Asymptotic first boundary value problem for elliptic operators
aa r X i v : . [ m a t h . C V ] J a n ASYMPTOTIC FIRST BOUNDARY VALUE PROBLEM FORELLIPTIC OPERATORS
JAVIER FALC ´O AND PAUL M. GAUTHIER
Abstract.
In 1955, Lehto showed that, for every measurable function ψ onthe unit circle T , there is a function f holomorphic in the unit disc, having ψ as radial limit a.e. on T . We consider an analogous problem for solutions f of homogenous elliptic equations P f = 0 and, in particular, for holomor-phic functions on Riemann surfaces and harmonic functions on Riemannianmanifolds. Introduction
In 1955, O. Lehto [4] showed that given an arbitrary measurable function ψ onthe interval [0 , π ) , there exists a function f holomorphic in the unit disc D ⊂ C such that lim ρ → f ( ρe iθ ) = ψ ( θ ) , for a.e. θ ∈ [0 , π ) . Lehto’s theorem shows that the radial boundary values of holomorphic functions inthe unit disc can be prescribed almost everywhere on the boundary of the disc. Onthe other hand, any attempt to prescribe angular boundary values fails dramaticallydue to the Luzin-Privalov uniqueness theorem [6]. This result asserts that if ameromorphic function f in the unit disc D has angular limit 0 at each point of asubset of the boundary having positive linear measure, then f = 0 . For p ∈ C and r > , we denote by B ( p, r ) the open disc of center p and radius r . We will denote the Lebesgue 2-measure by m . Our main result is the following. Theorem 1.
Given an arbitrary measurable function ψ on the interval [0 , π ) , whose restriction to some closed subset S ⊂ [0 , π ) is continuous, there exists afunction f holomorphic in D , and for every θ ∈ S and a.e. θ ∈ [0 , π ) , there is aset E θ closed in D , such that f ( z ) → ψ ( θ ) as z → e iθ , z ∈ E θ , and lim r → m (cid:0) B ( e iθ , r ) ∩ E θ (cid:1) m (cid:0) B ( e iθ , r ) ∩ D (cid:1) = 1 . Thus, although we may not prescribe angular approximation, we may prescribeapproximation at almost every point of the boundary, from within a set E θ whosecomplement D \ E θ at every point of the unit circle is asymptotically negligible withrespect to the Lebesgue measure.More generally, we shall present such a result for solutions of elliptic equationson manifolds. Our result applies in particular to harmonic functions on Riemannianmanifolds and to holomorphic functions on Riemann surfaces. Mathematics Subject Classification.
Primary: 58J99 Secondary: 31C12, 30F99.
Key words and phrases. elliptic equations, manifolds, approximation in measure, Dirichletproblem.First author was supported by MINECO and FEDER Project MTM2017-83262-C2-1-P. Secondauthor was supported by NSERC (Canada) grant RGPIN-2016-04107.
Let M be an oriented real analytic manifold with countable base. We shalldenote by ∗ the ideal point of the one-point compactification M ∗ of M. Fix adistance function d on M and a positive Borel measure µ for which open setshave positive measure and compact sets have finite measure. Then µ is regular(see [7, Theorem 2.18]). On M the Lebesgue measure of a measurable set is notwell-defined, but since M is smooth, Lebesgue measure zero is invariant underchange of coordinates, so the notion of absolute continuity of the measure µ (withrespect to Lebesgue measure) is well-defined. We shall assume that our measure µ is absolutely continuous. Let U be an open subset of M, p a boundary point of U and F a closed subset of U. For α ∈ [0 , F ⊂ U has µ -density α at p relative to U, if µ U ( F, p ) := lim r → µ (cid:0) B ( p, r ) ∩ F (cid:1) µ (cid:0) B ( p, r ) ∩ U (cid:1) = α. Denote by ϑ the trivial vector bundle ϑ = M × R k . For a (Borel) measurablesubset S ⊂ M, denote by M ( S, ϑ ) , the family of measurable sections of ϑ over S. Thus, an element u ∈ M ( S, ϑ ) can be identified with a k -tuple u = ( u , . . . , u k ) ofmeasurable functions u j : S → R , j = 1 , . . . , k. For an open set U ⊂ M, we denoteby C ∞ ( U, ϑ ) the family of smooth sections on U endowed with the topology ofuniform convergence on compact subsets of all derivatives. For u ∈ C ∞ ( U, ϑ ) and x ∈ U, we denote | u ( x ) | = max {| u ( x ) | , . . . , | u k ( x ) |} . Let P : ϑ → η be an ellipticoperator on M with analytic coefficients, where η is a real analytic vector bundleon M of the same rank k. With this notation we have the following result.
Theorem 2.
Let
M, d, µ, ϑ, η, P be as above and suppose that P annihilates con-stants. Let U ⊂ M be an arbitrary open subset and ϕ ∈ M ( ∂U, ϑ ) an arbitraryBorel measurable section on the boundary ∂U, whose restriction to some closed sub-set S ⊂ ∂U is continuous. Then, for an arbitrary regular σ -finite Borel measure ν on ∂U, there exists e ϕ ∈ C ∞ ( U, ϑ ) with P e ϕ = 0 , such that, for ν -almost every p ∈ ∂U , and for every p ∈ S, e ϕ ( x ) → ϕ ( p ) , as x → p outside a set of µ -density 0at p relative to U. Consider the two extremal situations, where S is empty and S is the entireboundary ∂U respectively. If S = ∅ , then Theorem 2 solves an asymptotic measur-able first boundary value problem. If S = ∂U, then Theorem 2 solves an asymptotic continuous first boundary value problem. The following two corollaries simply statethat Theorem 2 applies in particular for harmonic functions of several variables andto holomorphic functions of a single complex variable. Corollary 3.
Let M be a Riemannian manifold and let µ be the associated volumemeasure on M. Let U ⊂ M be an arbitrary open subset and ϕ an arbitrary Borelmeasurable function ϕ on the boundary ∂U, whose restriction to some closed subset S ⊂ ∂U is continuous. Then, for an arbitrary regular σ -finite Borel measure ν on ∂U, there exists a harmonic function e ϕ on U, such that, for ν -almost every p ∈ ∂U and for every p ∈ S, e ϕ ( x ) → ϕ ( p ) , as x → p outside a set of µ -density 0 at p relative to U. Corollary 4.
Let M be an open Riemann surface, π : M → C a holomorphicimmersion and µ the associated measure on M. Let U ⊂ M be an arbitrary opensubset and ϕ an arbitrary Borel measurable function ϕ on the boundary ∂U, whoserestriction to some closed subset S ⊂ ∂U is continuous. Then, for an arbitraryregular σ -finite Borel measure ν on ∂U, there exists a holomorphic function e ϕ on U, such that, for ν -almost every p ∈ ∂U and for every p ∈ S, e ϕ ( x ) → ϕ ( p ) , as x → p outside a set of µ -density 0 at p relative to U. Proof.
Although the theorem is for real vector bundles and the ∂ -operator on aRiemann surface is generally considered as an operator between complex vectorbundles of rank 1, we may also consider it as an operator between real vectorbundles of rank 2 (see [5, Remark 3.10.10, Theorem 3.10.11]). (cid:3) Remark. Riemann surfaces are complex manifolds of dimension 1. Our proofdoes not allow us to prove an analogue of Corollary 4 for holomorphic functionson higher dimensional complex manifolds because the proof of Theorem 2 is basedon the Malgrange-Lax Theorem [5] which is for differential operators P : ξ → η between bundles of equal rank. On complex manifolds, the Cauchy-Riemannoperator ∂ maps forms of type ( p, q ) to forms of type ( p, q + 1) . Thus, ∂ : E p,q →E p,q +1 . For ∂ : E p, → E , , it is elliptic and in particular it is elliptic for the case ∂ : E , → E , , mapping functions to forms of type (0 , . On a complex manifoldof dimension n, this is a map between bundles of respective (complex) ranks 1 and n (see [5, 3.10.10]). Thus, in order to have an operator between bundles of equalrank, we must restrict our attention to complex manifolds of dimension 1 , that is,Riemann surfaces.When S = ∂U, corollaries 3 and 4 were proved in [2] and [1] respectively. When M = C , U = D and S = ∅ , Corollary 4 gives Theorem 1.2.
Runge-Carleman approximation
A closed subset E of M is said to satisfy the open K − Q -condition if, for everycompact K ⊂ M there is a compact Q ⊂ M such that K ⊂ Q ◦ and E ∩ Q iscontained in Q ◦ .An exhaustion ( K j ) ∞ j =1 of M is said to be regular if, for each n , the sets K n arecompact, K n ⊂ K ◦ n +1 , M ∗ \ K n is connected and M = ∪ ∞ n =1 K ◦ n . We say that anexhaustion ( K j ) ∞ j =1 of M is open compatible with a closed subset E of M if, forevery j = 1 , , . . . , E ∩ K j is contained in K ◦ j . Lemma 5.
Let E be a closed subset of M, satisfying the open K − Q -condition,then there exists a regular exhaustion of M which is open compatible with E. Proof.
Let ( K j ) ∞ j =1 be a regular exhaustion of M . We shall define recursively anexhaustion ( Q n ) ∞ n =1 of M with certain properties. From the K − Q condition, wechoose a compact set Q , such that K ⊂ Q ◦ and E ∩ Q ⊂ Q ◦ . Now, we may choosea compact set Q , such that K ∪ Q ⊂ Q ◦ and E ∩ Q ⊂ Q ◦ . Suppose we haveselected compact sets Q , . . . , Q n , such that K j ∪ Q j ⊂ Q ◦ j +1 and E ∩ Q j +1 ⊂ Q ◦ j +1 , for j = 1 , . . . , n − . We may choose a compact set Q n +1 , such that K n ∪ Q n ⊂ Q ◦ n +1 and E ∩ Q n +1 ⊂ Q ◦ n +1 . Thus, we have inductively constructed an exhaustion( Q n ) ∞ n =1 such that, for each n, K n ∪ Q n ⊂ Q ◦ n +1 and E ∩ Q n ⊂ Q ◦ n . We denoteby Q cn, ∗ the connected component of M ∗ \ Q n that contains the point ∗ and put L n = M \ Q cn, ∗ . Then, ( L n ) ∞ n =1 is a regular exhaustion of M (see [5, p. 224]).Furthermore, for each n, E ∩ ∂Q n = ∅ , so E ∩ L n ⊂ L ◦ n . Thus, the exhaustion( L n ) ∞ n =1 is open compatible with E. (cid:3) A closed set E ⊂ M is said to be a set of Runge-Carleman approximation, if forevery open neighbourhood U of E, every section f ∈ C ∞ ( U, ϑ ) , with P f = 0 , andevery positive continuous function ε on E, there is a global section u ∈ C ∞ ( M, ϑ ) , with P u = 0 , such that | u − f | < ε on E. Theorem 6.
Let E be a closed subset of M satisfyting the open K − Q condition,with M ∗ \ E connected. Then E is a set of Runge-Carleman approximation.Proof. By lemma 5, let ( L n ) ∞ n =1 be a regular exhaustion of M which is open com-patible with E and set L = ∅ . Fix an open neighbourhood U of E, and a section J. FALC ´O AND P. M. GAUTHIER f ∈ C ∞ ( U, ϑ ) , with P f = 0. Consider ε a continuous and positive function on E, which we may assume is continuous and positive on all of M and set ε n = min { ε ( x ) : x ∈ L n } > , n = 1 , , . . . . Now we construct recursively a sequence ( u n ) ∞ n =0 of sections u n ∈ C ∞ ( M, ϑ )such that | u n − u n − | < ε n / n on L n − and | u n − f | < ε n / n on E ∩ ( L n \ L n − ).Consider u = 0. For n = 1 we only need to check the second condition on u since the first condition is void. Note that U = L ◦ ∩ U is an open set containing E ∩ L such that M \ U has no compact connected components and P f = 0 on U . By the Malgrange-Lax theorem, see [5], there exists a section u ∈ C ∞ ( M, ϑ )with
P u = 0 , such that | u − f | < ε / E ∩ L . Assume now that we have fixed ( u n ) N − n =1 satisfying the required conditions. Con-sider two open sets V N , W N such that E ∩ ( L N \ L N − ) ⊂ V N ⊂ ( U ∩ ( L N \ L N − )) ,L N − ⊂ W N ,V N ∩ W N = ∅ . Since M ∗ \ E is connected and ( L n ) ∞ n =1 is a regular exhaustion, without loss of gen-erality, we can assume that M \ ( V N ∪ W N ) has no compact connected components.Define g ∈ C ∞ ( G, ϑ ) , by putting g = u N − on W N and g = f on V N . Set K = L N − ∪ ( E ∩ ( L N \ L N − ))and U N = ( V N ∪ W n ) ∩ U . Then, P g = 0 on U N and M \ U N has no compactconnected components. By the Malgrange-Lax theorem again, there exists a section u N ∈ C ∞ ( M, ϑ ) with
P u N = 0 , such thatmax x ∈ K | u N ( x ) − g ( x ) | < ε N N . The section u N has the required properties, which completes the inductive con-struction of the sequence ( u n ) ∞ n =0 . For every x ∈ M, the sequence { u n ( x ) } ∞ n =0 is Cauchy and hence u convergespointwise to a section ϑ. Let u ( x ) = lim n →∞ u n ( x ) for every x ∈ M . Since forevery natural number j, the sequence ( u n ) ∞ n = j converges to u in C ∞ ( L ◦ j , ϑ ) and P u n = 0 on M, we have that u ∈ C ∞ ( L ◦ j , ϑ ) and also P u = 0 on L ◦ j . Since thisholds for every j = 1 , , . . . , we have that u ∈ C ∞ ( M, ϑ ) and
P u = 0 . To finish we show that | u ( x ) − f ( x ) | ≤ ε ( x ) on E . Fix x ∈ E. Then, there existsa unique natural number n = n ( x ) , such that x ∈ L n \ L n − . We have | u ( x ) − f ( x ) | ≤ | u ( x ) − u n ( x ) | + | u n ( x ) − f ( x ) |≤ ∞ X k = n | u k +1 ( x ) − u k ( x ) | ! + ε n n ≤ ∞ X k = n ε k k < ǫ n n − ≤ ε ( x )2 n − ≤ ε ( x ) . (cid:3) Corollary 7.
Let E be a subset of M that is a union of a locally finite familyof disjoint continua and suppose that M ∗ \ E is connected. Then E is a set ofRunge-Carleman approximation.Proof. Notice that E is closed, since it is the union of a locally finite family ofclosed sets. We only need to show that E satisfies the open K − Q condition. Forthis, fix a compact set K in M . We denote the connected components of E by E j and we may assume that they are ordered so that E , · · · , E m are the connectedcomponents which meet K. Set L = K ∪ E ∪ · · · ∪ E m and let Q be a compactneighbourhood of L disjoint from the closed set E m +1 ∪ E m +2 ∪· · · . Then Q satisfiesthe required conditions. (cid:3) Proof of Theorem 2
Lemma 8.
Let U be a proper open subset of a manifold M and Q and K be disjointcompact subsets of ∂U. Then, for each ε > , there exists δ > and an open set V δ that is a δ -neighbourhood of K in M disjoint from Q such that µ (cid:0) B ( p, r ) ∩ U ∩ V δ (cid:1) µ (cid:0) B ( p, r ) ∩ U (cid:1) < ε, ∀ p ∈ Q, ∀ r > . Furthermore, δ can be chosen so that B ( p, r ) ∩ V δ = ∅ for all p ∈ Q and r < δ .Proof. Set r = d ( Q, K ) / >
0. We claim that(1) ρ := min p ∈ Q µ (cid:0) B ( p, r ) ∩ U (cid:1) > . Assume this were not the case and we have that ρ = 0. Then, by the compactnessof Q , we could find a sequence of points ( p n ) ∞ n =1 ⊂ Q convergent to a point p ∈ Q so that µ (cid:0) B ( p n , r ) ∩ U (cid:1) < /n and d ( p n , p ) < r /
2. But then we would have that µ (cid:0) B ( p , r / ∩ U (cid:1) ≤ µ (cid:0) B ( p n , r ) ∩ U (cid:1) < /n for every natural number n . Hence µ (cid:0) B ( p , r / ∩ U (cid:1) = 0 contradicting the fact that µ has positive measure on opensets. Thus equation (1) holds.Consider V δ a δ -neighbourhood of K in M with δ < r and µ ( U ∩ V δ ) < ερ . Itis clear that for such δ we have that V δ and Q are disjoint and B ( p, r ) ∩ V δ = ∅ , for all p ∈ Q and all r < r . Thus, the last statement of the lemma is obvious bychoosing δ = r . Also, for any r ≥ r we have that µ (cid:0) B ( p, r ) ∩ U ∩ V δ (cid:1) µ (cid:0) B ( p, r ) ∩ U (cid:1) ≤ µ ( U ∩ V δ ) µ (cid:0) B ( p, r ) ∩ U (cid:1) < ερµ (cid:0) B ( p, r ) ∩ U (cid:1) ≤ ε. (cid:3) The following lemma was stated in [2, Lemma 4] for volume measure on a Rie-mannian manifold, but the same proof yields the following more general version.
Lemma 9.
Let U be a proper open subset of a manifold M and C a connectedcompact subset of U with µ ( C ) = 0 . Then, for each ǫ > there is a connected openneighbourhood R of C in U such that µ (cid:0) R ∩ B ( p, r ) (cid:1) µ (cid:0) U ∩ B ( p, r ) (cid:1) < ǫ, for every p ∈ ∂U and every r > . We recall that an open subset W of a real manifold M is an open parametric ball if there is a chart ϕ : W → B , where B is an open ball in the Euclidean space and ϕ ( W ) = B . A subset H ⊂ M is a closed parametric ball , if there is a parametricball ϕ : W → B and a closed ball B ⊂ B , such that H = ϕ − ( B ) . Lemma 10.
Under the hypotheses of Theorem 2, there exists a set F, with S ⊂ F ⊂ ∂U and ν ( ∂U \ F ) = 0 , and u ∈ C ( U, ξ ) on U, such that, for every p ∈ F , u ( x ) → ϕ ( p ) , as x → p outside a set of µ -density 0 at p relative to U. Proof.
We start by showing that there exists a subset F ⊂ ∂U containing S of theform F = S ˙ ∪ (cid:0) ˙ ∪ ∞ n =1 Q n (cid:1) , with Q n compact so that the restriction of ϕ is continuous on Q n and ν ( ∂U \ F ) = 0.First we assume that ν ( ∂U \ S ) < + ∞ . By Lusin’s theorem (see [3] and [8,Theorem 2]), there exists a compact set Q in ∂U \ S such that ν (( ∂U \ S ) \ Q ) < − and the restriction of ϕ to Q is continuous. Now, again by Lusin’s theorem,we can find a compact set Q in ( ∂U \ S ) \ Q with ν ( (cid:0) ( ∂U \ S ) \ Q ) \ Q (cid:1) < J. FALC ´O AND P. M. GAUTHIER − so that the restriction of ϕ to Q ˙ ∪ Q is continuous. By induction we canconstruct a sequence of compact sets ( Q n ) ∞ n =1 so that Q n ⊂ ( ∂U \ S ) \ ∪ n − j =1 Q j , ν (( ∂U \ S ) \ ∪ nj =1 Q j ) < − n and the restriction of ϕ to Q ˙ ∪ · · · ˙ ∪ Q n is continuousfor n = 1 , , , . . . . We set F = S ˙ ∪ (cid:0) ˙ ∪ ∞ n =1 Q n (cid:1) . It is obvious that ( Q n ) ∞ n =1 is a family of pairwise disjoint compact sets and ν ( ∂U \ F ) = 0.If ν ( ∂U \ S ) is not finite, by the σ -finiteness of the measure ν , there existsa pairwise disjoint sequence of measurable sets ( R l ) ∞ l =1 with ν ( R l ) < + ∞ and ∂U \ S = ˙ S ∞ l =1 R l . By the previous argument applied to the section ϕ restricted tothe set R l we can find a pairwise disjoint sequence of compact sets ( Q n,l ) ∞ n =1 of R l , so that the restriction of ϕ is continuous on Q n,l and ν ( R l \ ˙ ∪ ∞ n =1 Q n,l ) = 0 . Then, F = S ˙ ∪ (cid:0) ˙ ∪ ∞ n =1 ˙ ∪ ∞ l =1 Q n,l (cid:1) , satisfies the desired result.We now begin to extend the section ϕ . For this we shall construct inductivelya sequence of increasing sets ( E n ) ∞ n =1 and a sequence of sections ( f n ) ∞ n =1 . We canwrite S = ∪ ∞ n =1 S n and F = S ˙ ∪ (cid:0) ˙ ∪ ∞ n =1 Q n (cid:1) , with S n and Q n compact, S n increasingand Q n pairwise disjoint, so that the restriction ϕ n of ϕ to F n = S ∪ Q ˙ ∪ · · · ˙ ∪ Q n is continuous. By Lemma 8, for l = 2 , , . . . , there is an open δ ,l -neighbourhood V ,l of Q l in M such that(2) µ (cid:0) B ( p, r ) ∩ U ∩ V ,l (cid:1) µ (cid:0) B ( p, r ) ∩ U (cid:1) < l , ∀ p ∈ Q ∪ S , ∀ r ≤ ,µ (cid:0) B ( p, r ) ∩ U ∩ V ,l (cid:1) = 0 , ∀ p ∈ Q ∪ S , ∀ r < δ ,l . By the compactness of Q l , the sets V ,l can be chosen to be a finite union of openparametric balls in M . Let E = U \ ∪ ∞ l =2 V ,l . Since S ∪ Q is closed in E ∪ S ∪ Q , by the Tietze extension theorem we canextend the section ϕ to a continuous section f on E ∪ S ∪ Q . Set E = ∅ and assume that for j = 1 , . . . , n, we have fixed positive constants δ j,l < /j, for l ≥ j + 1 , sets E j = U \ ∪ ∞ l = j +1 V j,l with V j,l being an open δ j,l -neighbourhood of Q l in M \ E j that is a finite union of open parametric balls in M and sections f j continuous on E j ∪ F j such that f j ( x ) = (cid:26) f j − ( x ) , x ∈ E j − ,ϕ j ( x ) = ϕ ( x ) , x ∈ F j , and(3) µ (cid:0) B ( p, r ) ∩ U ∩ V j,l (cid:1) µ (cid:0) B ( p, r ) ∩ U (cid:1) < l , ∀ p ∈ ∪ jk =1 ( S k ∪ Q k ) , ∀ r ≤ ,µ (cid:0) B ( p, r ) ∩ U ∩ V j,l (cid:1) = 0 , ∀ p ∈ ∪ jk =1 ( S k ∪ Q k ) , ∀ r < δ j,l , for j = 1 , . . . , n and l = j + 1 , j + 2 , . . . . For the step n + 1, using Lemma 8 again wehave that for every natural number l > n + 1 there is an open δ n +1 ,l -neighbourhood V n +1 ,l of Q l in M \ E n such that µ (cid:0) B ( p, r ) ∩ U ∩ V n +1 ,l (cid:1) µ (cid:0) B ( p, r ) ∩ U (cid:1) < l , ∀ p ∈ ∪ n +1 k =1 ( S k ∪ Q k ) , ∀ r ≤ ,µ (cid:0) B ( p, r ) ∩ U ∩ V n +1 ,l (cid:1) = 0 , ∀ p ∈ ∪ n +1 k =1 ( S k ∪ Q k ) , ∀ r < δ n +1 ,l . Without loss of generality we can assume that δ n +1 ,l < / ( n + 1), and, by thecompactness of Q l , the sets V n +1 ,l can be chosen to be a finite union of openparametric balls in M . Set E n +1 = U \ ∪ ∞ l = n +2 V n +1 ,l . Note that E n ∪ F n +1 is relatively closed in E n +1 ∪ F n +1 . Furthermore, the section f n +1 defined as f n +1 = f n on E n and f n +1 = ϕ n +1 = ϕ on F n +1 is continuouson the set E n ∪ F n +1 since E n ∩ V n,n +1 = ∅ . Therefore, by the Tietze extensiontheorem we can extend the section f n +1 to a continuous section on E n +1 ∪ F n +1 that we denote in the same way.Note also that ∪ ∞ n =1 E n = U . Indeed, if x ∈ U , since U is open there exists r x > B ( x, r x ) ⊂ U . Fix a natural number l so that l < r x . Then, for every l > l , since V n,l is a l -neighbourhood of Q l ⊂ ∂U in M , we have that x / ∈ V n,l forevery natural number n . Thus, for n ≥ l we have that x ∈ E n = U \ ∪ ∞ l = n +1 V n,l .Then, the section u, defined on U as u ( x ) = f n ( x ) if x ∈ E n , is continuous at x. Since x is arbitrary we have the u is continuous on U .There only remains to show that, for every p ∈ F , u ( x ) → ϕ ( p ), as x → p outside a set of µ -density 0 at p relative to U. For this, fix p ∈ F. Then we can finda natural number n so that p ∈ F n . Note that f n is continuous on E n ∪ F n and wehave defined u = f n on E n and f n = ϕ n = ϕ on F n . Therefore, for every p ∈ F , u ( x ) → ϕ ( p ) as x → p in E n . By construction, U \ E n has µ -density 0 at p relativeto U. (cid:3) Before we continue, we introduce some terminology. A compact subset K ⊂ M is a parametric Mergelyan set if there is an open parametric ball ϕ : W → B , with K ⊂ W , and a compact set Q ⊂ B , such that B \ Q is connected and K = ϕ − ( Q ) . A subset E of a manifold M is a Mergelyan chaplet , which we simply call a chaplet,if it is the countable disjoint union of a (possibly infinite) locally finite family E j of pairwise disjoint parametric Mergelyan sets E j . We denote the chaplet by E = ( E j ) j . By Corollary 7, a chaplet is a Runge-Carleman set.
Proof of Theorem 2.
By Lemma 10, there exists a set F, with S ⊂ F ⊂ ∂U and ν ( ∂U \ F ) = 0 , and u ∈ C ( U, ξ ) on U, such that, for every p ∈ F , u ( x ) → ϕ ( p ), as x → p outside a set of µ -density 0 at p relative to U. Let S = { S l } ∞ l =1 be a locally finite family of smoothly bounded compact para-metric balls S l in U such that U = ∪ l S l and | S l | < dist ( S l , ∂U ) , where | S l | denotesthe diameter of S l . Assume also that none of these balls contains another. We mayalso assume that the balls become smaller as we approach ∂U so that the oscillation ω l = ω l ( u ) of u on S l is less than 1 /l, for each l. Let s l = ∂S l for l ∈ N . Since µ is absolutely continuous, Lemma 9 tells us that there is an open neighbourhood R j of s j in U such that(4) µ (cid:0) R j ∩ B ( p, r ) (cid:1) µ (cid:0) U ∩ B ( p, r ) (cid:1) < j , ∀ p ∈ ∂U, ∀ r > . Without loss of generality we may assume that each R j is a smoothly boundedshell. That is, that in a local coordinate system, R j = { x : 0 < ρ j < k x k < } . Bythe local finiteness of S we may also assume that if s j ∩ s l = ∅ , then R j ∩ R l = ∅ .Consider the closed set A = U \ S ∞ k =1 R k . Then, denoting by H j = S j ∩ A and A j = H j \ S j − k =1 S k we have that A = ∞ [ j =1 ( S j ∩ A ) = ∞ [ j =1 H j = ˙ [ ∞ j =1 H j \ j − [ k =1 S k ! = ˙ [ ∞ j =1 A j . J. FALC ´O AND P. M. GAUTHIER
For each j, the set H j is a parametric Mergelyan set in S j and the family ( H j ) ∞ j =1 is locally finite, but they may not be disjoint. However, the ( A j ) ∞ j =1 form a locallyfinite family of disjoint compacta and hence A = ( A j ) ∞ j =1 is a Mergelyan chaplet.Let us fix now a continuous function ε : A → (0 ,
1] so that ε ( x ) → , when x → ∂U . Since ( A j ) ∞ j =1 is a locally finite family of compacta, we may construct afamily ( V j ) ∞ j =1 of disjoint open neigbouhoods A j ⊂ V j , j = 1 , , . . . . For each A j of A we choose a point x A j ∈ A j and define a function g on V = ∪ j V j as g ( x ) = P j u ( x A j ) χ V j ( x ) . Since the function g is constant in each connected component V j and P annihilates constants we have that P g = 0 on V .We claim that U ∗ \ A = ∪ k R k is connected. Choose some R j and let R j be theconnected component of ∪ k R k containing R j . Then R j is the union of a subfamilyof ( R k ) ∞ k =1 . Let us show that this subfamily connects R j with ∗ in U ∗ . Supposethis is not the case. Conside the sets V = ∪ s l ⊂R j S ◦ l and W = ∪ s l j S ◦ l . Bothsets are open because they are the union of open sets. Note that if s k ∩ s l = ∅ ,then R k ∩ R l = ∅ . Now, for every set s l , either s l intersects some s k ⊂ R j or it isdisjoint from every s k ⊂ R j , in which case R ℓ is disjoint from every R k ⊂ R j . Inthe second case, R ℓ cannot be in the bounded complementary component of any R k with R k ⊂ R j , for then S ℓ would be a subset of S k which is forbidden. Therefore R ℓ and consequently S ℓ lies in the unbounded complementary component of every R k with R k ⊂ R j . This means that S ℓ ∩ S k = ∅ . We have shown that, if s ℓ
6⊂ R j , then S ℓ ∩ S k = ∅ , for every s k ⊂ R j and consequently V ∩ W = ∅ . If, as wesupposed, R j is bounded in U, both V and W are non-empty and this contradictsthe assumption that U is connected. Thus, every R j is unbounded in U. Since U ∗ \ A = ∪ j ( R j ∪ {∗} ) is the union of a family of connected sets having point ∗ incommon, it follows that U ∗ \ A is connected as claimed.By Corollary 7, there exists a function e ϕ ∈ C ∞ ( U, ϑ ) with P e ϕ = 0 , such that | e ϕ − g | < ε on A . We show now that(5) | e ϕ ( x ) − u ( x ) | → , as x → p ∈ ∂U, x ∈ A. If ( x n ) ∞ n =1 is a sequence of points in A tending to p ∈ ∂U , then ( x A jn ) ∞ n =1 is alsoa sequence of points in A tending to p ∈ ∂U , where x A jn is the previously fixedpoint in the A j n of A containing x n . Indeed this follows automatically since d ( x A jn , x n ) ≤ | A j n | ≤ d ( A j n , ∂U ) ≤ d ( x n , ∂U ) → , when n goes to infinity.Also,lim sup n →∞ | e ϕ ( x n ) − u ( x n ) | ≤ lim sup n →∞ (cid:0) | e ϕ ( x n ) − g ( x n ) | + | g ( x n ) − u ( x n ) | (cid:1) ≤ lim sup n →∞ (cid:0) ε ( x n ) + | u ( x A jn ) − u ( x n ) | (cid:1) ≤ lim sup n →∞ (cid:0) ε ( x n ) + ω j n ( u ) (cid:1) = 0 . We now show that A satisfies that,(6) µ U ( A, p ) = lim inf r → µ (cid:0) B ( p, r ) ∩ A (cid:1) µ (cid:0) B ( p, r ) ∩ U (cid:1) = 1 . For this we shall show thatlim sup r → µ ( B ( p, r ) ∩ ( U \ A )) µ ( B ( p, r ) ∩ U ) = 0 . For fixed ε > j ε so that X j ≥ j ε − j < ε. Consider r ε > B ( p, r ε ) is disjoint from the neigbourhoods R j of thesets s j for j ≤ j ε . Then, for all r < r ε , since U \ A = ∪ j R j , we have that µ ( B ( p, r ) ∩ ( U \ A )) µ ( B ( p, r ) ∩ U ) = µ ( B ( p, r ) ∩ ( ∪ j R j )) µ ( B ( p, r ) ∩ U ) ≤ X j>j ε µ ( B ( p, r ) ∩ R j ) µ ( B ( p, r ) ∩ U ) ≤ X j>j ε − j < ε. (by (4))Thus, the µ -density of U \ A relative to U at p is at most ε. Since p and ε arearbitrary, this proves (6).Note that the function u has all of the properties desired in the theorem, exceptthat of satisfying the differential equation P u = 0 . The function e ϕ does satisfy theequation P e ϕ = 0 and also satisfies the desired properties, because of (5) and (6). (cid:3) References [1]
J. Falc´o and P. M. Gauthier . An asymptotic holomorphic boundary problem on arbitraryopen sets in Riemann surfaces. Journal of Approximation Theory (2020), 105451.[2]
J. Falc´o and P. M. Gauthier . Approximation in measure: Dirichlet problem, universalityand the Riemann hypothesis. Izv. Ross. Akad. Nauk Ser. Mat. (to appear) . arXiv:2002.10129.[3]
P. H. Halmos . Measure Theory. D. Van Nostrand Company, Inc., New York, N. Y., 1950.[4]
Lehto, O . On the first boundary value problem for functions harmonic in the unit circle.Ann.Acad. Sci. Fenn. Ser. A. I. 1955 (1955), no. , 26 pp.[5]
Narasimhan, R . Analysis on Real and Complex Manifolds. Reprint of the 1973 edition. North-Holland Mathematical Library, . North-Holland Publishing Co., Amsterdam, 1985.[6] Noshiro, K . Cluster Sets. Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft Springer-Verlag, Berlin-G¨ottingen-Heidelberg 1960.[7]
Rudin, W . Real and Complex Analysis. Third edition. McGraw-Hill Book Co., New York,1987.[8]
Wage, M. L . A generalization of Lusin’s theorem. Proc. Amer. Math. Soc. 52 (1975), 327–332.(Javier Falc´o)
Departamento de An´alisis Matem´atico, Universidad de Valencia, DoctorMoliner 50, 46100 Burjasot (Valencia), Spain.
Email address : [email protected] (Paul M. Gauthier)
D´epartement de math´ematiques et de statistique, Universit´e deMontr´eal, Montr´eal, Qu´ebec, Canada H3C3J7.
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