 Application to Exchange Flow

**Lemma 9.1**

- (i)
\((\mathscr {D},\mathscr {E})\) is closable in \(L^2(B,\,\mathscr {L}^1)\) with closure denoted \((D(\mathscr {E}),\,\mathscr {E})\);

- (ii)
\((D(\mathscr {E}),\,\mathscr {E})\) is a symmetric Dirichlet form in \(L^2(B,\,\mathscr {L}^1)\);

- (iii)
\(D(\mathscr {E})=W^{1,2}_0(B)\).

*Proof*

*u*on

*B*such that \(|u|<\infty \) \(\mathscr {L}^1\)-a.e. and there exists an \(\mathscr {E}\)-Cauchy sequence \((u_h)\) of functions in \(D(\mathscr {E})\) such that \(u_h\rightarrow u\) \(\mathscr {L}^1\)-a.e. on

*B*. By [12, Lemma 1.5.5], \((D(\mathscr {E})_e,\mathscr {E})\) is a Hilbert space. The identity (9.1) extends to \(D(\mathscr {E})_e\) and \(D(\mathscr {E})_e\subset L^1(B,\psi ^{-1}\mathscr {L}^1)\).

*V*be the measure \(V:=\psi \mathscr {L}^1\) on

*B*. We work with the symmetric Dirichlet formin \(L^2(B,V)\) with domain

**Proposition 9.2**

*Proof*

*B*gives

*E*extends to \(D(\mathscr {E})\) by density of \(\mathscr {D}\) in \(D(\mathscr {E})\) as does (9.2). Let \(u\in D(\mathscr {E})_e\) and choose a sequence \((u_h)\) in \(D(\mathscr {E})\) such that \(u_h\rightarrow u\) \(\mathscr {L}^1\)-a.e. on

*B*as \(h\rightarrow \infty \) and \((u_h)\) is a \(\mathscr {E}\)-Cauchy sequence. Then \((\overline{u}_h)\) is a Cauchy sequence in \((D(\hat{\mathscr {E}}),\hat{\mathscr {E}})\) with limit \(v\in D(\hat{\mathscr {E}})\) say. Define \(Eu:=v\). This definition is well-defined and the identity (9.2) holds on \(D(\mathscr {E})_e\). In particular, the mapping

*E*is injective.

We show that *E* is a surjection. Let \(v\in D(\hat{\mathscr {E}})\). We put \(u:=\psi v\) and claim that \(u\in D(\mathscr {E})_e\). Then there exists a sequence \((w_h)\) in \(C^\infty (\overline{\check{B}})\) such that \(v_h:=w_h\circ R\rightarrow v\) in \((D(\hat{\mathscr {E}}),\hat{\mathscr {E}})\) as \(h\rightarrow \infty \). Put \(u_h:=\psi v_h\) for each *h*. Now \(u_h\in C^1(\overline{B})\) and \(u_h=0\) on \(\partial B\) for each *h*. This means that \((u_h)\) is a sequence in \(D(\mathscr {E})\). By (9.2), \((u_h)\) is a \(\mathscr {E}\)-Cauchy sequence. As \((v_h)\) converges to *v* in \(L^2(B,V)\) we may assume that \((v_h)\) converges to *v* *V*-a.e. on *B* by selecting a subsequence if necessary. Thus, \((u_h)\) converges to *u* \(\mathscr {L}^1\)-a.e. on *B*. This shows that \(u\in D(\mathscr {E})_e\) and \(Eu=v\).\(\square \)

**Theorem 9.3**

For any \(\mathscr {L}^1\)-measurable set *A* in *B*, \(J(A)\le J(A^\bigstar )\).

*Proof*

*A*be an \(\mathscr {L}^1\)-measurable set in

*B*and put \(u:=u_A\in D(\mathscr {E})\) and \(v:=u_{A^\bigstar }\in D(\mathscr {E})\). By Theorem 6.6, Proposition 9.2, the Cauchy–Schwarz inequality and Corollary 7.10,

**Theorem 9.4**

Suppose that *A* is an \(\mathscr {L}^1\)-measurable set in *B* such that \(J(A)= J(A^\bigstar )\). Then *A* is \(\mathscr {L}^1\)-a.e. equivalent to an open interval in *B* abutting a boundary point of *B*.

*Proof*

We may assume that \(0<V(A)<V(B)\). From the chain of inequalities in the proof of Theorem 9.3 we derive that \(\hat{\mathscr {E}}(\overline{u}^\bigstar ,\overline{u}^\bigstar )=\hat{\mathscr {E}}(\overline{u},\overline{u})\). Put \(v:=\overline{u}\). From Corollary 8.7 we infer that \(\widetilde{v}\) is monotone on *B*. Put \(p:=V(A)/V(B)\in (0,1)\). By Theorem 9.3, *A* is an optimal configuration for the problem (1.2) for the data (*B*, *p*). By Proposition 3.3, \(V(A\Delta \{\widetilde{v}>c\})=0\) for some \(c\in (0,1)\). As \(\widetilde{v}\) is monotone, \(\{\widetilde{v}>c\}\) is an open interval in *B* abutting an end-point of *B*. This leads to the result.\(\square \)

We may now characterise optimal configurations for the problem (1.2).

**Theorem 9.5**

- (i)
The sets \((-1,F^{-1}(pV(B)))\) and \((F^{-1}((1-p)V(B)),1)\) are optimal configurations for the problem (1.2) with data (

*B*,*p*). - (ii)
If \(E\subset B\) is an optimal configuration for the data (

*B*,*p*) then*E*is \(\mathscr {L}^1\)-a.e. equivalent to one of the sets in (i).

Let *U* be a bounded open connected set in \(\mathbb {R}^n\) (\(n\ge 1\)).

**Proposition 9.6**

- (i)
For \(\lambda \in (-1,1)\), \(\gamma (U,\lambda ) = 2 I(U,p)\) where \(p=(1-\lambda )/2\).

- (ii)
\(\gamma (U) = 2\sup _{p\in (0,1)} I(U,p)\).

*Proof*

*A*be an open set in

*U*. Suppose

*u*satisfies (1.3) and the condition \((u,1)=0\). Put \(f:=-(\lambda + 1)\chi _A-(\lambda - 1) \chi _{U\setminus A}\); then \(u=Gf\). From the flux-balance condition and symmetry of the Green operator,

**Lemma 9.7**

The mapping \((0,1)\rightarrow \mathbb {R};p\rightarrow I(B,p)\) has a unique global maximum at \(p=1/2\).

*Proof*

**Theorem 9.8**

- (i)
The sets \((-1,F^{-1}(pV(B)))\) and \((F^{-1}((1-p)V(B)),1)\) are optimal configurations for the problem (1.5) with data \((U,\lambda )\) where \(p=(1-\lambda )/2\).

- (ii)
If \(E\subset B\) is an optimal configuration for the problem (1.5) with data \((U,\lambda )\) then

*E*is \(\mathscr {L}^1\)-a.e. equivalent to one of the sets in (i). - (iii)
The sets \((-1,0)\) and (0, 1) are optimal configurations for the problem (1.4).

- (iv)
If \(E\subset B\) is an optimal configuration for the problem (1.4) then

*E*is \(\mathscr {L}^1\)-a.e. equivalent to one of the sets in (iii).

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