Affine Orlicz Pólya-Szegö principles and their equality cases
aa r X i v : . [ m a t h . M G ] O c t Affine Orlicz P´olya-Szeg¨o principles and theirequality cases
Youjiang Lin ∗ and Dongmeng Xi † Abstract . The conjecture about the Orlicz P´olya-Szeg¨o principle posed in [43] isproved. The cases of equality are characterized in the affine Orlicz P´olya-Szeg¨o principlewith respect to Steiner symmetrization and Schwarz spherical symmetrization.
AMS Subject Classification 2010
Keywords and Phrases.
P´olya-Szeg¨o principle; Orlicz-Sobolev space; Steinersymmetrization; Schwarz spherical symmetrization; Affine isoperimetric inequality.
The classical P´olya-Szeg¨o principle states that the L p norm of the gradient of anyreal-valued function from a certain class does not increase under an appropriate rear-rangement. Schwarz spherical symmetrization about a point and Steiner symmetriza-tion about a hyperplane are probably the most popular symmetrizations in the liter-ature. P´olya-Szeg¨o inequalities for these symmetrizations play a fundamental role inthe solution to a number of variational problems in different areas such as isoperimet-ric inequalities, optimal forms of Sobolev inequalities, and sharp a priori estimates ofsolutions to second-order elliptic or parabolic boundary value problems; see, for exam-ple, [13,15–17,22,26,58,60] and the references therein. In recent years, many importantgeneralizations and variations have been obtained (see, e.g., [2, 10, 11, 59, 65])It is a remarkable discovery of Zhang [69] that the Petty projection inequality,extended to a suitable class of nonconvex sets, can replace the isoperimetric inequality ∗ Research of the first named author is supported by the funds of the Basic and Advanced ResearchProject of CQ CSTC cstc2015jcyjA00009, cstc2018jcyjAX0790 and Scientific and Technological Re-search Program of Chongqing Municipal Education Commission (KJ1500628). † Research of the second named author is supported by Shanghai Sailing Program 16YF1403800,NSFC 11601310, and Chinese Post-doctoral Innovation Talent Support Program BX201600035. E p ( f ⋆ ) ≤ E p ( f ) , (1.1)where f ∗ denotes the Schwarz spherical symmetrization of f and E p denotes the L p affineenergy of f (see [52], or take φ ( t ) = | t | p in (2.9)). It was proved by Lutwak, Yang andZhang [52] for 1 ≤ p < n and by Cianchi, Lutwak, Yang and Zhang [19] for all p ≥ L p affine energy replaces thestandard L p norm of the gradient leading to an inequality which is significantly strongerthan its classical Euclidean counterpart. Moreover, Lutwak et al. [52] and Cianchiet al. [19] obtained new sharp affine Sobolev, Moser-Trudinger and Morrey-Sobolevinequalities by applying their affine P´olya-Szeg¨o principle, thereby demonstrating thepower of this new affine symmetrization inequality. Later, Haberl, Schuster, and Xiao[39] proved a remarkable asymmetric version of the affine P´olya-Szeg¨o-type inequalitywhich strengthens and implies the affine P´olya-Szeg¨o principle of Cianchi et al. [19].About the affine isoperimetric inequalities and their functional versions, also see [12,20, 27, 28, 32, 37, 38, 41, 44–48, 51, 54, 66, 68].The affine L p P´olya-Szeg¨o-type principle is closely related to the L p Brunn-Minkowskitheory of convex bodies (see, e.g., [5, 49, 50]). Based on the the seminal work of Lut-wak, Yang and Zhang [55, 56], now the L p Brunn-Minkowski theory has been extendedto the Orlicz-Brunn-Minkowski theory. The Orlicz-Brunn-Minkowski theory has ex-panded rapidly (see e.g., [4, 30, 31, 36, 37, 55, 56, 67, 70, 71]). It is natural to considerthe affine P´olya-Szeg¨o-type principle in Orlicz-Sobolev spaces. In [43], using functionalSteiner symmetrization, the first named author proved an affine Orlicz P´olya-Szeg¨oprinciple for log-concave functions, which includes the affine L p P´olya-Szeg¨o principleas special case. The case of equality of the affine Orlicz P´olya-Szeg¨o principle for log-concave functions is also characterized. In [43], the first named author conjecturedthat the principle can be extended to the general Orlicz-Sobolev functions. In thispaper, we confirm this conjecture and characterize the case of equality. An affine Or-licz P´olya-Szeg¨o principle with respect to Orlicz-Sobolev functions is formulated andproved.In this paper, we mainly make use of Steiner symmetrization of one-dimensionalrestrictions of Sobolev functions and Fubini’s theorem to prove our results. The proof2as the advantage of providing us with information about functions yielding equal-ity. The technique exploited in this paper differs from those of previous papers onaffine P´olya-Szeg¨o type inequalities, that make substantial use of fine results from theBrunn-Minkowski theory of convex bodies. The proof of the symmetric affine P´olya-Szeg¨o principle in [19] mainly relies on the L p Petty projection inequality from [51]and the solution of the normalized L p Minkowski problem [53]. The proof of the asym-metric affine P´olya-Szeg¨o principle [38] mainly relies on a generalization of the L p Petty projection inequality established by Haberl and Schuster [35] and the solutionof the normalized L p Minkowski problem [53]. The techniques for proving the affine L p P´olya-Szeg¨o principle could not be adapted to establish the affine Orlicz P´olya-Szeg¨o principle. One of the reasons is that the function φ defining the Orlicz-Sobolevspaces is usually not multiplicative, i.e., φ ( xy ) = φ ( x ) φ ( y ) for x, y ∈ R . Moreoverthe Orlicz Minkowski problem has not been completely solved. Our approach is basedon the functional Steiner symmetrization and makes use of a result for Steiner sym-metrization with approximation of Schwarz symmetrization by sequences of Steinersymmetrizations. Moreover, we prove the affine Orlicz P´olya-Szeg¨o principle and itscase of equality not only with respect to Schwarz spherical symmetrization but alsowith respect to Steiner symmetrization. The affine Orlicz P´olya-Szeg¨o principle forSteiner symmetrization is new even in the L p setting.In the remarkable paper [17], Cianchi and Fusco proved a beautiful P´olya-Szeg¨o-type inequality and analyzed the cases of equality in Steiner symmetrization inequalitiesfor Dirichlet-type integrals. The ideas and techniques of Cianchi and Fusco play acritical role throughout this paper, especially in the proofs of the affine Orlicz P´olya-Szeg¨o principle with respect to Steiner symmetrization and its case of quality. It wouldbe impossible to overstate our reliance on their work.As pointed out in [17], investigations on the cases of equality in P´olya-Szeg¨o typeprinciples are more recent, and typically require an additional delicate analysis. Such adescription has first been the object of the series of papers [1, 6, 12, 14, 17, 26, 27, 59] andhas been recently extended and simplified by new contributions, including [2, 8–10, 15,18,22,23,65]. An impulse to the study of this delicate issue was given by the paper [40],where the symmetry of PS-extremals for Schwarz and Steiner symmetrizations wasestablished, by classical techniques, in special classes of functions and domains.In [6] Brothers and Ziemer characterized the equality cases in the P´olya-Szeg¨oinequality for the Schwarz rearrangement of a Sobolev function under the minimalassumption that the set of critical points of the rearranged function has zero Lebesgue3easure (see also [26] for an interesting alternate proof). A version of this result in theframework of functions of bounded variation can be found in [16]. A Brothers-Ziemertype theorem for the affine P´olya-Szeg¨o principle and a quantitative affine P´olya-Szeg¨oprinciple were established by Wang [65]. The main goal of the second part of this paperis to prove a Brothers-Ziemer type theorem for the affine Orlicz P´olya-Szeg¨o principlewith respect to Schwarz symmetrization. Since the approach exploited in the paperrelies on the result dealing with cases of equality for Steiner symmetrization, we assumethat the domain of function is of finite perimeter. In view of the available result for theEuclidean P´olya-Szeg¨o principle, our assumption that the domain of function is a set offinite perimeter is probably unnecessary. However, if we remove such an assumption,then this would require the use of a different method to prove our result, that wouldgo beyond the scope of the paper.The paper is organized as follows. In Section 2 we state and comment the mainresults and in Section 3 we collect some background material on the Brunn-Minkowskitheory and the theory of Sobolev functions. Section 4 is devoted to the affine OrliczP´olya-Szeg¨o principle with respect to Steiner symmetrization while Section 5 deals withthe case of Schwarz spherical symmetrization. We begin with some definitions and elementary facts about Steiner symmetrizationof sets and functions. Steiner symmetrization is a classical and very well-known device,which has seen a number of remarkable applications to problems of geometric andfunctional nature, see, e.g., [3, 7–9, 12, 42, 62, 63].Given two sets E and F , we denote the symmetric difference by E △ F := ( E ∪ F ) \ ( E ∩ F ). Given two open sets Ω ′ ⊂ Ω we write Ω ′ ⋐ Ω if Ω ′ is compactly containedin Ω, i.e., cl Ω ′ ⊂ Ω, here cl Ω ′ denotes the closure of Ω ′ . A point x in the Euclideanspace R n , n ≥
2, will be usually labeled by ( x ′ , y ), where x ′ = ( x , . . . , x n − ) ∈ R n − and y ∈ R ; similarly, when x ∈ R n +1 , we shall write x as ( x ′ , y, t ). To emphasizethe different roles of the variables y and t , we shall also write R n = R n − × R y and R n +1 = R n − × R y × R t . Consistent notations will be used for subsets of R n and R n +1 . Let L m denote the outer Lebesgue measure in R m . Throughout this paper, for A, B ⊂ R n , A is equivalent to B means that L n ( A △ B ) = 0.4iven any measurable subset E of R n , define, for x ′ ∈ R n − , E x ′ = { y ∈ R : ( x ′ , y ) ∈ E } (2.1)and ℓ E ( x ′ ) = L ( E x ′ ) . (2.2)Then, we define the Steiner symmetral E s of E about the hyperplane { y = 0 } as E s = { ( x ′ , y ) ∈ R n : | y | < ℓ E ( x ′ ) / } . When E ⊂ R n − × R y × R t , its Steiner symmetral E s about { y = 0 } is definedanalogously, after replacing (2.1) and (2.2) by corresponding definitions of E x ′ ,t and ℓ E ( x ′ , t ).Let π n − (Ω) denote the orthogonal projection of Ω ⊂ R n onto R n − . Let Ω be ameasurable subset of R n and let f be a nonnegative measurable function in Ω suchthat, for L n − -a.e. x ′ ∈ π n − (Ω), L ( { y ∈ Ω x ′ : f ( x ′ , y ) > t } ) < ∞ for every t > . (2.3)The Steiner rearrangement f s of f is the function from R n to [0 , + ∞ ] given by f s ( x ′ , y ) = inf { t > µ f ( x ′ , t ) ≤ | y |} for ( x ′ , y ) ∈ R n − × R y , where µ f ( x ′ , t ) = L ( { y ∈ R : f ( x ′ , y ) > t } ) , the distribution function of f ( x ′ , · ), and f denotes the continuation of f to R n whichvanishes outside Ω. Note that f s = 0 L n -a.e. in R n \ Ω s .The notions of Steiner symmetral of a set and Steiner rearrangement of a functionare clearly related. Actually, if f : Ω → [0 , + ∞ ) is as above, and S f = { ( x ′ , y, t ) ∈ R n +1 : ( x ′ , y ) ∈ Ω , < t < f ( x ′ , y ) } , (2.4)the subgraph of f , then ( S f ) s is equivalent to S f s . (2.5)Moreover, for the level set of f defined by[ f ] t = { ( x ′ , y ) ∈ Ω : f ( x ′ , y ) > t } , (2.6)5e have [ f ] st is equivalent to [ f s ] t for every t > . (2.7)Let N be the class of convex functions φ : R → [0 , ∞ ) such that φ (0) = 0 andsuch that φ is strictly decreasing on ( −∞ ,
0] or φ is strictly increasing on [0 , ∞ ). Thesubclass of N consisting of those φ ∈ N that are strictly convex will be denoted by N s . Throughout the paper, we always set Φ( t ) := max { φ ( t ) , φ ( − t ) } , t ∈ [0 , ∞ ). It iseasily checked that Φ( t ) is a convex function and strictly increasing on [0 , ∞ ).We always assume that Ω is a bounded open subset of R n . Let W , Φ (Ω) be the firstorder Orlicz-Sobolev space (see Section 3 for the precise definition) corresponding toΦ. Let W , Φ0 (Ω) denote the subspace of W , Φ (Ω) of those functions whose continuationby 0 outside Ω belongs to W , Φ ( R n ). For v ∈ S n − and f ∈ W , Φ0 (Ω), we define k v k f,φ = k∇ v f k φ = inf (cid:26) λ > | Ω | Z Ω φ (cid:18) ∇ v fλ (cid:19) dx ≤ (cid:27) , (2.8)where ∇ v f is the directional derivative of f in the direction v . The definition im-mediately provides the extension of k · k f,φ from S n − to R n . Now ( R n , k · k f,φ ) isthe n -dimensional Banach space that we shall associate with f . And its unit ball B φ ( f ) = { x ∈ R n : k x k f,φ ≤ } is a convex body in R n . An important fact is thatits volume | B φ ( f ) | is invariant under affine transformations of the form x Ax + x ,with x ∈ R n and A ∈ SL ( n ). We call the unit ball B φ ( f ) the Orlicz-Sobolev ball of f .We call E φ ( f ) := | B φ ( f ) | − n = (cid:18) n Z S n − k∇ v f k − nφ dv (cid:19) − n (2.9)the Orlicz-Sobolev affine energy of f .In this paper, we will prove an affine Orlicz P´olya-Szeg¨o principle with respect toSteiner symmetrization. Theorem 2.1.
Let Ω be a bounded open subset of R n and let f be a nonnegativefunction from W , Φ0 (Ω) . Then for every Steiner rearrangement f s of f , E φ ( f s ) ≤ E φ ( f ) . (2.10)Using Theorem 2.1 and the convergence property of Steiner symmetrization, wecan obtain the affine Orlicz P´olya-Szeg¨o principle with respect to Schwarz sphericalsymmetrization. 6 heorem 2.2. Let Ω be a bounded open subset of R n and let f be a nonnegativefunction from W , Φ0 (Ω) . Then for every Schwarz spherical symmetrization f ⋆ of f , E φ ( f ⋆ ) ≤ E φ ( f ) . (2.11)When φ ( t ) = (1 − λ )( t ) p + + λ ( t ) p − , where p > λ ∈ [0 , t ) + := max { t, } and ( t ) − := max {− t, } , the affine Orlicz P´olya-Szeg¨o principle becomes the generalaffine P´olya-Szeg¨o-type principle established in [59]. The symmetric affine P´olya-Szeg¨oprinciple [19] and the asymmetric affine P´olya-Szeg¨o principle [39] correspond to thecases of λ = 1 / λ = 0, respectively.In order to state our result about the equality case in (2.10), we need some assump-tions on f and Ω. Consider f first, and set M f ( x ′ ) = inf { t > µ f ( x ′ , t ) = 0 } for x ′ ∈ π n − (Ω) . Obviously, M f ( x ′ ) agrees with ess sup { f ( x ′ , y ) : y ∈ Ω x ′ } for L n − -a.e. x ′ ∈ π n − (Ω).Moreover, M is a measurable function in π n − (Ω) with M f ( x ′ ) < ∞ for L n − -a.e. x ′ ∈ π n − (Ω), owing to (2.3). We demand that, for L n − -a.e. x ′ ∈ π n − (Ω), M f ( x ′ ) > f ( x ′ , · ) is L -a.e. different from 0 in the setwhere f ( x ′ , · ) < M f ( x ′ ). This is equivalent to the condition L n ( { ( x ′ , y ) ∈ Ω : ∇ y f ( x ′ , y ) = 0 } ∩ { ( x ′ , y ) ∈ Ω : M f ( x ′ ) = 0or f ( x ′ , y ) < M f ( x ′ ) } ) = 0 . (2.12)As far as Ω is concerned, we require that π n − (Ω) is connected , (2.13)and that, in a sense, the reduced boundary ∂ ∗ Ω of Ω is almost nowhere parallel tothe y -axis inside the open cylinder π n − (Ω) × R y . A precise formulation of the lastcondition readsΩ has locally finite perimeter in π n − (Ω) × R y and H n − (cid:0) { ( x ′ , y ) ∈ ∂ ∗ Ω : v Ω y ( x ′ , y ) = 0 } ∩ ( π n − (Ω) × R y ) (cid:1) = 0 , (2.14)where H k stands for k -dimensional Hausdorff measure, and v Ω y denotes the compo-nent along the y -axis of the generalized inner normal v Ω to Ω (see Section 3.2.2 fordefinitions).We are now ready to state our result about the equality case in (2.10).7 heorem 2.3. Let Ω be a bounded open subset of R n fulfilling (2.13)–(2.14) and let f be a nonnegative function from W , Φ0 (Ω) satisfying (2.12) and φ ∈ N s . Then E φ ( f ) = E φ ( f s ) (2.15) if and only if there exist A ∈ SL ( n ) and x ∈ R n such that f ( x ) = f s ( Ax + x ) f or L n - a.e. x ∈ Ω . (2.16)By Theorem 2.3 and some delicate analyses, we can characterize the case of equalityin (2.11). In particular we establish a Brothers-Ziemer type theorem for the affine OrliczP´olya-Szeg¨o principle. We demandΩ is a set of finite perimeter in R n (2.17)and L n ( { x ∈ Ω : ∇ f ( x ) = 0 and 0 ≤ f ( x ) < ess sup f } ) = 0 . (2.18) Theorem 2.4.
Let Ω be a bounded and connected open subset of R n fulfilling (2.17) and let f be a nonnegative function from W , Φ0 (Ω) satisfying (2.18) and φ ∈ N s . Then E φ ( f ) = E φ ( f ⋆ ) (2.19) if and only if there exist A ∈ SL ( n ) and x ∈ R n such that f ( x ) = f ⋆ ( Ax + x ) f or L n - a.e. x ∈ Ω . (2.20) In this section we fix our notation and collect basic facts from convex geometricanalysis. General references for the theory of convex bodies are the books by Gardner[29], Gruber [34], Schneider [61].We write K n for the set of convex bodies (compact convex subsets) of R n . We write K no for the set of convex bodies that contain the origin in their interiors. For K ∈ K n ,let h ( K ; · ) = h K : R n → R denote the support function of K ; i.e., h ( K ; x ) := max { x · z : z ∈ K } . K ∈ K no , its gauge function g K : R n → [0 , ∞ ) is defined by g K ( x ) := k x k K = inf { λ > x ∈ λK } . (3.1)If K ∈ K no , then the polar body K ∗ is defined by K ∗ := { x ∈ R n : x · z ≤ z ∈ K } . If K ∈ K no , it is well-known that g K = h K ∗ . (3.2)By (3.1), for x ∈ R n and K ∈ K no , it follows immediately that g K ( x ) = 1 if and only if x ∈ ∂K. (3.3)For K, L ∈ K n , the Hausdorff distance of K and L is defined by δ ( K, L ) := sup u ∈ S n − | h K ( u ) − h L ( u ) | . (3.4)When considering the convex body K ∈ K no as K ⊂ R n − × R y , the Steiner sym-metral , K s , of K in the direction e n is given by K s = (cid:26)(cid:18) x ′ , y + 12 y (cid:19) ∈ R n − × R : ( x ′ , y ) , ( x ′ , − y ) ∈ K (cid:27) . (3.5)In this paper, we shall make use of the following fact that follows directly from (3.5)and (3.3). Lemma 3.1.
Suppose
K, L ∈ K no and consider K, L ⊂ R n − × R . Then K s ⊂ L, if and only if k ( x ′ , η ) k K = 1 = k ( x ′ , − η ) k K , with η = − η = ⇒ k ( x ′ , η / η / k L ≤ . In addition, if K s = L , then k ( x ′ , η ) k K = 1 = k ( x ′ , − η ) k K with η = − η impliesthat k ( x ′ , η / η / k L = 1 . In this section, we review some basic definitions and facts about Sobolev functionsand functions of bounded variation on R n . Good general references for this are Ambro-sio, Fusco and Pallara [1], Cianchi and Fusco [17], Evans and Gariepy [24], Maz ′ ya [57]and Ziemer [72]. 9 .2.1 On Orlicz-Sobolev functions Let Ω be an open subset of R n and let φ ∈ N . Let Φ( t ) = max { φ ( t ) , φ ( − t ) } , t ∈ [0 , ∞ ). The Orlicz space L Φ (Ω) is defined as L Φ (Ω) := { f : f is a Lebesgue measurable real valued function on Ωsuch that Z Ω Φ (cid:18) | f ( x ) | λ (cid:19) dx < ∞ for some λ > (cid:27) . (3.6)The Luxemburg norm k f k L Φ (Ω) is defined as k f k L Φ (Ω) := inf (cid:26) λ > Z Ω Φ (cid:18) | f ( x ) | λ (cid:19) dx ≤ (cid:27) . (3.7)The space L Φ (Ω), equipped with the norm k · k L Φ (Ω) , is a Banach space. Note that, ifΦ( s ) = s p and p >
1, then L Φ (Ω) = L p (Ω), the usual L p space, and k·k L Φ (Ω) = k·k L p (Ω) .Usually, we write k · k Φ instead of k · k L Φ (Ω) .The first order Orlicz-Sobolev space W , Φ (Ω) is defined as W , Φ (Ω) = { f ∈ L Φ (Ω) : f is weakly differentiable and |∇ f | ∈ L Φ (Ω) } . (3.8)Here, ∇ denotes the approximate gradient (see the definition in (3.15)). By W , Φloc (Ω)we denote the space of those functions which belong to W , Φ (Ω ′ ) for every open setΩ ′ ⋐ Ω.The space W , Φ (Ω), equipped with the norm k f k W , Φ (Ω) = k f k Φ + k|∇ f |k Φ , (3.9)is a Banach space. Clearly, W , Φ (Ω) = W ,p (Ω), the standard Sobolev space, if Φ( s ) = s p with p > Lemma 3.2. If φ ∈ N , then for a, b ∈ R and a = 0 , the function Ψ( t ) := φ ( at − b ) + φ ( − at − b ) , t > is increasing. In addition, if φ ∈ N s , then Ψ( t ) is strictly increasing. Since Ω is a bounded open set and Φ is a convex function and strictly increasingon [0 , ∞ ), we have the following easily-established result. Lemma 3.3. If f ∈ W , Φ0 (Ω) , then f ∈ W , (Ω) .
10y [2, Lemma 2.7] and [17, Theorem 2.1], we have the following lemma.
Lemma 3.4.
Let Ω be a bounded open subset of R n . If f ∈ W , (Ω) , then f s ∈ W , (Ω s ) . For Sobolev spaces W ,p ( R n ) (1 ≤ p < ∞ ), Burchard [8, Proposition 7.1] provedthe following proposition on the approximation of Schwarz spherical symmetrizationby Steiner symmetrizations. Proposition 3.5. (Convergence of the W ,p -norm) . Let f be a nonnegative functionin W ,p ( R n ) , n ≥ and p ≥ , that vanishes at infinity. There exists a sequence ofsuccessive Steiner symmetrizations { f k } k ≥ of f so that f k ⇀ f ⋆ weakly in W , ( R n ) . Remark 3.1.
For Orlicz-Sobolev spaces W , Φ ( R n ), there does not exist a result similarto that of Proposition 3.5. Thus we consider the problem in W , ( R n ). By Lemma3.3, f ∈ W , (Ω) for f ∈ W , Φ0 (Ω). Thus, for f ∈ W , Φ0 (Ω), there exists a sequence ofsuccessive Steiner symmetrizations { f k } k ≥ of f so that f k ⇀ f ⋆ weakly in W , ( R n ) . The space of functions of bounded variation in Ω is denoted by BV (Ω). Recall thata function f ∈ L (Ω) is said to be of bounded variation in Ω if its distributional gradient Df is a vector-valued Radon measure in Ω whose total variation | Df | is finite in Ω(see the precise definitions in [24, p.196]). The space BV loc (Ω) is defined accordingly.Given a measurable set E in R n and a point x ∈ R n , the density of E at x is definedby D ( E, x ) = lim r → L n ( E ∩ B r ( x )) L n ( B r ( x )) , provided that the limit on the right-hand side exists. Here, B r ( x ) denotes the ball,centered at x , having radius r . The essential boundary of E is the Borel set ∂ M E = R n \ { x ∈ R n : D ( E, x ) = 0 or D ( E, x ) = 1 } . (3.11)One has ∂ M ( E ′ ∪ E ′′ ) ∪ ∂ M ( E ′ ∩ E ′′ ) ⊂ ∂ M E ′ ∪ ∂ M E ′′ (3.12)11or any measurable sets E ′ and E ′′ in R n .For any measurable function f in an open set Ω ⊂ R n , the approximate upper and lower limit of f at a point x are defined as f + ( x ) = inf { t : D ( { f > t } , x ) = 0 } and f − ( x ) = sup { t : D ( { f < t } , x ) = 0 } , (3.13)respectively. The function f is said to be approximately continuous at x if f + ( x )and f − ( x ) are equal and finite. The common value of f + ( x ) and f − ( x ) at a pointof approximate continuity x is called the approximate limit of f at x and is denotedby ˜ f ( x ). By C f we denote the Borel set of all points at which f is approximatelycontinuous. The precise representative f ∗ of f is defined as f ∗ ( x ) = f + ( x ) + f − ( x )2 if f + ( x ) and f − ( x ) are both finite , . (3.14)Clearly, f ∗ ≡ ˜ f in C f . A locally integrable function f in Ω is said to be approximatelydifferentiable at x ∈ C f if there exists a vector ∇ f ( x ) in R n , called the approximategradient of f at x , such thatlim r → L ( B r ( x )) Z B r ( x ) | f ( z ) − ˜ f ( x ) − h∇ f ( x ) , z − x i| r dz = 0 . (3.15)The set of all points x ∈ C f where f is approximately differentiable is a Borel setdenoted by D f . The subset of D f where ∇ f = 0 and the subset where ∇ f = 0 will bedenoted by D + f and D − f , respectively. If f ∈ BV (Ω), then L n (Ω \ D f ) = 0. Moreover,denoting by D a f and by D s f the absolutely continuous part and the singular part,respectively, of Df with respect to L n , we have that ∇ f agrees L n -a.e. with thedensity of D a f with respect to L n , and that | D s f | ( D f ) = 0. Thus, in particular, W , (Ω) can be identified with the subspace of BV (Ω) of those functions in BV (Ω)such that | Df | ( B ) = 0 for every Borel set B ⊂ Ω satisfying L n ( B ) = 0.A measurable subset E of R n is said to be of finite perimeter in an open set Ω ⊂ R n if Dχ E is a vector-valued Radon measure with finite total variation in Ω, where χ E denotes the characteristic function of E . The perimeter of E in a Borel subset B of Ωis defined by P ( E ; B ) = | Dχ E | ( B ) . When B = R n , we shall simply write P ( E ) instead of P ( E ; R n ). If χ E ∈ BV loc (Ω),then we say that E has locally finite perimeter in Ω.12he following theorem (see [33, Sect. 4.1.5, Theorem 1]) completely characterizesfunctions of bounded variation in terms of their subgraphs. Let us remark that aslightly different notion of subgraph is needed here. Given a function f : Ω ⊂ R n → R ,we set S − f := { ( x, y, t ) ∈ R n +1 : ( x, y ) ∈ Ω , t < f ( x, y ) } . Theorem 3.6.
Let Ω be a bounded open subset of R n and let f be a nonnegativefunction from L (Ω) . Then S − f is a set of finite perimeter in Ω × R t if and only if f ∈ BV (Ω) . Moreover, in this case, P ( S − f ; B × R t ) = Z B p |∇ f | + | D s f | ( B ) for every Borel set B ⊂ Ω . Let E be a set of locally finite perimeter in an open subset Ω of R n and let D i χ E denote the i -th component of the distributional gradient Dχ E . We denoteby v Ei , i = 1 , . . . , n , the derivative of the measure D i χ E with respect to | Dχ E | .The reduced boundary ∂ ∗ E of E is the set of all points x ∈ Ω such that the vector v E ( x ) = ( v E ( x ) , . . . , v En ( x )) exists and satisfies | v E ( x ) | = 1 (see the precise definitionsin [24, p.221]). The vector v E ( x ) is called the generalized inner normal to E at x . Theorem 3.7. [17, Theorem B] Let Ω be an open subset of R n and let E ′ and E ′′ be setsof locally finite perimeter in Ω . Then v E ′ ( x ) = ± v E ′′ ( x ) for H n − -a.e. x ∈ ∂ ∗ E ′ ∩ ∂ ∗ E ′′ . If f ∈ W , (Ω) and g : Ω → [0 , + ∞ ) is a Borel function, then coarea formula forSobolev functions can be written as Z Ω g |∇ f | dx = Z + ∞−∞ dt Z Ω ∩ ∂ ∗ { f>t } gd H n − = Z + ∞−∞ dt Z { f ∗ = t } gd H n − . (3.16)The following proposition is a special case of the coarea formula for rectifiable sets(see [1, Theorem 2.93]). Proposition 3.8.
Let Ω ⊂ R n be an open set and let E be a set of finite perimeter in Ω . Let g : Ω → [0 , + ∞ ] be a Borel function. Then Z ∂ ∗ E ∩ Ω g ( x ) | v En ( x ) | d H n − ( x ) = Z π n − (Ω) dx ′ Z ( ∂ ∗ E ∩ Ω) x ′ g ( x ′ , y ) d H ( y ) . (3.17)The next theorem links the approximate gradient of a function of bounded variationto the generalized inner normal to its subgraph (see [33, Sect. 4.1.5, Theorems 4 and5]). 13 heorem 3.9. Let Ω be an open subset of R n and let ∇ i f denote the i -th componentof ∇ f . Then for f ∈ BV (Ω) , v S − f ( x, t ) = ∇ f ( x ) p |∇ f ( x ) | , · · · , ∇ n f ( x ) p |∇ f ( x ) | , − p |∇ f ( x ) | ! (3.18) for H n -a.e. ( x, t ) ∈ ∂ ∗ S − f ∩ ( D f × R t ) and v S − f n +1 ( x, t ) = 0 f or H n - a.e. ( x, t ) ∈ ∂ ∗ S − f ∩ [(Ω \ D f ) × R t ] . In particular, if f ∈ W , (Ω) , then (3.18) holds for H n -a.e. ( x, t ) ∈ ∂ ∗ S − f ∩ (Ω × R t ) . In what follows, the essential projection of a set E ⊂ R n +1 onto R n − × R t is definedas π n − ,n +1 ( E ) + = { ( x ′ , t ) ∈ R n − × R t : ℓ E ( x ′ , t ) > } . The essential projection π n − ( E ) + onto R n − is defined similarly.Finally, we give a theorem concerning one-dimensional sections of sets of finiteperimeter. The result is due to Vol’pert [64]. In the present form, it can be easilydeduced from [1, Theorem 3.108]. Theorem 3.10.
Let E be a set of finite perimeter in Ω . Then, for L n − -a.e. x ′ ∈ π n − (Ω) , E x ′ has f inite perimeter in Ω x ′ , (3.19)( ∂ ∗ E ∩ Ω) x ′ = ∂ ∗ ( E x ′ ) ∩ Ω x ′ , (3.20) v En ( x ′ , y ) = 0 f or every y such that ( x ′ , y ) ∈ ∂ ∗ E ∩ Ω , (3.21) lim z → y + χ ∗ E ( x ′ , z ) = 1 , lim z → y − χ ∗ E ( x ′ , z ) = 0 if v En ( x ′ , y ) > , lim z → y + χ ∗ E ( x ′ , z ) = 0 , lim z → y − χ ∗ E ( x ′ , z ) = 1 if v En ( x ′ , y ) < . (3.22) In particular, there exists a Borel set Ω E ⊂ π n − ( E ) + ∩ π n − (Ω) satisfying L n − ( π n − ( E ) + ∩ π n − (Ω) \ Ω E ) = 0 and such that (3.19)-(3.22) hold for every x ′ ∈ Ω E . .3 On the Orlicz-Sobolev balls Let Ω be a bounded open subset of R n . By (3.8), if f ∈ W , Φ0 (Ω), then |∇ f | ∈ L Φ (Ω). Thus by (3.6), there exists some λ > Z Ω Φ (cid:18) |∇ f ( x ) | λ (cid:19) dx < ∞ . Since φ ∈ N and Φ( t ) = max { φ ( t ) , φ ( − t ) } , t ∈ [0 , ∞ ), there exists some λ > u ∈ S n − , Z Ω φ (cid:18) u · ∇ f ( x ) λ (cid:19) dx ≤ Z Ω Φ (cid:18) |∇ f ( x ) | λ (cid:19) dx < ∞ . (3.23)Thus, we can define the Orlicz-Sobolev ball B φ ( f ) of f ∈ W , Φ0 (Ω) as the unit ball ofthe n -dimensional Banach space whose norm is given by k z k f,φ := inf (cid:26) λ > | Ω | Z Ω φ (cid:18) z · ∇ f ( x ) λ (cid:19) dx ≤ (cid:27) , z ∈ R n . (3.24)And the volume of the Orlicz-Sobolev ball is given by | B φ ( f ) | = 1 n Z S n − k v k − nf,φ dv, (3.25)where dv denotes the spherical Lebesgue measure.Since f ∈ W , Φ0 (Ω), it is impossible that there exists some u ∈ S n − such that ∇ f ( x ) · u ≥ x ∈ Ω. Since φ is strictly increasing on [0 , ∞ ) or strictlydecreasing on ( −∞ , z = 0 the function λ | Ω | Z Ω φ (cid:18) z · ∇ f ( x ) λ (cid:19) dx is strictly decreasing in (0 , ∞ ). Thus, we have the following lemma. Lemma 3.11.
Let f ∈ W , Φ0 (Ω) and z ∈ R n \{ } . Then | Ω | Z Ω φ (cid:18) z · ∇ f ( x ) λ (cid:19) dx = 1 if and only if k z k f,φ = λ . The following Lemma 3.12, Lemma 3.13 and Lemma 3.14 demonstrate the affineinvariance of E φ ( f ), the non-negativity and boundedness of k · k f,φ , respectively. Sincetheir proofs are the same as the proofs of Lemma 4.2–Lemma 4.4 in [43], we omit theirproofs. 15 emma 3.12. ( [43, Lemma 4.2]) If f ∈ W , Φ0 (Ω) , then E φ ( f ) is invariant under SL ( n ) transformations and translations. Lemma 3.13. ( [43, Lemma 4.3]) If f ∈ W , Φ0 (Ω) , then k · k f,φ defines a norm on theBanach space ( R n , k · k f,φ ) . In particular, k v k f,φ > for any v ∈ S n − . Lemma 3.14. ( [43, Lemma 4.4]) If f ∈ W , Φ0 (Ω) and c φ = max { c > { φ ( c ) , φ ( − c ) } ≤ } , (3.26) then for any v ∈ S n − , we have R Ω f ( x ) dxc φ | Ω | diam(Ω) ≤ k v k f,φ ≤ sup {|∇ f ( x ) | : x ∈ Ω } c φ , (3.27) where diam(Ω) := sup {| x − y | : x, y ∈ Ω } denotes the diameter of Ω . The following lemma shows that the Orlicz-Sobolev ball operator B φ : W , Φ0 (Ω) →K no is in some sense weakly continuous. Lemma 3.15.
Let f i ∈ W , Φ0 (Ω i ) , i = 0 , , , . . . . If f i ⇀ f , weakly in W , ( R n ) , (3.28) then there exists a subsequence of { B φ ( f i ) } ∞ i =1 , denoted by { B φ ( f i ) } ∞ i =1 as well, and aconvex body K such that o ∈ K , lim i →∞ δ ( B φ ( f i ) , K ) = 0 (3.29) and K ⊂ B φ ( f ) . (3.30) Proof.
For u ∈ S n − , let k u k f i ,φ = λ i , (3.31)and note that Lemma 3.14 gives0 < R Ω i f i ( x ) dxc φ | Ω i | diam(Ω) ≤ λ i . (3.32)Moreover, by (3.28), we havelim i →∞ R Ω i f i ( x ) dxc φ | Ω i | diam(Ω) = R Ω f ( x ) dxc φ | Ω | diam(Ω ) > , (3.33)16hich implies that there exists a real number m > k u k f i ,φ > m for any u ∈ S n − and any positive integer i . Thus the radial functions ρ ( B φ ( f i ) , u ) < m forany u ∈ S n − and any positive integer i . By the Blaschke selection theorem (see [61,Theorem 1.8.7]), there exists a subsequence of { B φ ( f i ) } ∞ i =1 , denoted by { B φ ( f i ) } ∞ i =1 aswell, that converges to a convex body K ∈ K n . By Lemma 3.13, k u k f i ,φ > u ∈ S n − and any positive integer i . Thus o ∈ K .Let k u k K = λ ∗ . By (3.29) and (3.31), we havelim i →∞ λ i = λ ∗ . (3.34)Let ¯ f i denote the continuation of f by 0 outside Ω i and ˜ f i = ¯ f i /λ i . Since λ i → λ ∗ and ¯ f i ⇀ ¯ f weakly in W , ( R n ), we have˜ f i ⇀ ¯ f /λ ∗ weakly in W , ( R n ) . (3.35)The fact that k u k f i ,φ = λ i , together with Lemma 3.11, shows that1 | Ω i | Z R n φ (cid:16) u · ∇ ˜ f i ( x ) (cid:17) dx = 1 for all i. (3.36)Since φ is a convex function, by [25, Theorem 1 in P.19], the convex gradient integral1 | Ω | Z R n φ (cid:0) u · ∇ ¯ f ( x ) (cid:1) dx is lower semicontinuous with respect to weak convergence in W , ( R n ). By (3.35) and(3.36), we have1 | Ω | Z R n φ (cid:18) u · ∇ ¯ f ( x ) λ ∗ (cid:19) dx ≤ lim i →∞ | Ω i | Z R n φ (cid:16) u · ∇ ˜ f i ( x ) (cid:17) dx = 1 . This, together with the definition (3.24) yields k u k f ,φ ≤ λ ∗ = k u k K . (3.37)By (3.37) and the arbitrariness of u ∈ S n − , we have K ⊂ B φ ( f ). Lemma 4.1. [17, Proposition 2.3.] Let Ω be a bounded open subset of R n . Let f be anonnegative function from W , (Ω) . Then for L n − -a.e. x ′ ∈ π n − (Ω) , L ( { y : ∇ y f ( x ′ , y ) = 0 , t < f ( x ′ , y ) < M f ( x ′ ) } )= L ( { y : ∇ y f s ( x ′ , y ) = 0 , t < f s ( x ′ , y ) < M f ( x ′ ) } ) (4.1) for every t ∈ (0 , M f ( x ′ )) . emma 4.2. [17, Lemma 4.1] Let Ω be a bounded open subset of R n , and let f bea nonnegative function from f ∈ W , (Ω) . Then µ f ∈ BV ( π n − (Ω) × R + t ) , and, for L n − -a.e. x ′ ∈ π n − ( S f ) + , ∇ t µ f ( x ′ , t ) = − Z ∂ ∗ { y : f ( x ′ ,y ) >t } |∇ y f | d H , (4.2) ∇ i µ f ( x ′ , t ) = Z ∂ ∗ { y : f ( x ′ ,y ) >t } ∇ i f |∇ y f | d H i = 1 , . . . , n − , (4.3) for L -a.e. t ∈ (0 , M f ( x ′ )) . Corollary 4.3.
Let Ω be a bounded open subset of R n , and let f be a nonnegativefunction from f ∈ W , (Ω) . Then for L n − -a.e. x ′ ∈ π n − ( S f ) + and L -a.e. t ∈ (0 , M f ( x ′ )) , ∇ t µ f ( x ′ , t ) = − |∇ y f s ( x ′ , y ) | and ∇ i µ f ( x ′ , t ) = 2 ∇ i f s ( x ′ , y ) |∇ y f s ( x ′ , y ) | , (4.4) where y ∈ ∂ ∗ { y : f s ( x ′ , y ) > t } .Proof. By Lemma 3.4, the function f s ∈ W , (Ω s ). Moreover, by (2.5), π n − ( S f ) + isequivalent to π n − ( S f s ) + . Since µ f s = µ f , an application of Lemma 4.2 to f s yields(4.4). Lemma 4.4.
Let Ω be a bounded open subset of R n . If f is a nonnegative functionfrom W , Φ0 (Ω) and n ( x ′ , t ) := H ( ∂ ∗ { y ∈ Ω x ′ : f ( x ′ , y ) > t } ) , (4.5) then for L n − -a.e. x ′ ∈ π n − (Ω) such that M f ( x ′ ) > , n ( x ′ , t ) is an even number for L -a.e. t ∈ (0 , M f ( x ′ )) .Proof. For f ∈ W , Φ0 (Ω), by Lemma 3.3, f ∈ W , (Ω). By [72, Theorem 2.1.4], f has arepresentative ¯ f that is absolutely continuous on almost all line segments in Ω parallelto the coordinate axes. Since ¯ f is absolutely continuous, { y ∈ Ω x ′ : ¯ f ( x ′ , y ) > t } is anopen set. Moreover, since f vanishes on the boundary of Ω, { y ∈ Ω x ′ : ¯ f ( x ′ , y ) > t } = ∞ [ k =1 ( a k , b k ) (4.6)18or L n − -a.e. x ′ ∈ π n − (Ω) such that M f ( x ′ ) > L -a.e. t ∈ (0 , M f ( x ′ )), where a k and b k are the points on the reduced boundary of { y ∈ Ω x ′ : ¯ f ( x ′ , y ) > t } . For thesame x ′ and t , since { y ∈ Ω x ′ : ¯ f ( x ′ , y ) > t } is equivalent to { y ∈ Ω x ′ : f ( x ′ , y ) > t } , ∂ ∗ { y ∈ Ω x ′ : f ( x ′ , y ) > t } = ∂ ∗ { y ∈ Ω x ′ : ¯ f ( x ′ , y ) > t } . (4.7)Moreover, since Ω be a bounded open subset of R n and f ∈ W , (Ω), by Theorem 3.6, S − f is a set of finite perimeter in Ω × R t . Since (cid:0) S − f (cid:1) x ′ ,t is equivalent to { y ∈ Ω x ′ : f ( x ′ , y ) > t } for L n − -a.e. x ′ ∈ π n − (Ω) such that M f ( x ′ ) > L -a.e. t ∈ (0 , M f ( x ′ )) andby (3.19) in Theorem 3.10, n ( x ′ , t ) is finite. By (4.6) and (4.7), n ( x ′ , t ) is an evennumber. Proposition 4.5.
Let Ω be a bounded open subset of R n . If f is a nonnegative functionfrom W , Φ0 (Ω) , then ( B φ ( f )) s ⊂ B φ ( f s ) . (4.8) Proof.
Let ( x ′ , η ) , ( x ′ , − η ) ∈ R n − × R y and k ( x ′ , η ) k f,φ = 1 and k ( x ′ , − η ) k f,φ = 1 , with η = − η . By Lemma 3.11, this means that1 | Ω | Z Ω φ (( x ′ , η ) · ∇ f ( x )) dx = 1 (4.9)and 1 | Ω | Z Ω φ (( x ′ , − η ) · ∇ f ( x )) dx = 1 . (4.10)By Lemma 3.1, the desired inclusion (4.8) will be established if we can show that k ( x ′ , η / η / k f s ,φ ≤ . (4.11) Step 1:
We assume here that f is nonnegative function from W , (Ω) such that L ( { y : ∇ y f ( x ′ , y ) = 0 } ∩ { y : 0 < f ( x ′ , y ) < M f ( x ′ ) } ) = 0 (4.12)for L n − -a.e. x ′ ∈ π n − (Ω) such that M f ( x ′ ) >
0. By Lemma 4.1, (4.12) is fulfilledwith f replaced by f s as well. By [17, Theorem E], df s ( x ′ , y ) dy = ∇ y f s ( x ′ , y ) for L - a.e. y ∈ Ω sx ′ . (4.13)19ence, by (4.13) and the coarea formula (3.16), for L n − -a.e. x ′ ∈ π n − (Ω) such that M f ( x ′ ) >
0, we have Z { y : f s ( x ′ ,y ) > } φ (cid:18)(cid:18) x ′ , η + η (cid:19) · ∇ f s ( x ′ , y ) (cid:19) dy = Z M f ( x ′ )0 dt Z ∂ ∗ { y : f s ( x ′ ,y ) >t } |∇ y f s | φ (cid:18)(cid:18) x ′ , η + η (cid:19) · ∇ f s ( x ′ , y ) (cid:19) d H . (4.14)Moreover, by (2.5), for L n − -a.e. x ′ ∈ π n − (Ω) and L -a.e. t ∈ (0 , M f ( x ′ )), thereexist two real numbers y ( x ′ , t ) and y ( x ′ , t ) such that { y : f s ( x ′ , y ) > t } is equivalent to ( y ( x ′ , t ) , y ( x ′ , t )) (4.15)and ∇ y f s ( x ′ , y ( x ′ , t )) > ∇ y f s ( x ′ , y ( x ′ , t )) < . (4.16)Thus, Eqs. (4.4), (4.15) and (4.16) ensure that, for L n − -a.e. x ′ ∈ π n − (Ω) such that M f ( x ′ ) > Z ∂ ∗ { y : f s ( x ′ ,y ) >t } |∇ y f s | φ (cid:18) x ′ · ( ∇ f s , . . . , ∇ n − f s ) + η + η · ∇ y f s (cid:19) d H = 1 |∇ y f s | φ (cid:18) x ′ · ( ∇ f s , . . . , ∇ n − f s ) + η + η · ∇ y f s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ ,y ( x ′ ,t )) + 1 |∇ y f s | φ (cid:18) x ′ · ( ∇ f s , . . . , ∇ n − f s ) + η + η · ∇ y f s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ ,y ( x ′ ,t )) = − ∇ t µ f ( x ′ , t ) φ (cid:18) x ′ · (cid:18) ∇ µ f ( x ′ , t ) −∇ t µ f ( x ′ , t ) , · · · , ∇ n − µ f ( x ′ , t ) −∇ t µ f ( x ′ , t ) (cid:19) + η + η · −∇ t µ f ( x ′ , t ) (cid:19) − ∇ t µ f ( x ′ , t ) φ (cid:18) x ′ · (cid:18) ∇ µ f ( x ′ , t ) −∇ t µ f ( x ′ , t ) , · · · , ∇ n − µ f ( x ′ , t ) −∇ t µ f ( x ′ , t ) (cid:19) + η + η · ∇ t µ f ( x ′ , t ) (cid:19) = − ∇ t µ f ( x ′ , t ) φ (cid:18) x ′ · ( ∇ µ f ( x ′ , t ) , · · · , ∇ n − µ f ( x ′ , t )) + ( η + η ) −∇ t µ f ( x ′ , t ) (cid:19) − ∇ t µ f ( x ′ , t ) φ (cid:18) x ′ · ( ∇ µ f ( x ′ , t ) , · · · , ∇ n − µ f ( x ′ , t )) − ( η + η ) −∇ t µ f ( x ′ , t ) (cid:19) . (4.17)Let n ( x ′ , t ) := H ( ∂ ∗ { y : f ( x ′ , y ) > t } ) . (4.18)By Lemma 4.4, n ( x ′ , t ) is an even number for L n − -a.e. x ′ ∈ π n − (Ω) and L -a.e. t ∈ (0 , M f ( x ′ )). Let k ( x ′ , t ) := n ( x ′ , t ), then k ( x ′ , t ) is an integer and k ( x ′ , t ) ≥ ∂ ∗ l { y : f ( x ′ , y ) > t } be the subset of ∂ ∗ { y : f ( x ′ , y ) > t } satisfying v S f y ( x ′ , y, t ) > ∂ ∗ r { y : f ( x ′ , y ) > t } be the subset of ∂ ∗ { y : f ( x ′ , y ) > t } satisfying v S f y ( x ′ , y, t ) < . By k ( x ′ , t ) ≥
1, Lemma 3.2 and Lemma 4.2, for L n − -a.e. x ′ ∈ π n − (Ω) such that M f ( x ′ ) >
0, we have the last expression of (4.17) ≤ − ∇ t µ f ( x ′ , t ) φ (cid:18) x ′ · ( ∇ µ f ( x ′ , t ) , · · · , ∇ n − µ f ( x ′ , t )) + k ( x ′ , t )( η + η ) −∇ t µ f ( x ′ , t ) (cid:19) − ∇ t µ f ( x ′ , t ) φ (cid:18) x ′ · ( ∇ µ f ( x ′ , t ) , · · · , ∇ n − µ f ( x ′ , t )) − k ( x ′ , t )( η + η ) −∇ t µ f ( x ′ , t ) (cid:19) = 12 Z ∂ ∗ { ... } d H |∇ y f | φ x ′ · (cid:16)R ∂ ∗ { ... } ∇ f |∇ y f | d H , · · · , R ∂ ∗ { ... } ∇ n − f |∇ y f | d H (cid:17) + k ( x ′ , t )( η + η ) R ∂ ∗ { ... } d H |∇ y f | + 12 Z ∂ ∗ { ... } d H |∇ y f | φ x ′ · (cid:16)R ∂ ∗ { ... } ∇ f |∇ y f | d H , · · · , R ∂ ∗ { ... } ∇ n − f |∇ y f | d H (cid:17) − k ( x ′ , t )( η + η ) R ∂ ∗ { ... } d H |∇ y f | = φ R ∂ ∗ l {··· } ( x ′ ,η ) · ( ∇ f, ··· , ∇ n − f, |∇ y f | ) |∇ y f | d H + R ∂ ∗ r {··· } ( x ′ , − η ) · ( ∇ f, ··· , ∇ n − f, −|∇ y f | ) |∇ y f | d H R ∂ ∗ { ... } d H |∇ y f | + φ R ∂ ∗ l {··· } ( x ′ , − η ) · ( ∇ f, ··· , ∇ n − f, |∇ y f | ) |∇ y f | d H + R ∂ ∗ r {··· } ( x ′ ,η ) · ( ∇ f, ··· , ∇ n − f, −|∇ y f | ) |∇ y f | d H R ∂ ∗ { ... } d H |∇ y f | · Z ∂ ∗ { ... } d H |∇ y f | (4.19)for L -a.e. t ∈ (0 , M f ( x ′ )). Here ∂ ∗ { . . . } is a shorthand for ∂ ∗ { y : f ( x ′ , y ) > t } . Since21 is a convex function, Jensen’s inequality ensures that the last expression ≤ Z ∂ ∗ l { y : f ( x ′ ,y ) >t } |∇ y f | φ (( x ′ , η ) · ( ∇ f, · · · , ∇ n − f, |∇ y f | )) d H + 12 Z ∂ ∗ r { y : f ( x ′ ,y ) >t } |∇ y f | φ (( x ′ , − η ) · ( ∇ f, · · · , ∇ n − f, −|∇ y f | )) d H + 12 Z ∂ ∗ l { y : f ( x ′ ,y ) >t } |∇ y f | φ (( x ′ , − η ) · ( ∇ f, · · · , ∇ n − f, |∇ y f | )) d H + 12 Z ∂ ∗ r { y : f ( x ′ ,y ) >t } |∇ y f | φ (( x ′ , η ) · ( ∇ f, · · · , ∇ n − f, −|∇ y f | )) d H = 12 Z ∂ ∗ { y : f ( x ′ ,y ) >t } |∇ y f | φ (( x ′ , η ) · ( ∇ f, · · · , ∇ n − f, ∇ y f )) d H + 12 Z ∂ ∗ { y : f ( x ′ ,y ) >t } |∇ y f | φ (( x ′ , − η ) · ( ∇ f, · · · , ∇ n − f, ∇ y f )) d H . (4.20)Combining (4.17), (4.19) and (4.20) leads to Z ∂ ∗ { y : f s ( x ′ ,y ) >t } |∇ y f s | φ (cid:18) x ′ · ( ∇ f s , . . . , ∇ n − f s ) + η + η · ∇ y f s (cid:19) d H ≤ Z ∂ ∗ { y : f ( x ′ ,y ) >t } |∇ y f | φ (( x ′ , η ) · ( ∇ f, · · · , ∇ n − f, ∇ y f )) d H + 12 Z ∂ ∗ { y : f ( x ′ ,y ) >t } |∇ y f | φ (( x ′ , − η ) · ( ∇ f, · · · , ∇ n − f, ∇ y f )) d H (4.21)for L n − -a.e. x ′ ∈ π n − (Ω) such that M f ( x ′ ) > L -a.e. t ∈ (0 , M f ( x ′ )).Integrating (4.21) first with respect to t over (0 , M f ( x ′ )), and then with respect to x ′ over π n − (Ω), by Fubini’s theorem and (4.14), one gets Z π n − (Ω) × R y φ (cid:18)(cid:18) x ′ , η + η (cid:19) · ∇ f s ( x ′ , y ) (cid:19) dx ′ dy = Z π n − (Ω) dx ′ Z { y : f s ( x ′ ,y ) > , ∇ y f =0 } φ (cid:18)(cid:18) x ′ , η + η (cid:19) · ∇ f s ( x ′ , y ) (cid:19) dy = Z π n − (Ω) dx ′ Z M f ( x ′ )0 dt Z ∂ ∗ { y : f s ( x ′ ,y ) >t } |∇ y f s | φ (cid:18)(cid:18) x ′ , η + η (cid:19) · ∇ f s ( x ′ , y ) (cid:19) d H ≤ Z π n − (Ω) dx ′ Z M f ( x ′ )0 dt Z ∂ ∗ { y : f ( x ′ ,y ) >t } |∇ y f | φ (( x ′ , η ) · ∇ f ( x ′ , y )) d H + 12 Z π n − (Ω) dx ′ Z M f ( x ′ )0 dt Z ∂ ∗ { y : f ( x ′ ,y ) >t } |∇ y f | φ (( x ′ , − η ) · ∇ f ( x ′ , y )) d H = 12 Z π n − (Ω) × R y φ (( x ′ , η ) · ∇ f ( x ′ , y )) dx ′ dy + 12 Z π n − (Ω) × R y φ (( x ′ , − η ) · ∇ f ( x ′ , y )) dx ′ dy. (4.22)22ote that here we have made use of (4.14) and of an analogous equality for f . Step 2:
Let f be any nonnegative function from W , (Ω) and let ω := π n − (Ω).Lemma 4.5. in [17] ensures that there exists a sequence { f h } of nonnegative Lipschitzfunctions, with compact support in R n , satisfying (4.12) and converging strongly to f in W , ( ω × R y ). Assume, for a moment, that φ satisfies0 ≤ φ ( x ) ≤ C (1 + | x | ) for x ∈ R (4.23)for some positive constant C . Then for x ∈ R n , setting F ( x ) := φ (cid:18)(cid:18) x ′ , η + η (cid:19) · x (cid:19) F ( x ) := φ (( x ′ , η ) · x )and F ( x ) := φ (( x ′ , − η ) · x ) , we obtain that F , F and F are globally Lipschitz continuous, and hence F i ( ∇ f h )converges to F i ( ∇ f ) in L ( ω × R y ) for i = 1 , ,
3. On the other hand, since Steinerrearrangement is continuous in W , (see e.g. [8, Theorem 1.]), f sh converges to f s in W , ( ω × R y ). Thus, by Fatou’s lemma and by (4.22), we get Z π n − (Ω) × R y F ( ∇ f s ( x ′ , y )) dx ′ dy ≤ lim inf h →∞ Z π n − (Ω) × R y F ( ∇ f sh ( x ′ , y )) dx ′ dy ≤ lim inf h →∞ Z π n − (Ω) × R y F ( ∇ f h ( x ′ , y )) dx ′ dy + lim inf h →∞ Z π n − (Ω) × R y F ( ∇ f h ( x ′ , y )) dx ′ dy = 12 Z π n − (Ω) × R y F ( ∇ f ( x ′ , y )) dx ′ dy + 12 Z π n − (Ω) × R y F ( ∇ f ( x ′ , y )) dx ′ dy. (4.24)Let us now remove assumption (4.23). Since φ is nonnegative and convex, there existsequences { a j } of R and { b j } of R such that φ ( x ) = sup j ∈ N { a j x + b j } = sup j ∈ N { ( a j x + b j ) + } for every x ∈ R . (4.25)Set, for N ∈ N , φ N ( x ) := sup ≤ j ≤ N { ( a j x + b j ) + } for x ∈ R . (4.26)23bviously, φ N ( x ) converges monotonically to φ ( x ) for every x ∈ R . Since φ N satisfies(4.23), then (4.24) holds with φ replaced by φ N . Inequality (4.24) then follows bymonotone convergence.By Step 1 and Step 2, (4.22) is established for f ∈ W , (Ω). By the definition(3.24), (4.22), (4.9) and (4.10), (4.11) is established. Proof of Theorem 2.1 . By Proposition 4.5 and (2.9), Theorem 2.1 is established.
Proof of Theorem 2.2 . By Theorem 2.1, Remark 3.1 and Lemma 3.15, Theorem 2.2is established.Next, we prove Theorem 2.3.
Lemma 4.6.
Let φ ∈ N s . Let Ω be a bounded open subset of R n satisfying (2.13) andlet f be a nonnegative function from W , Φ0 (Ω) satisfying (2.12) and (2.15). Then, for L n -a.e. ( x ′ , t ) ∈ π n − ,n +1 ( S f ) + , there exist y ( x ′ , t ) , y ( x ′ , t ) ∈ R such that y ( x ′ , t )
12 ( y ( x ′ , t ) + y ( x ′ , t )) = b ( x ′ ) f or L - a.e. t ∈ (0 , M f ( x ′ )) . (4.36) Moreover, b ( x ′ ) ∈ W , ( π n − (Ω)) . Lemma 4.8.
Let φ , Ω and f be given as in Theorem 2.3, and let b : π n − (Ω) → R bethe function defined in Lemma 4.7. If there exist z ′ ∈ R n − and y ∈ R such that b ( x ′ ) = z ′ · x ′ + y f or L n − - a.e. x ′ ∈ π n − (Ω) , (4.37) then there exist A ∈ SL ( n ) and x ∈ R n such that f ( x ) = f s ( Ax + x ) f or L n - a.e. x ′ ∈ Ω . (4.38) Proof.
Let A = E n − − z ′ ! and x = A − y ! , (4.39)where E n − denotes the ( n − × ( n −
1) unit matrix.Since the level set [ f s ( Ax + x )] h = A − [ f s ] h − A − x and for any ( x ′ , y ) ∈ [ f s ] h , A − x ′ y ! − A − x = x ′ z ′ x ′ + y + y ! = x ′ b ( x ′ ) + y ! , we have [ f s ( Ax + x )] h = [ f ] h for every h > . (4.40)By (4.40) and the layer cake representation of a non-negative, real-valued measur-able function f , we get f ( x ) = f s ( Ax + x ) for L n -a.e. x ∈ Ω. Proof of Theorem 2.3.
By the affine invariance of E φ ( f ), i.e., Lemma 3.12, thesufficiency is established. Now we prove the necessity. Let y ( x ′ , t ) and y ( x ′ , t ) bedefined as in Lemma 4.6, and let b be the function defined as in Lemma 4.7. Let us set z ( x ′ , t ) = b ( x ′ ) − µ f ( x ′ , t ) , z ( x ′ , t ) = b ( x ′ ) + 12 µ f ( x ′ , t ) (4.41)26or ( x ′ , t ) ∈ π n − (Ω) × R + t . Then, by Lemma 4.2 and Lemma 4.7, z i ∈ BV loc ( π n − (Ω) × R + t ), i = 1 ,
2. By (4.27) and (4.30), for L n − -a.e. x ′ ∈ π n − (Ω) and L -a.e. t ∈ (0 , M f ( x ′ )), we have L ( { y : f ( x ′ , y ) > t } ) = y ( x ′ , t ) − y ( x ′ , t ) . (4.42)Thus, by (4.41), (4.36) and (4.42), a set N ⊂ π n − ,n +1 ( S f ) + exists such that L n ( π n − ,n +1 ( S f ) + \ N ) = 0 and z i ( x ′ , t ) = y i ( x ′ , t )for ( x ′ , t ) ∈ N . Therefore, thanks to Lemma 4.6, the set S f is equivalent to the set E defined by E = { ( x ′ , y, t ) : ( x ′ , t ) ∈ π n − ,n +1 ( S u ) + , z ( x ′ , t ) < y < z ( x ′ , t ) } . (4.43)Now, define E = { ( x ′ , y, t ) : ( x ′ , t ) ∈ π n − (Ω) × R + t , y > z ( x ′ , t ) } E = { ( x ′ , y, t ) : ( x ′ , t ) ∈ π n − (Ω) × R + t , y < z ( x ′ , t ) } . Observe that E is equivalent to E ∩ E . By Theorem 3.6, the sets E , E and E are offinite perimeter in U × R y for every bounded open set U ⋐ π n − (Ω) × R + t , and hence,by Theorem 3.10, Borel sets Ω E , Ω E and Ω E exist such that L n ( π n − ,n +1 ( E ) + \ Ω E ) = 0 , L n (( π n − (Ω) × R + t ) \ Ω E i ) = 0 , i = 1 , , and (3.19)-(3.22) hold. In particular,( ∂ ∗ E ) x ′ ,t = ∂ ∗ ( E x ′ ,t ) = { z ( x ′ , t ) , z ( x ′ , t ) } for every ( x ′ , t ) ∈ Ω E (4.44)( ∂ ∗ E i ) x ′ ,t = ∂ ∗ ( E i ) x ′ ,t = { z i ( x ′ , t ) } for every ( x ′ , t ) ∈ Ω E i , i = 1 , . (4.45)By Theorem 3.7 and by (3.22) of Theorem 3.10, a Borel set S exists such that H n ( S ) = 0and v E ( x ′ , y, t ) = v E i ( x ′ , y, t )for ( x ′ , y, t ) ∈ [( ∂ ∗ E ∩ ∂ ∗ E i ) \ S ] ∩ [(Ω E ∩ Ω E i ) × R y ] . (4.46)We next claim that a subset R of π n − ,n +1 ( E ) + exists such that L n ( π n − ,n +1 ( E ) + \ R ) = 0 and v Ei ( x ′ , z ( x ′ , t ) , t ) | v Ey ( x ′ , z ( x ′ , t ) , t ) | = v Ei ( x ′ , z ( x ′ , t ) , t ) | v Ey ( x ′ , z ( x ′ , t ) , t ) | + z ′ , i = 1 , . . . , n − ,v Ey ( x ′ , z ( x ′ , t ) , t ) v Et ( x ′ , z ( x ′ , t ) , t ) = − v Ey ( x ′ , z ( x ′ , t ) , t ) v Et ( x ′ , z ( x ′ , t ) , t ) , (4.47)27or ( x ′ , t ) ∈ R , where z ′ ∈ R n − is a constant vector. To verify this claim, recall fromTheorem 3.9 that, since f ∈ W , (Ω), a subset V of ∂ ∗ E ∩ (Ω × R + t ) exists such that H n ([ ∂ ∗ E ∩ (Ω × R + t )] \ V ) = 0 (4.48)and v E ( x ′ , y, t ) = ∇ f ( x ′ , y ) p |∇ f | , · · · , ∇ n − f ( x ′ , y ) p |∇ f | , ∇ y f ( x ′ , y ) p |∇ f | , − p |∇ f | ! (4.49)for every ( x ′ , y, t ) ∈ V . Set Q = π n − ,n +1 ([ ∂ ∗ E ∩ (Ω × R t )] \ V ). Eq. (4.48) entails that L n ( Q ) = 0. It is easy to observe that( x ′ , z i ( x ′ , t ) , t ) ∈ V for L n - a.e. ( x ′ , t ) ∈ π n − ,n +1 ( E ) + \ Q. (4.50)Eqs. (4.47) follow from (4.50) and (4.49) and from (4.28)–(4.29) of Lemma 4.6.Finally, from Eq. (3.18) applied to z and z , and from (4.44) we deduce that a set T ⊂ π n − (Ω) × R + t exists such that L n (( π n − (Ω) × R + t ) \ T ) = 0 and v E i ( x ′ , z i ( x ′ , t ) , t )= ( − i ∇ z i ( x ′ , t ) p |∇ z i | , · · · , ∇ n − z i ( x ′ , t ) p |∇ z i | , − p |∇ z i | , ∇ t z i ( x ′ , t ) p |∇ z i | ! ,i = 1 , , (4.51)for ( x ′ , t ) ∈ T . Now, set Z = [ π n − ,n +1 ( E ) + ∩ N ∩ Ω E ∩ Ω E ∩ Ω E ∩ R ∩ T ] \ π n − ,n +1 ( S ) , and note that L n ( π n − ,n +1 ( E ) + \ Z ) = 0. Combining (4.44)–(4.47) and (4.51) we inferthat ∇ x ′ z ( x ′ , t ) + ∇ x ′ z ( x ′ , t ) = − z ′ and ∇ t z ( x ′ , t ) + ∇ t z ( x ′ , t ) = 0for ( x ′ , t ) ∈ Z , and hence for L n -a.e. ( x ′ , t ) ∈ π n − ,n +1 ( S u ) + . Consequently, ∇ x ′ b ( x ′ ) = − z ′ for L n − -a . e . x ′ ∈ π n − (Ω) . (4.52)Thus, since b ∈ W , ( π n − (Ω)) and satisfies (4.52), and π n − (Ω) is assumed to beconnected, then a constant y ∈ R exists such that (see e.g. [72, Corollary 2.1.9]) b ( x ′ ) = − z ′ · x ′ + y for L n − - a.e. x ′ ∈ π n − (Ω) . (4.53)By Lemma 4.8, the necessity is established.28 Proof of Theorem 2.4
For u ∈ S n − , let u ⊥ denote the n -dimensional linear subspace orthogonal to u in R n . For a Lebesgue measurable set E ⊂ R n and x ′ ∈ u ⊥ , let E x ′ ,u := { x ′ + su : s ∈ R } ∩ E. (5.1)Let F ⊂ R n +1 denote a bounded Lebesgue measurable set. Let π u ( F ) denote theorthogonal projection of F onto u ⊥ and let π u,t ( F ) denote the orthogonal projectionof F onto u ⊥ × R t . Similar to (5.1), for u ∈ S n − and ( x ′ , t ) ∈ u ⊥ × R t , let F ( x ′ ,t ) ,u := { ( x ′ , t ) + su : s ∈ R } ∩ F. (5.2)For u ∈ S n − and K ⊂ R n , let π u ( K ) × R u := { x ′ + su : x ′ ∈ π u ( K ) , s ∈ R } . For u ∈ S n − and f ∈ W , Φ0 (Ω), let f su denote the Steiner symmetrization of f withrespect to u . For fixed x ′ ∈ π u (Ω) + , let M u ( x ′ ) := ess sup { f ( x ′ + su ) : x ′ + su ∈ Ω } and D f,u : = { x ′ + su ∈ Ω : x ′ ∈ π u (Ω) , ∇ u f ( x ′ + su ) = 0 }∩{ x ′ + su ∈ Ω : x ′ ∈ π u (Ω) , M u ( x ′ ) = 0 or f ( x ′ + su ) < M f ( x ′ ) } . (5.3) Lemma 5.1.
For a bounded Lebesgue measurable set E ⊂ R n let E := (cid:26) x ∈ R n : lim ε → + L n ( E ∩ C ( x, ε )) L n ( C ( x, ε )) = 1 (cid:27) , (5.4) where C ( x, ε ) is a cube centered at x and whose side length is ε . Then L n ( E △ E ) = 0 . (5.5) Proof.
We will use Lebesgue’s density theorem: If A is a Lebesgue measurable subsetof R n and ¯ A := (cid:26) x ∈ A : lim ε → + L n ( A ∩ C ( x, ε )) L n ( C ( x, ε )) = 1 (cid:27) , (5.6)29hen L n ( A \ ¯ A ) = 0.Let E c denote the complement of E . Let E c and ¯ E be the sets defined as in (5.6).On the one hand, if x ∈ E \ E , then x ∈ E c \ E c . Since L n ( E c \ E c ) = 0, L n ( E \ E ) = 0.On the other hand, if x ∈ E \ E , then x ∈ E \ ¯ E . Since L n ( E \ ¯ E ) = 0, L n ( E \ E ) = 0.In summary, L n ( E △ E ) = 0.The following lemma was proved in Lemma 2.2 of the paper [21], here we give adifferent proof. Lemma 5.2.
Let E ⊂ R n be a bounded measurable set. If there is a dense set T of directions in S n − such that for every u ∈ T , for L n − -a.e. x ′ ∈ π u ( E ) , E x ′ ,u isequivalent to a closed line segment, then E is a convex body up to an L n -negligible set.Proof. Let E be defined as in (5.4). By Lemma 5.1, we only need to prove that E is a convex set. Suppose that E is not convex, then there exist x , x ∈ E such thatthere exists a point z ∈ ( x , x ) but z / ∈ E .By (5.4), there exist 0 < ε < ε i > i = 1 , , . . . , such thatlim i →∞ ε i = 0 (5.7)and for any i , L n ( E ∩ C ( z, ε i )) L n ( C ( z, ε i )) ≤ − ε (5.8)and there exits i such that i ≥ i L n ( E ∩ C ( x k , ε i )) L n ( C ( x k , ε i )) ≥ − ε , k = 1 , , (5.9)where ¯ w := x − x | x − x | is a common normal vector of one of the ( n − C ( z, ε i ), C ( x , ε i ) and C ( x , ε i ) , and the orthogonal projections of C ( z, ε i ), C ( x , ε i )and C ( x , ε i ) onto ¯ w ⊥ are same.For i ≥ i , we aim to rotate C ( z, ε i ), C ( x , ε i ) and C ( x , ε i ) around z, such that: z ′ , x ′ and x ′ are collinear, z ′ = z and w ′ := x ′ − x ′ | x ′ − x ′ | is the common normal vector ofthe corresponding ( n − C ′ ( z ′ , ε i ), C ′ ( x ′ , ε i ) and C ′ ( x ′ , ε i ) andthe projections of C ′ ( z ′ , ε i ), C ′ ( x ′ , ε i ) and C ′ ( x ′ , ε i ) onto ( w ′ ) ⊥ are same (see Figure1, where dashed areas denote symmetric differences).Actually, by the denseness of T and the continuity of the Lebesgue measure, thereexists a rotation Ψ ∈ SO ( n ) , such that the points x ′ = z + Ψ( x − z ) , x ′ = z + Ψ( x − z ) , w ′ := x ′ − x ′ | x ′ − x ′ | x zx ′ z ′ x ′ C ( x , ε i ) C ′ ( x ′ , ε i ) C ( z, ε i ) C ′ ( z ′ , ε i ) C ( x , ε i ) C ′ ( x ′ , ε i )Figure 1. The rotations of cubes satisfy ω ′ ∈ T and L n ( C ( z, ε i ) △ C ′ ( z ′ , ε i )) ≤ ε L n ( C ′ ( z ′ , ε i )) , (5.10) L n ( C ( x k , ε i ) △ C ′ ( x ′ k , ε i )) ≤ ε L n ( C ′ ( x ′ k , ε i )) , k = 1 , . (5.11)By (5.8) and (5.10), we have L n ( C ′ ( z ′ , ε i ) ∩ E ) = L n ( C ′ ( z ′ , ε i ) ∩ ( C ( z, ε i ) ∪ C ( z, ε i ) c ) ∩ E )= L n ([( C ′ ( z ′ , ε i ) ∩ C ( z, ε i )) ∪ ( C ′ ( z ′ , ε i ) ∩ C ( z, ε i ) c )] ∩ E ) ≤ L n ( C ( z, ε i ) ∩ E ) + L n ( C ′ ( z ′ , ε i ) ∩ C ( z, ε i ) c ) ≤ (cid:16) − ε (cid:17) L n ( C ′ ( z ′ , ε i )) . (5.12)Thus, L n ( C ′ ( z ′ , ε i ) \ E ) ≥ ε L n ( C ′ ( z ′ , ε i )) . (5.13)Similarly, by (5.9) and (5.11), for k = 1 ,
2, we have L n ( C ′ ( x ′ k , ε i ) ∩ E ) = L n ( C ′ ( x ′ k , ε i ) ∩ ( C ( x k , ε i ) ∪ C ( x k , ε i ) c ) ∩ E )= L n ([( C ′ ( x ′ k , ε i ) ∩ C ( x k , ε i )) ∪ ( C ′ ( x ′ k , ε i ) ∩ C ( x k , ε i ) c )] ∩ E )= L n ( C ′ ( x ′ k , ε i ) ∩ C ( x k , ε i ) ∩ E ) + L n ( C ′ ( x ′ k , ε i ) ∩ C ( x k , ε i ) c ∩ E )= L n ( C ( x k , ε i ) ∩ E ) − L n ( C ′ ( x ′ k , ε i ) c ∩ C ( x k , ε i ) ∩ E )+ L n ( C ′ ( x ′ k , ε i ) ∩ C ( x k , ε i ) c ∩ E ) ≥ L n ( C ( x k , ε i ) ∩ E ) − L n (( C ′ ( x ′ k , ε i ) △ C ( x k , ε i )) ∩ E ) ≥ (cid:16) − ε (cid:17) L n ( C ′ ( x ′ k , ε i )) . (5.14)31et w ′ R := { rw ′ : r ∈ R } and D := { x ′ ∈ π w ′ ( C ′ ( z ′ , ε i )) : L (( x ′ + w ′ R ) ∩ ( C ′ ( z ′ , ε i ) \ E )) > } . (5.15)Then L n − ( D ) ≥ ε L n − ( π w ′ ( C ′ ( z ′ , ε i ))). Otherwise, L n ( C ′ ( z ′ , ε i ) \ E ) < ε L n ( C ′ ( z ′ , ε i )),a contradiction. Thus, there exists D ⊂ D such that L n − ( D ) > x ′ ∈ D , L (( x ′ + w ′ R ) ∩ C ′ ( x ′ , ε i ) ∩ E ) > L (( x ′ + w ′ R ) ∩ C ′ ( x ′ , ε i ) ∩ E ) > . (5.16)Otherwise, if for L n − -a.e. x ′ ∈ D , either L (( x ′ + w ′ R ) ∩ C ′ ( x ′ , ε i ) ∩ E ) = 0 or L (( x ′ + w ′ R ) ∩ C ′ ( x ′ , ε i ) ∩ E ) = 0, then L n (( C ′ ( x ′ , ε ) ∪ C ′ ( x ′ , ε )) \ E ) ≥ ε L n ( C ′ ( x ′ , ε i )) . (5.17)Therefore, we have L n ( C ′ ( x ′ , ε i ) ∩ E ) + L n ( C ′ ( x ′ , ε i ) ∩ E )= L n (( C ′ ( x ′ , ε i ) ∩ E ) ∪ ( C ′ ( x ′ , ε i ) ∩ E ))= L n (( C ′ ( x ′ , ε i ) ∪ C ′ ( x ′ , ε i )) ∩ E ) ≤ (2 − ε L n ( C ′ ( x ′ , ε i )) , (5.18)which contradicts L n ( C ′ ( x ′ , ε i ) ∩ E ) + L n ( C ′ ( x ′ , ε i ) ∩ E ) ≥ (2 − ε L n ( C ′ ( x ′ , ε i )) . (5.19)Since (5.13), (5.16) and L n − ( D ) > E must beconvex. Lemma 5.3.
Let K ∈ K n be a convex body. If there is a dense set T of directions in S n − such that for each u ∈ T , the midpoints of chords of K parallel to u lie in anaffine subspace of R n , then for any u ∈ S n − , the midpoints of chords of K parallel to u lie in an affine subspace of R n .Proof. For fixed x ∈ int K , let ρ ( x, u ) := max { r > x + ru ∈ K } denote the radial function of K with respect to x . It is clear that ρ ( x, u ) is continuouswith respect to u ∈ S n − . Therefore, the midpoint of ( x + u R ) ∩ K , denoted by m ( x, u ),satisfies that m ( x, u ) = x + ρ ( x, u ) − ρ ( x, − u )2 u m ( x, u ) is continuous with respect to u ∈ S n − . By the denseness of T , there existsa sequence of vectors { u i } ∞ i =1 ⊂ T such that lim i →∞ u i = u . By the assumptions, thereexists a sequence of vectors { v i } ∞ i =1 ⊂ T and α i ∈ R such that for any x ∈ int K ,( m ( x, u i ) , − · ( v i , α i ) = 0 . (5.20)Since v i ∈ S n − and | α i | < sup { h K ( u ) : u ∈ S n − } , there exist convergent subsequence { v i j } of { v i } and { α i j } of α i such thatlim j →∞ v i j = v and lim j →∞ α i j = α . (5.21)By (5.20) and (5.21), we have for any x ∈ int K ,( m ( x, u ) , − · ( v , α ) = 0 . (5.22)Since x ∈ int K is arbitrary, the midpoints of chords of K parallel to u lie in an affinesubspace of R n .By a classical characterization of ellipsoids (see [61, Theorem 10.2.1]) and Lemma5.3, we obtain the following theorem of characterization of ellipsoids. Theorem 5.4. (Characterization of ellipsoids.) A convex body K ∈ K n is an ellipsoidif and only if there exists a dense set T of directions in S n − such that for each u ∈ T ,the midpoints of chords of K parallel to u lie in an affine subspace of R n . Lemma 5.5.
Let E , E ⊂ R n be origin-centered ellipsoids. If there exists a dense set T of directions in S n − such that for all u ∈ T , the midpoints of chords of E and E parallel to u lie in the same hyperplane, then there exists r > such that E = rE . Proof.
First, we prove that for any u ∈ S n − , the midpoints of chords of E and E parallel to u lie in a hyperplane. Since T is dense in S n − , there exists a sequenceof vectors { u i } ∞ i =1 ⊂ T such that lim i →∞ u i = u . By assumptions and the proof ofLemma 5.3, there exist v and α such that for any x ∈ { z ∈ R n : z is the midpoint of chord of E i parallel to u , i = 1 , } , ( x, − · ( v , α ) = 0 . (5.23)Thus, the midpoints of chords of E and E parallel to u lie in a hyperplane. Bythe arbitrariness of u ∈ S n − and [43, Lemma 5.3], there exists r > E = rE . 33 emma 5.6. Let Ω be a bounded connected open subset of R n . Let f ∈ W , Φ0 (Ω) be anonnegative function fulfilling (2.18). Then there exist A ∈ SL ( n ) and x ∈ R n suchthat f ( x ) = f ⋆ ( Ax + x ) f or L n - a.e. x ∈ Ω (5.24) if and only if there exists a dense set T of directions in S n − and for any u ∈ T , thefollowing statements hold:(i). for L n -a.e. ( x ′ , t ) ∈ π u,t ( S f ) + , ( S f ) ( x ′ ,t ) ,u is equivalent to a closed line segment;(ii). the midpoints of all closed line segments obtained in (i) lie in an affine subspaceof R n +1 parallel to e n +1 .Proof. The necessity of the conditions is clear. Now we prove the sufficiency. Onthe one hand, by (i) and Lemma 5.2, the level set [ f ] h is a convex body up to an L n -negligible set for L -a.e. h ∈ (0 , ess sup f ). Therefore, by (ii), the arbitrariness of u ∈ T and Theorem 5.4, the level set [ f ] h is an ellipsoid up to an L n -negligible setfor L -a.e. h ∈ (0 , ess sup f ). By (ii), the arbitrariness of u ∈ T and Lemma 5.5,the level sets [ f ] h and [ f ] h are homothetic ellipsoids with a common center up to an L n -negligible set for L -a.e. h , h ∈ (0 , ess sup f ).Therefore, there exist A ∈ SL ( n ) and x ∈ R n such that for L -a.e. h ∈ (0 , ess sup f ),[ f ] h = (cid:18) | [ f ] h | ω n (cid:19) n A − B n − A − x , (5.25)up to an L n -negligible set.On the other hand, by the definition of Schwarz spherical symmetrization, for h ∈ (0 , ess sup f ), [ f ∗ ( Ax + x )] h is also an ellipsoid and[ f ∗ ( Ax + x )] h = (cid:18) | [ f ] h | ω n (cid:19) n A − B n − A − x . (5.26)By (5.25), (5.26) and the layer cake representation of a non-negative, real-valued mea-surable function f , we get f ( x ) = f ⋆ ( Ax + x ) for L n -a.e. x ∈ Ω. Lemma 5.7.
Let D ⊂ R n be a bounded measurable set satisfying L n ( D ) > . If thereexists an uncountable subset I of R such that A i ⊂ D and L n ( A i ) > for every i ∈ I ,then there exist i, j ∈ I and i = j such that L n ( A i ∩ A j ) > .Proof. Suppose L n ( A i ∩ A j ) = 0 for any i, j ∈ I and i = j . On the one hand, since L n ( D ) is finite, for any positive integer k , { i ∈ I : L n ( A i ) > k } is finite. On theother hand, for any i ∈ I , there exists a positive integer k such that L n ( A i ) ≥ k .Therefore, I is countable, a contradiction.34 emma 5.8. Let ( X, Σ , µ ) be a measure space with µ ( X ) < ∞ . Let D u ∈ Σ be ameasurable subset of X for every u ∈ S n − . If there exists a Borel set S ⊂ S n − such that H n − ( S ) > and µ ( D u ) > for every u ∈ S , then there exist n linearlyindependent vectors u , u , . . . , u n ∈ S such that µ n \ i =1 D u i ! > . (5.27) Proof.
Step 1: Since µ is a finite measure, we observe the basic property in measuretheory that any index set I satisfying µ ( D i ) > µ ( D i ∩ D j ) = 0 , ∀ i, j ∈ I, i = j, is countable.Step 2: We will prove a more general result by induction: for k = 1 , . . . , n and H k is a k -dimensional linear subspace of R n , if S k ⊂ S n − ∩ H k and H k − ( S k ) > , (5.28)and µ ( D u ) > u ∈ S k , (5.29)then there are k linearly independent vector u , u , . . . , u k ∈ S k such that µ k \ i =1 D u i ! > . (5.30)If k = 1, this conclusion is clear. Suppose k = 2. H ( S ) > S isuncountable. By Step 1, we can find two linearly independent vectors u , u ∈ S suchthat µ \ i =1 D u i ! > . (5.31)Assume that (5.30) is established for k = m , and m ∈ { , . . . , n − } , i.e. there exist m linearly independent vectors u m , . . . , u mm ∈ S m such that µ m \ i =1 D u mi ! > . (5.32)35ext, we consider the case k = m + 1. Since S m +1 ⊂ S n − ∩ H m +1 and H m ( S m +1 ) >
0, for any 2-dimensional linear subspace ¯ H of H m +1 , there exists a set δ ⊂ S n − ∩ ¯ H such that H ( δ ) > H m − ( S m +1 ∩ v ⊥ ) > v ∈ δ. (5.33)The subset δ can be obtained by using the generalized spherical coordinates to compute H m ( S m +1 ).By the assumption (5.32), for any v ∈ δ , there exist m linearly independent vectors u mv, , . . . , u mv,m ∈ S m +1 ∩ v ⊥ such that µ m \ i =1 D u mv,i ! > . (5.34)By H ( δ ) > v , v ∈ δ satisfying (5.34) and µ m \ i =1 D u mv ,i ! ∩ m \ i =1 D u mv ,i !! > . (5.35)Since u mv ,i and u mv ,i , i = 1 , . . . , m , lie in two different subspaces v ⊥ and v ⊥ , there mustbe a u mv ,i such that u mv , , . . . , u mv ,m and u mv ,i are m + 1 linearly independent vectors.By (5.35) µ m \ i =1 D u mv ,i ∩ D u mv ,i ! > . (5.36)By induction with respect to the dimension, (5.30) is established for k = 1 , . . . , n .Let S = S n , we get the desired result. Lemma 5.9.
Let Ω be a set of finite perimeter in R n and let D u = { x ∈ ∂ ∗ Ω : u · v Ω ( x ) = 0 } . (5.37) Then there exists a set T ⊂ S n − such that H n − ( S n − \ T ) = 0 and H n − ( D u ) = 0 for any u ∈ T .Proof. Let T := { u ∈ S n − : H n − ( D u ) > } . We would like to apply Lemma 5.8, and take µ to be the restriction of H n − to ∂ ∗ Ω. If H n − ( T ) >
0, then there exist n linearly independent vectors u , u , . . . , u n ∈ T suchthat H n − ( T ni =1 D u i ) >
0. Let D = T ni =1 D u i . Then for any x ∈ D , we have u i · v Ω ( x ) = 0 , for i = 1 , . . . , n. v Ω ( x ) = 0. This is a contradiction, since | v Ω ( x ) | = 1. Let T := S n − \ T . Then T satisfies the conclusion. Proof of Theorem 2.4.
On the one hand, if f ( x ) = f ⋆ ( Ax + x ) for L n -a.e. x ∈ Ωwith A ∈ SL ( n ) and x ∈ R n , by the the affine invariance of E φ ( f ), i.e., Lemma 3.12,we have E φ ( f ) = E φ ( f ⋆ ).On the other hand, suppose that f ( x ) = f ⋆ ( Ax + x ) for L n -a.e. x ∈ Ω is notestablished for any A ∈ SL ( n ) and x ∈ R n , then by Lemma 5.6, there exist u ∈ S n − and δ > u ∈ B ( u , δ ) ∩ S n − ,either (i) or (ii) in Lemma 5 . . (5.38)Then there exists some ¯ u ∈ B ( u , δ ) ∩ S n − ∩ T , where T is given as in Lemma 5.9,such that L n ( D ¯ u ) = 0 . (5.39)Otherwise, if L n ( D u ) > u ∈ B ( u , δ ) ∩ S n − ∩ T . Thus by Lemma 5.8, thereexist n linearly independent vectors u , u , . . . , u n ∈ B ( u , δ ) ∩ S n − ∩ T such that L n ( T ni =1 D u i ) >
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