Regularity Theory. Lihe Wang

Transcription

1 Regularity Theory Lihe Wang

2 2

3 Contents Schauder Estimates 5. The Maximum Principle Approach Energy Method Compactness Method Boundary Estimates on Flat Boundaries Estimates Near the Boundary Barrier Functions Hopf Lemma and Slope Estimates C α Estimates on Lipschitz Domains Boundary Estimates on Curved Domains Patching of Interior Estimates with Boundary Estimates Examples L p Estimates Introduction The geometry of functions and sets Geometry of Hölder spaces L p spaces W 2,p Estimates W,p estimates Harmonic Maps Introduction and Statement of Results Weakly Harmonic Maps Stationary Harmonic Maps C,γ Regularity

4 4 CONTENTS

5 Chapter Schauder Estimates The main topic of this chapter is to show that, due to Schauder, if u = f in B (.) and if f is Hölder continuous, then u is C loc 2,α, i.e. for any i, j n, u ij is Hölder continuous with the same exponent. This is the so-called the Schauder estimates. We will give three closely related proofs. The most elementary and geometric one is the maximum principle approach, followed by the energy method and the most important, flexible compactness method. Let us examine Schauder s theorem more closely. Clearly, it is a perfect theorem: if u was C 2,α, its second order derivatives and hence its Laplacian are C α. Next let us examine the estimates with respect to scaling: Hence it is expected that u(rx) r 2 = f(rx) u(rx) u(0) r u(0) x r 2 f(0). We see that u(rx) u(0) r u(0) x so does the second order derivative satisfies an r 2 equation with right hand side close to a constant and the closeness is getting better as how small the r is. All the proofs are based on this observation. This leads us to look at the questions how we can say a function is C 2,α. Let us recall the definition of continuity: a function is continuous if it is 5

6 6 CHAPTER. SCHAUDER ESTIMATES more and more close to a constant in a smaller and smaller neighborhood, where the closeness is characterized by the fact that the error, which is the popular o(), is smaller than any positive constant. Similarly, the definition of differentiability goes like that a function is differentiable if it is more and more close to a linear function in smaller and smaller neighborhoods, where the closeness to a linear function is characterized by saying that the error, which is the popular o( x ), between the function and the linear function is smaller than any linear function. Now let us recall the definition of C α continuity. A function u is C α at 0 if u(x) u(0) C x α. Here C x α is of course smaller than any constants. It is little bit more than continuity. It gives a qualitative control of the speed of how the function approaches the constant. It seems puzzling about why we use C α and C 2,α. The reason is that like most of mathematics, it is hard to say non qualitative things. Continuity is a very unstable process. However C α is a very stable measurement: any uniform limit of functions with the same Hölder continuity is still enjoying the same Hölder continuity. Similarly we can define C,α, C 2,α. A function is C 2,α at 0, for example, if there is a second order polynomial P (x) such that u(x) P (x) C x 2+α. It says that u can be approximated by a second order polynomial and the approximation is getting better and and better as x goes to 0. P is also the Taylor polynomial at 0 and the main thing is that the qualitative control of error of the approximation as C x 2+α. Now let us examine the difficulty. The main difficulty is that it is hard to get any information about the second order derivatives of the solutions. The equation says and only says that the sum of some second order derivatives is under control. This information certainly is not sufficient for us to determine the second order derivatives. It can be seen later that the second order derivatives actually depend, given the equation, on the boundary values. However, our problem is not to find them. We just want to show that the solution has a second order polynomial approximation and to show the order of approximation is like x 2+α. The trivial and psychologically suggestive case is when f = 0. In this case u is harmonic and we can take P as the Taylor polynomial of u. We

7 .. THE MAXIMUM PRINCIPLE APPROACH 7 have seen from the previous chapter that any order derivatives of a harmonic function are under control. Now consider the case when f is small. First we need a good guess about the second order polynomial. This leads us to the conclusion if f is small, we should able to say the solution is almost harmonic. That is the solution is close to a harmonic function. Then we have the first approximation, which we simply take as the second order polynomial of the harmonic function. Here we has not talk about in what sense that f is small and in what sense the solution is close to harmonic. Actually it turns out that no matter how we measure the smallness or closeness, we always obtain the same conclusion at the end. This equivalence is exactly the content of the Morrey or Campanato embedding theorem. All theorems are equivalent. We will use L norm to measure the smallness when we use the maximum principle approach, H for energy approach and L 2 for the compactness.. The Maximum Principle Approach In this section we give the most elementary proof for Schauder estimates. One of the drawback of this approach is that we have to assume solvability of Dirichlet problem and assume that the solution is C 2. However, we actually show that if u is C 2 than u is C 2,α and moreover that its C 2,α norm has a bound which is independent of of its C 2 norm. In other words, we obtain a C 2,α aprori estimate. We can overcome this shortcoming, if we use the existence theory of viscosity solutions and the corresponding maximum principle. However, in doing that we have to develop a lot of machinery, which we will postpone to later chapters. Lemma. Suppose u is a C 2 solution of u = f(x) in B. Then u(x) sup u + x 2 sup(f) for x B. (.2) B 2n B Proof. Apply the maximum principle to the subharmonic function u(x) sup u x 2 sup(f). B 2n B

8 8 CHAPTER. SCHAUDER ESTIMATES Corollary.2 Suppose u is a C 2 solution of (4.) in B. Then u L (B ) u L ( B ) + 2n f L (B ). (.3) Proof. Apply Lemma. to u, we obtain, u(x) inf u + x 2 inf(f). (.4) B 2n B Clearly (4.3) follows from (4.2) and (4.4). This Corollary says that the solution is in L norm is controlled by a multiple of the L norm of its Laplacian and the L of its boundary data. Now we start to prove the C 2,α estimates. We will show the discrete version first. Lemma.3 For any 0 < α <, there exist constants C 0, 0 < λ <, ε 0 > 0 such that for any function f and solution of u = f in B with u and f L (B ) ε 0, there is a second order harmonic polynomial such that and where C 0 is a universal constant. p(x) = 2 xt Ax + Bx + C u(x) p(x) λ 2+α for x λ A + B + C C 0, Let us first explain why we need the above statement. The essense of the theorem about C 2,α regularity of the solution is that the solution is almost a second order polynomial. We want to show that the solution, after subtract a second order polynomial, decays as C x 2+α. These are clearly infinitely many inequalities, for all x B. Now it comes one of the most important techniques in modern analysis. Instead of showing these uncountablely many inequalities, we show these inequality in countable rings or balls with sizes as a geometric series. We first show that the solution is close to a second polynomial such that the error, measured only in some smaller ball B λ, decays as C x 2+α. The beauty lies in that we show this only up to a fixed size of balls, not all the way up to the origin 0. In this way, we reduce the burden of the proof a lot.

9 .. THE MAXIMUM PRINCIPLE APPROACH 9 Let us explain it further. The following is clearly equivalent: () For all x B, (2) For k = 0,,..., u(x) P (x) C x 2+α. u(x) P (x) Cλ k(2+α) for x λ k. The key is to show (2) for k = first and reduce the general k to k = by scaling. Mathematically () means that the second order Taylor polynomial of u is exactly P at 0. It is very rigid. It is very hard to find such P for general u. The advantage of (2) is that it behaviors well under small errors in P, where as () is rigid. In practice, we will always to prove (2) in the following form, (3) For k = 0,,..., u(x) P k (x) Cλ k(2+α) for x λ k, provided that we could provide the convergence of P k, which in general is the case since the fast convergence of P k to u. P k is ususally born to be convergent or as P k (x) P k (x) 2Cλ k(2+α) for x λ k. We will find P k is an approximation of u in B λ k where f is very much approximated by a constant. Once f is also like a constant, the solution will close to a harmonic function and (3) is proved by taking P k as the Taylor polynomial of the harmonic function. This approximation of P k frees us from the rigidity P and gives us enough freedom so we can find P k. Then the estimates for B λ k(0) gave better and better approximation. These approximations are improved iteratively, which is much easier to an one step shot of (). This key lemma gives approximation in B λ. Later we will repeat the same procedure to estimate B λ 2, B λ 3 and so on. This requires countably many steps. Then there is another beautiful feature of PDEs. They behave nicely under scaling. Scaling, we mean to consider the equation in a blow-up the coordinates v(x) = u(λ k x). In general we can find a similar equation for v as that for u. The estimates from B λ k to B λ k+ is exactly the same as those from B to B λ. This scaling

10 0 CHAPTER. SCHAUDER ESTIMATES consideration reduces the burden of the proof further from countably many steps to one step. Proof of Lemma.3 Let h be the harmonic function such that h = u on B. From Corollary.2, u(x) h(x) 2n sup B f(x). Clearly h in B (0) by the maximum principle and u. Hence its second order Taylor polynomial at 0 p(x) = 2 xt Ax + B x + C has universal bounded coefficients and more over it is harmonic. By the mean value theorem, we have for x B, 2 h(x) p(x) 6 x 3 sup B 2 (x) 3 h. By Theorem 3.8, we have sup 3 h(x) C for some universal constant B 2 (x) C. Finally, we get u(x) p(x) h(x) p(x) + 2n sup f(x) B (0) C 6 x 3 + sup f. 2n Now, taking λ small enough, such that the first term is less or equal to and then, taking ɛ 0 such that the lemma follows. 2 λ2+α for x λ 2n sup f B (0) 2n ɛ 0 λ2+α 2,

11 .. THE MAXIMUM PRINCIPLE APPROACH Theorem.4 Suppose u is a C 2 solution of (4.) and suppose f is Hölder continuous at 0, i.e., f(x) f(0) [f] C α(0) = sup x x α < +. Then u(x) is C 2,α at 0, i.e. there is a second order polynomial P (x) = 2 xt Ax + B x + C such that u(x) P (x) D x 2+α for x, with D C 0 ([f] C α(0) + f(0) + u L (B ) ) and ) A + B + C C 0 ([f] C α (0) + f(0) + u L (B ), where C 0 is a universal constant depending only on the dimension. Proof. We first discuss the normalization of the estimates. First, we may assume f(0) = 0 otherwise, let v(x) = u(x) f(0) 2n x n. Then v = f(x) f(0). Clear the estimates of v will translates to that for u. Second, we can assume u L (B ) and [f] C α (0) ε 0, where ε 0 is the constant in Lemma.3. There are two ways of realizing this normalization. The linear method: consider u v(x) = ε 0. u L (B ) + [f] C α (0) Clearly we have v L (B ) and [ v] C α (0) ε 0. Again the estimate of v translates to that for u. The nonlinear method: consider v(x) = u(rx) u L (B ) +. Actually, the division of u L (B )+ is also depends on the linear structure. We say this is a nonlinear method because of the scaling u(rx). We will see later that many nonlinear equations behaves well under this scaling.

12 2 CHAPTER. SCHAUDER ESTIMATES Hence We see that v = r 2 f(rx) u L (B ) +. [ v] C α (0) r 2 α [f] C α (0) which can be smaller than ε 0 be taking fixed and small r. Now, we prove the following inductively: there are harmonic polynomials, p k (x) = 2 xt A k x + B k x + C k such that and u(x) p k (x) λ (2+α)k for x λ k A k A k+ C λ αk B k B k+ C λ (α+)k C k C k+ C λ (α+2)k. First we see that k = 0 is the normalization condition on u and k = is exactly the previous lemma. Let us assume it is true for k. Let Then w(x) = (u p k)(λ k x) λ (2+α)k. w = f(λk x) λ αk and w in B. Now, we apply the previous lemma on w. There is a harmonic polynomial p with bounded coefficients, such that provided Now, we scale back, w(x) p(x) λ 2+α for x λ, f(λ k x) sup x λ αk [f] C α(0) ε 0. u(x) p k (x) λ (2+α)k p 0 ( x λ k ) λ (k+)(2+α).

13 .2. ENERGY METHOD 3 Clearly we proved the (k + )-th step by letting p k+ (x) = p k (x) + λ (2+α)k p 0 ( x λ k ). Finally, it is elementary to see that A k, B k, C k converge to A, B, C respectively and the limiting polynomial satisfies p (x) = 2 xt A x + B x + C p k (x) p (x) Cλ k(2+α) for any x λ k. Hence for λ k+ x λ k, u(x) p (x) u(x) p k (x) + p k (x) p (x) λ k(2+α) + C x 2+α = λ 2+α λ(k+)(2+α) + C x 2+α λ 2+α x 2+α + C x 2+α = ( λ 2+α + C) x 2+α..2 Energy Method This section is parallel to 4.. Since we will use energy method, our methods in this section apply to weak solutions. Consequently, this yields a regularity theory for weak solutions, which is stronger than a-priori estimates. However, it is parallel to the previous section. The only difference between these two sections are the ways we measure the approximations: we measure it by L for u and f in the maximum principle approach while we use H semi-norm for u and L 2 norm for f in this section. We see that, by Theorem.2 and.3, the Hölder continuity does not depend on the way we measure the approximation of u from a second order polynomial once we prove the estimates in a domain. Lemma.5 Suppose u is a H weak solution of { u = f in B (0) u = 0 on B (0).

14 4 CHAPTER. SCHAUDER ESTIMATES Then u 2 C f 2. B B Lemma.6 For any 0 < α <, there exist C 0, 0 < λ < and ε 0 > 0 such that for \ B u 2 and u is a solution of (4.), there is a second order harmonic polynomial p(x) = 2 xt Ax + B x + C such that λ 2 \ Bλ (0) (u p) 2 λ 2(2+α) and A + B C 0 a universal constant, provided \ B f 2 ε 2 0. Proof. Let v be the solution of { v = f v = 0 on B (0). By the previous lemma, we have v 2 C f 2. B B Let h = u v. Then h is harmonic. Clearly, we have h 2 2 B u B v 2 2 B, B and consequently, h(x) h(x) 2 C h 2 Cfor B x 2 and 2 p + p (0) C 0 for some universal constant C 0. Now, let p(x) be the second order Taylor polynomial of h at 0. We have (h p)(x) C 0 x for x 2.

15 .2. ENERGY METHOD 5 Therefore, we obtain, λ 2 \ Bλ (u p(x)) 2 dx 2λ 2 \ Bλ (u h) 2 dx + 2λ B 2 (h p) 2 dx λ 2λ 2 n C0 2 f 2 dx + 2λ 2 Cλ 4 B λ. B Now, taking λ small such that the second term is less than or equal to 2 λ2(2+α) B λ, and then taking ε 0 such that the first term is less than or equal to the lemma follows. 2 λ2(2+α) B λ, Theorem.7 Suppose u is a weak solution of (4.) and suppose f is Hölder continuous at 0 in the L 2 sense, i.e. [f] L 2;α(0) =: sup 0<r r α \ f(x) f(0) 2 < +, Br where f(0) is defined as f(0) =: lim r 0 \Br f. Then u(x) is C 2,α at 0 in H semi-norm, i.e., [u] H ;2+α(0) = inf sup r α \ 0 P Br (u(x) P 2 (x)) 2 < +, 2 0<r where P 2 is taken over the set of second order polynomials. Moreover the coefficients of P 2 and [u] H ;2+α(0) are bounded by [f] L 2;α(0) and u 2 dx 0 B as [u] H ;2+α(0) C([f] L 2;α(0) + ( u 2 dx) 2 ). 0 B Proof. It is exactly the translation of the proof of Theorem.4, while we have to use Lemma.6 repeatedly. See also the proof of Theorem.. We can do the normalization as before so that [f] L 2;α(0) is small and \ B u 2 dx. Now, we prove the following inductively: there are harmonic polynomials, p k (x) = 2 xt A k x + B k x + C k

16 6 CHAPTER. SCHAUDER ESTIMATES such that λ 2k \ Bλ (u p k ) 2 λ 2k(2+α) k and A k A k+ C λ αk B k B k+ C λ (α+)k. Let us assume it is true for k. Let Then k = 0 is the condition on u. k = is exactly the previous lemma. w(x) = (u p k)(λ k x) λ (2+α)k. w = f(λk x) λ αk and \ B w 2 dx Now, we apply the previous lemma on w. There is a harmonic polynomial p with bounded derivatives, such that λ 2 \ Bλ (w p) 2 λ 2(2+α) provided Now, we scale back, \ B f(λ k x) 2 λ 2αk ([f] L 2;α(0)) 2 ε 2 0. ( x λ 2(k+) \ Bλ (u(x) p k (x) λ (2+α)k p 0 )) 2 k+ λ k λ 2(k+)(2+α). Clearly we proved the (k + )-th step by letting p k+ (x) = p k (x) + λ (2+α)k p( x λ k ). Finally, it is elementary to see that A k, B k converge to A, B respectively and the limiting polynomial p (x) = 2 xt A x + B x

17 .3. COMPACTNESS METHOD 7 satisfies λ 2k) \ Bλ (P k p ) 2 Cλ 2k(2+α). k Hence for λ k+ r λ k, r 2 \ Br (u p) 2 2r 2 \ Br (u P k ) 2 + (P k p) 2 2 \ Bλ (u P k ) 2 + (P k p) 2 k ( + C)λ 2k(2+α) = + C λ + C λ 2(2+α) r2(2+α)..3 Compactness Method 2(2+α) λ(k+)(2+α) In this section, we discuss the most important method in nonlinear analysis the compactness method. It has wide applications. It is an extremely powerful tool for the regularity theory. This method was discovered by De- Giorgi in the 60 s when he studied Plateau problem. The method presented here applies to weak solutions and hence gives a regularity theory, which is somewhat better than a priori estimates. The big advantage of compactness method is its potential application to nonlinear equations. The maximum principle and energy methods have their limitations, namely we have to use the solvability of Dirichlet problems and more seriously, the equation of the difference of the solution and its approximation. We will come back to this point when we deal with nonlinear equations. Lemma.8 Suppose u = f in B. Then for any smooth function η with compact support in B, we have η 2 u 2 dx B η 2 f 2 dx + B (4 η 2 + η 2 )u 2 dx. B Proof. This is Caccoippoli type inequality. Multiplying the equation by η 2 u, we have η 2 u( u) = η 2 ufdx.

18 8 CHAPTER. SCHAUDER ESTIMATES As before, integrating by parts, we have η 2 u 2 = 2 η 2 uf 2 η 2 f η η u u η 2 u η 2 u η 2 u 2. Cancelling 2 η 2 u 2, we obtain the inequality immediately. The following theorem uses the so-called compactness argument. It is also called the indirect method or blow-up method. The major advantage of this method compared with the maximum principle method and the energy method is that it requires no sovabilty of the Dirichlet problem. Theorem.9 For any ε > 0, there exists a δ = δ(ε) > 0such that for any weak solution of u = f in B with \ B u 2 dx and \ B f 2 δ 2, there exists a harmonic function h such that B 2 u h 2 ε 2. Remark Later it will be clear that the argument of this lemma does not depends on the linear structure of the equation. It can be generalized to solutions of different equations if the two equation are close. Proof. We prove it by contradiction. Suppose there exist ε 0 > 0, u n and f n with u n = f n \ B u 2 n, \ B f n 2 n, so that for any harmonic function h in B, 2 B 2 u n h 2 dx ε 2 0. By the previous lemma, B 2 u n 2 dx C.

19 .3. COMPACTNESS METHOD 9 Hence {u n } has a subsequence, which we still denoted as u n, such that u n u weakly in H (B ) and 2 u n u strongly in L 2 (B ). 2 We will show that u itself is harmonic in B 3, which is a contradiction. 4 Actually, for any test function ϕ C0 (B 3 ), 4 ϕ u n = ϕf n dx. Letting n go to infinity, we have ϕ u = 0. Hence u is harmonic! The rest of the proof for C 2,α is almost the same as that of the previous section. We state it as the following. Lemma.0 For any 0 < α <, there exist constants C 0, 0 < λ <, δ 0 > 0 such that for any function f and solution of u = f in B with \ B u 2 dx and \ B f 2 dx δ0, 2 there is a second order harmonic polynomial p(x) = 2 xt Ax + Bx + C such that \ Bλ u p(x) 2 λ 2(2+α) and where C 0 is a universal constant. A + B + C C 0, Proof. The proof is the same as that of Lemma.6. Let h be the harmonic function of the previous lemma with some ɛ < to be determined. Hence we have h 2 2 B 2 u B u h ε 2 4. B 2 By the estimates of harmonic functions, we have 3 h(x) 2 C \ B u 2 C for x 4.

20 20 CHAPTER. SCHAUDER ESTIMATES Now, let p(x) be the second order Taylor polynomial of h at 0. We have (h p)(x) C 0 x 3 for x 4. Therefore for each 0 < λ < 4, we have, \ Bλ u p 2 dx 2 \ Bλ u h 2 dx + 2 2ε2 B λ + 2C2 0λ 6. \ Bλ h p 2 dx Now, taking λ small such that the second term is less than or equal to 2 λ2(2+α), and then taking δ = δ 0 so that ε so small that the first term is less than or equal to 2 λ2(2+α), and the lemma follows. Theorem. For each 0 < α < and the dimension n, there is a constant C 0 so that for all weak solution u of u = f in B for [f] L 2;α(0) < then there exists a second order polynomial P (x) = 2 xt Ax + Bx + C such that \ Br u P (x) 2 C r 2(2+α) where the constant C depends on B u 2 dx and [f] L 2;α(0). Moreover and P = f(0) ) C 2 + A 2 + B 2 + C 2 C 0 ([f] L 2;α(0) 2 + f(0) 2 + u 2 dx, B where f(0) is defined as before. Proof. We can do the normalization as before so that f(0) = 0 and [f] L 2;α(0) δ 0 and \ B u 2 dx. Now, we prove the following inductively: there are harmonic polynomials, p k (x) = 2 xt A k x + B k x + C k

21 .3. COMPACTNESS METHOD 2 such that \ Bλ u p k 2 λ 2k(2+α), k and A k A k+ C λ αk B k B k+ C λ (α+)k C k C k+ C λ (α+2)k. Let us assume it is true for k. Let Then k = 0 is the condition on u. k = is exactly the previous lemma. w(x) = (u p k)(λ k x) λ (2+α)k. w = f(λk x) λ αk and \ B w 2 dx Now, we apply the previous lemma. There is a harmonic polynomial p with bounded coefficients, such that \ Bλ u p 2 λ 2(2+α) provided f(λ \ k x) 2 B λ 2αk [f] L 2;α(0) 2 δ0 2. Now, we scale back, ( x ) \ Bλ u(x) p k (x) λ (2+α)k p k+ λ k 2 λ (k+)(2+α). Clearly we proved the (k + )-th step by letting p k+ (x) = p k (x) + λ (2+α)k p 0 ( x λ k ). Finally, it is elementary to see that A k, B k, C k converge to A, B, C respectively and the limiting polynomial p (x) = 2 xt A x + B x + C

22 22 CHAPTER. SCHAUDER ESTIMATES satisfies p k (x) p (x) Cλ k(2+α) for any x λ k. Hence for λ k+ r λ k, \ Br u(x) p (x) 2 2 \ Br u(x) p k (x) 2 + P k (x) p (x) 2 2 \ Bλ u(x) P k (x) 2 + P k (x) p(x) 2 k ( + C)λ 2k(2+α) + C λ 2(2+α) r2(2+α). We remark that sometimes the following definition is also convenient. Definition where ( [f] L 2;α(0) = sup 0<r r α \ Br f f Br 2) 2, f Br = \ Br f. Now we have showed C 2,α estimates by three methods all of which are the second order polynomial approximation. We can use the same method to prove C α, C,α and any C k,α estimates for the solution if we replace the second order polynomial approximation to constant, linear and k th order polynomial approximations. Definition 2 [f] L 2; +α(0) = sup 0<r ( \ r +α Br f 2) 2. We remark that we don t have to normalized by averages or f(0) in our defintion. Such average doesn t converge when [f] L 2; +α(0) <. We also remark that from the Hölder inequality we have that if f is in L p for p > n, then [f] L 2; +α(0) f L p (B ), for α = n p. Let me state two of the theorem that our methods apply.

23 .4. BOUNDARY ESTIMATES ON FLAT BOUNDARIES 23 Theorem.2 Suppose u is a weak solution of u = f in B and [f] L 2, +α(0) <, then there exists a linear function L(x) = Bx + C such that \ Br u L(x) 2 C 0 r 2(+α) where the constant C 0 depends on B u 2 dx and [f] L 2, +α(0). Moreover ( ) C 0 + B 2 + C 2 C [f] 2 C +α + u 2 dx. L 2 B Definition 3 [f] L 2; 2+α(0) = sup 0<r ( \ r 2+α Br f 2) 2. We remark that if f is in L p for n > p > n 2, then for α = 2 n p. [f] L 2; 2+α(0) f L p (B ), Theorem.3 Suppose u is a weak solution of u = f in B then there exists a constant A such that \ Br u A 2 C 2 0r 2α where the constant C 0 depends on B u 2 dx and [f] L 2; 2+α(0). Moreover ) C0 2 + A 2 C ([f] 2 L 2; 2+α + u 2 dx. B We also remark that we can prove similar theorems by using [f] L p; 2+α(0) for any p. In order to carry out the argument the only estimates that we need is an estimate of u by the L p norm of u..4 Boundary Estimates on Flat Boundaries Let us start with an example of boundary estimates when the boundary is flat. We will use the L approach since it has the best geometric intuitions.

24 24 CHAPTER. SCHAUDER ESTIMATES We will denote x = (x, x n ) and R n + = {x : x n 0}, T r = {x R n : x < r} T r (x 0) = T r + x 0 for x 0 R n B r + = B (0) R n + B r + (x 0 ) = B r + (x 0, 0). Lemma.4 Suppose u in weak solution of u = 0 in B + and u = 0 on T and u(x) in C +. Then u is smooth in B+. Moreover 2 for x B +. 2 D α u(x) C α Proof. The conclusion is trivially proved by the odd extension, which is also weakly and therefore classically harmonic in B. Lemma.5 Suppose u = 0 in B + u L (B + ) u(x, 0) = ϕ(x ) on T. (.5) Then there exist constants a and a universal constant C such that C x 2 + inf T ϕ u(x) ax n C x 2 + sup T ϕ. Proof. Let v be the harmonic function (provided by the Poisson integral formula) v = 0 v T = sup ϕ. T v +(B + ) = u Clearly v u and u v sup ϕ + inf ϕ. T T

25 .5. ESTIMATES NEAR THE BOUNDARY 25 The lemma follows by applying the previous lemma to the function v sup ϕ and the fact that T v(0) = (0,, 0, n v(0)) =: (0,, 0, a). Corollary.6 For any 0 < α < there are universal positive constants ε 0 and λ < so that for any solution of u = 0in B + u L (B + ) there are constants a and b such that provided osc T ϕ ε 0. u(x, 0) = ϕ(x ) on T, u(x) ax n b L (B + λ ) λ+α An iteration of this corollary proves the following theorem. Theorem.7 Suppose u = 0 in B + (0) and ϕ(x ) = u(x, 0) is C,α at 0. Then u(x) is C,α at 0. Moreover ( ) [u] C,α(0) C [ϕ] C,α(0) + u L (B ). We can similarly prove C k,α, for k >, estimates at the boundary..5 Estimates Near the Boundary In this section, we introduce the basic tools for boundary estimates. We will also prove C α boundary estimates on Lipschitz domain, which is a very important class of domain since they are invarant under scaling..5. Barrier Functions Let us start out this section with some discussion on the most important function for elliptic equations: the Newtonian potential, x 0 (n 3) x n 2 p(x) = ln x x 0 n = 2.

26 26 CHAPTER. SCHAUDER ESTIMATES As seen in previous chapter, p(x) = C n δ 0 (x), where δ 0 (x) is the delta function with pole at 0. We will use the following function repeatedly: 0 < r, R <. h r,r (x) = p(x) p(r) p(r) p(r). Clearly, h r,r is harmonic in between B R and B r and h r,r = 0 on B R and (.6) h r,r = on B r. (.7) It is also clear that, if r < R, h r,r is convex in the radial direction and h r,r n C r,r < 0. (.8) As showed in the previous section that harmonic functions takes boundary data as regularly as the boundary data..5.2 Hopf Lemma and Slope Estimates The following important theorem (according to tradition we call it Lemma) says that the normal derivative of a harmonic function depends on the values of u far away in a qualitative way. This also tells us the information between the values of a harmonic function is infinitely fast and the first order derivative of a harmonic function is lifted if we lift the function a little bit somewhere else. Lemma.8 (Hopf boundary point lemma) Suppose u = 0 in B, u 0 in B. If u(x 0 ) = 0 for x 0 B then u(x) C x x 0 u(0) along the radial direction of x 0 for some C depending only on the dimension. Proof. From Harnack inequality, we have u(x) cu(0) for x 2. Hence, we have u(x) cu(0)h,(x) 2 by applying the maximum principle to the annulus { 2 x }. The lemma follows immediately from the property of h,(x). 2

27 .5. ESTIMATES NEAR THE BOUNDARY 27 Lemma.9 Suppose Ω is a domain such that B Ω = and x 0 B Ω. Let u = 0 in Ω with u in Ω with u 0 on Ω B 2. Then u(x) C x x 0 for x, where C is a constant depending only on the dimension. Proof. Applying the maximum principle to the function on the domain Ω B 2, we have u(x) h 2, (x) u(x) h 2, (x) in Ω B 2. The lemma follows immediately since h 2, (x) is a smooth function. Lemma.8 and.9 the control of the bebavior of the harmonic function near the boundary. One immediate consequence of Lemma.9 is the following gradient estimates. Lemma.20 Suppose Ω is a domain such that B Ω = and x 0 B Ω. Let u = 0 in Ω with u in Ω with u = 0 on Ω B 2. Then if u is differentiable at x 0, u(x 0 ) C. Proof. We apply Lemma.9 to u and u to obtain The estimates follows. u(x) C x x 0 for x B 2 Ω. A nice application of Lemma.20 is that it can easier adapted to situations that a domain is a small perturbation from a hyperplane and the boundary is also approximately constatant. Lemma.2 Suppose u and subharmonic in B 3 Ω. If the domain is flat as B 3 {x n ɛ} B 3 Ω B + 3, then there is a universal constant C so that u(x) sup u + C( ɛ + x n ). B 3 Ω

28 28 CHAPTER. SCHAUDER ESTIMATES Proof. Without lose of generality, we may assume that sup B3 Ω = 0. For each y T = B {x n = 0}, we see that B (y, ) is outside Ω. Now let t be the maximum of 0 t so that the balls B (y, t) is outside Ω. From the flatness condition we see that 0 t ɛ and B (y, t) Ω contains at least one point, say y 0 with y o y C ɛ. Now we apply Lemma.20 (with B replaced by B (y, t)) to obtain u(x) C x y 0. Therefore u(0, x n ) C( x n (y 0 ) n + y 0 ) C(x n + ɛ). Exercise Consider the domain C α = {(x, y): angle((x, y), (, 0)) < π α }. Suppose { u(x) = 0 in Cα B u(x) = 0 on ( C α ) B. Prove that. u is Lipschitz at 0 if and only if α. 2. u is C k (0) if and only if α is an integer or α k..5.3 C α Estimates on Lipschitz Domains Theorem.22 Suppose Ω is a Lipschitz domain. Let 0 Ω. Suppose ϕ(x) is C α at 0: Then any solution of [ϕ] C α(0) = ϕ(x) ϕ(0) sup 0< x x α < +. x Ω { u = 0 in Ω u = ϕ(x) on ( Ω) B is Hölder continuous at 0 with exponent β = β 0 α for some β 0 depending on the Lipschitz character of the domain. The only thing we need to show for C α regularity is that the oscillation of the solution is smaller if we restrict the solution to a smaller domain. Proof. Without loss of generality, we may suppose Ω is a Lipschitz graph in the x n direction and Ω is in the x + n direction. Suppose its Lipschitz

29 .5. ESTIMATES NEAR THE BOUNDARY 29 constant is K. From trigonometry, B r0 (0, /3) Ω c for r 0 = 3 K 2 +. Next, as usual, we may suppose, u(0) = 0, [ϕ] C α ( Ω)(0) ε 0 and u(x) in B (0). Applying the maximum principle to the function on the domain u(x) sup u h r0,2/3(x (0, /3)) ( Ω) B Ω B 2/3 ((0, /3)), we obtain u(x) sup u + h r0, 2 (x (0, Ω B 3 3 )). By a symmetric argument, we can obtain Hence where u(x) If Osc[ϕ] γ 2, then inf u h r0, 2 (x, (0 Ω B 3 3 )). u(x) Osc[ϕ] + γ for x 6, γ = sup h r0, 2 (x) <. x u(x) + γ 2 Now, letting γ 2 = max( 6 α, γ ) we will prove sup B and the theorem follows. We will prove it inductively as 6 k (0) u γ sup u osc B (0) 6 k B (0) 6 k [ϕ] C α(0) =: γ < k 2 [ϕ] + γ sup B 6 k (0) + γγk 6α(k ) It follows immediately by if [ϕ] C α(0) is small enough. u (.9) 2 (.0) ([ϕ] C α(0) + γ)γ k 2. (.)

30 30 CHAPTER. SCHAUDER ESTIMATES.6 Boundary Estimates on Curved Domains A combination of the methods in. to.6 provides different proof for the following theorems. Theorem.23 Suppose Ω is a Lipschitz domain, 0 Ω. ϕ is C,α at 0 and Ω is C,α at 0. Let u be a solution of { u = f(x) in B Ω u = ϕ on ( Ω) B. Then u is C,α at 0 if [f] C,α (0)<. Moreover ( ) [u] C,α(0) C [f] C,α(0) + [ϕ] C,α(0) + u L (B Ω), for some C depending only on [ Ω] C,α and the dimension. Theorem.24 Suppose Ω is a Lipschitz domain, 0 Ω. ϕ is C 2,α at 0 and Ω is C 2,α at 0. Let u be a solution of { u = f(x) in B Ω u = ϕ on ( Ω) B. Then u is C 2,α at 0 if [f] C α(0) <. Moreover ( ) [u] C 2,α(0) C [f] C α(0) + [ϕ] C 2,α(0) + [u] L (B Ω), for some C depends only on [ Ω] C 2,α and the dimension. Here we will outline the proof using the compactness method. Lemma.25 If u is a weak solution of { u = f(x) in B Ω u = ϕ on ( Ω) B. Then for any η C0 (B ) we have η 2 u 2 η 2 f 2 + C B Ω B Ω B Ω ( η 2 + η 2 ) u 2 + C ϕ 2 H 2 ( Ω B ).

31 .6. BOUNDARY ESTIMATES ON CURVED DOMAINS 3 Lemma.26 For any ɛ > 0 there is a δ > 0 so that if u is a solution of { u = f(x) in B Ω u = ϕ on ( Ω) B. then there is a harmonic function h in B + with h(x, 0) = 0 so that 2 u h 2 ɛ 2, B 2 Ω provided the following set of smallness condition f 2 δ 2, ϕ 2 H 2 δ, ( Ω B ) B Ω and a flatness condition on the boundary as B Ω B + and {x B : x n δ} Ω. The proof of this is an application of the compactness argument. C 2,α and C,α estimates will follow by approximations as before. One can also use the maximum principle approach by the estimates in Lemma.2. We prove the key approximation. Lemma.27 Suppose u is a solution of { u = f(x) in B3 Ω u = ϕ on ( Ω) B 3. with u and f δ and the domain is flat as B 3 Ω B + 3 and {x B 3 : x n ɛ} Ω. then there is a harmonic function h in B + with h(x, 0) = 0 so that u h C( ɛ + δ) in B Ω. Proof. From Lemma.2 applied to u ± δ 2n x 2, we see that u C(x n + ɛ + δ). (.2)

32 32 CHAPTER. SCHAUDER ESTIMATES (We notice that this inequality will be enough for C α estimates). In order to improve the estimates, we consider the harmonic h with boundary conditions as h = 0in B h(x) = C( ɛ + δ) for x B + \Ω h(x) = u for x B Ω, and by the maximum principle and with estimate.2, we have h C h, (x (y, 2 2 )) C( ɛ + δ) in B (y, 2 ) B+, for some constant C and each y. Therefore h C ɛ for x n = ɛ. We apply the maximum principle to u h in the domain B {x n ɛ} to yield the estimates for x n ɛ. u h 2C( ɛ + δ),.7 Patching of Interior Estimates with Boundary Estimates We only discuss estimates in L -norm. For L 2 and H norms, there are corresponding patching theorems. Theorem.28 Let S be a set of functions defined on a closed bounded set Ω. Suppose the following hold: () (Boundary Estimates) For any y Ω, there exists a polynomial P y of degree k such that u(z) P y (z) E z y k+α, z Ω. with σ k D σ P (z) E. (2) (Interior Estimates) For any x Ω 0, letting d x = d(x, Ω), then there exists a polynomial P of degree k such that D σ p(x) Cd σ x u L (B dx2 ) + Bd k σ 2+α x (.3) (u p x )(z) ( u L (b dx2 ) ) A d k+α + D z x k+α for z x dx x 2 (.4).

33 .8. EXAMPLES 33 (3) (Invariance) For any u S, P x in (), v = u P also satisfy the estimates of (2). Then S C k,α (Ω). In fact S B M C k+α (Ω) for some M depending on A, B, C, D and E. Proof. For any x Ω 0, there exists an y Ω such that d(x, y) = d x. Use (2) for y, we have u(z) P y (z) E z y k+α for z Ω. Applying () to u P y (z), we have u(z) P y (z) P x (z) (AE + D) z x k+α for z x < dx 2. Clearly, for z x dx 2, u(z) P y (z) P x (z) E y z k+α + P x (z) (.5) C (dx) k+α (.6) C z x k+α. (.7).8 Examples Let us review our technique in this section. Example Let u be a solution of { u = f(x) in Ω u Ω = ϕ(x) on Ω.. If ϕ C α ( Ω), Ω C, f L < +, then u C α (Ω). 2. If ϕ C,α ( Ω), Ω C,α, f L (Ω), then u C,α (Ω).

34 34 CHAPTER. SCHAUDER ESTIMATES 3. If ϕ C 2,α ( Ω), Ω C 2,α, f C 2,α (Ω), then u C 2,α (Ω). Proof. These are immediate consequences of scaled interior estimates, boundary estimates and the technique of patching. Example 2 Let u be a solution of { u = f(u) u Ω = ϕ(x) if u L (Ω) C and ϕ, f C then u C (Ω). Proof. Homework. Exercise Suppose { u = f(x) L (B ) u L (B ) < +. Show for any 0 r. ( ) u L (B r) C r u L (B ) + r f L be the charac- Exercise (Hu and Wang) Let Ω B be a C,α domain χ Ω teristic function of Ω. Suppose { u = χ Ω in B u L (B ) < +. Then u C 2,α (Ω) and u C 2,α loc (B Ω). Hint Using u L (B ) C u L (B ) if u B = 0. Most of the techniques for the Dirichlet problem can be extended to Neumann boundary value problems. However, the following problem is a

35 .8. EXAMPLES 35 negative result for mixed value problems. Exercise Find a function u in B + such that u = 0 in B + u = 0 on T {x n 0} u n = 0 on T {x n > 0} and u C 2 (B + ) but not in C 2 +ε (B + ) for any ε >

36 36 CHAPTER. SCHAUDER ESTIMATES

37 Chapter 2 L p Estimates In this chapter we will show a proof of the classical Calderón Zygmund estimates established in [2] 952. The estimates are among the fundamental estimates for elliptic equations. 2. Introduction In this paper, we will mainly discuss the theorem, Theorem 2. (Calderón Zygmund) If u is a solution of then B D 2 u p C u = f in B 2 (2.) ( ) f p + B 2 u p B 2 for any < p < +. (2.2) The analytical tools in this chapter are the maximal function, energy estimates and Vitali Covering Lemma. We recall the basic notations that we use. B r = {x R n : x < r}, Q r = {x = (x,..., x n ) R n : r < x i < r} and B r (x) = B r +x, Q r (x) = Q r +x. For any measurable set A, A is its measure. For any integrable function u, we denote the average of u as ū A = u = u. A A A The classical proof of Calderón Zygmund estimates, uses the singular integrals 2 u (x) = w ij (y) f (x y) dy (2.3) x i x j R n 37

38 38 CHAPTER 2. L P ESTIMATES where w ij is a homogeneous function of degree n with cancellation conditions. The approach involves an L 2 L 2 estimate and an L to weak-l estimate. See details in the book of Stein [?]. Our approach is more elementary. It gives an unified proof for elliptic, parabolic and subelliptic operators. Our proof is built upon geometrical intuitions. Our basic tools in this approach are the standard estimates for the equations, the Vitali covering lemma and Hardy Littlewood maximal function. Our approach is very much influenced by [4] and the early works in [3] and [29], in which the Calderón Zygmund decompositions were used. Here we will use the Vitali covering lemma. Analytically the difference between the Calderón Zygmund decomposition and Vitali covering lemma is not quite essential but subtle. One is on cubes and the latter is on balls. However we hope that Vitali covering lemma can easily adapted to more complicated situations where balls can be easily defined. 2.2 The geometry of functions and sets 2.2. Geometry of Hölder spaces We should start out with a geometric description of Hölder space which is the key to visualize the estimates. First of all, the geometry of u L (B ) is that the graph of u is in the box B [, ]. This gives a very mild control of u. The Hölder norm of u is actually very geometrical. Let us recall that u is Hölder with [u] C α (B ) if u (x) u (y) x y α for all x, y inb. (2.4) Geometrically, the graph of u is not a box anymore rather than a surface which is away from spikes: x α. That is, if (x 0, y 0 ) is on the graph of u, then all the points (x, y) with y y 0 > x x 0 α or y y 0 < x x 0 α are not on the graph. This actually leads us to say that if u is C α at x 0 if L p spaces u (x) u (x 0 ) C x x 0 α. The information carried by the L p norm of a function is not as local as by the Hölder norms. In contrast to Hölder spaces, there is no way to say a function is in L p function at a point.

39 2.2. THE GEOMETRY OF FUNCTIONS AND SETS 39 In order to examine the information carried by L p norm of a function, let us recall the formula, If Ω Ω u p dx = p λ p {x Ω : u > λ} dλ. (2.5) 0 u p dx =, one will have that {x Ω : u > λ} λ p, (2.6) i.e., the measure {x : Ω : u > λ} decays for λ large. This tells us that if we randomly choose a point x, then the probability for u(x) > λ is small for λ large. The identity (2.5) shows the decay of {x : Ω : u > λ} in a precise way and this decay is the only information carried by the L p norm. We also observe that the faster this probability decays the bigger the p is. Now let us discuss how we can show a function is in L p. First we see that one has to prove the decay of { u > λ}. As in the Hölder estimates, we should prove this decay inductively. A reasonable expectation of this sort is to prove: { u > λ 0 } ε { u > }. (2.7) The smaller ɛ or λ 0 is, the faster the decay is. Here one should realize that this estimate should be scaled to { u > λ 0 λ} ε { u > λ} (2.8) with proper conditions on the data. As in the Hölder space case, one should expect that an inductive argument proves the decay. The W 2,p theory of (2.) says that D 2 u is in L p if u is. Hence a reasonable expectation of an inductive estimate could be } ( { ) { D 2 u > λ 0 ε D 2 u > } + { f > δ0 }. (2.9) Here we can scale (2.9) to { } ( { ) D 2 u > λ 0 λ ε D 2 u > λ} + { f δ0 λ} (2.0)

40 40 CHAPTER 2. L P ESTIMATES which is the so-called good-λ inequality. It is elementary to see that (2.9) implies the estimate. This expectation (2.9), however, is not true. One reason for the failure of (2.9) is that the condition D 2 u(x 0 ) (2.) is unstable in the setting of W 2,p theory. Although (2.9) is not true, its modification (2.5) is true. The key modification is provided by one of the treasures in analysis: the Hardy Littlewood maximal function. For a locally integrable function v defined in R n, its maximal function is defined as Mv (x) = sup v dl n. (2.2) r>0 B r(x) We also use M Ω v (x) = M(vχ Ω ) (x), if v is not defined outside Ω or equivalently we replace or extend v by 0 outside Ω. We will drop the index Ω if Ω is understood clearly in the context. We can also define the maximal function by taking the supremum in cubes. Mv (x) = sup v dl n. (2.3) Q r(x) Q r(x) It is clear that, Mv C Mv CMv. We will use the maximal function Mv defined in (2.2) on balls in this paper. The basic theorem for Hardy Littlewood maximal function is the following: Theorem 2.2 M (v) (x) L p (Ω) C v L p (Ω) for any < p +. {x Ω : Mv(x) λ} C λ v L (Ω). The first inequality is call strong p p estimates and the second is call weak estimates. This theorem says that the measures of { v(x) > λ} and {Mv(x) > λ} decay roughly in the same way. However Mu(x) is much more stable and geometrical than u(x) if u is merely an L p function. The reason is that Mu is invariant with respect to scaling. Another aspect

41 2.2. THE GEOMETRY OF FUNCTIONS AND SETS 4 of the maximal function is that {Mu λ} and { u λ} have roughly the same measure. Likewise we will replace (2.) by ( M D 2 u 2) (x). (2.4) If M D 2 u 2 (x 0 ), one would see that D 2 u (x) is really at x 0 in all scales in the sense of L 2. In fact we will show that B { M D 2 u 2 > λ 2 0} ε ( B { ) M D 2 u 2 > } + B {M(f 2 ) > δ0} 2 (2.5) where δ 0 can be taken as small as possible since it is about the data. Now comes the question, how can we prove (2.5)? Let s first examine a limiting case that { M D 2 u 2 > } has big measure, for example B { M D 2 u 2 > } = B, i.e., { M D 2 u 2 } has measure 0 or is a point, say. This further reduces the question to the following. Under what condition on the data f, and M D 2 u 2 at some point, can we conclude that { M D 2 u 2 > λ 2 0} has small measure? Let us again look at a limiting case of this question in which f = 0. However the answer is clear in this case since now u is harmonic. The condition that M D 2 u 2 at some point, provides a bound, say N 0 on D 2 u and hence { M D 2 u 2 > N0 2 } = which has measure 0. If f is small, then { M D 2 u 2 > N0 2 + } will have small measure by the standard energy estimates. Namely that (N0 2 + ) {x : M D 2 u 2 > N0 2 + } C D 2 u 2 C f 2 <<, which is exactly what we want. Let us go back to the general case of (2.5). First of all one should understand Lemma 2.7 by its scaling invariant form: which says that if the density of the set { M D 2 u 2 > λ 0 } in a ball B: { M D 2 u 2 > λ 0 } B B ɛ, then B { M D 2 u 2 > }. One immediately observes that if one can cover the set by balls (or by cubes as in the Calderón-Zygmund decomposition) so that the density { M D 2 u 2 > λ 0 } B B = ɛ

42 42 CHAPTER 2. L P ESTIMATES and at the same time these balls are disjoint, then clearly B is ɛ times as big as { M D 2 u 2 } > λ 0 B in this situation. We conclude that (2.5) is true by taking sum over these balls. Of course the above expectation, that is the set is covered by disjoint balls with the exact density in each of these balls, is not true. However, it is almost true. There are two ways of arranging these coverings. One is called Calderón Zygmund decomposition and the other is called Vitali covering lemma. The Calderón Zygmund decomposition arranges cubes in an almost optimal way so that in each cube the density is almost ɛ and the cubes are disjoint. In Vitali covering lemma, one can arrange the balls so they are disjoint and the balls cover a major portion of the set. Lemma 2.3 (Vitali) Let C be a class of balls in R n with bounded radius. Then there is a finite or countable sequence B i C of disjoint balls such that B C B i 5B i, where 5B i is the ball with the same center as B i and radius five times big. We will use the following in this paper. Theorem 2.4 (Modified Vitali) Let 0 < ε < and let C D B be two measurable sets with C < ε B and satisfying the following property: for every x B with C B r (x) ε B r, B r (x) B D. Then D 20 n ε C. Proof. Since C < ε B, we see that for almost every x C, there is an r x < 2 so that C B rx (x) = ε B rx and C B r (x) < ε B r for all 2 > r > r x. By Vitali covering lemma, there are x, x 2,, so that B rx (x ), B rx2 (x 2 ), are disjoint and k B 5rxk (x k ) B C. From the choice of B rxk, we have C B 5rxk (x k ) < ɛ B 5rxk (x k ) = 5 n ɛ B rxk (x k ) = 5 n C B rxk (x k ). We also notice that since x k B and r xk 2. B rxk (x k ) 4 n B rxk (x k ) B

43 2.3. W 2,P ESTIMATES 43 Putting everything together, C = k B 5rxk (x k ) C k B 5r xk (x k ) C 5 n k ε B r xk (x k ) 20 n k ε B r xk (x k ) B = 20 n ε B rxk (x k ) B 20 n ε D. This finishes the proof. The reader can compare the above theorem with the following so called Calderón Zygmund Decomposition. Lemma 2.5 (Calderón Zygmund Decomposition) Let C D Q be measurable such that. C < ε, 2. if Q C ε Q for a cube then 3Q D. Then C ε D. Notice that this lemma is from a scaling invariant micro-condition to A ε B, a global estimate. The details are provided in the next section. We remark that one can prove Theorem 2.2 using either Theorem 2.4 or Lemma W 2,p Estimates Now we prove Theorem 2.. We only need to prove it for p > 2 since the statement for p < 2 follows from the standard duality argument. The starting point of the estimates is the following classical estimates. See [9], page 37. Lemma 2.6 If { u = f in B, u = 0 on B, then B D 2 u 2 C B f 2.

44 44 CHAPTER 2. L P ESTIMATES Lemma 2.7 There is a constant N so that for any ε > 0, δ = δ(ε) > 0 and if u is a solution of (2.) in a domain Ω B 4, with { ( M f 2) δ 2} { M D 2 u 2 } B (2.6) then { M D 2 u 2 } > N 2 B < ε B. (2.7) Proof. From condition (2.6), we see that there is a point x 0 B so that D 2 u 2 and f 2 δ 2, (2.8) B r(x 0 ) B r(x 0 ) for all B r (x 0 ) Ω and consequently we have D 2 u 2 2 n and f 2 2 n δ 2. B 4 Then B 4 B 4 u u B4 2 C. Let v be the solution of the following equation { v = 0 ( ) v = u u x ū B4 on B 4. B 4 Then by the minimality of harmonic function with respect to energy in B 4, v 2 2 u u B4 C. B 4 B 4 Now we can use the local C, estimates that there is a constant N 0 so that D 2 v 2 N L 0 2. (2.9) (B 3 ) At the same time we have, D 2 (u v) 2 C f 2 Cδ 2. B 4 B 3 From the weak estimate, λ {x B 3 : M B3 D 2 (u v) 2 (x) > λ} C B 3 D 2 (u v) 2 C B 4 f 2 Cδ 2.

45 2.3. W 2,P ESTIMATES 45 Consequently, Now we claim that {x B : M B3 D 2 (u v) 2 (x) > N 2 0 } Cδ 2. {x B : M D 2 u 2 > N 2 } {x B : M B3 D 2 (u v) 2 > N 2 0 }, where N 2 = max(4n 2 0, 2n ). Actually if y B 3, then D 2 u(y) 2 = D 2 u(y) 2 2 D 2 v(y) D 2 v(y) 2 2 D 2 u(y) D 2 v(y) 2 + 2N 2 0. Let x be a point in { x B : M B3 D 2 (u v) 2 (x) N 2 0 }. If r 2 we have B r (x) B 3 and sup D 2 u 2 2M B3 ( D 2 (u v) 2 )(x) + 2N0 2 4N0 2. r 2 B r(x) Now for r > 2, since x 0 B r (x) B 2r (x 0 ), we have B r(x) D 2 u 2 D 2 u 2 2 n, B r B 2r (x 0 ) where we have used (2.8). This says that M ( D 2 u 2) (x) N 2. This establishes the claim. Finally, we have {x B : M( D 2 u 2 ) > N 2 } {x B : M B3 ( D 2 (u v) 2 > N 2 0 } C N 2 0 f 2 Cδ2 N 2 0 < Cδ 2 = ε B, by taking δ satisfying the last identity above. This completes the proof. An immediate consequence of the above lemma is the following corollary. Corollary 2.8 { Assume ( u is a solution in a domain Ω and a ball B so that 4B Ω. If x : M D 2 u 2) } > N 2 B ε B, then { ( D B x : M 2 u 2) } (x) > {Mf 2 > δ 2 }.

46 46 CHAPTER 2. L P ESTIMATES { ( The moral of Corollary 2.8 is that the set x : M D 2 u 2) } > is much { ( bigger than the set x : M D 2 u 2) } > N 2 modulo {M(f 2 ) > δ 2 } if { ( D x : M 2 u 2) } > N 2 B = ε B.{ As said ( in the previous section, we will cover a good portion of the set x : M D 2 u 2) } > N 2 by disjoint balls so that in each of the balls the density of the set is ε. As an application { ( of Corollary 2.8, we will show the decay of the measure of the set x : M D 2 u 2) } > N 2. The covering is a careful choice of balls as in Vitali covering lemma. Corollary 2.9 { Assume that ( u is a solution in a domain Ω B 4, with the condition that x B : M D 2 u 2) } > N 2 ε B. Then for ε = 20 n ɛ, { (. B M D 2 u 2) } ( { > N 2 B ε M ( D 2 u ) } 2 (x) > { + B M ( f 2) > δ 2} ). { ( 2. B M D 2 u 2) ( { > N 2λ2} B ε x B : M ( D 2 u ) 2 } > λ 2 { + B M ( f 2) > δ 2 λ 2} ). { ( 3. B M D 2 u 2) > ( N 2 ) } k k { ε i ( B M f 2) ( ) } k i > δ 2 N 2 i= { ( B M D 2 f 2) > }. +ε k Proof. () is a direct consequence of Corollary 2.8 and Theorem 2.4 on C D { ( = B M D 2 u 2) } > N 2, { ( = B M D 2 u 2) } > { M ( f 2) > δ 2}. (2) is obtained by applying () to the equation ( λ u ) = λ f. (3) is an iteration of (2) by λ = N, (N ) 2,.... Theorem 2.0 If u = f in B 4 then D 2 u p C f p + u p. B 6 B Proof. Without lose of generality, we may assume that f p is small and the measure {x B : M D 2 u 2 > N 2} ε B by multiplying the

47 2.4. W,P ESTIMATES 47 function by a small constant. We will show that M( D 2 u 2) L p 2 (B ) from( which it follows that D 2 u L p (B ). M f 2) L p 2 with small norm. Suppose i= f L p (B 4 ) = δ. Since f L p, we have that Then + (N ) ip {M ( f 2) > δ 2 (N ) 2i} pn p δ p (N ) f p L p (B ) C f p L p (B 2 ) C. Hence D 2 u B p = p p p ( ( D M 2 u 2)) p 2 dx B λ p {x B : M D 2 u 2 λ 2} dλ 0 ( B + { + k= (N ) kp x B : M D 2 u 2 > (N ) 2k} ) ( B + + i= N kp k { ε i x B : M f 2 δ 2 N 2(k i) } i= } ) + k= N kp εk {x : M ( D 2 u 2) p B + C, + k= + N ip εi i= k i { N kp εk B N (k i)p { B } ) M D 2 u 2 M f 2 δ 2 N 2(k i) if we take ε so that N p ε < and the theorem follows. We remark that our methods can be adapted to prove the same result by using the Caldron Zygmund decomposition. The advantage of Vitali covering lemma is that it holds on any manifolds whereas the Caldron Zygmund decomposition requires cubes which give clean cut in Euclidean spaces which are rare to find on manifolds. 2.4 W,p estimates One can easily adapt the methods in the preceding section to obtain W,p estimates of the following type: }