The order completion method for systems of nonlinear PDEs revisited

The Order Com pletio n Metho d for System s o f N onlinea r PDEs Revisited Jan Harm v an der W alt Dep artment of Mathematics and Applie d M athematics University of Pr etoria Abstract. In th is p ap er we p resents furth er developments regarding th e enrichment of the basic Theory of Order Completion as presen ted in (Ob erguggenberger and Rosinger, 1994). In p articular, spaces of generalized functions are constructed that contain generalized solutions to a large class of systems of conti nuous, nonlinear PDEs. I n terms of the existence and uniquen ess results previously obtained for such systems of equations (v an d er W alt, 2008 [2]), one may interpret the existence of generalized solutions presented here as a regularity result. Keywords: Nonlinear PDEs, Order Completion, Uniform Conv ergence Space Mathematics Sub ject Clasifications (2000): 34A34, 54A20, 06B30, 46E05 1. In tro duction Consider a p ossibly nonlinear PDE, of order at most m , of the form T ( x, D ) u ( x ) = f ( x ) , x ∈ Ω ⊆ R n (1) with the r igh thand term f a con tinuous function of x ∈ Ω , and the partial different ial op erator T ( x, D ) defined by some jointly contin uous mapping F : Ω × R M → R through T ( x, D ) u ( x ) = F ( x, u ( x ) , ..., D α u ( x ) , ... ) , | α | ≤ m (2) It is well know th at an equation of the form (1) th r ough (2) may , in general, fail to h a ve a classical solution u ∈ C m (Ω). Moreo ver, there is in fact a physic al inter est in s olutions to (1) that are not classical. F rom there the int erest in generalized solutions to nonlinear PDEs. A long established idea in analysis is to obtain the existence of generalized solutions to (1) b y asso ciating with the partial differenti al op e rator T ( x, D ) a m ap p ing T : X ∋ u 7→ T u ∈ Y (3) where X is a relativ ely small sp ace of classica l functions on Ω, and Y is some su itable space of f unctions w ith f ∈ Y . Ap propriate topological structur es, typically a norm or lo cally con vex top ology , are defined on X and Y so that the mapping T is uniform ly con tinuous with r esp ect to these cus tomary structures. Generalized solutions to (1) are obtained by constructing the completions X ♯ and Y ♯ of X and Y , resp ectiv ely , and extending the mapping T to a mappin g T ♯ : X ♯ → Y ♯ (4) A s olution of the equation T ♯ u ♯ = f (5) where the unkno wn u ♯ ranges o v er X ♯ , is considered a generalized solution of (1). c  2021 Kluwer A c ademic Publishers. Printe d in the Netherlands. PDECSIII.t ex; 13/08/2021; 7:53; p.1 2 J H v an der W alt As menti oned, the customary structures on the spaces X and Y in (3) are t ypically lo cally con v ex linear sp ace top ologies, or ev en normable top ologies. How ev er, su c h meth- o ds, in v olving the customary linear top ological sp aces of generalized fun ctions, app ea r ineffectiv e in pro viding a g ener al and typ e indep endent th eory for the existence and reg- ularit y of solutions to n onlinear PDEs. This app aren t failure of the u sual metho ds of linear functional analysis in the study of nonlinear PDEs is ascrib ed to the ‘complicated geometry of R n ’ (Arn old, 2004). Moreo ver, in view of the ab ov e men tioned inabilit y of linear fun ctional analysis and other customary m etho ds to yield suc h a general app roac h , it is widely held that it is in fact imp ossible , or at the ve ry b est highly unlikely , that such a theory exists. This, as will b e seen in the sequel, is in fact a misund erstanding. In this r egard, w e s h ould ment ion that there are currentl y t w o gener al and typ e indep endent theories for the existence and regularit y of generalized solutions of nonlinear PDEs. The Central Theory of PDEs (Neub er ger, 199 7) through (Neub erger, 2005) is based on a generaliz ed metho d of steep est descent in suitably constructed Hilb ert spaces. This metho d is fully t y p e indep endent , that is, the particular form of the op erator that defines the equation is not u sed, and rather general, but as of yet it is n ot un iv ersally applicable. In those cases where the m etho d has b een applied, it has resulted in impressive numerical r esults. The Or d er Completion Metho d (Ob erguggen b erger and Rosinger, 1994) and (Anguelov and Rosinger, 2005), on th e other hand, constru cts generalized s olutions to a large class of nonlinear PDEs in the Dedekind order completion of su itable spaces of functions. Th e essen tial feature of b oth metho ds is that the spaces of generalized functions are tied to the particular nonlinear partial differenti al op erator T ( x, D ). Moreo ver, the u nderlying ideas up on whic h they are based apply to situations that are far more general than PDEs, th is b eing exactly the reason for their resp ectiv e type indep endent p ow er. Recen tly , (v an der W alt, 2008 [2]) the Or der Completion Method was recast in the setting of uniform con v ergence spaces (Beattie and Butzmann, 2002). F or a system of K nonlinear PDEs, eac h of order at most m , in K un k n o wn f unctions of the form T ( x, D ) u ( x ) = f ( x ) , x ∈ Ω ⊆ R n , (6) where Ω is op en, and f is a cont inuous, K -dimensional vec tor v alued fu nction on Ω with comp onent s f 1 , ..., f K : Ω → R , generalize d s olutions are constructed as the elemen ts of the completion of a suitable uniform con v ergence space. I n particular, sub ject to a mild assumption on the PDE (6), namely ∀ x ∈ Ω : f ( x ) ∈ in t { F ( x, ξ ) : ξ ∈ R M } (7) where F : Ω × R M → R K is the join tly con tin uous function that defin es the system of PDEs (6) th rough T ( x, D ) u ( x ) = F ( x, u 1 ( x ) , ..., u K ( x ) , ..., D α u i ( x ) , ... ) , | α | ≤ m and i = 1 , ..., K, (8) w e obtain the existence and uniqu eness of generalized solutions to (6). Moreo v er, the generalized solution satisfies a b lank et regularit y as it ma y b e assimilated with nearly finite normal lo w er semi-con tin uous functions. In particular, there is an uniformly con tin uous em b eddin g from the sp ace of generalized solutions in to the sp ace of nearly fi n ite normal lo wer semi-con tinuous fun ctions. It should b e n oted that the assum ption (7) is hardly a restriction on the class of PDEs to whic h the metho d applies. Indeed, ev ery linear PDEs, as we ll as most n on lin ear PDEs of ap p licativ e inte rest satisfy it trivially since, in these cases, { F ( x, ξ ) : ξ ∈ R M } = R K . PDECSIII.t ex; 13/08/2021; 7:53; p.2 The Order Completion Metho d F or Systems Of Nonlinear PDEs 3 Therefore, the O rder Completion Metho d (Ob erguggen b erger and Rosinger, 1994) and th e pseudo-top ological v ersion of th e th eory (v an der W alt, 2008 [2]) wh ic h w e briefly d iscuss here, is to a large exten t universally app licable. Nev erth eless, one ma y notice that there remains a large scop e for p ossible enr ichmen t of the basic theory . In p articular, the space of generaliz ed solutions ma y dep end on the nonlinear partial differentia l op erator (8). Moreo v er, there is no differentia l structure on the space of generalized functions asso ciated with the op erator T ( x, D ). The aim of this p ap er is to resolv e these issu es. This is ac hiev ed b y setting u p app ropriate uniform con v ergence spaces (Beattie and Butzmann, 2002), somewhat in the spirit of Sob olev, whic h d o not dep end on the particular op erator T ( x, D ). The p ap er is organized as follo ws. Section 2 introdu ces th e r elev an t spaces of fun ctions up on whic h the appropr iate spaces of generalized functions are constructed. In Section 3 w e discu s s the app ro ximation results u nderlying the O rder Completion Metho d , and in tro duce a suitable condition on th e system of n onlinear PDEs that allo ws the existence of generalized solutions in appropriate spaces of generalized fun ctions. These r esults are th en applied in S ection 4 w h ere we p ro v e the existence of generalized solutions. Th e structure of the spaces of generalized f u nctions, and that of the generalized functions that are their elemen ts, is discussed in Section 5. 2. F unction Spaces and Their Completions Recall (v an d er W alt, 2008 [1]), (Anguelo v et. al, 2006), (Dilw orth, 1950) that an extended real v alued function u : Ω → R is normal lo wer semi-con tinuous if ( I ◦ S ) ( u ) ( x ) = u ( x ) , x ∈ Ω (9) where I ( u ) ( x ) = sup { inf { u ( y ) : y ∈ Ω, k x − y k < δ } : δ > 0 } (10) and S ( u ) ( x ) = inf { sup { u ( y ) : y ∈ Ω , k x − y k < δ } : δ > 0 } (11) are th e the Low er and Upp er Baire Op erators resp ectiv ely , see (Anguelo v, 2004) and (Baire, 1905 ). These op erators, as w ell as th eir co mp osition, are monotone w ith resp ect to the p oin t w ise ord ering of functions u : Ω → R , and idemp otent. A normal low er semi-con tinuous function is said to b e nearly finite if { x ∈ Ω : u ( x ) ∈ R } op en and d ense The sp ace of all nearly finite normal lo wer semi-con tin uous on Ω is denoted by N L (Ω). These fu nctions satisfy the f ollo wing well kno w prop erty of con tin uous real v alued func- tions, namely , ∀ u, v ∈ N L (Ω) : ∀ D ⊆ Ω dense : u ( x ) ≤ v ( x ) , x ∈ D ⇒ u ( x ) ≤ v ( x ) , x ∈ Ω (12) Moreo ve r, for ev ery u ∈ N L (Ω) there is a set B ⊂ Ω of first Baire Catego ry suc h that u ∈ C (Ω \ B ). Clearly , eac h fu nction that is con tinuous is also normal lo we r semi-con tinuous. F or 0 ≤ l ≤ ∞ , the sub s pace of N L (Ω) consisting of functions that are con tinuous with PDECSIII.t ex; 13/08/2021; 7:53; p.3 4 J H v an der W alt con tinuous partial deriv ativ es up to order l on s ome op en and sense subs et of Ω is denoted ML l (Ω). That is, ML l (Ω) =  u ∈ N L (Ω) ∃ Γ ⊂ Ω closed nowhere dens e : u ∈ C l (Ω \ Γ )  (13) The space N L (Ω), ordered in a p oint wise wa y , is a f ully distributive lattice (v an der W alt, 2008 [1]), and con tains ML 0 (Ω) as a sub lattice. Th er efore (v an der W alt, 2006) the order con vergence of sequences (Luxem burg and Zaanen, 1971) on ML 0 (Ω), whic h is defined through ( u n ) order con v erges to u ⇔   ∃ ( λ n ) , ( µ n ) ⊂ ML 0 (Ω) : 1) λ n ≤ λ n +1 ≤ u n +1 ≤ µ n +1 ≤ µ n , n ∈ N 2) sup { λ n : n ∈ N } = u = inf { µ n : n ∈ N }   (14) is induced by a conv ergence structure (Beattie and Butzmann, 2002) . In fact, the un i- form con vergence structure J o (v an der W alt, 2008 [1]) indu ces the order con ve rgence of sequences, and is defined as follo ws. DEFINITION 1. A filter U on ML 0 (Ω) × ML 0 (Ω) b elongs to the family J o whenever ther e exists k ∈ N such that, for e v ery i = 1 , ..., k , ther e exists nonempty or der intervals  I i n  n ∈ N such that fol lowing c onditions ar e satisfie d: 1) I i n ⊇ I i n +1 for e very n ∈ N . 2) If V ⊆ Ω is op en, then \ n ∈ N I i n | V = ∅ , or ther e exists u i ∈ ML 0 ( V ) such that \ n ∈ N I i n | V = { u i } . 3)  I 1 × I 1  \ ... \  I k × I k  ⊂ U Her e I i = [ { I i n : n ∈ N } ] and I i n | V c onsists of al l functions u ∈ I i n , r estricte d to V . The uniform conv ergence structure J o is uniformly Hausd orff and fir st coun table. More- o ver, a filter F on ML 0 (Ω) conv erges to u ∈ ML 0 (Ω) with resp ect to J o if and only if ∃ ( λ n ) , ( µ n ) ⊂ ML 0 (Ω) : 1) λ n ≤ λ n +1 ≤ µ n +1 ≤ µ n , n ∈ N 2) su p { λ n : n ∈ N } = u = inf { µ n : n ∈ N } 3) [ { [ λ n , µ n ] : n ∈ N } ] ⊆ F (15) The completion of the uniform conv ergence sp ace ML 0 (Ω) is obtained as the sp ace N L (Ω) equipp ed with an approp r iate un iform conv ergence stru cture J ♯ o , s ee (v an der W alt, 2008 [1]). In p articular, the uniform conv ergence structure J ♯ o induces the order con v ergence structure (15). The usu al partial differen tial op erators on C l (Ω), w ith l ≥ 1, may b e extended to ML l (Ω) th rough D α : ML l (Ω) ∋ u 7→ ( I ◦ S ) ( D α u ) ∈ ML 0 (Ω) (16) Therefore, somewhat in the spirit of S ob olev, we equip the sp ace ML l (Ω), where l ≥ 1, with the in itial uniform con vergence stru cture J D with r esp ect to the family of mappings  D α : ML l (Ω) → ML 0 (Ω)  | α |≤ l (17) PDECSIII.t ex; 13/08/2021; 7:53; p.4 The Order Completion Metho d F or Systems Of Nonlinear PDEs 5 That is, for an y filter U on ML l (Ω) × ML l (Ω), w e ha ve U ∈ J D ⇔  ∀ | α | ≤ l ( D α × D α ) ( U ) ∈ J o  (18) Since the family of mappings (17) separates the elemen ts of ML l (Ω), that is, ∀ u, v ∈ ML l (Ω) : ∃ | α | ≤ l : D α u 6 = D α v , it follo ws th at J D is uniformly Hausdorff. A filter F on ML l (Ω) is a Cauc hy filter if and only if D α ( F ) is a Cauch y filter in ML 0 (Ω) for eac h | α | ≤ l . In particular, a fi lter F on ML l (Ω) con v erges to u ∈ ML l (Ω) if and only if D α ( F ) con v erges to D α u in ML 0 (Ω) for eac h | α | ≤ l . In view of the results (v an d er W alt, 2007 [2]) on the completion of un iform con vergence spaces, th e completion of ML l (Ω) is realized as a su bspace of N L (Ω) M , for an appr opriate M ∈ N . In analogy with the case l = 0 we denote the completio n of ML l (Ω) by N L l (Ω). The structure of the sp ace N L l (Ω) and its elemen ts will b e discus s ed in Section 5. With a s ystem of PDEs of the form (6) w e ma y asso ciate a mappin g T : ML m (Ω) K → ML 0 (Ω) K (19) Indeed, w e can write the system (6) comp onent wise as T 1 ( x, D ) u ( x ) = f 1 ( x ) . . . . . . . . . T j ( x, D ) u ( x ) = f j ( x ) . . . . . . . . . T K ( x, D ) u ( x ) = f K ( x ) (20) where, for eac h j = 1 , ..., K the comp onen t T j ( x, D ) of T ( x, D ) is defi ned through the comp onent F j of th e mapping F by T j ( x, D ) u ( x ) = F j ( x, u 1 ( x ) , ..., u K ( x ) , ..., D α u i ( x ) , ... ) , | α | ≤ m and i = 1 , ..., K (21) The mappings (21) are then extended to ML m (Ω) through T j : ML m (Ω) K ∋ u 7→ ( I ◦ S ) ( F j ( · , u 1 , ..., u K , ..., D α u i , ... )) ∈ ML 0 (Ω) , (22) yielding the comp onents of the mapping (19). An equiv alence relation is ind uced on ML m (Ω) K b y the mapping T through ∀ u , v ∈ ML m (Ω) K : u ∼ T v ⇔ T u = Tv (23) The resulting quotient sp ace ML m (Ω) K / ∼ T is denoted ML m T (Ω). With the m ap p ing (19) w e ma y asso ciate in a canonical wa y an inje ctive mapping b T : ML m T (Ω) → ML 0 (Ω) K (24) so th at the diagram PDECSIII.t ex; 13/08/2021; 7:53; p.5 6 J H v an der W alt ML m (Ω) K ✲ ML 0 (Ω) K T ❄ q T id ML m T (Ω) ✲ b T ML 0 (Ω) K ❄ comm u tes. Here q T denotes the ca nonical quotien t map asso ciated with the equiv alence relation (23), and id is the iden tit y on ML 0 (Ω) K . The pro du ct sp aces ML 0 (Ω) K and ML m (Ω) K will carry , naturally , the pro du ct uni- form con v ergence structures J K o and J K D , r esp ectiv ely . That is, U ∈ J K o ⇔  ∀ i = 1 , ..., K : ( π i × π i ) ( U ) ∈ J o  (25) and U ∈ J K D ⇔  ∀ i = 1 , ..., K : ( π i × π i ) ( U ) ∈ J D  (26) Here π i denotes the p ro jection on the i th co ordinate. The completion of ML 0 (Ω) K is N L (Ω ) K equipp ed with the p r o duct u niform conv ergence structure ind uced by J ♯ o . Similarly , the completion of ML m (Ω) K is N L m (Ω) K . The s p ace ML m T (Ω) carries the initial u niform conv ergence str u cture J b T with r esp ect to the m apping b T . That is, U ∈ J b T ⇔ b T ( U ) ∈ J K o (27) Since b T is inje ctive , it follo w s that ML m T (Ω) is uniformly isomorph ic to the subspace b T  ML m T (Ω)  of ML 0 (Ω) K . In particular, b T is a un iformly con tinuous em b ed d ing. In view of the general r esults on the completion of unif orm con v ergence sp aces (v an d er W alt, 2007 [2]), the completion N L T (Ω) of ML m T (Ω) is homeomorph ic to a s u bspace of N L (Ω) K . Indeed, b T extends to a u n iformly con tin uous emb edding of N L T (Ω) int o N L (Ω) K . 3. Appro ximation Results In this section we consider the appro ximation results underlying th e Order Comp letion Metho d (Ob erguggen b erger and Rosinger, 1994) and th e p seudo-top ological version of that theory develo p ed in (v an der W alt, 2008 [2]). In this regard we again consider a system of nonlinear PDEs of the form (6) thr ough (8). Recall (v an der W alt, 2008 [2]) that, sub ject to the mild assumption (7), w e obtain the follo wing lo c al appro ximation result. W e include the p ro of as an illustratio n of the tec hnique used. Moreo ve r, it serves to clarify the argumen ts that lead to a refinement of these results. PDECSIII.t ex; 13/08/2021; 7:53; p.6 The Order Completion Metho d F or Systems Of Nonlinear PDEs 7 PR OPOSIT ION 2. Consider a system of P D Es of the form (6) thr ough (8) that also satisfies (7). Then ∀ x 0 ∈ Ω : ∀ ǫ > 0 : ∃ δ > 0 , P 1 , ..., P K p olynomial in x ∈ R n : x ∈ B ( x 0 , δ ) ∩ Ω , 1 ≤ i ≤ K ⇒ f i ( x ) − ǫ < T i ( x, D ) P ( x ) < f i ( x ) (28) Her e P i s the K -dimensional ve ctor value d function with c omp onents P 1 , ..., P K . Pro of. F or any x 0 ∈ Ω and ǫ > 0 sm all enough it follo ws b y (7) that there exist ξ iα ∈ R with i = 1 , ..., K and | α | ≤ m suc h that F i ( x 0 , ..., ξ iα , ... ) = f i ( x 0 ) − ǫ 2 No w tak e P 1 , ..., P K p olynomials in x ∈ R n that satisfy D α P i ( x 0 ) = ξ iα for i = 1 , ..., K and | α | ≤ m Then it is clear that T i ( x, D ) P ( x 0 ) − f i ( x 0 ) = − ǫ 2 where P is the K -dimen sional ve ctor v alued function on R n with comp onen ts P 1 , ..., P K . Hence (28) follo ws b y the con tin uit y of the f i and the F i . It should b e observe d that, in contradistincti on to the usual fu nctional analytic metho ds, the lo cal lower solution in Prop osition 2 is constru cted in a particularly sim p le w a y . In d eed, it is obtained b y nothing but a straigh tforw ard application of the con tinuit y of the mapping F . Using exactly these same tec h niques, one ma y pro v e the existence of the corresp ond in g appro ximate upp er solutions . PR OPOSIT ION 3. Consider a system of P D Es of the form (6) thr ough (8) that also satisfies (7). Then ∀ x 0 ∈ Ω : ∀ ǫ > 0 : ∃ δ > 0 , P 1 , ..., P K p olynomial in x ∈ R n : x ∈ B ( x 0 , δ ) ∩ Ω , 1 ≤ i ≤ k ⇒ f i ( x ) < T i ( x, D ) P ( x ) < f i ( x ) + ǫ Her e P i s the K -dimensional ve ctor value d function with c omp onents P 1 , ..., P K . The glob al appr o xim ations, corresp ondin g to the lo cal appro ximation constructed in Prop o- sitions 2 and 3, ma y b e formula ted as follo w s. THEOREM 4. Cons ider a system of PDE s of the form (6) thr ough (8) that also satisfies (7). F or every ǫ > 0 ther e exists a close d nowher e dense set Γ ǫ ⊂ Ω , and functions U ǫ , V ǫ ∈ C m (Ω \ Γ ǫ ) K with c omp onents U ǫ, 1 , ..., U ǫ,K and V ǫ, 1 , ..., V ǫ,K r esp e ctively, suc h that f i ( x ) − ǫ < T i ( x, D ) U ǫ ( x ) < f i ( x ) < T i ( x, D ) V ǫ ( x ) < f i ( x ) + ǫ , x ∈ Ω \ Γ ǫ (29) PDECSIII.t ex; 13/08/2021; 7:53; p.7 8 J H v an der W alt Once again, and as w as ment ioned in connection with Prop osition 2, th e app ro ximation result ab o v e is based solely on th e existence of a compact tiling of an y op en sub set Ω of R n , the prop erties of compact sub sets of R n and the cont inuit y of usual r eal v alued functions, see for instance (v an der W alt, 2008 [2], Theorem 4). Hence it make s no use of so called advanc e d mathematics . In particular, tec h niques from fun ctional analysis are not used at all. Instead, the relev ant tec hniques b elong rather to the classical theory of real functions. As an immediate application of Theorem 4, w e ma y construct a sequ ence ( u n ) in ML m (Ω) K so that ( T u n ) con v erges to f in ML 0 (Ω) K . Ho w ev er, Theorem 4 mak es n o claim concernin g the con verge nce of the sequence ( u n ), or lac k thereof, in ML m (Ω) K . Indeed, assumin g only th at (7) is satisfied, it is p ossible, in almost all cases of applicativ e in terest, to construct a sequence ( U n ) that satisfies Theorem 4, and is unbou n ded on ev er y neigh b orho o d of ev er y p oin t of Ω. This follo ws easily from the fact that, in general, f or a fixed x 0 ∈ Ω, the sets { ξ ∈ R M : F ( x 0 , ξ ) = f ( x 0 ) } ma y b e unboun ded. In view of the ab o ve remarks, it app ears that a str onger assump tion than (7) may b e required in order to construct generalized s olutions to (6) in N L m (Ω) K . When formula ting suc h an appropriate condition on the system of PDEs (6), on e sh ould k eep in mind that the Order C ompletion Metho d (Ob erguggen b erger and Rosinger, 1994), and in particular the pseudo-top ological ve rsion of the theory d ev elop ed in (v an der W alt, 2008 [1]) and (v an der W alt, 2008 [2]), is based on some b asic top ological p ro cesses, namely , th e completion of uniform conv ergence sp aces, and the simple condition (7), wh ich is form ulated en tirely in terms of the u sual real mappings F an d f . In particular, (7) do es not in v olv e an y top ologica l structures on function spaces, or mappings on such spaces. F u rthermore, other than th e mere con tinuit y of the mapping F , (7) p laces no restriction on the typ e of equation treated. As such, it is then clear that an y fu rther assump tions that we ma y wish to imp ose on the system of equations (6) in order to obtain generalized solutions in N L m (Ω) K should in v olv e only basic top ological p r op erties of th e mapping F , and sh ould not inv olve any restrictions on the t yp e of equations. In form ulating suc h a cond ition on th e system of PDEs (6) that w ill ensu re the existence of a generalized solution in N L m (Ω) K , it is helpful to first u n derstand more completely the role of th e condition (7) in the p ro of of the lo cal approxima tion r esult Prop osition 2. In particular, and as is clear f rom the pr o of of Prop osition 2, the condition (7) r elates to the con tinuit y of the mapp ing F . F urthermore, and as has alrea dy b een men tioned, the appro ximations constructed in Prop osition 2 and Theorem 4 concern only conv ergence in the target space of the op erator T asso ciated with (6). Our inte rest here lies in constructing suitable approxi mations in the domain of T , and as suc h, pr op erties of the inv erse of the mapping F ma y p ro v e to b e particularly useful. In view of these remarks, w e in tro duce the follo wing condition. ∀ x 0 ∈ Ω : ∃ ξ ( x 0 ) ∈ R M : ∃ V ∈ V x 0 , W ∈ V ξ ( x 0 ) : 1) F ( x 0 , ξ ( x 0 )) = f ( x 0 ) : 2) F : V × W → R K op en (30) Note that the condition (30) ab ov e, although more restrictiv e than (7), allo ws for the treatmen t of a large class of equations. In particular, eac h equation of the form D t u ( x, t ) + G ( x, t, u ( x, t ) , ..., D α x u ( x, t ) , ... ) = f ( x, t ) PDECSIII.t ex; 13/08/2021; 7:53; p.8 The Order Completion Metho d F or Systems Of Nonlinear PDEs 9 with the mapping G merely contin uous, satisfies (30). Indeed, the mappin g F that defines the equation through (6) is b oth op en and sur j ectiv e. Other classes of equations that satisfy (30) can b e easily exhibited. 4. Existence of Genera lized Solutions The b asic existence result for the O rder Completion Method (v an der W alt, 2008 [2]) is an application of the global approximat ion result in Theorem 4, and the commutativ e diagram ML m T (Ω) ✲ ML 0 (Ω) K b T ❄ ✲ N L T (Ω) N L (Ω) K φ ϕ ❄ b T ♯ Here φ and ϕ are the un iformly con tinuous em b eddings asso ciated canonically with the completions N L T (Ω) and N L (Ω), and b T ♯ is the extension of b T ac hieved through un iform con tinuit y . T he approxima tion result, Th eorem 4, is used to construct a C auc hy sequ ence in ML m T (Ω) so that its image under b T con v erges to f . This deliv ers the existenc e of a solution to the generalize d equation b T ♯ U ♯ = f (31) whic h w e in terpret as a generalized solution to (6). Moreo v er, since b T is a un iformly con tinuous em b edd ing, s o is its extension b T ♯ and hence the solution to (31) is unique . THEOREM 5. F or every f ∈ C 0 (Ω) K that satisfies (7), ther e exists a u nique U ♯ ∈ N L T (Ω) so that b T ♯ U ♯ = f (32) The aim of this section is to obtain th e existence of generalized solutions to (6) in the space N L m (Ω) K . In order to formulat e an extended version of the equation (6) in this setting, the partial differential op erator T m ust b e extended to the completion of ML m (Ω) K . The mapp ing T must therefore b e u niformly contin uous with resp ect to the u niform con vergence structures on ML m (Ω) K and ML 0 (Ω) K . Th e pro of of this result is deferred to the app endix. THEOREM 6. The mapping T : ML m (Ω) K → ML 0 (Ω) K define d in (19) to (22) is uniformly c ontinuous. PDECSIII.t ex; 13/08/2021; 7:53; p.9 10 J H v an der W alt In view of Theorem 6 the mapping T extend s u niquely to a uniformly con tin uous mapp in g T ♯ : N L m (Ω) K → N L (Ω) K (33) so th at the diagram ML m (Ω) K ✲ ML 0 (Ω) K T ❄ ✲ N L m (Ω) K N L (Ω) K ψ ϕ ❄ T ♯ comm u tes, with ψ and ϕ the un iformly con tin uous embed d ings asso ciated with the comple- tions of ML m (Ω) K and ML 0 (Ω) K , resp ectiv ely . Th er efore we are justified in formulating the generalize d equation T ♯ u ♯ = f (34) where the unkno wn u ♯ ranges o v er N L m (Ω) K . THEOREM 7. F or every f ∈ C 0 (Ω) K so that the system of PDEs (6) satisfies (30), ther e is some u ♯ ∈ N L m (Ω) K so that T ♯ u ♯ = f (35) Pro of. Set Ω = [ ν ∈ N C ν (36) where, for ν ∈ N , the compact sets C ν are n -dimensional inte rv als C ν = [ a ν , b ν ] (37) with a ν = ( a ν, 1 , ..., a ν,n ), b ν = ( b ν, 1 , ..., b ν,n ) ∈ R n and a ν,j ≤ b ν,j for ev ery j = 1 , ..., n . W e also assume that the C ν , w ith ν ∈ N are lo cally fin ite, that is, ∀ x ∈ Ω : ∃ V ⊆ Ω a n eigh b orho o d of x : { ν ∈ N : C ν ∩ V 6 = ∅} is fin ite (38) W e also assume that the in teriors of C ν , with ν ∈ N , are pairwise disjoint. Let C ν b e arbitrary but fixed. In view of (30) and the con tinuit y of f , we h a ve ∀ x 0 ∈ C ν : ∃ ξ ( x 0 ) ∈ R M , F ( x 0 , ξ ( x 0 )) = f ( x 0 ) : ∃ δ , ǫ > 0 : 1) { ( x, f ( x )) : k x − x 0 k < δ } ⊂ in t  ( x, F ( x, ξ )) k x − x 0 k < δ k ξ − ξ ( x 0 ) k < ǫ  2) F : B δ ( x 0 ) × B 2 ǫ ( ξ ( x 0 )) → R K op en (39) PDECSIII.t ex; 13/08/2021; 7:53; p.10 The Order Completion Metho d F or Systems Of Nonlinear PDEs 11 F or eac h x 0 ∈ C ν , fix ξ ( x 0 ) ∈ R M in (39). Since C ν is compact, it follo ws from (39) that ∃ δ > 0 : ∀ x 0 ∈ C ν : ∃ ǫ > 0 : 1) { ( x, f ( x )) : k x − x 0 k < δ } ⊂ in t  ( x, F ( x, ξ )) k x − x 0 k < δ k ξ − ξ ( x 0 ) k < ǫ  2) F : B δ ( x 0 ) × B 2 ǫ ( ξ ( x 0 )) → R K op en (40) Sub divide C ν in to n -dimensional interv als I ν, 1 , ..., I ν,µ ν with diameter not exceeding δ suc h that th eir interiors are pairwise disjoin t. If a ν,j with j = 1 , ..., µ ν is the cent er of th e interv al I ν,j then b y (40) w e ha ve ∀ j = 1 , ..., µ ν : ∃ ǫ ν,j > 0 : 1) { ( x, f ( x )) : x ∈ I ν,j } ⊂ int  ( x, F ( x, ξ )) x ∈ I ν,j k ξ − ξ ( a ν,j ) k < ǫ ν,j  2) F : I ν,j × B 2 ǫ ν,j ( ξ ( a ν,j )) → R K op en (41) T ak e γ > 0 arbitrary bu t fixed. In view of Prop osition 2 and (41), w e ha v e ∀ x 0 ∈ I ν,j : ∃ U x 0 = U ∈ C m ( R n ) K : ∃ δ = δ x 0 > 0 : ∀ i = 1 , ..., K : x ∈ B δ ( x 0 ) ∩ I ν,j ⇒  1) D α U i ( x ) ∈ B ǫ ν,j ( ξ ( a ν,j )) , | α | ≤ m 2) f i ( x ) − γ < T i ( x, D ) U ( x ) < f i ( x )  As ab o v e, we ma y sub d ivide I ν,j in to pairwise disjoint, n -dimensional inte rv als J ν,j, 1 , ..., J ν,j,µ ν,j so th at for k = 1 , ..., µ ν,j w e ha v e ∃ U ν,j,k = U ∈ C m ( R n ) K : ∀ i = 1 , ..., K : x ∈ J ν,j,k ⇒  1) D α U i ( x ) ∈ B ǫ ν,j ( ξ ( a ν,j )) , | α | ≤ m 2) f i ( x ) − γ < T i ( x, D ) U ( x ) < f i ( x )  (42) Set V 1 = X ν ∈ N   µ ν X j =1 µ ν,j X k =1 χ J ν,j,k U ν,j,k !   where χ J ν,j,k is the c h aracteristic fun ction of J ν,j,k , and Γ 1 = Ω \   [ ν ∈ N   µ ν [ j =1 µ ν,j [ k =1 in t J ν,j,k !     . (43) Then Γ 1 is closed n o w here dense, and V 1 ∈ C m (Ω \ Γ 1 ) K . In view of (42) w e ha v e, for eac h i = 1 , ..., K f i ( x ) − γ < T i ( x, D ) V 1 ( x ) < f i ( x ) , x ∈ Ω \ Γ 1 F urthermore, f or eac h ν ∈ N , f or eac h j = 1 , ..., µ ν , eac h k = 1 , ..., µ ν,j , eac h | α | ≤ m an d ev er y i = 1 , ..., K w e hav e x ∈ int J ν,j,k ⇒ ξ α i ( a ν,j ) − ǫ < D α V 1 ,i ( x ) < ξ α i ( a j ) + ǫ PDECSIII.t ex; 13/08/2021; 7:53; p.11 12 J H v an der W alt Therefore the functions λ α 1 ,i , µ α 1 ,i ∈ C 0 (Ω \ Γ 1 ) d efined as λ α 1 ,i ( x ) = ξ α i ( a j ) − 2 ǫ ν,j if x ∈ in t I ν,j and µ α 1 ,i ( x ) = ξ α i ( a j ) + 2 ǫ ν,j if x ∈ in t I ν,j satisfies λ α 1 ,i ( x ) < D α V 1 ,i ( x ) < µ α 1 ,i ( x ) , x ∈ Ω \ Γ 1 and µ α 1 ,i ( x ) − λ α 1 ,i ( x ) < 4 ǫ ν,j , x ∈ in t I ν,j Applying (41), and pro ceeding in a fashion similar as ab o v e, w e may constr u ct, for eac h n ∈ N su c h th at n > 1, a closed nowhere den s e set Γ n ⊂ Ω, a fu n ction V n ∈ C m (Ω \ Γ n ) K and f u nctions λ α n,i , µ α n,i ∈ C 0 (Ω \ Γ n ) so that, for eac h i = 1 , ..., K and | α | ≤ m f i ( x ) − γ n < T i ( x, D ) V n ( x ) < f i ( x ) , x ∈ Ω \ (Γ n ∪ Γ n − 1 ) . (44) F urthermore, λ α n − 1 ,i ( x ) < λ α n,i ( x ) < D α V n,i ( x ) < µ α n,i ( x ) < µ α n − 1 ,i ( x ) , x ∈ Ω \ Γ n (45) and µ α n,i ( x ) − λ α n,i ( x ) < 4 ǫ ν,j n , x ∈ (in t I ν,j ) ∩ (Ω \ Γ n ) . (46) F or eac h n ∈ N and i = 1 , ..., K set v n,i = ( I ◦ S ) ( V n,i ). F urth ermore, for eac h | α | ≤ m set λ α n,i = ( I ◦ S )  λ α n,i  and µ ′ α n,i = ( I ◦ S )  µ α n,i  . Then, in view of (44) w e ha v e ∀ n ∈ N : ∀ i = 1 , ..., K : f i − γ n ≤ T i v n ≤ f i and f r om (45) it follo ws that ∀ n ∈ N : ∀ i = 1 , ..., K : ∀ | α | ≤ m : λ α n,i ≤ λ α n +1 ,i ≤ D α v n,i ≤ µ α n +1 ,i ≤ µ α n,i . Moreo ve r, in view of (46) it follo ws that µ α n,i ( x ) − λ α n,i ( x ) < 4 ǫ ν,j n , x ∈ in t I ν,j . Therefore ( v n ) is a Cauc hy sequen ce in ML m (Ω) K , and ( Tv n ) conv erges to f in ML 0 (Ω) K . The result no w follo ws by Theorem 6. PDECSIII.t ex; 13/08/2021; 7:53; p.12 The Order Completion Metho d F or Systems Of Nonlinear PDEs 13 5. The Structure of Generalized F unctions Recall (v an der W alt, 2008 [2]) that, in view of the abstract construction of the completion of a uniform con ve rgence space (Wyler, 1970), the unique solution to the generalized equation (31) ma y b e repr esented as the equiv alence class of Cauch y filters on ML m T (Ω) n F a filter on ML m T (Ω) : b T ( F ) con v erges to f o (47) That is, it consists of the totalit y of all filters F on ML m T (Ω) so that b T ( F ) conv erges to f in ML 0 (Ω) K . Moreo ver, eac h classical solution u ∈ C m (Ω) K , and also eac h nonclassical solution u ∈ ML m (Ω) K to (6) generates a Cauc h y filter in ML m T (Ω) w hic h b elongs to the equiv alence class (47). T herefore the generalized solution U ♯ ∈ N L T (Ω) is consistent with the us ual classical solutions as w ell as the n onclassical solutions in u ∈ ML m (Ω) K . This ma y b e represente d in the commuta tiv e diagram ML m (Ω) K ✲ ML 0 (Ω) K ML m T (Ω) T ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ q T b T           ✒ where q T is th e canonical qu otient map asso ciated with the equiv alence relation (23). In view of this diagram it app ears th at the mapping b T is nothing but a r e pr esentation of the usual nonlin ear partial differential op erator T . By virtue of the u niform con tinuit y of the mapping T , w e obtain a similar represen tation for the extended op erator T ♯ . N L m (Ω) K ✲ N L (Ω) K N L T ♯ (Ω) T ♯ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ q ♯ T b T ♯           ✒ PDECSIII.t ex; 13/08/2021; 7:53; p.13 14 J H v an der W alt In view of this d iagram, every generalized solution u ♯ ∈ N L m (Ω) K to (6) is mapp ed u n to the un ique generalized s olution in N L T ♯ (Ω). As su ch, these t w o concepts of generalized solution are consisten t. F urtherm ore, ∀ u ♯ , v ♯ ∈ N L m (Ω) K : T ♯ u ♯ = T ♯ v ♯ ⇒ q ♯ T u ♯ = q ♯ T v ♯ so th at the space N L T ♯ (Ω) con tains the quotien t space N L T ♯ (Ω) / ∼ T ♯ , w here ∀ u ♯ , v ♯ ∈ N L m (Ω) K : u ♯ ∼ T ♯ v ♯ ⇔ T ♯ u ♯ = T ♯ v ♯ Therefore, the generalized solution U ♯ ∈ N L T (Ω) constructed in Theorem 5 ma y b e represent ed as U ♯ = n u ♯ ∈ N L m (Ω) K : T ♯ u ♯ = f o (48) Regarding the stru cture of the space N L m (Ω) and its elemen ts, w e ma y r ecall that the uniform con v ergence structure J D on ML m (Ω) is the initial un iform conv ergence structure with resp ect to the family of mappings  D α : ML m (Ω) → ML 0 (Ω)  | α |≤ m As suc h, the mapping D : ML m (Ω) ∋ u 7→ ( D α u ) | α |≤ m ∈ ML 0 (Ω) M is a uniformly con tinuous em b eddin g. In p articular, for eac h | α | ≤ m , the diagram ML m (Ω) ✲ ML 0 (Ω) M ML 0 (Ω) D ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ D α π α           ✠ comm u tes, with π α the pro jection. Th is diagram amounts to a d ecomp osition of u ∈ ML m (Ω) in to its d ifferential comp onen ts. In view of th e uniform contin uity of the mapp in g D and its in v erse, D extends to an em b edd ing D ♯ : ML m (Ω) ♯ → N L (Ω) M Moreo ve r, since eac h m ap p ing D α is uniformly con tinuous, one obtains the comm u tativ e diagram PDECSIII.t ex; 13/08/2021; 7:53; p.14 The Order Completion Metho d F or Systems Of Nonlinear PDEs 15 N L m (Ω) ✲ N L (Ω) M N L (Ω) D ♯ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ D α♯ π ♯ α           ✠ where D α♯ is the extension through uniform con tin uit y of the partial differential op erator D α . S ince the mapping D ♯ is an em b eddin g, and in view of the comm utativ e diagram ab ov e, eac h generalized fun ction u ♯ ∈ N L m (Ω) ma y b e uniquely represen ted b y its differen tial comp onent s u ♯ 7→ D ♯ u ♯ =  D α♯ u ♯  | α |≤ m Moreo ve r, eac h differen tial comp onent D α♯ u ♯ of u ♯ is a nearly finite normal lo we r s emi- con tinuous fun ction. W e note, therefore, that th e set of singular p oint s of eac h u ♯ ∈ N L m (Ω), that is, the set  x ∈ Ω ∃ | α | ≤ m : D α♯ u ♯ not con tin uous at x  is at most a set of First Baire Category . T h at is, it is a top ologically small set. Ho we v er, this set ma y b e dense in Ω . F urth ermore, suc h a set ma y hav e arbitrarily large p ositive Leb esgue measure (Oxtob y , 1980). 6. Conclusion W e ha ve established the existence of generalized solutions to a large class of systems of non- linear PDEs. The space of generalized fun ctions that conta ins the solutions is constructed as the un iform con v ergence space completion of a space of piecewise smo oth f unctions, and is in dep end ent of the p articular p artial differenti al equation u nder consideration. Moreo v er, the generalized functions may b e represented through their differentia l comp onents as normal lo wer semi-cont inuous fu nctions. T o what exten t the generalized solutions may b e interpreted classical ly , that is, as cont inuously different iable functions that satisfy the equations classically , is still an op en problem. In terms of the previous existence and uniqueness results obtained throu gh the Order Completion Metho d, notably Theorem 5, w e ma y in terpret Theorem 7 as a regularit y result. In particular, it app ear that the generalized solution delivered b y Theorem 5 is nothing but the totality of all solutions in N L m (Ω) K . PDECSIII.t ex; 13/08/2021; 7:53; p.15 16 J H v an der W alt App endix This app end ix con tains th e p ro of of Theorem 6. It is based on the follo wing results whic h ma y b e found in (Anguelo v and v an der W alt, 200 5), (v an der W alt, 2008 [1]) and (v an der W alt, 2008 [3]). PR OPOSIT ION 8. F or any u ∈ N L (Ω) ther e exists se quenc e s ( λ n ) and ( µ n ) in ML 0 (Ω) so that ∀ n ∈ N : λ n ≤ λ n +1 ≤ u ≤ µ n +1 ≤ µ n (49) and ∀ x ∈ Ω : sup { λ n ( x ) : n ∈ N } = u ( x ) = inf { µ n ( x ) : n ∈ N } PR OPOSIT ION 9. Consider a set U ⊂ N L (Ω) so that ∃ B ⊆ Ω of First Bair e Cate gory : ∃ v : Ω \ B → R : x ∈ Ω \ B ⇒ u ( x ) ≤ v ( x ) , u ∈ U Then ther e is some w ∈ N L (Ω) so that ∀ u ∈ U : u ≤ w The c orr esp onding statement for sets b ounde d fr om b elow is also true. PR OPOSIT ION 10. L et L b e a lattic e with r esp e ct to a given p artial or der ≤ . 1. F or every n ∈ N , let the se quenc e ( u m,n ) in L b e b ounde d and incr e asing and let u n = sup { u m,n : m ∈ N } , n ∈ N u ′ n = sup { u m,n : m = 1 , ..., n } If the se quenc e ( u n ) is b ounde d f r om ab ove and incr e asing, and has supr emum in L , then the se quenc e ( u ′ n ) is b ounde d and incr e asing and sup { u n : n ∈ N } = su p { u ′ n : n ∈ N } 2. F or every n ∈ N , let the se quenc e ( v m,n ) in L b e b ounde d and de cr e asing and let v n = inf { v m,n : m ∈ N } , n ∈ N v ′ n = inf { v m,n : m = 1 , ..., n } If the se que nc e ( v n ) is b ounde d fr om b elow and de cr e asing, and has infimum in L , then the se quenc e ( v ′ n ) is b ounde d and de cr e asing and inf { v n : n ∈ N } = inf { v ′ n : n ∈ N } Pro of of Theorem 7. The m ap p ing T may b e represen ted through the diagram PDECSIII.t ex; 13/08/2021; 7:53; p.16 The Order Completion Metho d F or Systems Of Nonlinear PDEs 17 ML m (Ω) K ✲ ML 0 (Ω) K ML 0 (Ω) M T ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ D F           ✒ where D maps u to its vecto r of deriv ativ es, th at is, D : ML m (Ω) K ∋ u 7→ ( D α u i ) | α |≤ m,i ≤ K ∈ ML 0 (Ω) M (50) and F =  F i  i ≤ K is defined comp onent wise through F i : ML 0 (Ω) M ∋ u 7→ ( I ◦ S ) ( F i ( · , u 1 , ..., u M )) ∈ ML 0 (Ω) (51) Clearly the mapping D is uniform ly contin uous, so in view of the diagram it su ffices to sho w that F is uniformly con tin uous with resp ect to the pro d uct uniform conv ergence structure. In this regard, we consider sequences of ord er in terv als  I i n  in ML 0 (Ω), wher e i = 1 , ..., M , that satisfies condition 2) of Definition 1. W e claim ∀ n ∈ N : ∃ J 1 n , ..., J K n ⊆ ML 0 (Ω) ord er in terv als : F j  Q M i =1 I i n  ⊆ J j n , j = 1 , ..., K (52) T o v erify (52), observe that there is a closed nowhere dense set Γ n ⊆ Ω so that ∀ x ∈ Ω \ Γ : ∃ a ( x ) > 0 : ∀ i = 1 , ..., M : u ∈ I i n ⇒ | u ( x ) | ≤ a ( x ) (53) Since F j : Ω × R M → R is con tinuous, it follo ws fr om (53) that ∀ x ∈ Ω \ Γ : ∃ b ( x ) > 0 :  ∀ i = 1 , ..., M : u i ∈ I i n  ⇒ | F j ( x, u 1 ( x ) , ..., u M ( x )) | ≤ b ( x ) (54) Therefore, in view of Pr op osition 9, our claim (52) holds. In particular, since N L (Ω) is Dedekind complete (v an der W alt, 2008 [1]), we may set J j n = [ λ j n , µ j n ] where, for eac h n ∈ N and eac h j = 1 , ..., K λ j n = inf { F j u : u ∈ M Y i =1 I i n } PDECSIII.t ex; 13/08/2021; 7:53; p.17 18 J H v an der W alt and µ j n = sup { F j u : u ∈ M Y i =1 I i n } The sequen ce  λ j n  and  µ j n  are increasing and decreasing, resp ectiv ely . F or eac h j = 1 , ..., K we m a y consider sup { λ j n : n ∈ N } = u j ≤ v j = inf { µ j n : n ∈ N } W e claim that u j = v j . T o see this, we note that for eac h j = 1 , ..., K there is some w j ∈ N L (Ω) so that sup { l j n : n ∈ N } = w j = inf { u j n : n ∈ N } where I j n = [ l j n , u j n ]. App lying Lemma 11 and the conti nuit y of F j our claim is ve rified. Applying Prop ositions 8 and 9 w e obtain sequence  I j n  of order in terv als in ML 0 (Ω) that satisfies condition 2) of Definition 1 and F j M Y i =1 I i n ! ⊆ I j n This completes the pro of.  The pro of also r elies on the follo wing lemma. LEMMA 11. Consider a de cr e asing se qu enc e ( u ) in N L (Ω) that satisfies u = inf { u n : n ∈ N } ∈ N L (Ω) Then the f ol lows holds: ∀ ǫ > 0 : ∃ Γ ǫ ⊆ Ω close d nowh er e dense : x ∈ Ω \ Γ ǫ ⇒  ∃ N ǫ ∈ N : u n ( x ) − u ( x ) < ǫ , n ≥ N ǫ  The c orr esp onding statement for incr e asing se quenc es is also true. Pro of. T ake ǫ > 0 arbitrary bu t fixed . W e start with the set C =  x ∈ Ω ∀ n ∈ N : u n , u con tin uous at x  whic h must hav e complemen t a set of First Baire Catego ry , and hence it is dense. In v iew of (12), th e set of p oin ts C ǫ =  x ∈ C ∃ N ǫ ∈ N : u n ( x ) − u ( x ) < ǫ , n ≥ N ǫ  m ust b e dense in Ω. F rom the con tin uity of u and the u n on C it follo w s that ∀ x 0 ∈ C ǫ : ∃ δ x 0 > 0 : x ∈ C , k x − x 0 k < δ ⇒ x ∈ C ǫ PDECSIII.t ex; 13/08/2021; 7:53; p.18 The Order Completion Metho d F or Systems Of Nonlinear PDEs 19 Since C is dense in Ω, the result follo ws. References Anguelov, R . Dedekind order completion of C(X) by Hausdorff contin u ous functions, Quaestiones Mathematicae 27 , 153-170 (2004). Anguelov, R., Marko v, S. and S en dov, B. The S et of Hausdorff Continuous F unctions — the Largest Linear Space of Interv al F un ct ions, R eliable Computing 12 , 337-363 (2006). Anguelov, R. and Rosinger E. E. 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