# Symmetrization of exterior parabolic problems and probabilistic interpretation

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## Abstract

We prove a comparison theorem for the spatial mass of the solutions of two exterior parabolic problems, one of them having symmetrized geometry, using approximation of the Schwarz symmetrization by polarizations, as it was introduced in Brock and Solynin (Trans Am Math Soc 352(4):1759–1796, 2000). This comparison provides an alternative proof, based on PDEs, of the isoperimetric inequality for the Wiener sausage, which was proved in Peres and Sousi (Geom Funct Anal 22(4):1000–1014, 2012).

## Keywords

Schwarz symmetrization Polarization Parabolic equations Wiener sausage## Mathematics Subject Classification

35K20 35B51## 1 Introduction

In the present article we prove a comparison theorem for the spatial mass, at any time *t*, for the solutions of two parabolic exterior problems, the second being the “symmetrization” of the first one. In order to do so, we show that the spatial mass of the solution decreases under polarization, and since the Schwarz symmetrization is the limit of compositions of polarizations, we carry the comparison to the limit. This technique was introduced in [4].

Our result is motivated by a problem in probability theory. Namely, the isoperimetric inequality for the Wiener sausage, which was proved in [15]. The problem is the following. If \((w_t)_{t\ge 0}\) is a Wiener process in \(\mathbb {R} ^d\), one wants to minimize the expected volume of the set \(\cup _{t \le T} (w_t+A)\), for \(T\ge 0\), over “all” subsets *A* of \(\mathbb {R}^d\) of a given measure. It was proved in [15] that the minimizer is the ball (the result was for a more general setting, see Sect. 2 below). This was proved by obtaining a similar result for random walks by using rearrangement inequalities of Brascamp-Lieb-Luttinger type on the sphere, which were proved in [6], and then by Donsker’s theorem, the authors obtain the result for the Wiener process. It is known that the expected volume of the Wiener sausage up to time *t*, can be expressed as the integral over \(x\in \mathbb {R}^d\) of the probability that a Wiener process starting from \(x \in \mathbb {R}^d\) hits the set *A* by time *t*. It is also known that this collection of probabilities, as a function of (*t*, *x*), satisfies a parabolic equation on \((0,T) \times \mathbb {R}^d{\setminus } A\). For properties of these hitting times and applications to the Wiener sausage we refer the reader to [3] and references therein, and for the case of Riemannian manifolds, we refer to [11]. Therefore, we provide an alternative proof of the isoperimetric inequality for the Wiener sausage, based on PDE techniques.

Comparison results between solutions of partial differential equations and solutions of their symmetrized counterparts, were first proved in [16]. Since then, much work has been done in this area, for elliptic and parabolic equations, and we refer the reader to [2, 4, 13, 14] and references therein. The equations under consideration at these works, are on a bounded domain, with Dirichlet or Neumann boundary conditions. Our approach is based on the techniques introduced in [4].

*A*,

*B*subsets of \(\mathbb {R}^d\), we write

*H*be a closed half-space. If

*A*is measurable, |

*A*| will stand for the Lebesgue measure of

*A*. We will write \(\sigma _H(x)\) and \(A_H\) for the reflections of

*x*and

*A*respectively, with respect to the shifted hyperplane \(\partial H\). We will write \(\overline{A}\) and \(\underline{A}\) for the closure and the interior of

*A*respectively. We will use the notation \(P_HA\) for the polarization of

*A*with respect to

*H*, that is

*u*on \(\mathbb {R}^d\) we will write \(P_H u\) for the polarization of

*u*with respect to

*H*, that is

*H*such that \(0 \in H\). For positive functions

*f*and

*g*on \(\mathbb {R}^d\) and for \(H \in \mathcal {H}\), we will write \(f \lhd _H g\), if \(f(x)+f(\sigma _H(x) ) \le g(x)+g(\sigma _H(x) )\) for a.e. \(x\in H\). For a bounded set \(V \subset \mathbb {R}^d\), we will denote by \(V^*\) the closed, centered ball of volume |

*V*|. For a positive function

*u*on \(\mathbb {R}^d\) such that \(|\{u> r \}|< \infty \) for all \(r >0\), we denote by \(u^*\) its symmetric decreasing rearrangement. For an open set \(D \subset \mathbb {R}^d\) we denote by \(H^1(D)\) the space of all functions in \(u \in L_2(D)\) whose distributional derivatives \(\partial _i u:=\frac{\partial }{\partial x_i}u\), \(i=1,..,d\), lie in \(L_2(D)\), equipped with the norm

*D*) in \(H^1(D)\). We will write \(\mathbb {H}^1(D)\), and \(\mathbb {H}^1_0(D)\) for \(L_2((0,T);H^1(D))\), and \(L_2((0,T);H^1_0(D))\) respectively. Also we define , \(\mathscr {H}^1(D):=\mathbb {H}^1(D) \cap C([0,T];L_2(D))\) and \(\mathscr {H}^1_0(D):=\mathbb {H}^1_0(D) \cap C([0,T];L_2(D))\). The notation \((\cdot ,\cdot )\), will be used for the inner product in \(L_2(\mathbb {R}^d)\). Also, the summation convention with respect to integer valued repeated indices will be in use.

The rest of the article is organized as follows. In Sect. 2 we state our main results. In Sect. 3 we prove a version of the parabolic maximum principle, and some continuity properties of the solution map with respect to the set *A*. These tools are then used in Sect. 4 in order to prove the main theorems.

## 2 Main results

Let \((\varOmega , \mathscr {F}, \mathbb {P})\) be a probability space carrying a standard Wiener process \((w_t)_{t \ge 0}\) with values in \(\mathbb {R}^d\), and let *A* be compact subset of \(\mathbb {R}^d\). For \(T \ge 0\) we let us consider the expected volume of the Wiener sausage generated by *A*, that is, the quantity \(\mathbb {E}\left| \cup _{t \le T} \left( w_t+A\right) \right| \). In [15], the following theorem is proved.

### Theorem 1

The result in [15] is stated for open sets *A*, and the set *A* is allowed to depend on time. As it was mentioned above, this was proved by obtaining a similar inequality for random walks, using rearrangement inequalities of Brascamp-Lieb-Luttinger type on the sphere, which were proved in [6], and then by using Donsker’s theorem, the authors obtain the inequality for the Wiener process.

### Definition 1

*u*is a solution of the problem \(\varPi (A, \psi )\) if

- (i)
\(u \in \mathscr {H}^1(\mathbb {R}^d {\setminus } A)\),

- (ii)for each \(\phi \in C^\infty _c(\mathbb {R}^d {\setminus } A)\),for all \(t\in [0,T]\)$$\begin{aligned} (u_t,\phi )=(\psi , \phi )-\int _0^t\frac{1}{2}(\partial _iu_s , \partial _i \phi ) \ ds, \end{aligned}$$
- (iii)
\(v-\xi \in \mathbb {H}^1_0(\mathbb {R}^d{\setminus } A)\), for any \(\xi \in H^1_0(\mathbb {R}^d)\) with \(\xi =1\) on a compact set \(A'\), \(A \subset \underline{A'}\).

The following is very well known.

### Theorem 2

There exists a unique solution of the problem \(\varPi (A,\psi )\).

If \(\psi \in L_2(\mathbb {R}^d)\), then by \(\varPi (A, \psi )\) we obviously mean \(\varPi (A, \psi |_{\mathbb {R}^d {\setminus } A})\). Our two main results read as follows.

### Theorem 3

Let \(\psi \in L_2(\mathbb {R}^d)\) with \(0\le \psi \le 1\), \(\psi =1\) on *A*. Let *u*, *v* be the solutions of the problems \(\varPi (A,\psi )\) and \(\varPi (P_HA,P_H\psi )\), extended to 1 on *A* and \(P_HA\) respectively. Then for all \(t \in [0,T]\), we have \( v_t \lhd _H u_t. \)

### Theorem 4

*A*. Suppose that \(|A|>0\). Let

*u*,

*v*be the solutions of the problems \(\varPi (A, \psi )\) and \(\varPi (A^*, \psi ^*)\) respectively . Then for any \(t \in [0,T]\) we have

*A*and \(A^*\) respectively.

### Remark 1

*m*-dimensional Wiener process and \(\sigma \) is a measurable function from [0,

*T*] to the set of \(d\times m\) matrices such that \((\sigma _t \sigma ^\top _t)_{i,j=1}^d\) satisfies (5).

## 3 Auxiliary results

In this section we prove some tools that we will need in order to obtain the proof of our main theorems. Namely, we present a version of the parabolic maximum principle for functions that are not necessarily continuous up to the parabolic boundary. This result (Lemma 1 below) is probably well known but we provide a proof for the convenience of the reader. The maximum principle is the main tool used in order to show the comparison of the solution of the problem \(\varPi (A, \psi )\) and its polarized version. The reason that we need this version of the maximum principle is that, \(P_H A\) is not guaranteed to have any “good” properties, even if \(\partial A\) is of class \(C^ \infty \), and therefore one can not expect the solution of \(\varPi (P_HA, P_H \psi )\) to be continuous up to the boundary. We also present certain continuity properties of the solution map with respect to the set *A*, so that we can then iterate Theorem 3 in order to obtain Theorem 4.

### Lemma 1

*Q*be a bounded open set and let \(u\in \varPsi (Q)\). If there exists \(M\in \mathbb {R}\), such that \(u_0(x)\le M\) for a.e. \(x \in Q\) and \(\limsup _{(t,x) \rightarrow (t_0, x_0)} u_t(x) \le M\) for any \((t_0,x_0) \in (0,T] \times \partial Q\), then

### Proof

*u*(in \(L_2(Q)\)) we have

*u*solves the problem \(\varPi _0(A, \xi , f)\) if

- (i)
\(u \in \mathscr {H}_0^1(\mathbb {R}^d {\setminus } A)\), and

- (ii)for each \(\phi \in C^\infty _c(\mathbb {R}^d {\setminus } A)\),for all \(t\in [0,T]\).$$\begin{aligned} (u_t,\phi )=(\xi , \phi )+\int _0^t \left( (f_s, \phi )-(a^{ij}_s\partial _iu_s , \partial _j \phi ) \right) \ ds, \end{aligned}$$

### Assumption 1

- (i)
\(\xi ^n \rightarrow \xi \) weakly in \(L_2(\mathbb {R}^d)\)

- (ii)
\( f^n \rightarrow f \) weakly in \( L_2([0,T]; L_2(\mathbb {R}^d))\)

- (iii)
\(A_{n+1} \subset A_n\) For each \(n \in \mathbb {N}\), and \( \cap _n A_n=A\).

### Lemma 2

*u*be the solutions of the problems \(\varPi _0(A_n, \xi ^n,f^n)\) and \(\varPi _0(A, \xi ,f)\) respectively . Let us extend \(u^n\) and

*u*to zero on \(A_n\) and

*A*respectively. Then

- (i)
\(u^n \rightarrow u\) weakly in \(\mathbb {H}^1_0(\mathbb {R}^d)\) as \(n \rightarrow \infty \),

- (ii)
\(u^n_t \rightarrow u_t\), weakly in \(L_2(\mathbb {R}^d)\) as \(n \rightarrow \infty \), for any \(t \in [0,T]\).

### Proof

*N*depending only on \(d, K, \kappa \), and

*T*, such that for all

*n*

*C*in the above inequality, to obtain that there exists a subsequence \((u^{n_k})_{k=1}^\infty \subset \mathbb {H}^1_0(C)\), and a function \(v \in \mathbb {H}^1_0(C)\) such that \(u^{n_k} \rightarrow v\) weakly in \(\mathbb {H}^1_0(C)\).

*k*large enough \(\text {supp} (\phi ) \subset C_{n_k}\). Also, \(u^{n_k}\) solves \(\varPi _0(A_{n_k}, \xi ^{n_k},f^{n_k})\), and therefore

*v*belongs to the space \( \mathscr {H}^1_0(D)\) (by Theorem 2.16 in [12] for example), and is a solution of \(\varPi _0(A, \xi ,f)\). By the uniqueness of the solution we get \(u=v\) (as elements of \( \mathscr {H}^1_0(C)\)), and this proves (i).

Let us fix \(t \in [0,T]\). It suffices to show that there exists a subsequence \(u^{n_k}_t\) such that \(u^{n_k}_t \rightarrow u_t\) weakly in \(L_2(C)\) as \(k \rightarrow \infty \). Notice that by (9), there exists a subsequence \(u^{n_k}_t\) which converges weakly to some \(v' \in L_2(C)\). Again, for \(\phi \in C_c^\infty (C)\) and *k* large enough, we have that (10) holds. As \(k \rightarrow \infty \), the right hand side of (10) converges to the right hand side of (11) (for our fixed \(t \in [0,T]\)), which is equal to \((u_t,\phi )\), while the left hand side of (11) converges to \((v', \phi )\). Hence, \(v'=u_t\) on *C*, and since \(u^{n_k}_t\) converges weakly in \(L_2(C)\) to \(v'\), the lemma is proved. \(\square \)

### Corollary 1

Suppose that (i) and (iii) from Assumption 1 hold, and let \(u^n\) and *u* be the solutions of the problems \(\varPi (A_n,\psi ^n)\) and \(\varPi (A,\psi )\). Set \(u^n=1\) and \(u=1\) on \(A_n\) and *A* respectively. Then for each *t*, \(u^n_t \rightarrow u_t\) weakly in \(L_2(\mathbb {R}^d)\) as \(n \rightarrow \infty \).

### Proof

Let \(g \in C^\infty _c (\mathbb {R}^d)\) with \(g=1\) on a compact set *B* such that \(A_0 \subset \underline{B}\). Then \(u^n-g\) and \(u-g\) solve the problems \(\varPi _0(A_n,\psi ^n-g,-\frac{1}{2}\Delta g)\) and \(\varPi _0(A,\psi -g,-\frac{1}{2}\Delta g)\) and the result follows by Lemma 2. \(\square \)

### Remark 2

Let \(A \subset \mathbb {R}^d\) be compact such that \(\mathbb {R}^d {\setminus } A\) is a Carathéory set (i.e. \(\partial (\mathbb {R}^d {\setminus } A) = \partial \overline{(\mathbb {R}^d {\setminus } A})\)). If \(u \in H^1(\mathbb {R}^d)\) and \(u=0\) a.e. on *A*, then \(u \in H^1_0(\mathbb {R}^d {\setminus } A)\). To see this, suppose first that \(\text {supp}(u) \subset B_R\), where *R* is large enough, so that \(A \subset B_R\). It follows that \(B_R {\setminus } A\) is a Carathéodory set, and by Theorem 7.3(ii), p. 436 in [9], if \(u \in H^1_0(B_R)\), and \(u=0\) a.e. on *A*, then \(u \in H^1_0(B_R {\setminus } A)\), and therefore \(u \in H^1_0(\mathbb {R}^d {\setminus } A)\). For general *u* we can take \(\zeta \in C^\infty _c(\mathbb {R}^d)\), such that \(0\le \zeta \le 1\) and \(\zeta (x)=1\) for \(|x| \le 1\), and set \(\zeta ^n(x)=\zeta (x/n)\). Then by the previous discussion \(\zeta ^n u \in H^1_0(\mathbb {R}^d{\setminus } A)\) and since \(\zeta ^n u \rightarrow u\) in \(H^1(\mathbb {R}^d{\setminus } A)\) we get that \(u \in H^1_0(\mathbb {R}^d{\setminus } A)\).

### Assumption 2

- (i)
\(\xi ^n \rightarrow \xi \) weakly in \(L_2(\mathbb {R}^d)\)

- (ii)
\( f^n \rightarrow f \) weakly in \( L_2([0,T]; L_2(\mathbb {R}^d))\)

- (iii)
\(d(A,A_n) \rightarrow 0\), \( |A {\setminus } A_n|\rightarrow 0\), as \(n \rightarrow \infty \), and \(\mathbb {R}^d {\setminus } A\) is a Carathéodory set.

### Lemma 3

*u*be the solutions of the problems \(\varPi _0(A_n, \xi ^n,f^n)\) and \(\varPi _0(A, \xi ,f)\). Let us extend \(u^n\) and

*u*to 0 on \(A_n\) and

*A*respectively. Then

- (i)
\(u^n \rightarrow u\) weakly in \(\mathbb {H}^1_0(\mathbb {R}^d)\),

- (ii)
\(u^n_t \rightarrow u_t\) weakly in \(L_2(\mathbb {R}^d)\), as \(n \rightarrow \infty \), for any \(t \in [0,T]\).

### Proof

*N*depending only on \(d, \kappa , T\) and

*K*, such that for all \(n \in \mathbb {N}\)

*k*large enough \(\text {supp} (\phi ) \subset \mathbb {R}^d {\setminus } A_{n_k}\). Also, \(u^{n_k}\) solves \(\varPi _0(A_{n_k}, \xi ^{n_k},f^{n_k})\), and therefore

*k*large enough, we have that (13) holds. As \(k \rightarrow \infty \), the right hand side of (13) converges to the right hand side of (14), which is equal to \((u_t,\phi )\), while for our fixed

*t*, the left hand side of (11) converges to \((v', \phi )\). Hence, \(v'=u_t\) on \(\mathbb {R}^d{\setminus } A\). Also if \(\phi \in L_\infty (A)\)

*A*. This shows that \(v' =u_t\) on \(\mathbb {R}^d\) and the lemma is proved. \(\square \)

As with Lemma 2, we have the following corollary, whose proof is similar to the one of Corollary 1.

### Corollary 2

Suppose that (i) and (iii) from Assumption 2 hold and let \(u^n\) and *u* be the solutions of the problems \(\varPi (A_n,\psi ^n)\) and \(\varPi (A,\psi )\). Set \(u^n=1\) and \(u=1\) on \(A_n\) and *A* respectively. Then for each *t*, \(u^n_t \rightarrow u_t\) weakly in \(L_2(\mathbb {R}^d)\) as \(n \rightarrow \infty \).

## 4 Proofs of Theorems 3 and 4

### Proof of Theorem 3

Let us assume for now that \(\mathbb {R}^d {\setminus } A\) has smooth boundary, \(\psi \) is compactly supported and smooth. It follows under these extra conditions that \(u \in C^\infty ([0,T] \times \overline{\mathbb {R}^d {\setminus } A})\). Also, by the De Giorgi-Moser-Nash theorem *v* is continuous in \((0,T) \times (\mathbb {R}^d {\setminus } P_HA)\).

*A*and \(P_HA\) respectively so that they are defined on the whole \(\mathbb {R}^d\), and for a function

*f*let us use the notation \(\overline{f}(x):=f(\sigma _H(x))\). Clearly it suffices to show that for each \(t \in (0,T]\)

*A*and \(A_H\) are of measure zero, since they are smooth). On \(\varGamma _1\), by definition \(w_t =0\) for any \(t \in [0,T]\), and therefore (15) holds for some \(i \in \{2,3,4\}\). Suppose it holds for \(i=2\). Since the initial conditions are compactly supported, we can find an open rectangle

*R*with \(A\cup A_H \subset R\), such that

- (i)
\(\limsup _{(0,T)\times \varTheta \ni (t,x) \rightarrow (t_0,x_0)} \hat{w_t} \le 0, \, \) for any \((t_0,x_0) \in (0,T) \times \partial A_H\),

- (ii)
\(P_H\psi - \overline{\psi } \le 0\) on \(H^c\),

- (iii)
inequality (16) holds,

*u*up to the parabolic boundary. Then we have for all

*n*large

*A*and \(\psi \), let \(A_n\) be a sequence of compact sets such that for \(n \in \mathbb {N}\), \(\mathbb {R}^d{\setminus } A_n\) has smooth boundary, \(A \subset A_{n+1} \subset A_n\), and \(A = \cap _nA_n\) (see e.g. page 60 in [7]) . Let \(0 \le \psi ^n \le 1\) be smooth with compact support such that \(\psi ^n=1\) on \(A_n\), and \(\Vert \psi ^n- \psi \Vert _{L_2(\mathbb {R}^d)} \rightarrow 0\) as \(n \rightarrow 0\). Then we also have that \(\Vert P_H\psi ^n- P_H\psi \Vert _{L_2(\mathbb {R}^d)} \rightarrow 0\) as \(n \rightarrow 0\), \(P_HA \subset P_HA_{n+1} \subset P_HA_n\) for any \(n \in \mathbb {N}\), and \(P_HA = \cap _n P_HA_n\) (see [4]). Let \(u^n\) and \(v^n\) be the solutions of the problems \(\varPi (A_n,\psi ^n)\) and \(\varPi (P_HA_n, P_H\psi _n)\) respectively. By Lemma 2 we have that \(u^n_t\) and \(v^n_t\) converge to \(u_t\) and \(v_t\) weakly in \(L_2(\mathbb {R}^d)\). In particular \(z^n:=(u^n_t,v^n_t)\) converges weakly to \(z:=(v_t,u_t)\) in \(L_2(\mathbb {R}^d;\mathbb {R}^2) \). By Mazur’s lemma there exists a sequence \((g_k=(g^1_k,g^2_k))_{k \in \mathbb {N}}\) of convex combinations of \(z^n\) such that the convergence takes place strongly. Then we can find a subsequence \(g_{k(l)}\), \(l \in \mathbb {N}\), where the convergence takes place for a.e. \(x\in \mathbb {R}^d\). For each

*l*we have

### Remark 3

With \(\psi \) and *A* as in Theorem 3, it is easily seen that if u solves \(\varPi (A, \psi )\) then \(0\le u \le 1\), if for example \(\mathbb {R}^d {\setminus } A\) has Lipschitz boundary. For general *A*, we can take \(A_n\) compact such that for all *n*, \(\mathbb {R}^d {\setminus } A_n\) has smooth boundary, and \(A_n \downarrow A\). Then the corresponding solutions \(u^n\) satisfy \(0 \le u^n \le 1\), which by virtue of Corollary 1 implies that \(0\le u \le 1\).

### Proof of of Theorem 4

*n*, \(v^n\) is convex combination of elements from \((u^n_t)_{n=1}^\infty \), we have by (19)

## Notes

### Acknowledgments

The author would like to thank Takis Konstantopoulos and Tomas Juskevicius for the useful discussions.

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