A NOTE ON THE RELLICH FORMULA IN LIPSCHITZ DOMAINS

Let L be a symmetric second order uniformly elliptic operator in divergence form acting in a bounded Lipschitz domain Ω of RN and having Lipschitz coefficients in Ω. It is shown that the Rellich formula with respect to Ω and L extends to all functions in the domain D = {u ∈ H1 0 (Ω); L(u) ∈ L2(Ω)} of L. This answers a question of A. Chäıra and G. Lebeau.


Introduction
Let L = 1≤i,j≤N ∂ i (a ij ∂ j. ) be a uniformly elliptic operator in divergence form in R N , the coefficients a ij being (real) Lipschitz continuous functions in R N such that a ij = a ji for 1 ≤ i, j ≤ N .Let A denote the matrix {a ij }.
If Ω ⊂ R N is a bounded Lipschitz domain in R N , if V is a C 1 vector field in Ω and if u ∈ H 2 (Ω), then the following so-called Rellich formula holds (for references see Necas [N, p. 224]). (1) where ν is the unit exterior normal field along ∂Ω and ν L = A(ν) is the conormal field; ∂ U denotes the differentiation operator in the direction U , and we have let , the formula follows from the Stokes formula ∂Ω W.ν dσ = Ω div(W ) dx on taking W = du(V )A(∇u) − 1 2 ∇u 2 L V .The general case follows from an approximation argument.Of course the Lipschitz regularity of Ω is only needed in a neighborhood of supp(V ) ∩ ∂Ω.
In this note it is shown that the Rellich formula extends to all functions u in the domain of L, that is u ∈ H 1 0 (Ω) with L(u) ∈ L 2 (Ω); this amounts ( [N]) to a continuity property of the gradient of L-solutions with respect to perturbations of Ω (see Theorem 1 below).This extension of Rellich formula answers a question raised to me by A. Chaïra and G. Lebeau [CL] (see also [N,Problème 2.2,p. 258)] and is useful in some problems in control theory for the wave equation ( [C], [CL]).The proof relies on well-known results and methods of the Potential theory in Lipschitz domains (in particular [D], [A1], [JK1] and [A2]).

Notations and preliminaries
In this section we fix some notations and recall several basic properties of the Potential theory in Lipschitz domains with respect to elliptic second order operators.

2.1.
Let N be a fixed integer ≥ 2 and let ϕ : R N −1 → R be a function such that ϕ(0) = 0 and |ϕ(x) − ϕ(y)| ≤ k|x − y| for x, y ∈ R N −1 and a positive constant k.For x ∈ R N , we note x = (x , x N ) the decomposition of A(P, r) = (P , P N − 5kr) and In the sequel, the dependence on N of the various constants is not made explicit.We note δ

For
where a ij , b i and γ are bounded borel functions on R N such that when

Harnack boundary principle.
If L ∈ Λ(α, M ), if P ∈ Σ, if u and v are two positive L-solutions in ω(P, r) vanishing on Σ(P, r) and if for all x ∈ ω(P, r/2), where c = c(k, α, M, r 0 ) > 0 ( [A1], see [A3] and references there for other related results).More generally, under the same assumptions on L and u, if v is positive L 1 -harmonic on ω(P, r) for some L 1 ∈ Λ(α, M ) having on Σ(P, r) the same second order part than L, and if v = 0 on Σ(P, r), inequalities (5) hold on ω(P, r/2) for some 2.4.Ratios of positive harmonic functions near the boundary.The Harnack boundary principle (5) when combined with the maximum principle implies a stronger continuity statement for the ratios of harmonic functions [JK1].If u and v are positive L-solutions on ω(P, r), P ∈ Σ, r ≤ r 0 , vanishing on Σ(P, r), and if for x ∈ ω(P, rθ/2).Here c and β are > 0 constants (depending only on k, M , α and r 0 ).

Fatou's Theorem. Denote µ Ω
A the harmonic measure of A = A(P, r) in Ω = ω(P, r) with respect to L. If s is positive and L-superharmonic in Ω then s admits a fine limit at µ Ω A almost every point P ∈ Σ(P, r), this fine limit being zero µ Ω A -a.e. if s is a potential.If s is L-harmonic in Ω then s admits a non-tangential limit at µ Ω A almost every point P ∈ Σ(P, r).The last property is related to the first by the following fact.If U ⊂ Ω is the union of a sequence of balls B(x j , εδ(x j )) ⊂ Ω where ε > 0 is fixed and x j → Q ∈ Σ(P, r), then U is not minimally thin (in Ω) at P (ref.[A1]).

Density of harmonic measure.
Let Ω be a domain such that Ω ∩ T (P, r) = ω(P, r) for some P ∈ Σ and r ≤ r 0 and let A = A(P, r).Let L ∈ Λ(1, M) be formally self-adjoint.The L-harmonic measure µ Ω x of x ∈ Ω is equivalent on Σ(P, r) to the natural area-measure σ.In fact, on Σ = Σ(P, r/2), µ This follows from the Rellich formula (see also [D], [JK2], [A2]).Also, f x > 0 a.e. on Σ (the argument of [D] for L = ∆ is easily extended).The Harnack boundary principle shows that the density f A satisfies also a reverse Hölder inequality.For each a = (a , a d ) ∈ Σ and each positive t with t < 1 2 r: . The above extends to wider classes of divergence type elliptic operators (see [FKP] and references there), and also to every L in Λ(α, M ), 0 < α ≤ 1 ( [A4]), but this will not be needed here.

Non-tangential differentiability property
From now on (Section 3, 4, 5) we consider an operator 4) and ( 4) with α = 1.As a first step for the proof of Theorem 1 we prove the next lemma which is probably known but an explicit reference seems difficult to locate (see [KP] for L p estimates of the non-tangential maximal function of the gradient and a variant of L p convergence, compare also [A2]).We give a proof which relies on Fatou theorem (2.6 above).

Lemma 1.
Let Ω be a Lipschitz domain in R N .If u is a solution of Lu = 0 in Ω vanishing on an open subset S of ∂Ω, then ∇u admits a non-tangential limit at almost every point P ∈ S.
Proof: It is enough to consider the case where Ω = ω(0, r) and S = Σ(0, r/2), r > 0 (with the notations in 2.1 and with respect to some Lipschitz continuous function ϕ : R N −1 → R such that ϕ(0) = 0).We let Ω = ω(0, r/2), Ω = ω(0, 3r/4) and assume as we may that u is continuous and positive on Ω with u = 0 on Σ(0, r).Then u ∈ W 2,p loc (Ω) for all p < ∞ ( [LU,) and ) and let w be the solution to the problem L 0 w = 0 in Ω, w = 1 on ∂Ω \ Σ, and w = 0 on Σ (compare [A2]).Note that and positive (by the maximum principle).It follows from Harnack inequalities, the uniform decay property 2.5 and the interior gradient estimates that for By the boundary Harnack principle (5 The argument is now broken into three steps.First we note that the distribution L 0 (∂ k u) (which is defined as an element of Using the Hardy inequality (Remark 1.1) and where we have first used that for x ∈ Ω as above, and then a Whitney partition, Schwarz and Hardy's inequalities.In the same time we have also shown that ( , a well-known projection argument shows that there is a L 0 -supersolution p ∈ H 1 0 (Ω ) such that |v| ≤ p.By Fatou's theorem (2.6) applied to p and L 0 , v converges finely (w.r. to L 0 ) to zero at almost every point P ∈ S.
and h is hence a difference of two positive L 0 solutions in Ω .By 2.6, h converges finely (and non-tangentially) almost everywhere on S. Thus, ∂ k u converges finely at almost all point P ∈ S.
Third step: By [LU,p. 205], ∂ k u has also the following uniform continuity property: for x ∈ Ω , and ), for some constants α = α(M, r) ∈]0, 1] and C > 0. Therefore, by (9) and Harnack inequalities, It follows that if P ∈ S is such that ∂ N w is non-tangentially bounded at P and ∂ k u admits a fine limit at P , then ∂ k u converges nontangentially to at P .If not, a positive number ε and points x j ∈ Ω converging non-tangentially to P could be constructed such that for ) is thin at P , a contradiction (see 2.6 above).
We shall also need the following observation.
Lemma 2. If the function u in Lemma 1 is positive, then ∇u(P ) = 0 a.e. on S.
On S j , the harmonic measure of A = A(0, r 0 /2) w.r. to L 0 and Ω j is µ j = −∂ ν L G j (., A) .dσSj ; here ν L = A(ν) on S j , where ν is the exterior unit normal field along S j , and G j is the Green's function w.r. to L 0 in Ω j .This follows from the Stokes formula ∂Ωj W, ν dσ = Ωj div(W ) dx which is valid for each vector field W of class W 1,p (Ω j ), p > N; with W = ϕ A(∇G j (., A))−G j (., A) A(∇ϕ), where ϕ is smooth and of support in T (0, r 0 /4) one gets that ψ Remark 2. From 2.3 and the uniform bound f A L p (S) ≤ c(k, M, r) (with p = p(M, k) > 2, S = Σ(0, r/2)), it follows that every positive L-harmonic function u in ω(0, r) vanishing on Σ(0, r) verifies ∇u L p (S) ≤ c (k, M, r) u(A(0, r)).Note also that under the assumptions of Lemma 1, and if u ∈ H 1 (Ω), a simple limit argument shows that ∇u coincides on S with the weak gradient ∇u ∈ H
Proof: The first claim follows by the arguments used in the proof of Lemma 2. As before when x ∈ Ω .Hence if ε > 0 is so small that w(x) > ε for |x | ≤ r 0 /2 and x N ≤ −kr 0 , the implicit function theorem shows that the region U (w, ε) ∩ Ω is as required by the first claim above.Recall that by Schauder's theory w is locally of class C 2,1 in Ω.That ϕ ε → ϕ uniformly on D(r 0 /2) is then obvious, since w is continuous on Ω ∪ Σ(0, r 0 ).
The last part of the proposition follows now from Lemma 1, Lemma 2 (applied to w) and Lemma 4 below.If ϕ is differentiable at a ∈ D(r 0 /2) and if ∇w(x) admits a non tangential limit α at (a , ϕ(a )) = a with α = (α , α N ) = 0 (that is α N = 0 by ( 10)), then To see this, fix η > 0 and j, 1 ≤ j ≤ N − 1, and apply Lemma 4 below to the function It follows that for each small enough ε > 0, there is a point x (ε) ∈ D(r 0 /2) with the following properties: converges non tangentially to a in Ω as well as a ε = (a , ϕ ε (a )).Hence, since ∇w has a non tangential limit α = (α , α N ) at a such that α N = 0, 1 the lemma follows at once from the mean value theorem.

The Rellich formula for u in the domain of L
The previous constructions are now used to obtain the following strong L 2 approximation property.It is well-known that the later implies the desired extension of the Rellich formula.Notations and assumptions are as in the previous section.It is also assumed for sake of simplicity that γ ≤ 0. Recall that Ω = ω ϕ (0, r0 2 ) = ω(0, r0 2 ) and that Here, ∂ ν L u = A∇u, ν denotes the conormal derivative of u along Σ (and ν is the unit exterior normal), ∂ ν L u j denotes the conormal derivative of u j along Σ j = {(x , ϕ j (x ); x ∈ D(0, r 0 /2)}.Note that if the u j are ≥ 0, then simple convergence in Ω already implies uniform convergence on ω(0, r ), r < r 0 /2, by boundary Harnack property.Also an obvious decomposition of u j shows that to prove Theorem 1 we may restrict to the case where u j ≥ 0.
Proof of Theorem 1: Consider first the special case of the sequence v j = w − ε j with L = L 0 and denote f 0 j , f 0 , the corresponding functions f j and f .Then by Lemma 1 f 0 j (x ) → f 0 (x ) = ∇w(x), ν L (x) almost everywhere in D = D(r 0 /4) (where x = (x , ϕ(x ))).Since there is a uniform bound on f 0 j L p (D ) for some p > 2, it follows that f 0 j converge (strongly) to f 0 in L 2 (D ).And the proposition follows for the case at hand.It is then clear that g j (x ) = ∂ ν L v j (x , ϕ j (x )) tends to f 0 (x ) a.e. in D and in L 2 (D ) (note that f 0 In the general case (with u j ≥ 0, u > 0 in Ω ), consider h j = f j /f 0 j .By Lemma 5 below, this is a sequence of Hölder continuous functions on D which is bounded in C α (D ) for some α, 0 < α < 1. Moreover the function h = f/f 0 -which may be seen as a Hölder continuous function on D -is the unique cluster value of this sequence in L ∞ (D ).In fact, by Lemma 5, if H is such a cluster value and if η > 0 is small, both quantities where x ∈ D and A = (x , ϕ(x ) − η), are bounded by ≤ c η δ for some positive real δ.
Thus, h j → H in L ∞ (D ).Since f 0 j → f 0 almost everywhere on D and in L 2 (D ), it follows that f j → f in L 2 (D ) and almost everywhere on D which proves the theorem.
Recall that for a = (a , a Lemma 5.There are constants C > 1 and D ∈ (0, 1) with the following property.If P ∈ S, if h is positive L-harmonic in T (P, η) ∩ U ε vanishing on T P (η) ∩ ∂U ε , ε > 0, and if η > 0 is small, then We use a construction from [A2].Let s = f (w) where f (t) = t 0 e −θ α dθ, u = g(w) where g(t) = t 0 e θ α dθ and 0 < α < 1.It is immediately checked, using (9), that if α is small, then s (resp.u) is L-superharmonic (resp.L-subharmonic) in Ω {w < ε} for ε > 0 small.In fact, so that using ( 9) and Schauder interior estimates, we have on Ω near Σ, for some positive constants C and β, and where in the last line we have used 2.5.It follows that if we fix α in (0, β), then s is L-superharmonic near Σ in Ω .The subharmonicity of u = g(w) is obtained similarly. Let for some constants b > 0 and c > 0, where we have applied again 2.5.
For η > 0 and small, the function sε (resp.ũε ) is positive L-superharmonic (resp.Lsubharmonic) on ω = T (P, η)∩U (ε), vanishes on Σ ε = ∂U ε ∩T (P, η) and ũε ≤ sε in ω.Taking the smallest L-harmonic majorant of ũε in ω, we obtain a positive L-harmonic function h 1 on ω such that ũε ≤ h 1 ≤ sε on ω.Of course, h 1 vanishes on Σ ε and by the previous estimates, we have in ω Finally, if h is any positive L-harmonic function on ω vanishing on Σ ε , we know (see Section 2.4) that for some real β ∈ ). Combining ( 12) and ( 13) we obtain (11 ) the a ij satisfying (4) and ( 4) with α = 1.From Theorem 1, the desired generalization of Rellich formula (1) together with an extension of Theorem 1 itself are easily derived.In the next corollary notations are the same as in Theorem 1.
Recall (ref.[N]) that if W be a bounded Lipschitz region in R N , then for u ∈ H 1 0 (W ) such that L(u) = f ∈ L 2 (W ), the weak conormal derivative ∂ ν L (u) (defined as a member of H − 1 2 (∂W )) belongs to L 2 (∂Ω) and . This follows from a natural approximation argument combined with Rellich formula for functions in H 2 .By Theorem 1, if f = 0 in an open neighborhood V of P ∈ ∂W , then the weak and the strong conormal derivatives of v coincide in V ∩ ∂W .
Proof of Corollary 1: By decomposing f j into its positive and negative parts we may assume that f j ≤ 0 in Ω j , j ≥ 1.By Rellich formula there is a uniform estimate ) and a constant C independent of j.Thus by a standard approximation argument we may also assume that f j = 0 on a neighborhood V of Σ.Then, v j → v simply on V ∩ Ω and the result follows from Theorem 1.

Remarks.
5.1.It follows from Lemma 1 and Theorem 1 (and the obvious approximation argument) that in a given bounded region Ω the L-harmonic measure of x 0 ∈ Ω induces on a Lipschitz open piece S of ∂Ω the measure of density −∂ ν L (G(x 0 , .), if G denotes the Green's function of L in Ω.

5.2.
Corollary 1 is easily extended to the following case: v ∈ H 1 (Ω ) with L(v) = f ∈ L 2 (Ω ) in Ω and v = F on ∂Ω for some F ∈ H 2 (Ω ); the v j ∈ H 1 (Ω j ) are such that L(v j ) = f in Ω j and v j = F in ∂Ω j .One has just to look to v j = v j − F and to notice that ∇F (x , ϕ j (x )) → ∇F (x , ϕ(x )) a.s. in D(r 0 /4) and also in L 2 (D ) (in fact in H 1/2 (D )) since ∇F ∈ H 1 (Ω).
From Corollary 1 and Remark 5.2 above the extension of Rellich formula follows.
Corollary 2. Let L be as before, let V be a C 1 0 vector field in R N and let Ω be a domain in R N which is Lipschitz in a neighborhood of each point P of F = supp(V ) ∂Ω.If u ∈ H 1 (Ω) is such that L(u) ∈ L 2 (Ω) and if u = g in a neighborhood of F in ∂Ω for some g ∈ H 2 (Ω) then the Rellich formula (1) holds.
This follows from Rellich formula (Corollary 2) and the fact that in a Hilbert space (H, .) every weakly convergent sequence θ j such that lim j→∞ θ j = lim j→∞ θ j is strongly convergent.Corollary 3 means that for a bounded Lipschitz domain Ω with a given C 0,1 approximation (ref.[N]) by a sequence of Lipschitz domains Ω j the following holds: if v j ∈ H 1 0 (Ω j ), v ∈ H 1 0 (Ω) are such L(v j ) = f j ∈ L 2 (Ω j ) converges strongly (in the appropriate sense) to L(v) = f ∈ L 2 (Ω), then ∇v j |∂Ωj → ∇v |∂Ω in the appropriate strong L 2 sense.
ψ is the solution to L 0 (ψ) = 0 and ψ = ϕ on ∂Ω j .Thus, by the Harnack boundary principleC −1 |∂ N (w)|.σ≤ µ j ≤ C|∂ N (w)|.σ on S j ; in particular ∂ N w L 2 (Sj ) ≤ C for j large.Since the L 0 -harmonic measure µ of A in Ω is the weak limit of µ j and since the Lipschitz constants of the graphs S j are uniformly boundedC −1 |∂ N (w)|.σ≤ µ ≤ C |∂ N (w)|.σ on S for some constant C = C(M, k, r).Since the density f A of µ is > 0 a.e. on S it follows that ∂ N w = 0 a.e. in S.