LOCAL SOLVABILITY OF ELLIPTIC EQUATIONS OF EVEN ORDER WITH H¨OLDER COEFFICIENTS

. We consider elliptic equations of order 2 m with H¨older coeﬃcients. We show local solvability of the Dirichlet problem with m conditions on the boundary of the upper half space. First we consider local solvability in free space and then we treat the boundary case. Our method is based on applying the operator to an approximate solution and iterating in the H¨older spaces. A priori estimates for the approximate solution is the essential part of the paper.


Introduction
This paper is about local solvability of elliptic equations of order 2m with Hölder coefficients. We consider elliptic operators Lu(x) = |α|≤2m a α (x)D α u(x), where the coefficients a α (x) are Hölder continuous and complex valued.
The purpose of this paper is to solve in a direct way two types of problems. First, we treat the local problem in free space and we prove Theorem. Assume the operator L satisfies (2.3), (2.1) and (2.2) Then there exists > 0 depending on K 0 , λ and n so that given f ∈ C δ (B ), there exists u ∈ C 2m+δ (B ) such that Lu = f in B 2 . This is Theorem 3.8. And secondly, we treat the local problem in half space with m boundary conditions. Specifically, we prove Theorem. Let the operator L satisfy (2.3), (2.1) and (2.2). In addition assume that (2.7) holds. Then there exists > 0 depending on K 0 and λ so that given f ∈ C δ (B + ), there exists u ∈ C 2m+δ (B + ) such that Lu = f in B +  We have proved similar results in [7], where we treated the global problem via Schauder estimates and the continuity method with restrictions on the lower order terms.
One advantage of this direct method is that a solution is shown to exist locally with no restrictions on the lower order terms. Our motivation is that we can use these results to treat the problem in a bounded domain by localizing via partition of unity.
There is a very vast literature concerning this problem but our inspiration came basically from an old and classical paper of Agmon, Douglis and Nirenberg [1] in which they treat the Dirichlet problem in a bounded domain and they impose restrictions on the regularity of the lower order terms. Our work differs in the method of proof and in that our result imposes no restrictions on lower terms. Essential to their work is a representation formula for the constant coefficient equation. This representation formula comes in turn as a result of a Radon type transformation due to F. John [5]. We have also relied for some of the basic estimates on the book [6].
In all of the estimates we will use the letter C to denote a constant that depends only on structure. This means that C can depend on the ellipticity constant λ, on the dimension n, and on the fixed constants δ and K 0 introduced below.

Preliminaries
Let > 0. All our estimates will take place in B (0), from now on B . Throughout the paper, η will be a function in C ∞ 0 (B ) such that η = 1 in B 2 , while ϕ will be a function in C ∞ 0 (B 2 ) such that ϕ = 1 in B 1 . We will frequently consider the truncation ϕ ξ R with R → ∞. We fix two real numbers δ and β such that 0 < δ < β ≤ 1. We consider operators of the form where x ∈ R n and a α (x) ∈ C.

LOCAL SOLVABILITY OF ELLIPTIC EQUATIONS OF EVEN ORDER 3
Let us write the characteristic polynomial of the equation by We will assume the ellipticity condition for all x ∈ R n and for all ξ ∈ R n . A crucial consequence of the ellipticity assumption are the following estimates.
Lemma 2.1. There exists a constant C depending on K 0 , λ and |γ| such that for all x ∈ B , all ξ ∈ R n , and all γ.
The above estimates suffice to treat the problem in free space. We now introduce the concepts we need in order to treat the problem in half space.
We assume that the dimension n ≥ 2.

M. A. MUSCHIETTI AND F. TOURNIER
We emphasize the last coordinate by writing Note that |a(x)| = |p(x, 0, i)| ≥ λ for all x ∈ B . Notice that by ellipticity, the polynomial p(x, iξ , z) as a polynomial in the complex variable z has no pure imaginary roots.
Write p(x, iξ , z) = 0, z = z(x, ξ ) to denote the roots. By ellipticity and the K 0 bound it follows that each root satisfies Lemma 2.2. There exists a constant C depending on K 0 and λ such that for all ξ ∈ R n−1 and all x ∈ B we have Proof. Fix x ∈ B and ξ ∈ R n−1 and let p(z) := p(x, iξ , z). Assume p(z) = 0 and write z = a + ib.
Denote the roots of P (x, iξ , z, it) = 0 by Z k (x, ξ , t) and the roots of p(x, iξ , z) = 0 by z k (x, ξ ). Since, for t = 1, we have P (x, iξ, i) = p(x, iξ), we get It follows that m of the roots z k (x, iξ ) have positive real part. As above, let {z + k (x, ξ) : k = 1, . . . , m} be the m roots with positive real part and {z − k (x, ξ) : k = 1, . . . , m} the m roots with negative real part.
We can factor It follows that s +− k (x, ξ ) = S k (x, ξ , 1). Hence we can conclude: k (x, ξ ) are smooth functions of ξ for ξ ∈ R n−1 , and there exists a constant C depending on K 0 , λ and |γ| such that for all x ∈ B and for all ξ ∈ R n−1 .
In addition, we need to assume that there exists a constant C depending on K 0 such that We believe this hypothesis should be a consequence of a similar estimate for homogeneous polynomials with Hölder coefficients but we could not find a proof. It holds for all the examples we have at hand.
We give a few examples of operators satisfying our hypothesis. Let λ ≤ a j (x) ≤ Λ be C δ real valued functions, j = 1, . . . , n − 1 and a n = 1, and consider the operator The roots with positive real part are and the roots with negative real part are where r = (1 + A 2 ) 1 8 and γ = arccos The estimates (2.6) and (2.7) follow by direct computation. The referee proposed an example with real constant coefficients, Also, we have and again, the estimates (2.6) can be checked directly, and obviously the estimate (2.7) holds since the coefficients do not depend on x. A slight variation of the above is to assume that a k (x) and b k (x) are Hölder continuous real valued such that λ ≤ a k (x) ≤ Λ and λ ≤ b k (x) ≤ Λ for all x and for some positive constants λ and Λ, and let b k,l (x) = 2b k (x)b l (x) for k = l and b k,k (x) = (b k (x)) 2 , and consider As above we have and one can check (2.7) directly.

Local existence in free space
We will work on the Hölder spaces defined below.
Our goal in this section is to solve Given g ∈ C δ (B ), we define the approximate solution N (η g) by the following formula. For x ∈ B let We will find a solution to Lu = f of the form u = N (η g) for an appropriate g.
In fact, if the equation has constant coefficients, then u = N (η f ) already solves Lu = f .
We call u := N (η g)(x) and approximate solution to the equation Lu = g in B since we will show that the C δ norm of Lu − g is small in B .
Before we proceed with the main result we need to prove some auxiliary estimates.
3.1. Auxiliary integrals for free space. In this subsection we prove two auxiliary results to be used below.
First, let us a emphasize the basic calculations that will be used several times in the sequel.
Proof. Fix |α| ≤ 2m. We may assume n−2m+|α| > 0. Note that (−∆ ξ ) k e i(x−y).ξ = |x − y| 2k e i(x−y).ξ . Fix k such that 2k ≥ n + 1 and integrate by parts to get We estimate each integral as follows, using the first estimate in Lemma 2.1: , where X is the indicator function. It is also important to notice that J 3 → 0 as R → ∞, for x = y.
Next, we prove two important bounds that we need in order to continue. Notice that for |α| = 2m the integrand in the integral below is only bounded by 1 |x−y| n , which is not integrable. However, note that the polynomial is evaluated at x and this allows for a favorable estimate.
We have |I| ≤ C, where C depends only on K 0 and n.
Proof. To prove this, write and note that by Lemma 3.1 Similarly we have We have |I| ≤ C, where C depends only on K 0 and n.
Proof. To prove this, write We have, by Lemma 3.1, We can estimate again by Lemma 3.1: Next we show that N (η g) has derivatives up to order 2m.
Proof. Let |α| = 2m. We show that the above limit exists and is uniform in x. Write Let k be an integer such that 2k ≥ n + 1 and let First we show that the integrals converge. We claim that Integration by parts gives that Using (2.4) we get This proves the claim. It follows that The convergence of the integral II is proved in the same way, this time noting that and similarly for the integral III , this time using (2.5) to estimate Let us now show that III R → III as R → ∞. The other two are similar. First, we write We have and hence we can write To show that II R → II , first note that and then we proceed as before.

3.2.
Main results for free space.
Proof. We have shown that u has derivatives up to order 2m and for |α| ≤ 2m and let We have, using Lemma 3.1, that by Lemma 3.3. Next, and we have, by Lemma 3.1, finishing the estimation of B 1 . Now, and by Lemma 3.1 This finishes the estimation of the term A.
and we estimate and we estimate by Lemma 3.3. This finishes the estimation of C 1 . Now, This finishes the estimation of the term B. The lemma is proved.
We notice that and also Hence, we define Remark 3.6. If the coefficients a α are constant then u := N (gη ) solves Lu = gη in R n .
We can now state the main theorem of this section.
Theorem 3.7. There exists a constant C depending on K 0 , λ, and n such that Proof. For fixed R ≥ 1, let We write We estimate, using Lemma 3.1, and estimate by Lemma 3.1 and estimate by Lemma 3.3, and by Lemma 3.1 we have We also have thus finishing the estimation of A 2 and hence of A. To and we have by Lemma 3.1 We have by Lemma 3.2, and by Lemma 3.1 we have This finishes the estimation of the term B.
We are now ready to prove local existence of solutions of Lu = f when f ∈ C δ (B ). Let g 0 = f and g k+1 = f − T (g k η ). By Lemma 3.5 we have g k in C δ (B ), and by Theorem 3.7 we have Choosing small so that β−δ K ,β < 1, by the contraction mapping theorem we can conclude that there exists g ∈ C δ (B ) such that g = f − T (gη ), which gives

Local existence in half space. Boundary conditions
In this section we prove local existence of solutions in half space. We will show that there exists > 0 such that given f ∈ C δ (B + ) there exists u ∈ C 2m+δ (B + ) such that Lu = f in B + and such that u satisfies the m boundary conditions, for k = 0, . . . , m − 1.
Given g ∈ C δ (B ), we define the approximate solution N + (η g) by the following formula. For x ∈ B + let In order to achieve the boundary conditions, we define for where Φ(y, x, ξ ) is given by where γ + (ξ ) denotes any piecewise smooth contour on the right half plane enclosing the roots of p + (y, iξ , z). And γ − (ξ ) denotes any piecewise smooth contour on the left half plane enclosing the roots of p − (y, iξ , z).
Defining the function we will show in this section that In particular, for constant coefficient operators, we have Lu = η g. We will also show in this section that 4.1. Auxiliary integrals in half space for the operator N + . We proceed to prove estimates for the auxiliary integrals we need in half space related to the operator N + . For x ∈ R n , we write x = (x , x n ) with x ∈ R n−1 .

Lemma 4.1. Let
We have |A R (x, ξ )| ≤ C, for x ∈ R n + and ξ ∈ R n−1 . The constant C depends only on K 0 , λ and n.
Here α is an n − 1 multi-index.
A change of variables s = x n ξ n , ds = x n dξ n , gives Note that s, ξ ), and note that Hence, Notice that ds.

Lemma 4.2. Let
We have |I R | ≤ C |x − y | n−3 , with the constant C depending only on K 0 , λ and n.
To estimate B, first we claim that

M. A. MUSCHIETTI AND F. TOURNIER
To prove the claim, write We have, using Lemma 4.1, that proving the claim.
And hence we have finishing the proof of the lemma.
We use the estimate above to prove estimates for two auxiliary integrals in half space.

Lemma 4.3. Define
We have |I R | ≤ C, with C depending only on K 0 , λ and n. Proof. Write We have In the first inequality of the line above we have used Lemma 4.2. This proves the lemma.
We now consider a similar integral over a ball.
Then we have |I R | ≤ C.

Proof.
Write We have We can estimate We can estimate To estimate F R , consider first the casex n ≥ ρ 2 . Note thatx n − x n ≤ |x n − x n | ≤ |x −x| = ρ 4 . Therefore x n ≥x n − ρ 4 ≥ ρ 4 .
In this case, For the casex n ≤ ρ 2 , note that since r 2 + x 2 n = ρ 2 , we get Hence, in this case we write

Now use Lemma 4.2 to get
So the lemma is proved.
This finishes the estimation of the auxiliary integrals in half space for the operator N + .

Auxiliary integrals in half space for the operator H.
We now proceed to prove estimates for the operator H.
The following estimates follow directly from the definition.
We have for all ξ , y, θ and φ, that For all ξ , y, θ and φ, and any multi-index α, we have Using these estimates together with the estimates of (2.6) we will prove the following lemma.
Proof. To prove this claim, note that we can write Φ β (ξ , x, y) as a sum of four terms of the form for appropriate limits of integration in φ and θ that do not depend on ξ and where z = z(φ, ξ ) and w = w(θ, ξ ). Hence, it is enough to estimate Write the integrand as g1g2 g3 , where g 1 = e −ynz e xnw w |β1| (iξ ) β2 , g 2 = z φ w θ and g 3 = p + (y, iξ , z)p − (y, iξ , w)(w − z). By direct computation we obtain that Also note that by estimate (2.6) we have The claim is a direct consequence of the estimates above.
With the aid of Lemma 4.5 and the estimates (2.6) and (2.7) the following lemma holds.

Lemma 4.6. Let
We have We have The constant C in both cases depends on K 0 , γ and n.
Proof. To prove the first estimate note that Hence, The proof of the lemma is hence a direct application of Lemma 4.5.
In the second estimate we have to use (2.7).
To continue, we prove Lemma 4.7. Suppose s > 0, n − 2m + |β| > 0, and Proof. To prove this, write We can estimate Also, setting ζ = sξ , we get Since 2m − 1 − |β| < n − 1, we get Let us now estimate B: We may assume 1 s ≤ 1 |y | since otherwise we are done. Setting ζ = sξ , we get Noting that ρ := s |y | ≥ 1 we get and the lemma is proved. Now we combine the results in Lemmas 4.5 and 4.6 with Lemma 4.7 to prove Lemma 4.8. For any multi-index β such that n − 2m + |β| > 0 we have The constant C depends only on K 0 , λ and n.
Proof. The proof follows immediately.
We use the result of Lemma 4.8 to prove the following estimates.
Then we have
This finishes the estimation of I.
Finally, we consider an auxiliary integral over a ball in half space.

Lemma 4.10. Let
Then we have that Proof.

M. A. MUSCHIETTI AND F. TOURNIER
To estimate the term B, note that We have, again by Lemma 4.8, We are now ready for the main results of this section. Then, u ∈ C 2 (B + ).
Proof. The proof is a standard modification of the proof of Lemma 3.4. By using a cutoff function one shows that for |α| ≤ 2m, we have for x ∈ B + : By Lemma 4.11, we have Therefore, By repeating the same proof for free space, this time using the auxiliary integrals in half space, we can prove the following theorem.  Proof. By exactly the same proof as in Lemma 3.5 one shows that for any |α| ≤ 2m except α = 2me n , we have which is a sum of two functions in C δ (B + ) by Theorem 4.12.
Since a ≥ λ and a ∈ C δ (B + ), the result follows.
We now proceed to prove estimates for the operator H. Recall that where the last equality follows since f (w) := e xnw p + (y,iξ ,w) w−z is analytic inside γ − (ξ ) for each fixed z ∈ γ + (ξ ).
We now prove the main estimate for the operator H. Set F (gη ) = L(H(gη )).
Let x,x ∈ B + , and we may assume x n ≥x n . Write Let's estimate the term A. Write
For C 2 , we have We can write dw dz dξ dy, and estimate by the estimations of Lemma 4.10. Also, |x − y| δ (|x − y | + x n + y n ) n dy ≤ CK δ; |x −x| δ , again by Lemma 4.10 and using that x n ≥x n .
The term C 3 is estimated exactly as the term B 3 . This finishes the estimation of the term A.
We have e −ynz ex nw (p(x, iξ , w) − p(y, iξ , w)) a(y)(w − z)p + (y, iξ , z)p − (y, iξ , w) dw dz dξ dy, and dw dz dξ dy, and we estimate, using Lemma 4.8, In exactly the same way we get As for E 2 , we proceed in a similar fashion: dw dz dξ dy and dw dz dξ dy.
We have where dw dz dξ dy.
We have Using the argument at the beginning of the proof of Lemma 4.9 and using Lemma 4.8, we have where
We also have Proof. One shows using Lemmas 4.9 and 4.10 that for α = 2me n , we have D α u ∈ C 2m+δ (B + ).
Then observe that which is a sum of two functions in C δ (B + ) by Theorem 4.14.
Since a ≥ λ and a ∈ C δ (B + ), the result follows.
In order to analyze the boundary values we need two lemmas.  y, iξ , z) .
Proof. Let C R denote the circle of radius R centered at the origin parametrized counterclockwise. Let C − R denote the closed contour which is the part of C R on the left half plane followed by the vertical segment joining −iR with iR and let C + R denote the closed contour which is the vertical segment joining iR with −iR followed by the part of C R on the right half plane.
For k = 0, . . . , m − 1 we have that for large enough R, Hence, where the last equality follows from Cauchy's formula.
Lemma 4.17. Let r > 0 and assume y n > r and 0 < x n < r 2 . Then for any k and 2m + 2l − k ≥ n + 2 we have Proof. Write We have where we have used thatR ≥ R ≥ 1 and 2l ≥ n + 2 − 2m + k.
To finish the proof, we choose R ≥ 1 large enough depending on and r to make |A|, |D| < and then chooseR ≥ R, to make |C| < .
We will use the previous two lemmas to prove  = E 0 (r, x n ) + E 1 (r, x n ).
It follows from Lemmas 3.1 and 4.8 that for some C independent of x, we have |E 0 (r, x n )| ≤ Cr.
Next, we claim We have, using the same argument as in Lemma  , and using Lemma 4.16 we have Q = 0 and hence C = 0. Thus, the claim follows and the proof of the lemma is complete.
We are now ready to prove the main theorem of this section. For g ∈ C δ (B + )), it follows by Theorem 4.12 and Theorem 4.14 that |T (gη )| δ;B + ≤ CK δ, |g| δ;B + .
For to be chosen, define the sequence g k ∈ C δ (B + )) by g 0 = f and g k+1 = f − T (g k η ).
Note that Choosing small enough so that C β−δ K β, < 1, by the contraction mapping theorem we can conclude that there exists g ∈ C δ (B + )) such that g = f − T (gη ).