Valentin Gledel

Vesna Iršič

Sandi Klavžar

Maker-Breaker domination number

Keywords: Maker-Breaker domination game, Maker-Breaker domination number, domination game, perfect matching, tree, cycle, union of graphs AMS Subj. Class: 05C57, 05C69, 91A43

The Maker-Breaker domination game is played on a graph G by Dominator and Staller. The players alternatively select a vertex of G that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper we introduce the Maker-Breaker domination number γ MB (G) of G as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted γ MB (G).

Comparing the two invariants it turns out that they behave much differently than the related game domination numbers. The invariant γ MB (G) is also compared with the domination number. Using the Erdős-Selfridge Criterion a large class of graphs G is found for which γ MB (G) > γ(G) holds. Residual graphs are introduced and used to bound/determine γ MB (G) and γ MB (G). Using residual graphs, γ MB (T ) and γ MB (T ) are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given.

Introduction

Maker-Breaker games (as well as other positional games) have been introduced by Erdős and Selfridge in [START_REF] Erdős | On a combinatorial game[END_REF], and since then have been the subject of numerous studies, see [START_REF] Bednarska | Biased positional games for which random strategies are nearly optimal[END_REF][START_REF] Ben-Shimon | Local resilience and Hamiltonicity maker-breaker games in random regular graphs[END_REF][START_REF] Ferber | Efficient winning strategies in random-turn Maker-Breaker games[END_REF][START_REF] Hefetz | Fast winning strategies in Maker-Breaker games[END_REF]. Maker-Breaker games are played on hypergraphs by two players called Maker and Breaker. They take turns and at each turn the current player selects a new vertex. Maker wins if at some point of the game he has selected all vertices from one of the hyperedges, while Breaker wins if she can keep him from doing it. See [START_REF] Beck | Combinatorial Games[END_REF] and [START_REF] Hefetz | Positional Games[END_REF] for general introductions on this field.

Very recently, the Maker-Breaker domination game was introduced in [START_REF] Duchêne | Maker-Breaker domination game[END_REF]. The game is played on a graph G with two players named Dominator and Staller. These names were selected to emphasize the domination nature of the game and to be consistent with the usual domination game where these two names are standard by now. (The domination game was introduced in [START_REF] Brešar | Domination game and an imagination strategy[END_REF] and further studied in dozens of papers, cf. [START_REF] Cs | On the game domination number of graphs with given minimum degree[END_REF][START_REF] Dorbec | The domination game played on unions of graphs[END_REF][START_REF] Nadjafi-Arani | Characterisation of forests with trivial game domination numbers[END_REF][START_REF] Schmidt | The 3/5-conjecture for weakly S(K 1,3 )-free forests[END_REF][START_REF] Xu | On graphs with largest possible game domination number[END_REF].) The players alternatively select a vertex of G that was not yet chosen in the course of the game. Dominator wins if at some point, the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. Note that the Maker-Breaker domination game is a Maker-Breaker game. Indeed, if for a graph G we build a hypergraph F with the same set of vertices as G, and in which the hyperedges are the dominating sets of G, then Dominator wins the Maker-Breaker domination game on G if and only if Maker wins the Maker-Breaker game on F.

In several papers on Maker-Breaker games the authors were interested in the smallest number of moves needed for Maker to win, see [START_REF] Clemens | Fast strategies in Maker-Breaker games played on random boards[END_REF][START_REF] Clemens | How fast can Maker win in fair biased games?[END_REF][START_REF] Hefetz | Fast winning strategies in Maker-Breaker games[END_REF]. Also, in [START_REF] Duchêne | Maker-Breaker domination game[END_REF] it was emphasized that when dealing with the Maker-Breaker games, there are two natural questions: (i) which player has a winning strategy and (ii) what is the minimum number of moves if Dominator has a winning strategy. In the seminal paper question (i) is investigated, while in this paper we study (ii). For this sake we say that if G is a graph, then the Maker-Breaker domination number γ MB (G) of G is the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. Otherwise we set γ MB (G) = ∞. Similarly, γ MB (G) denotes is the minimum number of moves of Dominator in the game in which Staller plays first.

We proceed as follows. In the next section we list additional definitions and several known results needed in this paper, as well as prove some basic results on the Maker-Breaker domination number. In Section 3 we first compare γ MB (G) with γ MB (G) and find out that they behave totally different than the related game domination invariants. We also compare γ MB (G) with the domination number and using the Erdős-Selfridge Criterion prove that if the number of γ-sets of G is not too big, then γ MB (G) > γ(G). In Section 4 we introduce residual graphs, determine (resp. bound) γ MB (G) (resp. γ MB (G)) in terms of the residual graph, and determine γ MB (T ) and γ MB (T ) for an arbitrary tree. In the next two sections we obtain the invariants for cycles and bound them for union of graphs. We conclude with a list of open problems and directions for further investigations.

Preliminaries

Let G be a graph. A vertex of G adjacent to a leaf is a support vertex of G. A perfect matching of G is a set of pairwise independent edges that cover V (G). The order of G will be denoted with n

(G). If u is a vertex of G, then N [u] denotes the closed neighborhood of u. If v is another vertex then we set N [u, v] = N [u] ∩ N [v]. A set D ⊆ V (G) is a dominating set of G if ∪ u∈D N [u] = V (G). The domination number γ(G) is the size of a smallest dominating set of G. A dominating set of size γ(G) is called a γ-set of G.
The Maker-Breaker domination game is called a D-game (resp. S-game) if Dominator (resp. Staller) is the first to play a vertex. The sequence of vertices selected in a D-game will be denoted with d 1 , s 1 , d 2 , s 2 , . . ., and the sequence of vertices selected in an S-game with s 1 , d 1 , s 2 , d 2 , . . . Suppose that Dominator wins a D-game. Then the last vertex played is by Dominator, let it be d k . By the definition of the game, {d 1 , . . . , d k } is a dominating set of G. Similarly, if Dominator wins an S-game and the last vertex played by Dominator is d , then {d 1 , . . . , d } is a dominating set of G. We say that a move of Staller is a double threat if it creates two possibilities for her to win in the next move and consequently Dominator cannot prevent Staller to win.

Let G be a graph, k ≥ 1, and u 1 , . . . , u k , v 1 , . . . , v k pairwise different vertices of G. Then we say that

X = {{u 1 , v 1 }, . . . , {u k , v k }} is a pairing dominating set if k i=1 N [u i , v i ] = V (G) .
In the rest we will use this concept via the following interpretation proved in [START_REF] Duchêne | Maker-Breaker domination game[END_REF]Proposition 9]. To be self-contained, we give here an alternative, short proof.

Lemma 2.1 Let u 1 , . . . , u k , v 1 , . . . , v k be pairwise different vertices of a graph G, and let X = {{u 1 , v 1 }, . . . , {u k , v k }}.
Then X is a pairing dominating set if and only if every set {x 1 , . . . , x k }, where

x i ∈ {u i , v i }, i ∈ [k], is a dominating set of G. Proof. Suppose first that X is a pairing dominating set, that is, ∪ k i=1 N [u i , v i ] = V (G). Let {x 1 , . . . , x k } be an arbitrary set with x i ∈ {u i , v i }, i ∈ [k]. Then V (G) = ∪ k i=1 N [u i , v i ] ⊆ ∪ k i=1 N [x i ]. So {x 1 , . . . , x k } is a dominating set of G.
Conversely, consider a set {x 1 , . . . , x k }, where

x i ∈ {u i , v i }, i ∈ [k], and suppose that ∪ k i=1 N [u i , v i ] is a proper subset of V (G). Let w ∈ V (G) \ ∪ k i=1 N [u i , v i ]. Then for every i ∈ [k] we have w / ∈ N [u i , v i ]. Let y i ∈ {u i , v i } be such that w / ∈ N [y i ]. But then ∪ k i=1 N [y i ] is a proper subset of V (G), that is, {y 1 , . . . , y k } is not a dominating set. If X = {{u 1 , v 1 }, . . . , {u k , v k }} is a pairing dominating set such that u i v i ∈ E(G) holds for i ∈ [k]
, then we say that X is a dominating matching. The converse of Fact 2.2 does not hold in general. For instance, in [12, Figure 4] a chordal graph is presented on which Dominator has a winning strategy in both games but admits no pairing dominating set. On the other hand, the converse holds in the class of trees because if Dominator has a winning strategy on a tree T , then it was proved in [START_REF] Duchêne | Maker-Breaker domination game[END_REF] that T has a dominating matching. Moreover, the converse also holds for co-graphs. Lemma 2.3 (No-Skip Lemma) In an optimal strategy of Dominator to achieve γ MB (G) or γ MB (G) it is never an advantage for him to skip a move. Moreover, if Staller skips a move it can only be an advantage for Dominator.

Proof. Suppose a D-game or an S-game is played. Let Dominator and Staller play optimally until some point when Staller decides to skip a move. In that case, Dominator imagines an arbitrary move of Staller, say x, and replies optimally to this move. Since Dominator can always, no matter the way Staller selects vertices, finish the game in no more that γ MB (G) (resp. γ MB (G)) moves, this property is preserved after the imagined move x and the reply to it. Then Dominator proceeds until the end of the game with the same strategy. Note that it may happen that in the course of the game Staller selects a vertex which is not a legal move in the game Dominator is imagining. In that case Dominator imagines that yet some other legal move has been played by Staller. In this way the game on G will finish in no more than γ MB (G) (resp. γ MB (G)) moves.

With a strategy of Staller parallel to the above strategy of Dominator we also infer that it is never an advantage for Dominator to skip a move.

If G is a graph and S ⊆ V (G), then let G|S denote that graph G in which the vertices from S are declared to be already dominated, that is, Dominator is not obliged to dominate them in the rest of the game. Then we have the following Continuation Principle, a proof of which is much simpler that the corresponding principle for the domination game [START_REF] Kinnersley | Extremal problems for game domination number[END_REF].

Remark 2.4 (Continuation Principle) Let G be a graph with A, B ⊆ V (G). If B ⊆ A, then γ MB (G|A) ≤ γ MB (G|B) and γ MB (G|A) ≤ γ MB (G|B).
Indeed, the remark follows from the fact that Dominator can apply the same strategy in G|A as in G|B.

Suppose that γ MB (G) < ∞. Then in any winning strategy of Dominator, he will play at most half of the vertices (because Staller will play the other half) which in turn implies that

1 ≤ γ MB (G) ≤ n(G) 2 . ( 1
)
The bound is sharp, consider for instance the disjoint union of K 1 and several copies of K 2 . It is also easy to see that all the possible values from (1) can be realized by considering the disjoint union of a complete graph and an appropriate number of K 2 s. Similarly, for the S-game, assuming that γ MB (G) < ∞, we have

1 ≤ γ MB (G) ≤ n(G) 2 , (2)
where again all the values can be realized. Later we will apply the celebrated Erdős-Selfridge Criterion for Maker-Breaker games that reads as follows.

Theorem 2.5 (Erdős-Selfridge Criterion [START_REF] Erdős | On a combinatorial game[END_REF])

If F is a hypergraph, then A∈F 2 -|A| < 1 2 ⇒ F is a Breaker's win .
This theorem together with its proof can also be found in the book [16, Theorem 2.3.3].

Maker-Breaker domination numbers

In this section we first compare γ MB (G) with γ MB (G) and construct graphs for all possible values of the invariants. In the second part we compare γ MB (G) with the domination number and using the Erdős-Selfridge Criterion find a large class of graphs G for which γ MB (G) > γ(G) holds.

Realizations of Maker-Breaker domination numbers

One of the fundamental theorems on the domination game proved in [START_REF] Brešar | Domination game and an imagination strategy[END_REF][START_REF] Kinnersley | Extremal problems for game domination number[END_REF] asserts that |γ g (G) -γ g (G)| ≤ 1 holds for every graph G. The next result reveals that the situation with the Maker-Breaker domination number is dramatically different.

Theorem 3.1 If G is a graph, then γ(G) ≤ γ MB (G) ≤ γ MB (G). Moreover, for any integers r, s, t, where 2 ≤ r ≤ s ≤ t, there exists a graph G such that γ(G) = r, γ MB (G) = s, and γ MB (G) = t. Proof. The assertion γ(G) ≤ γ MB (G) is clear since {d 1 , d 2 , . . .} is a dominating set of G.
A D-game can be viewed as an S-game in which Staller has passed her first move. The No-Skip Lemma thus implies that γ MB (G) ≤ γ MB (G).

Let r, s, t be fixed integer where 2 ≤ r ≤ s ≤ t. Construct a graph G r,s,t as follows. Start with a path of length r -1 on consecutive vertices x 1 , . . . , x r . Attach t -r + 1 pendant triangles at x 1 and s -r + 1 pendant triangles at x 2 . Finally, at each (if any) of the vertices x 3 , . . . , x r attach a pendant vertex y 3 , . . . , y r , respectively. The construction should be clear with the aid of Fig. 1.

t -r + 1 s -r + 1 • • • • • • x 1 x 2 x 3 x r y 3 y r Figure 1: Graph G r,s,t
The t-r+1 triangle edges opposite to x 1 and the s-r+1 triangle edges opposite to x 2 together with the edges x 3 y 3 , . . . , x r y r form a dominating matching of G r,s,t . Hence by Fact 2.2, Dominator has a winning strategy in both games. We claim that γ(G r,s,t ) = r, γ MB (G r,s,t ) = s, and γ MB (G r,s,t ) = t, where the first assertion is clear.

Consider the D-game. We first describe the following strategy of Dominator. He starts the game with the move d 1 = x 1 . Then no matter how Staller plays, Dominator can proceed such that at most (r -2) + (s -r + 1) moves of him will be needed to dominate the graph. To do this, whenever Staller plays on x i or y i , i ≥ 3, Dominator replies with y i or x i , respectively. Also, Dominator proceeds along the same lines when Staller plays on a vertex that lies in a triangle attached to x 2 . Using this strategy Dominator ensures that he will play at most 1 + (r -2) + (s -r + 1) = s moves which means that γ MB (G) ≤ s. To prove the other inequality, consider the following strategy of Staller. If Dominator starts with d 1 = x 1 , then Staller sets s 1 = x 2 . Then Dominator will need at least (r -2) + (s -r + 1) additional moves to dominate G. On the other hand, if d 1 = x 1 , then Staller sets s 1 = x 1 , but then Dominator will need at least (r -2) + (t -r + 1) ≥ (r -2) + (s -r + 1) additional moves. In any case, γ MB (G) ≥ 1 + (r -2) + (s -r + 1) = s.

Consider next the S-game. The proof that γ MB (G r,s,t ) = t proceeds similarly as above. To show that γ MB (G) ≥ t, Staller can apply a strategy to start the S-game with s 1 = x 1 . In this way she can guarantee that Dominator will need to play at least (t -r + 1) + 1 + (r -2) = t moves. On the other hand, Dominator can play such that no more than t vertices will be selected by him. Whenever Staller plays x i or y i , i ≥ 3, he replies with y i or x i , respectively. Moreover, during the game he will be able to play x 1 or x 2 .

Note that if γ(G) = 1, then also γ MB (G) = 1. Hence Theorem 3.1 does not extend to the case r = 1. On the other hand, if G t , t ≥ 1, is the graph obtained from t disjoint triangles by identifying a vertex from each of the triangles (so that this new vertex is of degree 2t), then γ(G t ) = 1, γ MB (G t ) = 1, and γ MB (G t ) = t.

Theorem 3.1 extends also to highly connected graphs. To see this, consider the graphs H k,r,s,t , 2 ≤ r ≤ s ≤ t, k ≥ 1, that are schematically drawn in Fig. 2. Here, each vertex of a K k clique is adjacent to each vertex of the clique K k+r . Then by arguments similar to those from the proof of Theorem 3.1 one can see that γ(H k,r,s,t ) = r, γ MB (H k,r,s,t ) = s, and γ MB (H k,r,s,t ) = t. Moreover, H k,r,s,t is (k + 1)-connected.

x r+1 x k+r K k K k K k K k K k K k K k+r t -r + 1 s -r + 1 • • • • • • . . . . . . x 1 x 2 x 3 x r Figure 2: Graph H k,r,s,t

Relation with the domination number

As already observed above, γ(G) = 1 if and only if γ MB (G) = 1. In general it would be interesting to characterize the graphs

G such that γ MB (G) = γ(G) = k,
where k ≥ 2 is a fixed integer. For k = 2 the answer is simple:

Proposition 3.2 Let G be a graph with γ(G) = 2. Then γ MB (G) = γ(G) = 2 if
and only if G has a vertex that lies in at least two γ-sets of G. Conversely, let u be a vertex that lies in two γ-sets of G. Then Dominator plays d 1 = u, and then no matter which vertex is selected by Staller in her first move, Dominator can finish the game in his second move. Proposition 3.2 can be rephrased to hold for larger k also, but this would be more or less just rephrasing the definitions. It would be more interesting to find a structural characterization of the corresponding graphs. This task, however, seems difficult. On the other hand, the Erdős-Selfridge Criterion gives a sufficient condition for γ MB (G) > γ(G). Let X γ (G) be the number of γ-sets of a graph G, cf. [START_REF] Connolly | Bounds on the maximum number of minimum dominating sets[END_REF]. Then:

Proof. Suppose γ MB (G) = γ(G) = 2.
Proposition 3.3 If G is a graph and X γ (G) < 2 γ(G)-1 , then γ MB (G) > γ(G).
Proof. Let F be the hypergraph with V (F) = V (G) and whose hyperedges are the γ-sets of G. Then Theorem 2.5 asserts that

A∈F 2 -|A| < 1 2 ⇒ F is a Breaker's win . (3)
Since |E(F)| = X γ (G) and each of these hyperedges has size γ(G), we can estimate as follows:

A∈F 2 -|A| = A∈F 2 -γ(G) = X γ (G) • 2 -γ(G) < 2 γ(G)-1 • 2 -γ(G) = 1 2 .
Therefore, F is a Breaker's win by [START_REF] Ben-Shimon | Local resilience and Hamiltonicity maker-breaker games in random regular graphs[END_REF]. But this means that in G, Dominator is unable to win with γ(G) moves, thus γ MB (G) > γ(G).

Consider the cycles C 3k-1 , k ≥ 1. It is known and easy to see that γ(C 3k-1 ) = k. We now determine the number of γ-sets of C 3k-1 . Each vertex from a γ-set dominates itself and its two neighbors. As there are k such triplets and 3k -1 vertices in the graph, there is only one vertex that is dominated by two vertices from the γ-set, all others are dominated exactly once. Thus if the vertex that is dominated twice is fixed, then the γ-set of the cycle is uniquely determined. As there are 3k -1 choices for this vertex, we have

X γ (C 3k-1 ) = 3k -1. If k ≥ 5, then X γ (C 3k-1 ) = 3k -1 < 2 k-1 = 2 γ(C 3k-1
)-1 , and by Proposition 3.3, we conclude that γ MB (C 3k-1 ) > k = γ(C 3k-1 ). Actually, γ MB (C 3k-1 ) is much bigger than γ g (C 3k-1 ) as we will see in Section 5.

The converse of Proposition 3.3 does not hold as the following example shows.

If k ∈ {3, 4}, then X γ (C 3k-1 ) = 3k -1 > 2 k-1 = 2 γ(C 3k-1 )-1 , but as we will see in Section 5, γ(C 3k-1 ) = k < k + 1 = 3k-1 2 = γ MB (C 3k-1 ).

Residual graphs

In this section we study the Maker-Breaker domination number on a construction that might be of independent interest and that will be later used to determine the invariant for trees.

If G is a graph, then we say that the residual graph R(G) of G is the graph obtained from G by iteratively removing pendant paths P 2 until no such path is present. By a pendant P 2 we mean P 2 attached to G with an edge. Hence, when such a pendant P 2 is removed, exactly two vertices and two edges are removed. When G = P 2 , we can also remove it and obtain the empty graph.

Note that H = R(G) for some graph G if and only if H is the empty graph, H = K 1 , or each support vertex of H has degree at least 3. This is in particular true if H has no support vertices. We further observe:

Lemma 4.1 If G is a graph, then R(G) is unique (up to isomorphism).
Proof. Let P = xy be an arbitrary pendant P 2 of G, where x is a leaf of G and let z be the other neighbor of y. Then either P is removed at some point when pendant P 2 s are removed from G, or P is not removed at all. The latter possibility can only happen if after this removal process only the path induced with x, y, z remains, and then the pendant path P = yz is removed. In this case, however, we have R(G) = K 1 . In addition, by induction every pendant P 2 that appears during the removal process will either be eventually removed or will lead to the residual tree K 1 .

Note that the proof of Lemma 4.1 also reveals that if R(G) = K 1 , then G \ V (R(G)) is unique. To see that it is not unique in general, consider a path P 2k+1 , k ≥ 2, and different sequences of removing pendant P 2 s. Lemma 4.2 Let G be a graph and R(G) a residual graph of G. Then (i) G \ V (R(G)) is a forest that has a unique perfect matching, and (ii) G has a perfect matching if and only if R(G) has a perfect matching.

Proof. (i) G \ V (R(G)) is a forest since it is build from the empty graph by successively attaching to it pendant P 2 s. If x i y i , i ∈ I, are the pendant P 2 s that were removed when R(G) was obtained from G, then {x i y i } i∈I is a unique perfect matching of G -V (R(G)).

(ii) If G has a perfect matching, then its restriction to G \ V (R(G)) must be the unique perfect matching of G \ V (R(G)), hence R(G) has a perfect matching. Conversely, if R(G) has a perfect matching, then it can be extended to a perfect matching of G by means of (i).

For the proof of the main result of this section, we also need the following. Lemma 4.3 If T is a tree that admits a perfect matching and v ∈ V (T ), then Staller has a strategy for the S-game such that Dominator has to select at least n(T ) 2 vertices to dominate T and v is played by Staller in her last move.

Proof. We prove the claim by induction on n(T ). If T = P 2 and v ∈ V (P 2 ), then Staller can play on v and Dominator has to reply on the other vertex.

Let now n(T ) ≥ 4 and consider T as a BFS-tree rooted at an arbitrary vertex r. Let x be a leaf of this BFS-tree at the largest distance from r and let y be the neighbor of x. Then deg(y) = 2 because T has a perfect matching. Let z be the other neighbor of y. Set T = T \ {x, y}. As T has a perfect matching, xy belongs to it, hence T also has a perfect matching. If v ∈ V (T ), then Staller starts on y, Dominator has to reply on x (otherwise Staller would win) and then Staller applies her strategy on T (by the induction hypothesis). If v ∈ {x, y}, then she applies her strategy on T with her last move on z, and then plays v in her last move. Note that if Dominator plays on v while Staller is playing on T , then Staller wins the game as she can prevent Dominator from playing on one pair of vertices from the matching in T .

From the above strategy of Staller we conclude that the total number of Dominator's moves was n(T ) 2

+ 1 = n(T ) 2 .
Note that by the strategy from the proof of lemma 4.3, unless Staller wants to play on a leaf, she plays on the support vertex, forcing Dominator to reply on its neighboring leaf and separating this P 2 from the rest of the graph.

Theorem 4.4 Let R(G) be a residual graph of G and let

H = G \ V (R(G)). Then (i) γ MB (G) = n(H) 2 + γ MB (R(G)), (ii) n(H) 2 + γ MB (R(G)) -1 ≤ γ MB (G) ≤ n(H) 2 + γ MB (R(G)).
Proof. (i) H has a perfect matching and is a forest by Lemma 4.2(i). Let S-game be played on G and consider the following strategy of Staller. By Lemma 4.3 she can play on each tree of H and play last on the vertex of this tree adjacent to R(G). Dominator has to reply on the matching (otherwise Staller wins the game). Thus, Dominator makes (at least) n(H) 2 moves on H. Moreover, Staller plays on vertices adjacent to R(G), hence no vertex in R(G) will be dominated by the time Staller makes her first move in R(G). Next, Staller is the player to make the first move on R(G) and she follows her optimal strategy there to ensure at least γ MB (R(G)) moves of Dominator.

On the other hand, Dominator's strategy is to then reply wherever Staller plays, H or R(G), with its strategy on this graph. As H has a perfect matching, Dominator makes no more than n(H) 2 moves on H. Moreover, he makes at most γ MB (R(G)) moves on R(G). Hence, we have γ MB (G) = n(H) 2 + γ MB (R(G)). (ii) Suppose now that the D-game is played on G. To prove the upper bound, Dominator's strategy is to start on R(G) and then reply on R(G) or H if Staller plays there. As H has a perfect matching, Dominator makes no more than n(H) 2 moves on H. Moreover, he makes at most γ MB (R(G)) moves on R(G). Hence we get the upper bound γ MB (G) ≤ n(H) 2 + γ MB (R(G)). To prove the lower bound, consider the following strategy of Staller depending on the first move of Dominator. We will distinguish two cases, the second with two subcases, which are schematically depicted in Fig. 3.

Case 1: The first move of Dominator is on R(G).

Staller first applies her strategy from Lemma 4.3 on each tree of H, playing the vertex adjacent to R(G) as her last move on each of the trees. With this, she forces Dominator to play (at least) n(H) 2 moves on H. After that we have an ordinary D-game played on R(G), so at least γ MB (R(G)) moves are made on it by Dominator if Staller follows her strategy there.

Case 2: The first move of Dominator is on H.

Let d 1 be the vertex Dominator plays in his first move, let T be the connected component of H containing d 1 (recall that T is a tree), let P be the shortest path between d 1 and R(G) in T , and let M be the unique perfect matching of T (cf. Lemma 4.2(i)).

In this case, Staller first applies her strategy from Lemma 4.3 on all the other trees of H, playing the vertex adjacent to R(G) as her last move on each tree. Next, Staller applies her strategy from Lemma 4.3 on the edges from M , which are not incident with P . Additionally, she plays last on the vertices closest to P . After that, only R(G), P , and maybe some vertices adjacent to P , remain undominated.

R(G) d 1 H T 1 T 2 • • • T k G Case 1 R(G) d 1 Case 2.1 R(G) y 2 y 1 v x 1 x 2 d 1 Case 2.2
Case 2.1: At least one vertex adjacent to P is still undominated (see Fig. 3).

Let u be an undominated vertex adjacent to P . Staller plays on its neighbor on P , forcing Dominator to reply on u. Staller does so on each such vertex. After that, the only undominated vertices lie on P , moreover, up to now, at least one move of Dominator was played on each already completely dominated edge from M .

As long as there are some more undominated edges from M on P , at least one of them, say e ∈ M , is adjacent to a vertex s of P already played by Staller. Her strategy is to play on the vertex of e which is at distance 2 from s. Then Dominator has to reply on the other vertex of e, otherwise Staller wins by playing it. Hence, Staller can force Dominator to reply on all remaining edges.

Case 2.2: The only undominated vertices in H lie on P . Staller's strategy is to play on the vertex v of P at distance 3 from d 1 . Dominator has to reply on a neighbor of v, otherwise one of the neighbors of v is not dominated and Staller can win by playing that vertex and creating a double threat. Indeed, in this case, two undominated adjacent vertices are played by Staller, and no matter where Dominator answers, she can play another consecutive vertex and win the game. Let x i be a vertex at distance i from v on P in the direction of d 1 , and y i be a vertex at distance i from v on P in the direction of R(G) for all possible i ≥ 1, see Fig. 3 again. If Dominator replies on x 1 , then Staller's next move is y 2 . Now, Dominator has to reply on y 1 , otherwise Staller wins. Then Staller repeats this strategy until P is dominated, i.e., she plays on the vertices y 2k in the increasing order, and Dominator is forced to reply on y 2k-1 .

If Dominator replies on y 1 , then Staller replies on x 2 . After that, Dominator has to play x 1 . Next, Staller applies the same strategy as before, taking y 1 as the new d 1 .

In both cases, Dominator is forced to play at least one move on each edge of the matching M , hence at least n(T ) 2 moves are made on T . On H -T , at least

n(H-T ) 2
moves are made by Lemma 4.3.

After T is completely dominated, Staller follows her optimal strategy on R(G), but it might happen that one vertex u in R(G) is already dominated (by a move of Dominator in H close to R(G)). As Staller's strategy on H forces Dominator to answer on H, Staller will be the first player to play on R(G). But as she can imagine that Dominator's move was u, we have

γ MB (R(G)|u) ≥ γ MB (R(G)|N [u]) ≥ γ MB (R(G)) -1 ,
hence the total number of moves on R(G) is at least γ MB (R(G)) -1.

In either case, Dominator played at least n(H) 2 + γ MB (R(G)) -1 moves, which proves the lower bound.

Note that in the inequality γ MB (G|u) ≥ γ MB (G) -1 from the above proof, the equality can be attained. For example, consider the graph G on Appending to G some trees with perfect matchings, where at least one of them is attached to u, we get graphs that attain the lower bound from Theorem 4.4(ii).

To conclude the section we apply the residual construction to determine the Maker-Breaker domination number of trees. This contrasts the domination game where no such result is known, cf. [START_REF] Brešar | Domination game played on trees and spanning subgraphs[END_REF][START_REF] Henning | Domination game: extremal families for the 3/5-conjecture for forests[END_REF][START_REF] Henning | Trees with equal total domination and game total domination numbers[END_REF]. Theorem 4.5 If T a tree, then

γ MB (T ) =          n(T ) 2 ;
T has a perfect matching,

n(T )-1 2 ; R(T ) ∼ = K 1 , n(T )-k+1 2 ; R(T ) ∼ = K 1,k for k ≥ 3, ∞;
otherwise, and γ MB (T ) = n(T )

2 ; T has a perfect matching, ∞;

otherwise.

Proof. Let T be a tree. Note first that by Lemma 4. If R(T ) ∼ = K 1,k for some k ≥ 3, then T -K 1,k admits a perfect matching. Dominator can start the game on the center vertex of the star K 1,k and then follow the matching in the remaining part. Thus he makes at most 1 + n(T )-k-1 2 = n(T )-k+1 2 moves. On the other hand, Staller can assure that he cannot win in less moves. If his first move is indeed in the center of a star, then Lemma 4.3 assures that at least 1+ n(T )-k-1 2 Domintor's moves are needed. If his first move is elsewhere, then Staller can assure that Dominator follows her moves in the perfect matchings in subtrees of T -K 1,k while additionally she plays the last move on a vertex closest to K 1,k in all subtrees of at least two descendants of the star (again by Lemma 4.3). With this she creates a double threat on these descendants, hence Dominator is forced to play on the center of the star and Dominator makes at least 1 + n(T )-k-1 2 moves. This proves that γ MB (T ) = n(T )-k+1 2 .

Otherwise, we know by [START_REF] Duchêne | Maker-Breaker domination game[END_REF] that Staller wins on T (no matter which player starts the game), hence γ MB (T ) = γ MB (T ) = ∞.

Note that by Theorem 4.5, γ MB and γ MB of trees are polynomial.

Cycles

The D-game domination number and the S-game domination number of cycles are given with the following formulas:

γ g (C n ) = n 2 -1; n ≡ 3 mod 4, n 2 ; otherwise, γ g (C n ) = n-1 2 -1; n ≡ 2 mod 4, n- 1 2
; otherwise.

This fundamental result was first obtained in an unpublished manuscript [START_REF] Kinnersley | Game domination for grid-like graphs[END_REF]. The result appeared for the first time in press in the paper [START_REF] Košmrlj | Domination game on paths and cycles[END_REF], where an alternative proof is given. For the total domination game, parallel results were obtain in [START_REF] Dorbec | Game total domination for cycles and paths[END_REF]. The latter paper investigates the total domination game on paths and cycles only. So the (total) game domination number of cycles is far from being straightforward.

Here we determine the Maker-Breaker domination number of cycles, a task that turned out to be less involved.

Theorem 5.1 If n ≥ 3, then γ MB (C n ) = γ MB (C n ) = n 2 .
Proof. We begin by showing that γ

MB (C n ) ≤ n 2 and γ MB (C n ) ≤ n 2 .
If n is even, C n has a perfect matching (which is also a dominating matching), thus by Fact 2.2, it holds γ MB (C n ) ≤ n 2 = n 2 and γ MB (C n ) ≤ n 2 . Now consider the case when n is odd. In a D-game, let v be the first vertex played by Dominator. Clearly, among undominated vertices,

V (C n ) -N [v], there is a perfect matching. Thus γ MB (C n ) ≤ 1 + n-3 2 = n 2 .
In an S-game, suppose s 1 = u. Then Dominator should reply on a neighbor v of the vertex u. Now there is a perfect matching among

V (C n ) -N [v], so γ MB (C n ) ≤ 1 + n-3 2 = n 2 .
This proves the upper bounds. To find the lower bounds we need to find an appropriate strategy for Staller. Set for the rest of the proof that V (C n ) = {x 1 , . . . , x n }, where the adjacencies are natural.

We first show the lower bound for the S-game: γ MB (C n ) ≥ n 2 . Suppose, without loss of generality, that s 1 = x 1 . Notice that Dominator has to reply on a neighbor of x 1 , for otherwise Staller plays as s 2 the not yet dominated neighbor of x 1 . Then Dominator cannot in one move dominate s 1 and s 2 . Say he leaves s 2 undominated. Theorem 6.1 If G and H are graphs, then

(i) γ MB (G) + γ MB (H) ≤ γ MB (G ∪ H) ≤ min{γ MB (G) + γ MB (H), γ MB (G) + γ MB (H)} , (ii) max{γ MB (G)+γ MB (H), γ MB (G)+γ MB (H)} ≤ γ MB (G∪H) ≤ γ MB (G)+γ MB (H) .
Moreover, all the bounds are sharp.

Proof. (i) It follows directly from the results from [START_REF] Duchêne | Maker-Breaker domination game[END_REF]

that if γ MB (G) = ∞ or γ MB (H) = ∞, then γ MB (G ∪ H) = ∞.
Suppose then that both γ MB (G) and γ MB (H) are finite. We give a strategy for Staller such that when a D-game is played on G ∪ H, she can assure that Dominator will select at least γ MB (G) + γ MB (H) vertices. The strategy of Staller is the following: each time Dominator selects a vertex from G or from H, she answers optimally in G or H (with an optimal strategy restricted to G or H), respectively, as long as this is possible. Suppose without loss of generality that Dominator has first dominated G. Then Staller can reply either in G, provided she has a legal move in G available, or in H. In the first case a usual D-game will be played on H. In the second case we have a game on H in which Dominator has passed one move. By the No-Skip Lemma, Dominator will need to select at least γ MB (H) vertices from H. In any case, γ MB (G ∪ H) ≥ γ MB (G) + γ MB (H).

To prove the upper bound, suppose first that Dominator starts by playing his optimal move on G and then follows Staller in G or in H whenever she plays in G or in H. In this way (having in mind the No-Skip Lemma) Dominator achieves at most γ MB (G) moves in G and at most γ MB (H) moves in H, hence at most γ MB (G) + γ MB (H) vertices for G ∪ H. If instead he starts by playing his optimal move on H, then he can guarantee to play at most γ MB (G) + γ MB (H) vertices. By choosing the smaller of the two values, Dominator has a strategy such that he dominates G ∪ H with no more than min{γ MB (G) + γ MB (H), γ MB (G) + γ MB (H)} moves and so γ MB (G ∪ H) ≤ min{γ MB (G) + γ MB (H), γ MB (G) + γ MB (H)}.

(ii) This is done using similar arguments as in (i).

To demonstrate the sharpness of the bounds, consider the graphs X n,m , 1 ≤ m ≤ n, and Y k , k ≥ 1, as depicted in Fig. 5.

Observe first that γ MB (X n,m ) = m + 1, γ MB (X n,m ) = n + 1, γ MB (Y k ) = 1, and γ MB (Y k ) = k. Consider next the union X n,m ∪ Y k , where k ≤ m, and the union

Y k 1 ∪ Y k 2 ,
where k 1 ≤ k 2 . Then in the D-game as well in the S-game played in either of the unions, an optimal first move is to play a vertex of highest degree. Moreover, an optimal reply to this move is to play a vertex of second highest degree. Therefore,

• γ MB (X n,m ∪ Y k ) = 1 + m + 1, reaching the lower bound of (i),

• γ MB (X n,m ∪ Y k ) = n + 1 + k, reaching the upper bound of (ii),

• γ MB (Y k 1 ∪ Y k 2 ) = k 1 + 1, reaching the upper bound of (i), and

X n,m • • • • • • n m Y k • • • k

Concluding remarks

To conclude the paper we list several problems and directions for further investigation of the Maker-Breaker domination number.

1. For the upper bound in (1) we have provided examples of graphs that attain the equality. These examples are not connected and it is not difficult to achieve the equality with connected graphs of even order. However, we do not know of any connected graph of odd order (different from K 1 ) for which the equality in (1) is achieved. More generally, we ask for a characterization of the extremal graphs with respect to (1) and (2).

2. As we already mentioned, it would be interesting to find a structural characterization of the graphs G for which γ MB (G) = γ(G) = k holds, where k ≥ 2 is a fixed integer.

3. It would also be interesting to investigate γ MB (G H) and γ MB (G H), where G and H are arbitrary graphs and G H is the Cartesian product of G and H. In particular, it would be interesting to determine γ MB (P n P m ) (and γ MB (P n P m )), as well as γ MB (G K 2 ) (and γ MB (G K 2 )) for an arbitrary graph G.

4. If G is a cograph, then it is not difficult to determine whether Dominator or Staller wins the Maker-Breaker domination game [START_REF] Duchêne | Maker-Breaker domination game[END_REF]. On the other hand, it does not seem straightforward to determine the Maker-Breaker domination numbers of co-graphs.

5. In this paper we have considered the Maker-Breaker domination number which is an optimization problem from Dominator's point of view. It would likewise

Fact 2 . 2 [ 12 ,

 2212 Proposition 10] If G admits a pairing dominating set, then Dominator has a winning strategy on G in the D-game as well as in the S-game.

Figure 3 :

 3 Figure 3: Representations of the cases from the proof of Theorem 4.4

Fig. 4 .Figure 4 :

 44 Figure 4: The graph G with the property γ MB (G|u) = γ MB (G) -1.

  2 the cases in Theorem 4.5 are disjoint. If T admits a perfect matching, then γ MB (T ) ≤ n(T ) 2 since by playing once on every edge of the matching, Dominator dominates T . By Lemma 4.3, γ MB (T ) ≥ n(T ) 2 . Hence, γ MB (T ) = n(T ) 2 and by the same reasoning, γ MB (T ) = n(T ) 2 . If R(T ) ∼ = K 1 and v is the remaining vertex, then T -v admits a perfect matching by Lemma 4.3. In this case, Dominator first plays on a neighbor of v (which also belongs to one edge of the matching in T -v) and then follows the perfect matching in T -v. Thus he makes at most n(T )-1 2 moves. By Lemma 4.3, Staller can force him to play on the perfect matching except for his first move, hence γ MB (T ) = n(T )-1 2 .

Figure 5 :

 5 Figure 5: Representation of graphs X n,m and Y k

  After the moves d 1 and s 1 are played, Dominator has a strategy to win the game with the move d 2 . Then {d 1 , d 2 } is a γ-set of G. Moreover, Staller has an option to select s 1 such that {d 1 , s 1 } is a γ-set, hence d 1 must lie in at least two γ-sets.

be of interest to consider the Staller's point of view, that is, assuming that Staller wins on a graph G, what is the minimum number of moves with which she can achieve the goal?

Then Staller can play the other neighbor of s 2 and win the game as Dominator cannot dominate s 2 . So without loss of generality, Dominator replies with d 1 = x n . Staller's next move is s 2 = x 3 . In order to prevent Staller from winning, Dominator has to play d 2 = x 2 . Then Staller continues with the same strategy, forcing Dominator to play on (almost) every second move.

If n is even, the game ends after Staller plays x n-1 and Dominator replies on x n-2 . So in this case, Dominator plays all even labeled vertices, hence

If n is odd, the game ends after Staller plays x n-2 and Dominator replies on x n-3 (as x n-1 is already dominated by d 1 ). So Dominator again plays all even labeled vertices, thus

Assume without loss of generality that d 1 = x 1 . Staller replies on s 1 = x 4 (so at distance 3 from d 1 ). Again, Dominator has to reply on a neighbor of s 1 . If he replies on x 3 , then Staller can apply the above strategy by playing x 6 next, and then every second vertex along the cycle to ensure at least n 2 moves. If Dominator replies on d 2 = x 5 , then Staller plays s 2 = x 8 (again at distance 3 from d 2 in the same direction as before). But observe that at some point of the game Dominator will have to play on {x 2 , x 3 } to dominate the whole graph.

By repeating this strategy, Staller ensures that among every four consecutive vertices of the cycle, Dominator makes at least two moves (except maybe in the last one, two or three remaining vertices). We now distinguish four different cases.

Union of graphs

The Maker-Breaker domination game was in [START_REF] Duchêne | Maker-Breaker domination game[END_REF] studied on disjoint unions on graphs, the obtained results were in particular applied to cographs. In this section we complement their investigation with the following result.


Roberta Bianchini
email: roberta.bianchini@ens-lyon.fr

Laurent Gosse
email: l.gosse@ba.iac.cnr.it

AND Enrique Zuazua
email: enrique.zuazua@deusto.es

A TWO-DIMENSIONAL "FLEA ON THE ELEPHANT" PHENOMENON AND ITS NUMERICAL VISUALIZATION

Keywords: Asymmetric double-well potential, Lipschitz domains, Schrödinger equation, AMS subject classifications. 35P20, 35Q40, 65N06, 65N15

Localization phenomena (sometimes called "flea on the elephant") for the operator L ε = -ε 2 ∆u + p(x)u, p(x) being an asymmetric double-well potential, are studied both analytically and numerically, mostly in two space dimensions within a perturbative framework. Starting from a classical harmonic potential, the effects of various perturbations are retrieved, especially in the case of two asymmetric potential wells. These findings are illustrated numerically by means of an original algorithm, which relies on a discrete approximation of the Steklov-Poincaré operator for L ε , and for which error estimates are established. Such a two-dimensional discretization produces less mesh-imprinting than more standard finite-differences and captures correctly sharp layers.

Introduction and preliminaries.

The "flea on the elephant" is an expression coined by Barry Simon in [START_REF] Simon | Semiclassical Analysis of Low Lying Eigenvalues[END_REF] to describe the following counter-intuitive phenomenon: being V (x, y) a symmetric (smooth) two-well potential and V + δV an arbitrarily small (smooth) perturbation of it, then for 0 < ε ≪ 1, corresponding perturbed eigenfunctions will preferably localize in the largest well (see also [START_REF] Landsman | A Flea on Schrödinger's Cat[END_REF]).

Elementary "flea on the elephant" in 1D.

Given ε > 0, we start from the steady one-dimensional Schrödinger equation for some energy level E ε ,

-ε∂ xx ψ ε + V (x)ψ ε = E ε ψ ε (x),
x ∈ (0, 1),

where, for 0 < a < b < 1, the (square-wells) potential V (x) is piecewise constant,

V (x) = V χ x∈(a,b) + ∞ ( χ x<0 + χ x>1 ) , V > 0.
Dirichlet conditions correspond to infinite potential walls: ψ(0) = ψ(1) = 0,

k := k(E ε ) = √ E ε ε , κ := κ(E ε ) = √ V -E ε ε ,
describe bound-states, so that, = exp(κ(b -a)), then

A 2 cos 2 ka(κ 2 tan 2 ka -k 2 ) D 2 cos 2 k(1 -b)(κ 2 tan 2 k(1 -b) -k 2 ) = 1, (1.2)
so that, when a = 1 -b in (1.2), symmetry, D 2 = A 2 , easily follows. The asymmetric case is less straightforward in terms of algebra, even in 1D, see for instance [START_REF] Graffi | Tunnelling instability via perturbation theory[END_REF][START_REF] Jelic | The double well potential in quantum mechanics: a simple, numerically exact formulation[END_REF][START_REF] Landsman | A Flea on Schrödinger's Cat[END_REF][START_REF] Selg | Exactly solvable asymmetric double-well potentials[END_REF][START_REF] Song | Tunneling and energy splitting in an asymmetric double-well potential[END_REF] and [START_REF] Harrison | Quantum wells, wires and dots: theoretical and computational physics of semiconductor nano-structures[END_REF]Chapter 2]. Accordingly, numerical simulations of a 1D explicit example are displayed in Fig. 1.1. Finite-differences based on L-splines and borrowed from [START_REF] Gosse | L-Splines and Viscosity Limits for Well-Balanced Schemes Acting on Linear Parabolic Equations[END_REF] were set up in order to maximize both stability and accuracy in this context of a square-well potential: namely, we implemented both, a = 0.35, b = 0.65 on the left, a = 0.35, b = 0.675 on the right, with ε = 0.001, 2 7 grid points, and the following discretization,

r j± 1 2 = √ V (x j± 1 2 ) ε , - ε ∆x [ r j+ 1 2 sinh(r j+ 1 2 ∆x) ( ψ j+1 -cosh(r j+ 1 2 ∆x)ψ j ) - r j-1 2 sinh(r j-1 2 ∆x) ( cosh(r j-1 2 ∆x)ψ j -ψ j-1
) ]

= E ε ψ j .

2.

Gaussian eigenfunctions of the quadratic potential. Let a differential operator L ε (u) := -ε 2 ∆u + p(x)u have a potential with conditions [P]:

• p(x) ∈ C 2 (R N ), p(x) ≥ 0;

• p(x) has exactly two quadratic minima located in x i , for i = 1, 2 lim x→xi p(x) |x -x i | 2 = C i , for some constants C i .

More general potentials were considered in, e.g., [START_REF] Dancer | Semiclassical analysis of general second order elliptic operators on bounded domains[END_REF][START_REF] Harrison | Quantum wells, wires and dots: theoretical and computational physics of semiconductor nano-structures[END_REF][START_REF] Martinez | Effet tunnel et puits dégénérés[END_REF][START_REF] Reyes | An asymptotic eigenvalue problem for a Schrödinger type equation on domains with boundaries[END_REF]. In the whole space x ∈ R N , the eigenvalue problem for the harmonic oscillator reads,

p(x) = N 2 + |x| 2 4ε 2 , L ε (u) = -ε 2 ∆u + ( N 2 + |x| 2 4ε 2 ) u, ( 2
( |x| 2 4ε 2 ) w(x) inside L ε (u) = λu, ( 2.2)
yields a (steady) convection-diffusion equation,

Lε (w) := exp ( - |x| 2 4ε 2 ) L ε (u) = ( -ε 2 ∆w -x • ∇w ) = λw. (2.3)
Namely, in terms of the new variable w(x), the spectral problem (2.1) reads

-ε 2 ∆w(x) -x • ∇w(x) = λw(x), x ∈ R N .
(2.4)

Convection-diffusion problem (2.4) recasts as a drift-diffusion (conservative) one,

-ε 2 ∇ • (K ε (x)∇w) -λwK ε (x) = 0, K ε (x) = exp ( |x| 2 2ε 2
) , (2.5) so that nontrivial solutions to (2.4) are critical points of the following functional:

f → E ε (f ) = ε 2 2 ∫ R N |∇f | 2 K ε (x) dx - λ 2 ∫ R N |f (x)| 2 K ε (x) dx, (2.6)
for f belonging, for ε fixed, to the following "weighted Sobolev space",

H 1 (K ε ) = { f : R N → R : ∫ R N ( |f | 2 + ε 2 |∇f | 2 ) K ε (x) dx < +∞ } . (2.7)
Proposition 1. For K ε (x) being as in (2.5) and H 1 (K ε ) given by (2.7), the continuous embedding

H 1 (K ε ) ⊂ L 2 (K ε ) is compact, H 1 (K ε ) ⊂⊂ L 2 (K ε ) := { f : R N → R : ∫ R N |f | 2 K ε (x) dx < +∞ } .
Proof. The proof contained in [START_REF] Escobedo | Variational problems related to self-similar solutions of the heat equation[END_REF] applies, based on the following lemma:

Lemma 1. For any f ∈ C 1 c (R N ), the Poincaré inequality holds: ∫ R N |f | 2 ( N 2 + |x| 2 4ε 2 ) K ε (x) dx ≤ ε 2 ∫ R N |∇f | 2 K ε (x) dx. (2.8)
The differential operator Lε in (2.3) is set in the domain

D( Lε ) = { f ∈ H 1 (K ε ); Lε f ∈ L 2 (K ε ) } := H 2 (K ε ), (2.9)
in which it is self-adjoint and, by Proposition 1, D( Lε ) ⊂⊂ L 2 (K ε ) compactly embedded. This implies that Lε presents an increasing sequence of eigenvalues (λ k ) k≥0 , where λ 0 > 0 is guaranteed by inequality (2.8). Moreover, for any m ≥ 0,

if f ∈ L 2 (K ε ), then (1 + |ξ| 2 ) m 2 f ∈ L 2 (R N ), so that the Fourier transform f is smooth, ∀m ≥ 0, f ∈ H m (R N ), f ∈ C ∞ (R N ).
It remains to take the Fourier transform of (2.4),

ε 2 |ξ| 2 ŵ(ξ)+N ŵ(ξ)+ξ•∇ ŵ(ξ) = λ ŵ(ξ), (x j ∂ xj w)(ξ) = -ŵ(ξ)-ξ j ∂ ξj ŵ(ξ). (2.10)
Another change of variables yields,

ξ • ∇v(ξ) = (λ -N )v(ξ), v(ξ) = exp(ε 2 |ξ| 2 2 ) ŵ(ξ), thus, v(ξ) is a polynomial function of degree (λ -N ) = k -1, k ∈ N, namely v(ξ) = P k-1 (ξ), P k-1 (ξ) homogeneous polynomial of degree k -1,
and solutions to (2.10) satisfy ŵ(ξ) = exp(-ε 2 |ξ| 2 /2)P k-1 (ξ), see also [START_REF] Escobedo | Variational problems related to self-similar solutions of the heat equation[END_REF].

Theorem 2.1. The eigenvalues of Lε in (2.3), where D( Lε ) ⊂⊂ H 1 (K ε ), are

λ k = N + k -1, k ∈ N, (2.11)
with the ground-state and its related eigenfunctions,

ϕ ε 0 (x) := exp ( - |x| 2 2ε 2 ) , D α ϕ ε 0 (x), α = (α 1 , • • • , α N ), |α| = k -1.

Perturbation analysis of the two wells potential.

Hereafter, double well potentials with identical and different wells are considered, in order to show the role played by symmetry. In the symmetric case with ε ≪ 1, eigenfunctions accumulate equally in both wells; oppositely, for asymmetric potentials, the ground state fills mostly the widest well (see [START_REF] Holcman | Singular perturbation for the first eigenfunction and blow-up analysis[END_REF] for more of a differential geometry approach). We qualitatively analyze effects of symmetry breaking in the potential shape p(x):

• accumulation of solutions in the widest well;

• the role played by the position of the perturbation;

• how a perturbation affects the spectrum of the original problem. Consider the spectral problem associated with

D(L ε ) ∋ u → L ε (u) = -ε 2 ∆u + p(x) 4ε 2 u, where p(x) satifies [P], (3.1)
the following statements hold true:

• the ground state accumulates in the widest well;

• potential perturbations' affect the spectral problem as much as they are close to the wells; • eigenvalues and eigenfunctions' modifications occur according to the way the perturbation acts on the eigenfunctions of the original spectral problem.

Accumulation in the widest

well. Yet, pick 0 < ε ≪ 1, and consider the spectral problem (3.1), mostly when p(x) is a C 2 (R N ) has two wells at the same height but with possibly different widths. Its eigenvalues form an increasing sequence, whose minimum λ 1 is given by Rayleigh's quotient,

λ 1 = inf u∈D(L ε ) ∫ R N ε 2 |∇u| 2 + p(x) 4ε 2 |u| 2 dx ∫ R N |u| 2 dx . (3.2)
Defining the measure dp = p(x) dx, Rayleigh's quotient rewrites

λ 1 = inf u∈D(L ε ) ∫ R N ε 2 |∇u| 2 dx + 1 4ε 2 ∫ R N |u| 2 dp ∫ R N |u| 2 dx .
Lemma 2. Let x 0 be any point located at any positive and fixed distance from 0. Let d > 0: the first eigenvalue λ0 associated with the operator,

L ε = -ε 2 ∆ + p(x) 4ε 2 , p(x) = |x| 2 χ B R (0) + d|x -x 0 | 2 χ B R (x0) , (3.3)
satisfies, for a positive constant value c, the following expansion: .4) The shift between eigenvalues (3.4) and (2.11) comes from the factor N 2 in (2.1). Proof. The Rayleigh quotient reads:

λ0 - N 2 = O ( R 2 ε 2 ) exp ( - cR 2 ε 2 ) . ( 3
λ0 = min ψ∈H 1 (R 2 ) ∫ R 2 ε 2 |∇ψ(x)| 2 + |ψ(x)| 2 4ε 2 ( |x| 2 χ B R (0) + d|x -x 0 | 2 χ B R (x0) ) dx ∫ R 2 |ψ(x)| 2 dx . (3.5) From Section 2, ϕ 0 (x) = exp(- |x| 2 4ε 2 ) solves -ε 2 ∆ϕ 0 + |x| 2 4ε 2 ϕ 0 = N 2 ϕ 0 ,
where

N 2 = λ 0 = min ψ ∈H 1 (R 2 ) ∫ R 2 ε 2 |∇ψ(x)| 2 + |x| 2 4ε 2 |ψ(x)| 2 dx ∫ R 2 |ψ(x)| 2 dx = ∫ R 2 ε 2 |∇ϕ 0 (x)| 2 + |x| 2 4ε 2 |ϕ 0 (x)| 2 dx ∫ R 2 |ϕ 0 (x)| 2 dx . (3.6)
The explicit expression of λ 0 , provided by the Rayleigh quotient, can be used in λ0 in (3.5), in order to get the desired approximation. More precisely, by substituting ϕ 0 (x) in the previous expression, the upper bound is given by

λ0 ≤ N 2 + 1 2πε 2 ( ∫ R 2 |x| 2 exp(-|x| 2 2ε 2 ) 4ε 2 (χ B R (0) -1) dx + ∫ R 2 d|x -x 0 | 2 χ B R (x0) exp(-|x| 2 2ε 2 ) 4ε 2 dx
) .

On the other hand, the lower bound,

λ0 = min ψ ∈H 1 (R 2 ) ∫ R 2 ε 2 |∇ψ(x)| 2 + |ψ(x)| 2 4ε 2 ( |x| 2 χ B R (0) + d|x -x 0 | 2 χ B R (x0) ) dx ∫ R 2 |ψ(x)| 2 dx = min ψ ∈H 1 (R 2 ) ( ∫ R 2 ε 2 |∇ψ(x)| 2 + |x| 2 4ε 2 |ψ(x)| 2 dx ∫ R 2 |ψ(x)| 2 dx + ∫ R 2 |ψ(x)| 2 |x| 2 4ε 2 (χ B R (0) -1) dx ∫ R 2 |ψ(x)| 2 dx + ∫ R 2 |ψ(x)| 2 d|x -x 0 | 2 4ε 2 χ B R (x0) dx ∫ R 2 |ψ(x)| 2 dx ) ≥ min ψ ∈H 1 (R 2 ) ∫ R 2 ε 2 |∇ψ(x)| 2 + |x| 2 4ε 2 |ψ(x)| 2 dx ∫ R 2 |ψ(x)| 2 dx - ( ∫ R 2 | ψ(x)| 2 |x| 2 4ε 2 (χ B R (0) -1) dx ∫ R 2 | ψ(x)| 2 dx + ∫ R 2 | ψ(x)| 2 d|x -x 0 | 2 4ε 2 χ B R (x0) dx ∫ R 2 | ψ(x)| 2 dx ) ,
where the last inequality holds for ψ(x) ∈ H 1 (R 2 ), being all the addends in the previous line positive. By using again (3.6) and setting ψ(x) = ϕ 0 (x), one gets

λ0 ≥ N 2 - 1 2πε 2 ( ∫ R 2 |x| 2 exp(-|x| 2 2ε 2 ) 4ε 2 (χ B R (0) -1) dx + ∫ R 2 d|x -x 0 | 2 χ B R (x0) exp(-|x| 2 2ε 2 ) 4ε 2 dx
) .

Passing to polar coordinates,

1 2πε 2 ∫ R 2 |x| 2 4ε 2 (Id -χ B R (0) )|ϕ 0 (x)| 2 dx = 1 8πε 4 ∫ 2π 0 dθ ∫ ∞ R ρ 3 exp(- ρ 2 2ε 2 ) dρ = (R 2 + 2ε 2 ) 4ε 2 exp(- R 2 2ε 2 ), ∫ R 2 exp(- |x| 2 2ε 2 ) dx = 2πε 2 .
Remark 1. Lemma 2 implies that λ 0 = N 2 is a good approximation for the expression of the first eigenvalue when the potential only has two wells. Performing exactly the same computations with

ϕ d 0 (x) = exp(-d|x-x0| 2 4ε 2
), one gets λ d 0 = N d 2 . However, in this case the error term in the expansion is bigger, meaning that this approximation is less accurate than the one provided by the widest Gaussian.

The parameter d > 0 in (3.3) controls the convexity of the well centered in x 0 . If the potential p(x) has different wells, the ground state is known to accumulate only in one well, see [START_REF] Simon | Semiclassical analysis of low lying eigenvalues. IV. The flea on the elephant[END_REF][START_REF] Helffer | Multiple wells in semi-classical limit[END_REF][START_REF] Holcman | Singular perturbation for the first eigenfunction and blow-up analysis[END_REF]. From Lemma 2 (and Remark 1), the expression of the

first eigenvalue λ0 = N 2 + O ( R 2 ε 2 ) exp ( - cR 2 ε 2
) is reached by ϕ 0 (x). This suggests that the accumulation well should be the one provided by |x| 2 4ε 2 . Definition 1 (see [START_REF] Ralston | Approximate eigenfunctions of the Laplacian[END_REF][START_REF] Ralston | On the construction of quasi-modes associated with stable periodic orbits[END_REF]). For a fixed λ ∈ R, a quasi-eigenfunction is any function u(x) ∈ D(L ε ) such that the following "error function",

Err(u) := -ε 2 ∆u + p(x) 4ε 2 u -λu,
satisfies, for any C(ε) exponentially decreasing, the inequality:

∥Err(u)∥ L 1 (R N ) ≤ C(ε), ε ≪ 1.
In the next Lemma, we show that ϕ 0 (x) is a quasi-eigenfunction of L ε in (3.3), in the sense of Definition 1. Identical computations bring that

ϕ d 0 (x) := exp ( - d|x -x 0 | 2 4ε 2
) is a quasi-eigenfunction of the operator defined in (3.3), too. However, it turns out that ϕ 0 (x) is a better approximation of the ground-state of (3.3) than ϕ d 0 (x). The meaning of the last sentence is clarified below.

Lemma 3. Under the hypotheses of Lemma 2, let us fix d > 1.

Then

∥Err(ϕ 0 )∥ L 1 (R N ) < ∥Err(ϕ d 0 )∥ L 1 (R N ) , ϕ d 0 (x) = exp(- d|x -x 0 | 2 4ε 2
).

(3.7)

Proof. Thanks to Lemma 2, in terms of the Rayleigh quotient,

λ0 = N 2 + O ( R 2 ε 2 ) exp ( - cR 2 ε 2
) , is reached by ϕ 0 (x). This way, by comparing the two problems,

-ε 2 ∆ϕ 0 + p(x) 4ε 2 ϕ 0 = N 2 ϕ 0 + Err(ϕ 0 ), -ε 2 ∆ϕ d 0 + p(x) 4ε 2 ϕ d 0 = N 2 ϕ d 0 + Err(ϕ d 0 ), with p(x) in (3.3),
where

Err(ϕ 0 ) = [ |x| 2 4ε 2 (χ B R (0) -Id) + d|x -x 0 | 2 4ε 2 χ B R (x0) ] ϕ 0 , Err(ϕ d 0 ) = [ d|x -x 0 | 2 4ε 2 (χ B R (x0) -Id) + |x| 2 4ε 2 χ B R (0) + N (d -1) 2 ] ϕ d 0 ,
and integrating in space, one gets (3.7). After proving that • the best approximation for the first eigenvalue associated with

L ε in (3.3), λ0 ≈ N 2 , is provided by ϕ 0 (x) (Lemma 2), • ϕ 0 (x)
is the best quasi-eigenfunction (Lemma 3), we finally show that the solution to the spectral problem accumulates in |x| 2 4ε 2 , the widest well.

Proposition 2. Le p(x) be a double well potential satisfying conditions [P]. Let x 1 , x 2 be the centers of the two potential wells, and x 1 be the widest one. The (normalized) solution u(x) ∈ H 1 (K ε ) to the spectral problem associated with (3.3) accumulates in the widest well. More precisely, for a fixed ε > 0, let B R (x i ) be the disk of radius R > 0 and center x i , i = 1, 2, there exists ε 0 > 0 such that,

∀ε ≤ ε 0 , ∫ B R (x1) |u(x)| 2 dp > exp ( c|x 1 -x 2 | 2 ε 2 ) ∫ B R (x2) |u(x)| 2 dp.
Proof. Without loss of generality, we consider L ε in (3.3). The first eigenfunction is well approximated by ϕ 0 (x). This way,

∫ B R (0) |ϕ 0 (x)| 2 dp = ∫ B R (0) |ϕ 0 (x)| 2 |x| 2 4ε 2 dx = 2π ∫ R 0 exp(- ρ 2 2ε 2 ) ρ 3 dρ; ∫ B R (x0) |ϕ 0 (x)| 2 dp = ∫ B R (x0) |ϕ 0 (x)| 2 d|x -x 0 | 2 2ε 2 dx = d exp(- |x 0 | 2 2ε 2 ) ∫ 2π 0 dθ ∫ R 0 exp( -ρ 2 -2ρx 0 cos θ -2ρy 0 sin θ 2ε 2 ) ρ 3 dρ ≥ 2πd exp(- |x 0 | 2 ε 2 ) ∫ R 0 exp(- ρ 2 ε 2 ) ρ 3 dρ,
which ends the proof.

Distance from the wells.

Given a potential well and a localized perturbation: does the distance between each other affect the spectral problem ? If wells are far enough from each other, this is a local question which is addressed by perturbing the harmonic potential. Hereafter, ε = 1 and the term N 2 in (2.1) is ignored,

L(u) = -∆u + |x| 2 4 u, ϕ 0 (x) = exp(- |x| 2 4 ). (3.8)
Let a small perturbation be given by χ B R (x0) (x), the characteristic function of a disk centered in x 0 of radius R > 0, and consider a perturbed spectral problem,

L R (u) = -∆u + |x| 2 4 u + χ B R (x0) u = λu. ( 3.9)
The ground-state ϕ 0 (x) of (3.8) can be seen as an quasi-eigenfunction for the perturbed problem (3.9), as it almost satisfies the spectral equation for L R (u),

-∆ϕ 0 + |x| 2 4 ϕ 0 + χ B R (x0) ϕ 0 ≈ λϕ 0 ,
with an error

Err(x 0 ) = χ B R (x0) ϕ 0 (x) = χ B R (x0) exp(- |x| 2 4
).

In one dimension, N = 1, such an error is easy to quantify, for instance in L 1 ,

x 0 → ∥Err(x 0 )∥ L 1 (R) = ∫ x0+R x0-R exp(- |x| 2 4 ) dx.
so that its variations with respect to x 0 ∈ R satisfy:

d dx 0 ∥Err(x 0 )∥ L 1 (R) = exp(- x 0 + R 2 2 ) -exp(- x 0 -R 2 2 ) = { < 0, (x 0 > 0), > 0, (x 0 < 0).
This derivative shows that,

• the position x 0 = 0 is a local maximum for ∥Err(x 0 )∥ L 1 (R) ;

• the perturbation's effects on (3.8) decrease with both |x 0 | and R. This elementary argument can be extended to N dimensions; precise results on the role of the position of the perturbation are given in, e.g. [START_REF] Simon | Semiclassical analysis of low lying eigenvalues. IV. The flea on the elephant[END_REF].

Variations of the spectral components.

Coming back to a general potential p(x) in (3.1), we consider L(u) in (3.8), where λ, u are any eigenvalue and normalized eigenfunction. Let M be any linear bounded operator and δ ≪ 1 a constant small enough: again, we treat λ as a quasi-eigenvalue for the perturbed operator,

Lu + δM u = λu + Err δ (u),
where Err δ (u) = δM u.

(3.10)

We take the derivative of (3.10) w.r.t. δ, calculated in δ = 0,

Lu ′ + M u = λ ′ u + λu ′ + O(∥M u∥),
where

• ′ := d dδ δ=0
Namely,

(L -λ)u ′ = (λ ′ -M )u + O(∥M u∥). (3.11)
Since λ is an eigenvalue of L, associated with the eigenfunction u, then u ∈ Ker(L-λ).

Fredholm's Alternative implies that there exist solutions u ′ to (3.11) such that

(λ ′ -M )u + O(∥M u∥) ⊥ Ker(L -λ), i.e., denoting by < •, • > the L 2 scalar product, < (λ ′ -M )u, u >= O(∥M u∥), ∥u∥ 2 =< u, u >= 1, then, λ ′ = d dδ δ=0 λ(δ) =< M u, u > +O(∥M u∥), (3.12) namely |λ ′ | ≤ ∥M u∥∥u∥ + O(∥M u∥) = ∥M u∥ + O(∥M u∥),
so that the perturbation δM affects eigenvalues according to the way M affects the original eigenfunction u. Yet, we look at the variations of this eigenfunction by taking the derivative with respect to δ of the scalar product ∥u∥ 2 =< u, u >= 1.

1 2

d dδ δ=0 (∥u∥ 2 ) =< u, u ′ >= 0, i.e. u ′ ⊥ u. Orthogonality u ′ ⊥ u and (L -λ)u ′ ⊥ u, along with Fredholm's Alternative imply u ′ is proportional to (λ ′ -M )u + O(∥M u∥) = (L -λ)u ′ ,
so that, locally for δ ≃ 0, ∥u ′ ∥ ≤ C∥M u∥ for a given constant value C(λ).

4. Two-dimensional "Steklov numerical scheme". In this section, a derivation is recalled from [START_REF] Gosse | Viscous equations treated with L-splines and Steklov-Poincaré operator in two dimensions[END_REF], in order to establish several properties in close relation with the original problem of the "flea on the elephant". Hereafter, we shall always work on a uniform, Cartesian computational grid, with ∆x = ∆y. 4.1. Derivation of the numerical process. Consider the stationary, strictly elliptic and coercive, two-dimensional problem,

u n i,j u n i+1,j u n i,j+1 u n i,j-1 u n i-1,j ⊗ ⊗ ⊗ ⊗ ᾱi-1 2 ,j-1 2
{ -∆u + α 2 (x, y)u = 0, (x, y) ∈ Ω, u(x, y) = g(x), (x, y) ∈ ∂Ω, (4.1)
where Ω is a polygonal domain of R 2 . The numerical simulations, carried out with the "Steklov scheme" [START_REF] Gosse | Viscous equations treated with L-splines and Steklov-Poincaré operator in two dimensions[END_REF], and presented below mostly focus on the spectral analysis associated with the strictly elliptic operator L(u) := -∆u + α 2 (x, y)u. Accordingly, we shall work on the square (Lipschtiz) domain Ω = (0, 1) 2 ⊂ R 2 , so that the typical regularity for u should be at least H 2 (Ω), see [START_REF] Dahlke | Besov regularity for elliptic boundary value problems[END_REF][START_REF] Grisvard | Elliptic problems in non smooth domains[END_REF]. We refer to Appendix A for elementary regularity results for problems like (4.1) in a square domain: in particular, when the compatibility conditions (A.2) hold at each corner, the Hölder regularity of inhomogeneous Dirichlet data g passes to the solution u, see Theorem A.2.

The "Steklov scheme", see [START_REF] Gosse | Viscous equations treated with L-splines and Steklov-Poincaré operator in two dimensions[END_REF][START_REF] Gosse | L-Splines and Viscosity Limits for Well-Balanced Schemes Acting on Linear Parabolic Equations[END_REF] works in the following way: (see Fig. 4.1)

• at each "node", (x i-1 2 , y j-1 2 ), the potential is "frozen", so that

ᾱ2 i-1 2 ,j-1 2 := α 2 (x i-1 2 , y j-1 2 ), i, j ∈ Z 2 ; • in each disk D R (i -1 2 , j -1 2 ), centered at a node, D R (i - 1 2 , j - 1 2 ) = { |x -x i-1/2 | 2 + |y -y j-1/2 | 2 ≤ R 2 } , R = ∆x √ 2 ,
the following problem is explicitly solved in polar coordinates,

{ -∆v + ᾱ2 i-1 2 ,j-1 2 v = 0, (x, y) ∈ D R , v = g on the boundary, (x, y) ∈ C R := ∂D R , ( 4.2)
by means of Fourier-Bessel series involving "modified Bessel functions", I n (•).

Note that, as shown in Figure 4.1, in the discrete framework we only consider exactly three values of the trace of v at the boundary C R (i -1 2 , j -1 2 ), i.e., v(x i , y j ), v(x i-1 , y j ), v(x i , y j-1 ). However, at the moment let us consider the BVP (4.2) at the formal level, keeping in mind that precise trace estimates on the discrete approximation of g in (4.2) will be discussed in Section 4.2.

• The exact solution of the stationary BVP (4.2) reads,

v(r, θ) = A 0 I 0 (ᾱr) + ∑ n∈N (A n cos nθ + B n sin nθ)I n (ᾱr), r ≤ R, (4.3)
where ᾱ = ᾱi-1 2 ,j-1 2 , and coefficients are determined by the boundary data prescribed on the circle C R (i - 1 2 , j -1 2 ). More precisely, if we are given a boundary condition g ∈ H s (0, T = 2πR), then it admits a Fourier series,

g(x) = a 0 + ∑ n∈N a n cos( 2πn x T ) + b n sin( 2πn x T ),
so that, setting r = R in (4.3), and by uniqueness, we have

A 0 = a 0 I 0 (ᾱR) , A n = a n I n (ᾱR) , B n = b n I n (ᾱR) , ( 4.4)
meaning that the solution to (4.2) is completely known. • At this point, the main idea to derive the Steklov scheme, see [START_REF] Gosse | Viscous equations treated with L-splines and Steklov-Poincaré operator in two dimensions[END_REF][START_REF] Gosse | L-Splines and Viscosity Limits for Well-Balanced Schemes Acting on Linear Parabolic Equations[END_REF], is based on the discretization of the normal derivative of the solution v(r, θ) in (4.3), obtained by means of the Steklov-Poincaré operator. Since the domain is circular, the normal derivative reduces to the radial one, and the Steklov-Poincaré operator can be made explicit,

1 ᾱ ∂v ∂r (R, θ) = A 0 I 1 (ᾱR) + ∑ n∈N (A n cos nθ + B n sin nθ) I n-1 + I n+1 2 (ᾱR),
and, by using identities (4.4), this rewrites,

1 ᾱ ∂v ∂r (R, θ) = a 0 I 1 I 0 (ᾱR) + ∑ n∈N (a n cos nθ + b n sin nθ) I n-1 + I n+1 2I n (ᾱR). (4.5)
• In practice, as anticipated before, we never have all the Fourier coefficients at hand, because there are at most four discrete (grid) numerical values,

P k = R(cos θ k , sin θ k ), 0 ≤ k ≤ 2, θ k = kπ 2 , (4.6) on each circle C R (i ± 1 2 , j ± 1 2 ), so the former series are truncated at n ≤ 1, 1 ᾱ ∂v ∂r (R, θ) = a 0 I 1 I 0 (ᾱR) + (a 1 cos θ + b 1 sin θ) I 0 + I 2 2I 1 (ᾱR). (4.7)
The explicit expressions of coefficients a 0 , a

P 0 = v(R, θ = 0) = u n i,j = A 0 I 0 (ᾱR) + A 1 I 1 (ᾱR), P 1 = v(R, θ = π 2 ) = u n i-1,j = A 0 I 0 (ᾱR) + B 1 I 1 (ᾱR), P 2 = v(R, θ = -π 2 ) = u n i,j-1 = A 0 I 0 (ᾱR) -B 1 I 1 (ᾱR), yield:            a 0 = A 0 I 0 (ᾱR) = u n i-1,j + u n i,j-1 2 , a 1 = A 1 I 1 (ᾱR) = u n i,j - u n i-1,j + u i,j-1 2 , b 1 = B 1 I 1 (ᾱR) = u n i-1,j -u n i,j-1
2 .

(4.8)

Inserting (4.8) into (4.7),

1 ᾱ ∂v ∂r (R, θ) = { u n i,j I 0 + I 2 2I 1 cos θ + u n i-1,j [ I 1 2I 0 + I 0 + I 2 4I 1 (sin θ -cos θ) ] + u n i,j-1 [ I 1 2I 0 - I 0 + I 2 4I 1 (sin θ + cos θ) ]} (ᾱR).
(4.9) In the end, by using the regularity of the solution to the considered BVP, the Steklov scheme is derived by balancing the four normal (radial) derivatives

for each circle C R (i ± 1 2 , j ± 1 2 ), ∂v ∂r (R, 0)(ᾱ i-1 2 ,j-1 2 R) + ∂v ∂r (R, π 2 )(ᾱ i-1 2 ,j+ 1 2 R) + ∂v ∂r (R, π)(ᾱ i+ 1 2 ,j+ 1 2 R) + ∂v ∂r (R, 3π 2 )(ᾱ i+ 1 2 ,j-1 2 R) = 0, (4.10)
and so the following numerical scheme is obtained,

u i,j { F(ᾱ i-1 2 ,j-1 2 ) + F(ᾱ i+ 1 2 ,j-1 2 ) + F(ᾱ i+ 1 2 ,j+ 1 2 ) + F(ᾱ i-1 2 ,j+ 1 2 ) } + u i-1,j { G(ᾱ i-1 2 ,j-1 2 ) + G(ᾱ i-1 2 ,j+ 1 2 ) } + u i,j-1 { G(ᾱ i-1 2 ,j-1 2 ) + G(ᾱ i+ 1 2 ,j-1 2 ) } + u i+1,j { G(ᾱ i+ 1 2 ,j-1 2 ) + G(ᾱ i+ 1 2 ,j+ 1 2 ) } + u i,j+1 { G(ᾱ i+ 1 2 ,j+ 1 2 ) + G(ᾱ i-1 2 ,j+ 1 2 ) } = 0, (4.11)
where

F(α) = α I 0 + I 2 2I 1 (αR), G(α) = H(α) -F(α) 2 , H(α) = α I 1 I 0 (αR).
Standard results for I n (•) imply that both F(α) and G(α) behave like O(1/R) when ᾱR ≪ 1 and O(1) when ᾱR ≫ 1. Moreover, (4.11) was proved to be consistent with (4.1) and monotone (hence, positivity-preserving) in [START_REF] Gosse | Viscous equations treated with L-splines and Steklov-Poincaré operator in two dimensions[END_REF]. Yet, consider a N × N two-dimensional grid (N 2 internal points), and given the 2D array of values u i,j , define the following one-dimensional vector,

U N = [ u 1,1 , u 2,1 , • • • , u N,1 , u 1,2 , • • • , u N,2 , • • • • • • • • • , u 1,N , • • • , u N,N ]
. (4.12)

Identity (4.11) recasts in matrix form like,

P N U N = B, P N a (N 2 × N 2 ) matrix, (4.13)
where B stands for discretized boundary data. Lemma 4. Matrix P N in (4.13) is a symmetric M -matrix. Proof. Let C i,j (S i,j ) be the general coefficient of the term u i,j , where S i,j indicates in which grid's point x i,j relation (4.11) is used. It is straightforward to check that

• C i+1,j (S i,j ) = C i,j (S i+1,j ) = G(ᾱ i+1/2,j-1/2 ) + G(ᾱ i+1/2,j+1/2 ), i, j ≥ 1; • C i,j+1 (S i,j ) = C i,j (S i,j+1 ) = G(ᾱ i+1/2,j+1/2 ) + G(ᾱ i-1/2,j+1/2 ), i, j ≥ 1; • C i-1,j (S i,j ) = C i,j (S i-1,j ) = G(ᾱ i-1/2,j-1/2 ) + G(ᾱ i-1/2,j+1/2 ), i ≥ 2, j ≥ 1; • C i,j-1 (S i,j ) = C i,j (S i,j-1 ) = G(ᾱ i+1/2,j-1/2 ) + G(ᾱ i-1/2,j-1/2 ), i ≥ 1, j ≥ 2.
A sufficient condition for P N being a M -matrix is that:

• its principal diagonal is strictly positive,

• other entries are negative or null,

• it is diagonally-dominant. All these properties follow from (4.11) because, for any n, I n (x) ≥ 0 for x ≥ 0 and they are monotonically increasing.

Accordingly, P N has a positive inverse, uniformly in ∥α 2 ∥ ∞ R. A useful estimate is the minimum of its eigenvalues, because ∥(P N ) -1 ∥ 2 ≤ 1/ min(µ ℓ ). Such a number can be explicitly computed (see [11, pages 179-181] and [START_REF] Val | On mean value solutions for the Helmholtz equation on square grids[END_REF]) when the potential is a constant, α 2 ∈ R + : being

P N block-diagonal (with N × N blocks),    µ ℓ = 4 F(α) + 8 G(α) cos(π∆x), (P N ) -1 2 ≤ 1 4(F(α) + 2G(α) cos(π∆x)) . ( 4.14)
Before studying the truncation errors of (4.11), we stress that such a monotone discretization of (4.1) is a good candidate for problems which develop sharp layers. Indeed, the modified Bessel functions I n display an exponential behavior at infinity, so that (4.11) can be fairly considered a "two-dimensional exponential-fit scheme".

Truncated series and "localized sampling".

As mentioned before, in practice, it is necessary to truncate the Fourier-Bessel series of the exact solution v(r, θ), because only 4 grid points are available on each circle C R (i ± 1 2 , j ± 1 2 ) (instead of boundary data of co-dimension 1), and for the present scheme, we use only 3. In this section, we provide an estimate of the approximation due to this 3-points discretization on each one of the four circles of the stencil, by comparison with the 4points discretization, which results to be an application of the trapezoidal integration rule. Hereafter, consider h(θ) ∈ H s (0, 2π), 2π-periodic, that is approximated by two different trigonometric polynomials,

{ h 4 (θ) = a 0 + a 1 cos θ + b 1 sin θ + a 2 cos 2θ, h 3 (θ) = α 0 + α 1 cos θ + β 1 sin θ,
both of them being determined by imposing,

{ h 4 (0) = c, h 4 ( π 2 ) = a, h 4 (π) = d, h 4 (-π 2 ) = b, h 3 (0) = c, h 3 ( π 2 ) = a, h 3 (-π 2 ) = b.
Hence, given any 4 generic points on a circle C R , coefficients read:

a 0 = 1 4 (a + b + c + d), b 1 = 1 2 (a -b), a 1 = 1 2 (c -d), a 2 = c + d 4 - a + b 4 ,
so that,

a 0 = O(1), a 1 , b 1 = O(R), a 2 = O(R 2 ),
along with

α 0 = 1 2 (a + b), α 1 = c - 1 2 (a + b), β 1 = 1 2 (a -b).
The polynomial h 3 is sometimes referred to as to a "localized sampling" because, oppositely to h 4 , it interpolates only the 3 points which are the closest to the location where the radial derivative is meant to be computed. Despite their coefficients are distinct, their difference h 4 -h 3 depends only on a 2 : indeed,

(h 4 -h 3 )(θ) = (a 0 -α 0 ) + (a 1 -α 1 ) cos θ + a 2 cos 2θ = a 2 (1 -2 cos θ + cos 2θ). (4.15)
The fact that the gap h 3 -h 4 is like a 2 , a "trapezoidal approximation", is essential.

Local truncation error.

If the circular trace of a 2D function u(x, y), has a very small "mixed derivative", |∂ 2 xy u| ≪ 1, then a 2 ≃ 0. Accordingly, define

g(x) = h(Rθ), g : (0, T = 2πR) → R, g is T -periodic.
It isn't easy to estimate the (curvilinear) derivatives of g with respect to the abscissa x because of the circle's curvature, which equals 1 R :

dg dx = 1 R dg dθ = -∂ x u(R cos θ, R sin θ) sin θ + ∂ y u(R cos θ, R sin θ) cos θ, d 2 g dx 2 = - 1 R ( ∂ x u(R cos θ, R sin θ) cos θ + ∂ y u(R cos θ, R sin θ) sin θ ) + ...,
hence a "bad term" 1 R ∂u ∂r appears in the second derivative of g. This complicates the control of errors produced when approximating h with both h 4 and h 3 . To estimate the noise induced by the "localized sampling" of only 3 grid points,

∥g -g 3 ∥ L 2 (0,T ) ≤ ∥g -g 4 ∥ L 2 (0,T ) + ∥g 4 -g 3 ∥ L 2 (0,T ) ,
and we proceed thanks to the elementary observation:

∥g 4 -g 3 ∥ 2 L 2 (0,T ) = |a 2 | 2 ∫ T 0 1 -2 cos( x R ) + cos( 2x R ) 2 dx = T (1 + 4 + 1)|a 2 | 2 . Lemma 5. Let u ∈ H s (R 2 ), s ≥ N + 1 2
, and f : (0, T = 2πR) → R its trace on a circle of radius R > 0. Define the "N -points approximation" of its average f (0) as, Proof. See Appendix B. Remark 2. Theorem 7 in [START_REF] Birkhoff | Optimal few points discretizations of linear source problems[END_REF] states that the LTE of the classical 5-points scheme for Laplace's equation1 , being the limit ᾱ → 0 of (4.11, is of order ∆x 4 . Its proof draws on the harmonic polynomial, the real part of (x + iy) 4 -∆x 4 , see Fig. 4.2,p(x,y

fN (0) := 1 N N -1 ∑ j=0 f (j T N ), f (0) = 1 T ∫ T 0 f (x)dx = 1 2π ∫ 2π 0 f (θ)dθ, θ = x R , then, either • the method is exact, f (0) = fN (0) if f (k) ≡ 0 when |k| ≥ N ; • or its error satisfies | f (0) -fN (0)| ≤ ∑ k∈Z * | f (N k)| ≤ O(R N ).
) = x 4 -6x 2 y 2 + y 4 -∆x 4 , (x, y) ∈ (-∆x, ∆x) 4 ,
for which p(±∆x, ±∆x) = 0, but p(0, 0) = ∆x 4 ̸ = 0. Its trace on the circle centered at the origin, of radius ∆x, C(0, ∆x), is displayed on Fig. 4.2 (right) and reads

p(∆x cos θ, ∆x sin θ) = ∆x 4 ( cos 4 θ -6 cos 2 θ sin 2 θ + sin 4 θ -1 ) = ∆x 4 ( cos(4θ) -1
) .

Lemma 5 applies to that simple case because, for any harmonic function u ∈ H

9 2 (R 2 ), u(0, 0) = 1 2πR ∫ C(0,R) u(x, y) dℓ = 1 4 ( u(R, 0) + u(0, R) + u(-R, 0) + u(0, -R) ) + O(R 4 ),
and picking R = ∆x, one recovers both second-order accuracy for general solutions, and the exactness of the method if no frequencies higher than |k| = 3 are present. Such bounds apply to "discrete weighted means" [START_REF] Gartland | Discrete weighted mean approximation of a model convection-diffusion equation[END_REF] and "tailored methods" [START_REF] Han | A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain[END_REF], too. The discrepancy between boundary data which involves the exact Fourier coefficients ĝ(k) and the one involving

α 0 , α 1 , β 1 is: (since g is real, ĝ(k) = ĝ(-k) * ) |ĝ(0) -α 0 | 2 + |2ℜ(ĝ(1)) -α 1 | 2 + i ( ĝ(1) -ĝ(-1) ) -β 1 2 + ∑ |k|>1 |ĝ(k)| 2 .
Theorem 4.1. Consider the problem (4.1) with the coefficient α ≡ ᾱ being constant, and a corresponding solution u(x, y):

• if u ∈ C 4 , the global L 2 error of (4.11) is second-order as ∆x → 0;

• if ĝ(k) = 0 for |k| ≥ 2, the approximation (4.10) matches the exact one (4.5). Proof. We establish each statement independently for the sake of clarity:

• Given a point x i , y j , the local truncation error (LTE) is the discrepancy τ i,j := τ (x i , y j ) which remains when pointwise values of u, the exact solution to (4.1) are inserted into the scheme (4.11). By grouping terms, it comes,

0 = F(ᾱ) ( 4u i,j -u i±1,j±1 ) + H(ᾱ)u i±1,j±1 = F(ᾱR) ( -∆x 2 ∆u(x i , y j ) + O(∆x 4 ) ) + 4H(ᾱ) ( u(x i , y j ) + O(∆x 2 )
) , so that, by performing Taylor expansions of I n (ᾱR) with 0 ≤ ᾱR ≪ 1,

F(ᾱ) = 1 R + O(R), H(ᾱ) = ᾱ2 R 2 + O(R 3 ),
and the LTE is O(R 3 ). To deduce the global error of the scheme, the L 2 norm is computed by means of pointwise values,

∥u -u ∆x ∥ 2 L 2 = ∆x 2 ∑ i,j |u(x i , y j ) -u i,j | 2 . (4.16)
Thanks to the linearity of the scheme (4.11)-(4.13),

P N (u) -P N (u ∆x ) = τ = P N (u -u ∆x ),
being τ the formerly derived LTE. The global L 2 error is controlled by,

∥u -u ∆x ∥ 2 L 2 = ∥(P N ) -1 τ ∥ 2 L 2 ≤ (P N ) -1 2 2 ∥τ ∥ 2 L 2 ,
and the bound (4.14) on P N . Second-order accuracy follows from,

(P N ) -1 2 = O( 1 R ) when 0 ≤ ᾱR ≪ 1.
• We study the deviation between the exact derivatives (4.5) and the truncated ones (4.7), used in (4.10), which involve only the first 3 terms of the Fourier expansion (4.3). Since exact modified Bessel functions are used in the solution v(r, θ) to (4.2) along with its radial derivatives (4.9), we must control

I 0 + I 2 2I 1 (ᾱR) 2 ( |ĝ(1) + ĝ(-1) -α 1 | 2 + |i(ĝ(1) -ĝ(-1)) -β 1 | 2 ) + I 1 (ᾱR) I 0 (ᾱR) 2 |ĝ(0) -α 0 | 2 + ∑ |k|>1 I k-1 + I k+1 2I k (ᾱR) 2 |ĝ(k)| 2 ≤ I 0 + I 2 2I 1 (ᾱR) 2 ( |ĝ(1) + ĝ(-1) -α 1 | 2 + |i(ĝ(1) -ĝ(-1)) -β 1 | 2 + ∑ |k|>1 k 2 |ĝ(k)| 2 ) + I 1 (ᾱR) I 0 (ᾱR) 2 |ĝ(0) -α 0 | 2 ,
because standard properties of modified Bessel functions yield,

∀(x > 0, n ∈ N), I n-1 + I n+1 2I n (x) 2 ≤ n 2 I 0 + I 2 2I 1 (x) 2 .
Each term can be bounded as follows:

-|ĝ(0) -α 0 | = |(ĝ(0) -a 0 ) + (a 0 -α 0 )| = |ĝ(0) -a 0 + a 2 |; -|ĝ(1) + ĝ(-1) -α 1 | = |ĝ(1) + ĝ(-1) -a 1 + 2a 2 |; -|i(ĝ(1) -ĝ(-1)) -β 1 | = |i(ĝ(1) -ĝ(-1)) -b 1 |; - ∑ |k|>1 k 2 |ĝ(k)| 2 ≤ ∥g∥ 2 H 1 .
Since g is real-valued and a 0 , a 1 , b 1 , a 2 are computed by trapezoidal rule with N = 4, we use triangular inequalities in conjunction with Lemma 5,

a 0 -ĝ(0) = ∑ k∈Z * ĝ(4k), a 2 = 4ℜ(ĝ(2)) + 2 ∑ k̸ ={0,-1} ℜ ( ĝ(2 + 4k) ) , a 1 -2ℜ(ĝ(1)) = 2ℜ(ĝ(3)) + 2ℜ(ĝ(5)) + ∑ |k|>1 ĝ(1 + 4k) + ĝ(-1 + 4k), b 1 -2ℑ(ĝ(1)) = 2ℑ(ĝ(3)) + 2ℑ(ĝ(5)) -i ∑ |k|>1 ĝ(1 + 4k) -ĝ(-1 + 4k),
so that there is no error if ĝ(k) = 0 when |k| ≥ 2.

Theorem 4.1 expresses the fact that, while being only endowed with a 5-points stencil, the "Steklov scheme" (4.11) delivers a high accuracy if the exact solution is smooth, hence contains mostly low frequencies. Such regularity in a square domain requires smooth boundary data to be supplemented by compatibility conditions (A.2) at each corner, in order to apply Theorem A.2. This is a situation closely related to "well-balanced methods" in 1D, see also [START_REF] Bianchini | A truly two-dimensional discretization of drift-diffusion equations on Cartesian grids[END_REF][START_REF] Gosse | Dirichlet-to-Neumann mappings and finite-differences for anisotropic diffusion[END_REF][START_REF] Gosse | Aliasing and two-dimensional well-balanced for drift-diffusion equations on square grids[END_REF] for 2D considerations. In particular, the modified Bessel functions contained in (4.11), which numerically allow not to split between the Laplacian and the zero-order term, moreover can fit the sharp layers appearing in the solution in case the potential becomes stiff (like "exponentialfit" methods in 1D). However, our error bound doesn't directly extend to

-∆u + α 2 u = f (x, y), u ∂Ω = g,
because the 2D scheme approximates the source f as a piecewise constant function inside each disk D R (i -1 2 , j -1 2 ) (see [17, page 177]); see e.g. [START_REF] Birkhoff | Optimal few points discretizations of linear source problems[END_REF] for more details. Moreover, "compatibility conditions" involving f would be more intricate, see [START_REF] Azzam | On Differentiability Properties of Solutions of Elliptic Differential Equations[END_REF][START_REF] Han | Differentiability properties of solutions of the equation -ε 2 ∆u + ru = f (x, y) in a square[END_REF].

Two-dimensional numerical illustrations.

Validation of error estimates.

To assess practically the former estimates, the following exact solutions were considered in Ω = (-1, 1) 2 with ε = 0.75,

E 0 (x, y) = I 0 (r/ε), r 2 = x 2 + y 2 , tan θ = y/x, E 1 (x, y) = -E 0 (x, y) + I 1 (r/ε) (cos θ -0.5 sin θ) E 2 (x, y) = 2E 1 (x, y) + 0.5 I 2 (r/ε)(sin 2θ + cos 2θ).
These exact solutions to -ε 2 ∆u + u = 0 allow to check both the accuracy of (4.11) and the numerical features of a computational domain endowed with "corners". Indeed, boundary data of E 0 , E 1 , E 2 on ∂Ω display Lipschitz areas, where "compatibility conditions" (see Appendix A) aren't satisfied and most of the pointwise errors of the Steklov scheme accumulates: this is easily noticed with the radial solution E 0 . On the right column of Fig. 5.1, the convergence rates of standard finite-differences, discrete weighted means (DWM, see [START_REF] Gartland | Discrete weighted mean approximation of a model convection-diffusion equation[END_REF][START_REF] Han | A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain[END_REF][START_REF] Val | On mean value solutions for the Helmholtz equation on square grids[END_REF]) and (4.11) are compared. The red line, which indicates second-order accuracy, allows to easily check that both finite-differences and DWM are second-order methods, the size of the errors being very different, though. The Steklov scheme (4.11) behaves differently, in the sense that when the grid is coarse, it displays a fourth-order accuracy, which degrades to second-order when ∆x/ε goes below a certain value. This was to be expected as the modified Bessel functions behave like polynomials when their argument is very small (see also [START_REF] Val | On mean value solutions for the Helmholtz equation on square grids[END_REF]). The DWM (or "tailored", [START_REF] Han | A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain[END_REF][START_REF] Han | Multiscale tailored finite point method for second order elliptic equations with rough or highly oscillatory coefficients[END_REF]) methods, while being very accurate on E 0 , E 1 , E 2 , aren't endowed with such a fast convergence property. Next, in order to illustrate the behavior for smaller ε's, another (more complex) exact solution, E 3 (x, y) = 0.5 I 2 (r/ε)(sin 2θ + cos 2θ) + I 3 (r/ε) sin 3θ, ε = 0.1125 or 0.075, was computed by both the DWM and Steklov schemes; see Fig. 5.2. By inspecting boundary data, one sees that a seemingly small variation in ε can produce a strong increase of the "spikes" at the Ω = (-1, 1) 2 square's corners where compatibility conditions aren't met. Indeed, for smaller ε, most of the numerical error accumulates in these regions. Both numerical methods appear to be similar and to converge at second order in L 2 when ∆x/ε is moderate; when ∆x/ε is bigger, the Steklov scheme seems to behave a little bit better than the DWM (see Fig. 5.2, bottom).

Simple harmonic potential.

Consider now an eigenvalue problem like,

-ε 2 ∆u + V (x, y)u = λu, (x, y) ∈ R 2 ,
for which both eigenvalues λ's and eigenfunctions are explicitly known when posed in the whole space, see Theorem 2.1. When 0 < ε is small enough, this explicit spectrum can still be used in order to quantify pointwise errors of numerical schemes, even if the computational domain must be restricted to a bounded square in R 2 . Accordingly,

-ε 2 ∆u + V (x, y)u = λu, V (x, y) = 0.25 ( |x -0.5| 2 + |y -0.5| 2 ) , ( 5.1)
is now posed in x, y ∈ (0, 1) 2 and its numerical spectrum is sought by both centered finite-differences, and the "Steklov 2D scheme". The Gaussian ground-state ϕ ε 0 allows to measure practically the decay of L 2 norms (4.16) as ∆x → 0 for both numerical algorithms: on Fig. 5.3, errors appear to decay at an order higher than 3 as soon as ∆x/ε is small enough, so that the grid can efficiently represent the numerical approximation. The decay corresponding to third order is illustrated by a red line on Fig. 5.3. To minimize the influence of truncating the computational domain without restricting too much the support of eigenfunctions, a convenient value of ε was experimentally found to be ε ≃ 0.007, with a grid 32 × 32. Pointwise relative errors are displayed on Fig. 5.4: the left column displays relative errors coming from the "Steklov scheme" which suffer from less "mesh imprinting" (and are smaller) than the ones generated by standard centered finite differences (on the right column). Remark 3. An inspection of Fig. 5.3 suggests that there are three different regimes (instead of apparently two for finite differences) for the "Steklov scheme":

• a stiff one, ε ≪ ∆x, where the grid is really too coarse for a radial solution,

• an intermediate one for which ε ≃ ∆x where L 2 errors can decay very fast (faster than order 4) compared to centered finite-differences, • a fine-grid one, ε ≫ ∆x for which L 2 errors decay at a rate close to order 3.

(A-

)symmetric two-well potential. We illustrate this case with "Steklov 2D scheme" on the same computational grid, ε = 0.02 and the potentials,

V (x, y) = 1.01 -exp ( -30 ( |x -0.65| 2 + |y -0.35| 2 ) ) -exp ( -30 ( |x -0.35| 2 + |y -0.65| 2 ) ) , (V + δV )(x, y) = 1.01 -exp ( -30.5 ( |x -0.65| 2 + |y -0.35| 2 ) ) -exp ( -30 ( |x -0.35| 2 + |y -0.65| 2 ) ) ,
which share the same depth, but one is slightly wider than the other. On the left column of Fig. 5.5, usual symmetric eigenfunctions are generated by the scheme in presence of the double-Gaussian potential V (x, y), x, y ∈ (0, 1) 2 with Dirichlet boundaries and 2 5 × 2 5 grid points. As expected, the ground state has a definite sign, in accordance with the statements in Proposition 2. For the perturbed potential V +δV , a quite different picture emerges as the perturbed ground state is now strongly localized in the largest well, still being of a definite sign. The first perturbed excited state is localized in the narrow well and is endowed with small negative values. More excited eigenfunctions are less localized, but still, are never symmetric. Discrete orthogonality is checked in practice by looking at the L 2 scalar products: it was found to hold, up to machine accuracy (≃ 10 -16 ) for both the five first unperturbed eigenfunctions, and the perturbed ones (see also [START_REF] Kong | Asymptotic analysis and numerical method for singularly perturbed eigenvalue problems[END_REF] for other benchmarks).

5.4. Time-dependent problem with two asymmetric wells. The 2D Steklov scheme (4.11) can be recast as a time-marching algorithm, see [17, page 184], for

∂ t u + ε 2 ∆u + V (x, y)u = 0. (x, y) ∈ (0, 1) 2 , ( 5.2)
with Dirichlet boundary conditions. The potential was chosen as,

V (x, y) = 0.95 -exp ( -10 ( |x -0.75| 2 + |y -0.25| 2 ) ) -exp ( -25 ( |x -0.25| 2 + |y -0.75| 2 )
) .

(5.3)

Numerical results for ε 2 = 0.00005, with a 32 × 32 grid, and Gaussian initial data are displayed on Fig. 5.6. As expected from the spectral results derived in Section 3, the numerical solution accumulates more and more in the widest well, in agreement with the shape of the ground-state for asymmetric potentials, as displayed (for instance) on the right column of Fig. 5.5. The large-time behavior of (5.3) sees all the initial data's mass accumulate in the widest well, the narrow one being depleted (at a rate which slows down as time grows). This is what is illustrated on the bottom line of mass-accumulation in wider wells was justified with elementary arguments (see also [START_REF] Dauphine | Asymmetric wave functions from tiny perturbations[END_REF][START_REF] Graffi | Tunnelling instability via perturbation theory[END_REF][START_REF] Jelic | The double well potential in quantum mechanics: a simple, numerically exact formulation[END_REF][START_REF] Selg | Exactly solvable asymmetric double-well potentials[END_REF]). Then, in Section 4, the 2D scheme of [START_REF] Gosse | Dirichlet-to-Neumann mappings and finite-differences for anisotropic diffusion[END_REF][START_REF] Gosse | Viscous equations treated with L-splines and Steklov-Poincaré operator in two dimensions[END_REF] is derived in a way which allows to produce error bounds for the constant coefficient case (see [START_REF] Val | On mean value solutions for the Helmholtz equation on square grids[END_REF]). Such an algorithm, involving modified Bessel functions in 2D, is reminiscent of "discrete weighted means", see [START_REF] Gartland | Discrete weighted mean approximation of a model convection-diffusion equation[END_REF][START_REF] Han | A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain[END_REF][START_REF] Han | Multiscale tailored finite point method for second order elliptic equations with rough or highly oscillatory coefficients[END_REF][START_REF] Mayo | Fast High Order Accurate Solution of Laplace's Equation on Irregular Regions[END_REF][START_REF] Val | On mean value solutions for the Helmholtz equation on square grids[END_REF]. Accordingly, the expected localization behavior is retrieved in Section 5, mostly for rough computational grids for which ε/∆x ≪ 1, see also [START_REF] Trefethen | Computed eigenmodes of planar regions[END_REF], in both static and time-dependent contexts. As an outlook, we outline an extension to a fourth-order Cahn-Hilliard [START_REF] Cahn | Free energy of a nonuniform system. I. Interfacial free energy[END_REF][START_REF] Elliott | Numerical Studies of the Cahn-Hilliard Equation for Phase Separation[END_REF] type of equation:

∂ t u -∆ ( u(1 -u 2 ) -ε∆u ) = 0, x, y ∈ (0, 1) 2 . (6.1)
Following [START_REF] Sun | A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation[END_REF], we recast it as a system, except that, here, the nonlinearity is handled as a (locally) constant coefficient, (i.e. mass-conservation property may be lost)

∂ t u + ε∆v -α 2 v = 0, v = ∆u, ∆ ( u(1 -u 2 ) ) ≃ (1 -ū2 )∆u.
The "frozen parameter" α varies from disk to disk (see Fig. 4.1),

α 2 (t n , x i± 1 2 , y j± 1 2 ) = 1 -u 2 (t n , x i± 1 2 , y j± 1 2
). Starting from random initial data belonging to the interval (-1, 1), and fixing ε = 0.002 with a 32 × 32 computational grid, the expected phase-separation phenomenon, displayed on Fig. 6.1, was retrieved by using an explicit Euler time-integrator.

where ℓ = 0, 1, 2, 3 is the index of a vertex, and Λ stands for a linear functional acting on singular profiles ϕ localized around each corner of the domain Ω. These functional are linear combinations of differential operators acting on g, and the number of involved derivatives of g increases with respect to the index k. Accordingly, such a decomposition suggests that cancelling the Λ's should ensure that u can become as smooth as its remainder v. These functionals were studied in many papers, after early works by Nicol'ski, Volkov, Fufaev, see [34, Chap. III]: for (A.1) and x = y = 0,

Λ j (g) = d 2j g dx 2j (0 + , 0) -(-1) j d 2j g dy 2j (0, 0 + ) (A.2) - j ∑ i=1 (-1) i-1 ∂ 2(j-i) ∂x 2(j-i) ∂ 2(j-i) ∂y 2(i-1) ( r(0, 0)u(0, 0) ) , j = 0, 1, 2, ...
Following [6, eq. (2.1)], see also [1, eq. (1.8-9)], the first ones in x = y = 0 are:

Λ 0 (g) = g(0 + , 0) -g(0, 0 + ), Λ 1 (g) = d 2 g dx 2 (0 + , 0) + d 2 g dy 2 (0, 0 + ) -r(0, 0)u(0, 0), Λ 2 (g) = d 4 g dx 4 (0 + , 0) - d 4 g dy 4 (0, 0 + ) + ( ∂ 2 ∂x 2 - ∂ 2 ∂y 2
) ( r(0, 0)u(0, 0)

) .

If all the derivatives of both g and r vanish close to each corner, (A.2) holds. According to [24, Theorem 3.2] (see also [2, Theorem 2] and references therein), these "compatibility conditions" suffice for (A.1) to pass the Hölder regularity of g onto u: Theorem A.2. Let g ∈ C 2,λ on Γ minus its 4 vertices and Λ ℓ 0 (g) = 0 for ℓ = 0, 1, 2, 3: the solution of (A.1) u ∈ C 2,λ ( Ω). Moreover, for Ω = [0, 1] × [0, 1], u ∈ C 2k,λ ( Ω) ⇔ g ∈ C 2k,λ and Λ ℓ j (g) = 0 for j ≤ k and ℓ = 0, 1, 2, 3, and if, under the same conditions, g ∈ C 2k+1,λ , then u ∈ C 2k+1,λ ( Ω). Such regularity results can be formulated in other functional spaces, see for instance [START_REF] Konenkov | The Dirichlet Problem in Zygmund Spaces in a Square[END_REF][START_REF] Konenkov | Dirichlet and Neumann problems for Laplace and heat equations in domains with right angles[END_REF]. There are several variants of this statement, like in [START_REF] Andreev | On the Accuracy of Grid Approximations to Nonsmooth Solutions of a Singularly Perturbed Reaction-Diffusion Equation in the Square[END_REF][START_REF] Dosiyev | New Properties of 9-Point Finite Difference Solution of the Laplace Equation[END_REF][START_REF] Hell | Modification of dimension-splitting methodsovercoming the order reduction due to corner singularities[END_REF][START_REF] Kellogg | Some Simple Boundary Value Problems with Corner Singularities and Boundary Layers[END_REF]:

Corollary 1. Assume that g ∈ C 1,1 on Γ minus its 4 vertices and Λ ℓ 0 (g) = 0 for ℓ = 0, 1, 2, 3, then the following second derivatives of u are bounded in Ω, max

( ∂ 2 u ∂x 2 , ∂ 2 u ∂y 2
) ≤ C, but not the mixed one, ∂ 2 u ∂x ∂y .

Restricting (A.2) to k = 0 only brings g(0 + , 0) = g(0, 0 + ) at the vertex x = y = 0, so that condition Λ ℓ 0 (g) = 0 means that g is continuous at all four vertices of Ω. Appendix B. Trapezoidal rule and circular traces. Let γ R be the "circular trace operator" H s (R 2 ) → H s-1 2 (0, 2πR) for which,

γ R : u → γ R [u], γ R [u](θ) = u(R cos θ, R sin θ), f (x) = γ R [u](Rθ),
where θ ∈ (0, 2π). The proof of Lemma 5 is split into several steps:

• Being f : (0, T = 2πR) → R periodic, it rewrites as a Fourier series,

f (x) = ∑ k∈Z f (k) exp(ik 2πx T ), f (k) = 1 T ∫ T 0 f (x) exp(-ik 2πx T )dx.
Accordingly, the N -points (Nyström's) "method of rectangles" rewrites:

T N N -1 ∑ j=0 f (j T N ) = T N N -1 ∑ j=0 ( ∑ k∈Z f (k) exp(ik 2πj N ) ) = T N ∑ k∈Z f (k)   N -1 ∑ j=0 exp(ik 2πj N )   .
• An elementary observation is that

1 T ∫ T 0 exp(ik 2πx T )dx - 1 N N -1 ∑ j=0 exp(ik 2πj N ) = -χ(k ∈ N Z * ),
where χ stands for the indicator function and Z * , for Z without 0. Indeed, for k ∈ Z, being a geometric summation of reason exp(ik 2π N ),

N -1 ∑ j=0 exp(ik 2πj N ) = 1 -exp(ik 2π) 1 -exp(ik 2π N ) = N χ(k ∈ N Z),
so the numerator always vanishes. Indexes k ∈ N Z are such that the denominator cancels, too: a Taylor expansion yields the expected result. Yet,

f (0) -   1 N ∑ k∈Z f (k) N -1 ∑ j=0 exp(ik 2πj N )   = - ∑ k∈Z, k̸ =0 f (N k), (B.1)
and it remains to estimate the lacunary sum on the right-hand side. • To estimate the right-hand side of (B.1), we recall that f is the restriction on a circle of radius R > 0 and centered in (x 0 , y 0 ) of u(x, y) ∈ H s (R 2 ). Standard trace theory [START_REF] Miyazaki | Sobolev Trace Theorem and the Dirichlet Problem in the Unit Disk[END_REF] gives that f ∈ H s-1 2 (0, T ): if s > 1, Sobolev inequalities ensure that f is continuous. Moreover, for any index k ∈ N, Recalling that γ R [u](θ) = u(R cos θ, R sin θ), this yields:

γ R [u](k) = R 2k ( γ R [∂ x u](k -1) -γ R [∂ x u](k + 1) -i ( γ R [∂ y u](k -1) + γ R [∂ y u](k + 1)
) ) , so that, the smoothness of u induces Fourier coefficients of γ R [u] which are of the order of powers of R, the radius of the circle. By induction, this calculation can be repeated until, -either Fourier coefficients of index zero are reached, and integrating by parts isn't possible; -or the function u hasn't enough regularity (when |k| > s - 1 2 ). Finally, we get an estimate on the size of Fourier coefficients on the circle of radius R, which can obviously be used to control the error (B.1):

∀k ∈ Z, γ R [u](k) = O ( R min(|k|,s-1 2 )
) .

(B.2)

If s -1 2 ≥ |k|, then in general,

∀k ∈ Z, γ R [u](k) = O(R |k| ) γ R [D |k| u](0) ,
which is the "circular average" of D |k| u, and the value at the circle's center of its harmonic extension. This agrees with the approximate values found at §4.2.

AA 1 )

 1 sin kx, x ∈ (0, a) B exp(κ(x -a)) + C exp(κ(b -x)), D sin k(1 -x),x ∈ (b, 1), where (real) constants A, B, C, D are such that C 1 smoothness holds:sin ka = B + C exp(κ(b -a)) D sin k(1 -b) = B exp(κ(b -a)) + C } C 0 smoothness, kA cos ka = κ[B -C exp(κ(b -a))] -kD cos k(1 -b) = κ[B exp(κ(b -a)) -C]The vector of coefficients can be characterized by means of the kernel of a 4×4 matrix, sin ka+ k cos ka) D(κ sin k(1 -b) -k cos k(1 -b)) = exp(κ(a -b)), A(κ sin ka -k cos ka) D(κ sin k(1b) + k cos k(1 -b))

Figure 1 . 1 .

 11 Figure 1.1. Eigenfunctions for symmetric (left) vs. asymmetric (right) square wells.

. 1 )

 1 being | • | the Euclidean norm. Inserting a change of variables, ∀x ∈ R N , u(x) := exp

Figure 4 . 1 .

 41 Figure 4.1. Transmission conditions at u n i,j given by Steklov-Poincaré operators.

Figure 4 . 2 .

 42 Figure 4.2. Counter-example of [4, Thm. 7]: polynomial (left) and its trace on C(0, ∆x) (right).

Figure 5 . 1 .

 51 Figure 5.1. Boundary data (left), pointwise errors (middle) and L 2 convergence rates (right) for finite-differences, discrete weighted-means and Steklov schemes. Red line indicates second order.

Figure 5 . 2 .

 52 Figure 5.2. Boundary data (left), pointwise errors (middle) and L 2 convergence rates (right) for Tailored (discrete weighted-means) and Steklov schemes. Red line again indicates second order.

Figure 5 . 3 .

 53 Figure 5.3. Decay of L 2 norm (4.16): ε = 0.0066 (top, left), ε = 0.0033 (top, right), ε = 0.0022 (bottom, left) and ε = 0.0011 (bottom, right), for problem (5.1). Red line third order.

Figure 5 . 4 .

 54 Figure 5.4. Steklov scheme with Bessel functions (left) vs. centered FD (right).

Figure 5 . 5 .

 55 Figure 5.5. Steklov scheme: symmetric (left) vs. asymmetric (right) potential.

Fig. 5 . 6 , 6 .

 566 Fig.5.6, where the numerical steady-state is shown to have nearly all its mass inside the widest well. Beyond T ≃ 50, the rate of change is very slow, though.6. Conclusion and outlook.After recalling B. Simon's "flea on the elephant" phenomenon in 1D, see Fig.1.1, analytical bounds were derived in Section 3 dealing with various perturbations of the classical quadratic harmonic potential; in particular,

Figure 5 . 6 .

 56 Figure 5.6. Numerical solutions of time-dependent problem (5.2)-(5.3) at times T = 2, 3, 7, 20; numerical steady-state at T = 100 (bottom, left) and corresponding L 2 residues (bottom, right).

Figure 6 . 1 .

 61 Figure 6.1. Simulation of Cahn-Hilliard equation (6.1): steady-state (left) and residues (right).

(

  cos θ, R sin θ) exp(-ikθ)dθ = --sin θ ∂ x u + cos θ ∂ y u) i(k -1)θ) -exp(-i(k + 1)θ) i(k -1)θ) + exp(-i(k + 1)θ) )dθ.

which is exact for linear solutions of ∆u = 0. In

2D, there are only two linearly independent homogeneous harmonic polynomials of degree m : the real and imaginary part of (x + iy) m .

Acknowledgments. Enrique Zuazua's research was supported by the Advanced Grant DyCon (Dynamical Control) of the European Research Council Executive Agency (ERC), the MTM2014-52347 and MTM2017-92996 Grants of the MINECO (Spain) and the ICON project of the French ANR-16-ACHN-0014. L.G. thanks Profs. François Bouchut and Roberto Natalini for some technical discussions.

Appendix A. Elliptic regularity in a planar, square domain.

Let Ω = (0, 1) × (0, 1) be the (open) unit square and Γ stand for its boundary. Consider the following strictly elliptic operator in Ω,

where both r and g are smooth functions. Because the domain is convex, but endowed with "corners", the variational solution to (A.1) cannot be expected to be smoother than H 2 in general, see [START_REF] Grisvard | Elliptic problems in non smooth domains[END_REF] and, e.g., [START_REF] Hua | A regularity result for boundary value problems on Lipschitz domains[END_REF]Theorem 4.3]. Refined regularity estimates relying on Besov spaces theory were given in [8, Carollary 3.1]: Theorem A.1. Let r ≡ 0 in (A.1), which reduces to Laplace's equation, and g ∈ W s,p (Γ) with p < 2 1-ε and s ∈ (1 -ε, 1): the harmonic solution u ∈ W t,q (Ω) for

If g ∈ H 1 (Γ), then s = 1, p = 2 and u ∈ W t,q (Ω) with 1 3 < q ≤ 2 and 0 < t < 1 + 1 q . Another point of view consists in seeking under which restrictions on g the solution u can be as smooth as if (A.1) was solved in a smooth domain. Following [START_REF] Kellogg | Some Simple Boundary Value Problems with Corner Singularities and Boundary Layers[END_REF], let u(x, y) = 3 ∑ j ,ℓ=0 Λ ℓ j (g)ϕ i,j (x, y) + v(x, y), v smooth,