On conormal and oblique derivative problem for elliptic equations with Dini mean oscillation coefficients
aa r X i v : . [ m a t h . A P ] J u l ON CONORMAL AND OBLIQUE DERIVATIVE PROBLEM FOR ELLIPTICEQUATIONS WITH DINI MEAN OSCILLATION COEFFICIENTS
HONGJIE DONG, JIHOON LEE, AND SEICK KIMA bstract . We show that weak solutions to conormal derivative problem for ellip-tic equations in divergence form are continuously di ff erentiable up to the bound-ary provided that the mean oscillations of the leading coe ffi cients satisfy the Dinicondition, the lower order coe ffi cients satisfy certain suitable conditions, and theboundary is locally represented by a C function whose derivatives are Dini contin-uous. We also prove that strong solutions to oblique derivative problem for ellipticequations in nondivergence form are twice continuously di ff erentiable up to theboundary if the mean oscillations of coe ffi cients satisfy the Dini condition and theboundary is locally represented by a C function whose derivatives are double Dinicontinuous. This in particular extends a result of M. V. Safonov (Comm. PartialDi ff erential Equations 20:1349–1367, 1995).
1. I ntroduction and main results
Let Ω be a bounded domain in R n . We consider second-order elliptic operators L in divergence form Lu = n X i , j = D i ( a ij ( x ) D j u + a i ( x ) u ) + n X i = b i ( x ) D i u + c ( x ) u (1.1)and also second-order elliptic operators L in nondivergence form L u = n X i , j = a ij ( x ) D ij u + n X i = b i ( x ) D i u + c ( x ) u . (1.2)We assume that the principal coe ffi cients A = ( a ij ) ni , j = are defined on R n and satisfythe uniform ellipticity condition λ | ξ | ≤ n X i , j = a ij ( x ) ξ i ξ j , ∀ ξ = ( ξ , . . . , ξ n ) ∈ R n , ∀ x ∈ R n (1.3)and the uniform boundedness condition n X i , j = | a ij ( x ) | ≤ Λ , ∀ x ∈ R n (1.4) Mathematics Subject Classification.
Key words and phrases.
Dini mean oscillation, oblique derivative problem, conormal derivativeproblem.H. Dong was partially supported by the NSF under agreement DMS-1600593.S. Kim is partially supported by NRF Grant No. NRF-2016R1D1A1B03931680 and No. NRF-20151009350. for some positive constants λ and Λ . In the nondivergence case, we may assumethat A is symmetric (i.e. a ij = a ji ) as usual. We shall further assume that A is ofDini mean oscillation; i.e., its mean oscillation function ω A ( r ) : = sup x ∈ R n ? B ( x , r ) | A ( y ) − ¯ A x , r | dy ¯ A x , r : = ? B ( x , r ) A ! satisfies the Dini condition. We say that a function ω : [0 , → [0 , ∞ ) satisfies theDini condition if Z ω ( t ) t dt < + ∞ and that ω satisfies the double Dini condition if Z s Z s ω ( t ) t dt ds = Z ω ( t ) ln t t dt < + ∞ . We say that a function f is Dini continuous (resp. double Dini continuous) if itsmodulus of continuity satisfies the Dini condition (resp. double Dini condition).We write f ∈ C k , Dini (resp. f ∈ C k , Dini ) if D α f is Dini continuous (resp. double Dinicontinuous) for each multi-index α with | α | ≤ k ; refer to Section 2.1 for the moreprecise definitions.In the divergence case, we assume that ∂ Ω is C , Dini and consider the conormalderivative operator A ∇ u · ν + a u · ν + a u : = n X i , j = a ij ( x ) D j u ν i + n X i = a i u ν i + a u on ∂ Ω , where ν = ( ν , . . . , ν n ) denotes the outward unit normal vector, a = ( a , . . . , a n ) is ofDini mean oscillation, and a is Dini continuous. In the nondivergence case, weassume that ∂ Ω is C , Dini and consider the oblique derivative operator β u + β · ∇ u : = β ( x ) u + n X i = β i ( x ) D i u on ∂ Ω , where β and β = ( β , . . . , β n ) are in C , Dini ( Ω ) and β satisfies the obliquenesscondition | β · ν | ≥ µ | β | on ∂ Ω (1.5)for some positive constant µ .In this paper, we are concerned with the conormal derivative problem for di-vergence form equation Lu = div g + f in Ω , A ∇ u · ν + a u · ν + a u = g · ν + g on ∂ Ω , and the oblique derivative problem for nondivergence form equation L u = f in Ω , β u + β · ∇ u = g on ∂ Ω . For the conormal derivative problem, we shall show that u is continuously dif-ferentiable up to the boundary if the data g is of Dini mean oscillation, g is Dinicontinuous, and if the data f and the lower order coe ffi cients of L belong to L q with q > n . For the oblique derivative problem, we shall show that u is twicecontinuously di ff erentiable up to the boundary if the data f and the lower ordercoe ffi cients of L are of Dini mean oscillation, and the boundary data g , β , and β are of C , Dini . N OBLIQUE DERIVATIVE PROBLEM 3
A few remarks are in order. Very recently, under the same condition on A asimposed here, the first and third named authors [6] proved that any W , weaksolution of the equation div( A ∇ u ) = ff erentiable and that any W , strong solution of the equation tr( A D u ) = ff er-entiable. Later, the first and third named authors and Escauriaza [4] consideredgeneral elliptic equation with lower order coe ffi cients (as considered here) subjectto Dirichlet boundary condition and extended the interior estimates in [6] to thecorresponding C and C estimates up to the boundary. In this perspective, thispaper can be considered as a natural extension of [4] to conormal and obliquederivative boundary conditions. Regarding the oblique derivative problem, weare obliged to mention a paper by Safonov [18], where he proved a priori global C ,α estimates for solutions assuming that the coe ffi cients and domain satisfy theH ¨older condition, which was also established earlier by Lieberman [13] by a dif-ferent method. We borrowed some crucial technical details from [18] and adaptedto our setting.There are many other literature dealing with the oblique derivative problemand the conormal derivative problem. Among them, we point out that in [12,Theorem 5.1] a result similar to Theorem 1.7 below was proved for quasilinearelliptic equations under the uniform Dini continuity condition. In [14, Theorem5.4] a weighted C estimate was obtained for fully nonlinear elliptic equations withthe oblique derivative boundary condition under the uniform Dini condition. Wealso mention a book by Lieberman [15], which gives a comprehensive expositionon the theory of oblique derivative problems for elliptic equations. We ask readersinterested in history and applications of oblique derivative problems to consult[15] and references therein.Now we state the main results of the paper more precisely. We first consider theconormal derivative problem for a divergence structure elliptic equation. Condition 1.6. A = ( a ij ) and a = ( a , . . . , a n ) are of Dini mean oscillation in Ω , a isDini continuous in Ω , and b = ( b , . . . , b n ) and c belong in L q ( Ω ) with q > n . Theorem 1.7.
Let Ω have C , Dini boundary, the coe ffi cients of L in (1.1) satisfy theconditions (1.3) and (1.4) , and Condition 1.6. Suppose u ∈ W , ( Ω ) is a weak solution ofLu = div g + f in Ω , A ∇ u · ν + a u · ν + a u = g · ν + g on ∂ Ω , where g = ( g , . . . , g n ) are of Dini mean oscillation in Ω , g is Dini continuous in ∂ Ω ,and f ∈ L q ( Ω ) with q > n . Then we have u ∈ C ( Ω ) . We also consider the oblique derivative problem for nondivergence form ellipticequations.
Condition 1.8. A = ( a ij ), b = ( b , . . . , b n ), and c are of Dini mean oscillation in Ω . Condition 1.9. β and β = ( β , . . . , β n ) are in C , Dini ( Ω ), and β satisfies (1.5). Theorem 1.10.
Let Ω have C , Dini boundary, the coe ffi cients of L in (1.2) satisfy thecondition (1.3) and (1.4) , and Condition 1.8. Let β and β satisfy Condition 1.9. Supposeu ∈ W , ( Ω ) is a strong solution of the oblique derivative problem L u = f in Ω , β u + β · ∇ u = g on ∂ Ω , where f is of Dini mean oscillation in Ω and g ∈ C , Dini ( Ω ) . Then we have u ∈ C ( Ω ) . H. DONG, J. LEE, AND S. KIM
Remark . In [2] global Lipschitz estimates for certain quasilinear divergenceform elliptic equations were established under minimal conditions on the data,the nonlinearity, and the domains. In particular, the condition on the domain isweaker than the C , Dini condition in Theorem 1.7.The organization of the paper is as follows. In Section 2, we introduce somenotation, definitions, and lemmas used in the paper. Sections 3 and 4 are devotedto the proofs of our main results, Theorem 1.7 and Theorem 1.10, respectively. Inthe Appendix, we provide the proofs for some technical lemmas that are slightlymodified from those in Safonov’s paper [18].2. P reliminaries
Notation and definitions.
We follow the same notation as used in [4]. Forcompleteness, we reproduce most frequently used ones here. We denote by B ( x , r )the Euclidean ball centered at x with radius r and B r = B (0 , r ) , B + r = B r ∩ { x n > } and T (0 , r ) = B r ∩ { x n = } . Let us fix a smooth domain D satisfying B + / ⊂ D ⊂ B + (2.1)so that ∂ D contains a flat portion T (0 , ). For ¯ x ∈ ∂ R n + = { x n = } , we then set B + ( ¯ x , r ) = B + r + ¯ x , T ( ¯ x , r ) = T (0 , r ) + ¯ x , and D ( ¯ x , r ) = r D + ¯ x . Hereafter, we shall adopt the usual summation convention for repeated indices.Throughout the paper, we shall use the notation[ u ] k ; E : = sup x ∈ E | D k u ( x ) | and [ u ] k ,µ ; E : = sup x , y ∈ Ex , y | D k u ( x ) − D k u ( y ) || x − y | µ , (2.2)where k = , , , . . . , 0 < µ <
1, and E ⊂ R n . We also write | u | k ; E : = k X j = [ u ] j ; E and | u | k ,µ ; E : = | u | k ; E + [ u ] k ,µ ; E . (2.3) Definition 2.4.
Let E ⊂ R n and let f : E → R . The modulus of continuity of f isthe increasing function ̺ f : [0 , ∞ ) → [0 , ∞ ) defined by ̺ f ( t ) : = sup (cid:8) | f ( x ) − f ( y ) | : x , y ∈ E , | x − y | ≤ t (cid:9) . A function f is said to be Dini continuous (in E ) if ̺ f satisfies the Dini condition Z ̺ f ( t ) t dt < + ∞ ; f is said to be double Dini continuous (in E ) if ̺ f satisfies the double Dini condition(see [16, 17]) Z s Z s ̺ f ( t ) t dt ds = Z ̺ f ( t ) ln t t dt < + ∞ . For k = , , , .. , we denote by C k , Dini ( E ) (resp. C k , Dini ( E )) the set of all k -timescontinuously di ff erentiable functions f on E such that D α f is Dini continuous(resp. double Dini continuous) in E , for each multi-index α with | α | ≤ k . By the N OBLIQUE DERIVATIVE PROBLEM 5 C k , Dini characteristics of f in E , we mean | f | k ; E and ̺ D α f ( t ) with multi-index α with | α | = k . Definition 2.5.
Let Ω ( x , r ) : = Ω ∩ B ( x , r ). For any k = , , . . . , we say that theboundary ∂ Ω is C k , Dini (resp. C k , Dini ) if for each point x ∈ ∂ Ω , there exist r > x and a C k , Dini (resp. C k , Dini ) function γ : R n − → R such that (uponrelabeling and reorienting the coordinates axes if necessary) in a new coordinatesystem ( x ′ , x n ) = ( x , . . . , x n − , x n ), x becomes the origin and Ω (0 , r ) = { x ∈ B (0 , r ) : x n > γ ( x , . . . , x n − ) } , γ (0 ′ ) = , D γ (0 ′ ) = . Remark . By using the implicit function theorem and a partition of the unity, it iseasily seen that ∂ Ω is of C k , Dini (resp. C k , Dini ) if and only if there exists a C k , Dini (resp. C k , Dini ) function ψ : R n → R such that Ω = { x ∈ R n : ψ ( x ) > } and | D ψ | ≥ ∂ Ω . We call ψ a defining function of Ω . Clearly, the C k , Dini (resp. C k , Dini )characteristic of ψ is comparable to that of γ in Definition 2.5. In the sequel, weshall use these two equivalent definitions interchangeably. Definition 2.7.
We say that a function f : Ω → R is of Dini mean oscillation if itsmean oscillation function ω f defined by ω f ( r ) : = sup x ∈ Ω ? Ω ( x , r ) | f ( y ) − ¯ f Ω ( x , r ) | dy ¯ f Ω ( x , r ) : = ? Ω ( x , r ) f ! satisfies the Dini condition Z ω f ( r ) r dr < + ∞ . Remark . In [1, Proposition 1.13], it is proved that under the additional assump-tion that ω f is almost increasing and ω f ( r ) / r is almost decreasing, then ω f ( r ) iscomparable to sup x ∈ Ω ? Ω ( x , r ) | f ( y ) − ¯ f Ω ( x , r ) | p dy ! / p for any p ∈ (1 , ∞ ). However, it is unclear whether they are still comparable withoutthis additional assumption. On the other hand, an example in [6] shows that theDini mean oscillation condition is weaker than the usual Dini continuity condition.Finally, we adopt the usual summation convention over repeated indices. Also,for nonnegative (variable) quantities A and B , the relation A . B should be under-stood that there is some constant c > A ≤ cB . We write A ≃ B if A . B and B . A .2.2. Some preliminary lemmas.Lemma 2.9.
If f is uniformly Dini continuous and g is of Dini mean oscillation in Ω ,then f g is of Dini mean oscillation in Ω .Proof. For any x ∈ Ω and r >
0, we have ? Ω ( x , r ) (cid:12)(cid:12)(cid:12)(cid:12) f g − f g Ω ( x , r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ? Ω ( x , r ) (cid:12)(cid:12)(cid:12) f g − f ¯ g Ω ( x , r ) (cid:12)(cid:12)(cid:12) + ? Ω ( x , r ) (cid:12)(cid:12)(cid:12)(cid:12) f ¯ g Ω ( x , r ) − f g Ω ( x , r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup Ω ( x , r ) f · ω g ( r ) + ̺ f ( r ) · ? Ω ( x , r ) | g | , H. DONG, J. LEE, AND S. KIM where we used sup Ω ( x , r ) (cid:12)(cid:12)(cid:12)(cid:12) f ¯ g Ω ( x , r ) − f g Ω ( x , r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ̺ f ( r ) · ? Ω ( x , r ) | g | . Therefore, we get ω f g ( r ) ≤ k f k L ∞ ( Ω ) ω g ( r ) + k g k L ∞ ( Ω ) ̺ f ( r )and thus ω f g is a Dini function. (cid:4) Lemma 2.10.
Let ω : [0 , a ] → [0 , ∞ ) be a function satisfying the (double) Dini condition.Suppose there are constants c , c > such thatc ω ( t ) ≤ ω ( s ) ≤ c ω ( t ) (2.11) whenever t ≤ s ≤ t and ≤ t ≤ a. (It should be noted that the condition (2.11) isautomatically satisfied by ̺ f ( t ) and ω f ( t ) introduced in Definitions 2.4 and 2.7). Let β ∈ (0 , be given. Then, there is a function ˜ ω : [0 , a ] → [0 , ∞ ) such that ω ( t ) ≤ ˜ ω ( t ) for any t ∈ [0 , a ] and that t t − β ˜ ω ( t ) is decreasing on (0 , a ] . Moreover, ˜ ω ( t ) satisfies the(double) Dini condition and also satisfies the condition (2.11) .Proof. We set ˜ ω (0) = < t ≤ a , define˜ ω ( t ) = sup s ∈ [ t , a ] (cid:18) ts (cid:19) β ω ( s ) . Then, it is clear that ω ( t ) ≤ ˜ ω ( t ) and t ˜ ω ( t ) / t β is decreasing. Also, it is straight-forward to verify that ˜ ω satisfies (2.11) when ω satisfies (2.11). Finally, we referto the proof of [4, Lemma 2.9] for the fact that ˜ ω ( t ) satisfies the (double) Dinicondition. (cid:4) Lemma 2.12.
Let
D ⊂ R n be a smooth domain satisfying (2.1) . Let ¯ A = ( ¯ a ij ) be a constantmatrix satisfying (1.3) and (1.4) . For f ∈ L ( D ) , let u ∈ W , ( D ) be a weak solution of div( ¯ A ∇ u ) = div f in D , ¯ A ∇ u · ν = f · ν on ∂ D . Then there exists a constant C = C ( n , λ, Λ , D ) such that for any t > , we have |{ x ∈ D : | Du ( x ) | > t }| ≤ Ct Z D | f | . Proof.
Since u is unique up to a constant, we see that the map T : f Du is welldefined and is a bounded linear operator on L ( D ). We modify the proof of [6,Lemma 2.2] using [4, Lemma 4.1]. Let b ∈ L ( D ) be supported in B ( ¯ y , r ) ∩ D withmean zero, where ¯ y ∈ D and 0 < r < diam D . Suppose u ∈ W , ( D ) is a weaksolution (unique up to a constant) ofdiv( ¯ A ∇ u ) = div b in D , ¯ A ∇ u · ν = b · ν on ∂ D . By [4, Lemma 4.1], it is enough to show that Z D\ B ( ¯ y , r ) | Du | ≤ C Z B ( ¯ y , r ) ∩D | b | . For any R ≥ r such that D \ B ( ¯ y , R ) , ∅ and g ∈ C ∞ c (( B ( ¯ y , R ) \ B ( ¯ y , R )) ∩ D ) , let v ∈ W , ( D ) be a weak solution (unique up to a constant) ofdiv( ¯ A T ∇ v ) = div g in D , ¯ A T ∇ v · ν = g · ν on ∂ D . N OBLIQUE DERIVATIVE PROBLEM 7
Then, we have the following equality Z D Du · g = Z D b · Dv = Z B ( ¯ y , r ) ∩D b · ( Dv − Dv B ( ¯ y , r ) ∩D ) . Therefore we get, by the mean value theorem, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ( B ( ¯ y , R ) \ B ( ¯ y , R )) ∩D Du · g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k b k L ( B ( ¯ y , r ) ∩D ) k Dv − Dv B ( ¯ y , r ) ∩D k L ∞ ( B ( ¯ y , r ) ∩D ) ≤ r k b k L ( B ( ¯ y , r ) ∩D ) k D v k L ∞ ( B ( ¯ y , r ) ∩D ) . Note that div( ¯ A T ∇ v ) = B ( ¯ y , R ) ∩ D and r ≤ R . Since ¯ A is constant and theboundary ∂ D is smooth, we have k D v k L ∞ ( B ( ¯ y , r ) ∩D ) ≤ CR − − n k Dv k L ( B ( ¯ y , R ) ∩D ) ≤ CR − − n k Dv k L ( D ) ≤ CR − − n k g k L ( D ) = CR − − n k g k L (( B ( ¯ y , R ) \ B ( ¯ y , R )) ∩D ) . Therefore, by the duality, we have k Du k L (( B ( ¯ y , R ) \ B ( ¯ y , R )) ∩D ) ≤ CrR − − n k b k L ( B ( ¯ y , r ) ∩D ) and hence by H ¨older’s inequality we get k Du k L (( B ( ¯ y , R ) \ B ( ¯ y , R )) ∩D ) ≤ CrR − k b k L ( B ( ¯ y , r ) ∩D ) . Now let N > D ⊂ B ( ¯ y , N + r ). By taking R = r , r , . . . , N r in the above, we get Z D\ B ( ¯ y , r ) | Du | ≤ C N X k = − k k b k L ( B ( ¯ y , r ) ∩D ) ≤ C Z B ( ¯ y , r ) ∩D | b | . We note that C depends only on n , λ , Λ , and D . Thus, we see that T satisfies thehypotheses of Lemma 4.1 of [4], and the proof is complete. (cid:4) Lemma 2.13.
Let ¯ A = ( ¯ a ij ) be a constant symmetric matrix satisfying (1.3) and (1.4) . Forf ∈ L ( B + ) , let u ∈ W , ( B + ) be a strong solution of the mixed problem ¯ a ij D ij u = f in B + , u = on ∂ B ∩ { x n > } , D n u = on T (0 , . (2.14) Then there exists a constant C = C ( n , λ, Λ ) such that for any t > , we have (cid:12)(cid:12)(cid:12) { x ∈ B + : | D u ( x ) | > t } (cid:12)(cid:12)(cid:12) ≤ Ct Z B + | f | . Proof.
Without loss of generality, we may assume that ¯ a nn =
1. We introduce a newmatrix valued function ˆ A = ˆ A ( x n ) as follows. When i = j = n or i , j ∈ { , . . . , n − } ,ˆ a ij ( x n ) = ¯ a ij . When j = , . . . , n −
1, ˆ a nj ( x n ) = ˆ a jn ( x n ) = ¯ a nj if x n ≥ , − ¯ a nj if x n < . It is easy to check that ˆ A satisfies the conditions (1.3) and (1.4). Let ˆ f be an evenextension of f and let ˆ u ∈ W , ( B ) ∩ W , ( B ) be a unique solution ofˆ a ij D ij u = ˆ f in B , u = ∂ B . (2.15) H. DONG, J. LEE, AND S. KIM
See [7] for the solvability of (2.15). By the uniqueness, it is straightforward to seethat ˆ u is even with respect to x n coordinate, which implies that D n ˆ u = T (0 , u ≡ ˆ u in B + .Therefore, it is enough to show (cid:12)(cid:12)(cid:12) { x ∈ B : | D ˆ u ( x ) | > t } (cid:12)(cid:12)(cid:12) ≤ Ct Z B | ˆ f | . Fix ¯ y ∈ B , 0 < r < , and let b ∈ L ( B ) be supported in B ( ¯ y , r ) ∩ B with meanzero. Let ˜ u ∈ W , ( B ) ∩ W , ( B ) be a solution ofˆ a ij D ij u = b in B ; u = ∂ B , (2.16)the solvability of which is stated in [7, p. 6483].For any R ≥ r such that B \ B ( ¯ y , R ) , ∅ and g = ( g kl ) ∈ C ∞ c (( B ( ¯ y , R ) \ B ( ¯ y , R )) ∩ B ),let v ∈ W , ( B ) be a weak solution of D i ( ˜ a ij D j v ) = div g in B , v = ∂ B , where ˜ A = ( ˜ a ij ) is defined as follows˜ a nn =
1; ˜ a ij = ˆ a ij for i , j ∈ { , . . . , n − } ;˜ a nj = a nj and ˜ a jn = j = , . . . , n − . It is easy to check that ˜ A satisfies the ellipticity and boundedness conditions (1.3)and (1.4) (with new constants ˜ λ and ˜ Λ determined by λ and Λ ). Since g = B ( ¯ y , R ) ∩ B and r ≤ R /
2, we find D i ( ˜ a ij D j v ) = B ( ¯ y , R ) ∩ B , and thus, by the De Giorgi-Nash-Moser estimate (up to the boundary) we see that v is H ¨older continuous in B ( ¯ y , r ) ∩ B and[ v ] µ ; B ( ¯ y , r ) ∩ B ≤ CR − µ − n k v k L ( B ( ¯ y , R ) ∩ B ) (2.17)for some constants µ ∈ (0 ,
1) and C > n , λ , and Λ .On the other hand, observe that n X i , j = D i ( ˜ a ij D j v ) = n − X i , j = D i ( ˆ a ij D j v ) + n − X j = D n ( ˆ a nj D j v ) + D n ( D n v ) = n − X i , j = D ij ( ˆ a ij v ) + n − X j = D nj ( ˆ a nj v ) + D nn v = n X i , j = D ij ( ˆ a ij v ) . Here, we used that ˆ a ij = ˆ a ij ( x n ) and ˆ a nn =
1. Therefore, we see that v is also anadjoint solution of D ij ( ˆ a ij v ) = div g in B , v = ∂ B (2.18)and hence by [8, Lemma 2], we have k v k L ( B ) ≤ C k g k L ( B ) . (2.19)By (2.16) and (2.18) and the hypothesis on b , we have the identity Z B D ij ˜ u g ij = Z B vb = Z B ( ¯ y , r ) ∩ B b ( v − ¯ v B ( ¯ y , r ) ∩ B ) . N OBLIQUE DERIVATIVE PROBLEM 9
Then by using (2.17) and (2.19), we have Z ( B ( ¯ y , R ) \ B ( ¯ y , R )) ∩ B D ij ˜ u g ij ≤ k b k L ( B ( ¯ y , r ) ∩ B ) [ v ] µ ; B ( ¯ y , r ) ∩ B (2 r ) µ ≤ C (cid:18) rR (cid:19) µ R − n k b k L ( B ( ¯ y , r ) ∩ B ) k g k L (( B ( ¯ y , R ) \ B ( ¯ y , R )) ∩ B ) . The rest of the proof is almost the same as that of Lemma 2.12 and omitted. (cid:4)
Lemma 2.20.
Let ¯ A = ( ¯ a ij ) be a constant symmetric matrix satisfying (1.3) and (1.4) .Suppose u ∈ W , ( B + ) satisfies ¯ a ij D ij u = in B + , D n u = on T (0 , . Then for any p > , there exists a constant C = C ( n , λ, Λ , p ) such that k Du k L ∞ ( B + / ) ≤ C ? B + | u | p p . (2.21) Proof.
The estimate (2.21) can be deduced from [9, Theorem 6.26]. We give analternative proof here. Let ˆ u be an even extension of u (with respect to x n coordinate)and ˆ A be defined as in the proof of Lemma 2.13. Then ˆ u satisfiesˆ a ij D ij ˆ u = B . Since ˆ A = ˆ A ( x n ), we have the Lipschitz estimate (see [10]) k D ˆ u k L ∞ ( B / ) ≤ C k ˆ u k L ( B ) , from which (2.21) follows by standard argument. (cid:4)
3. P roof of T heorem C estimates. Proposition 3.1.
We have u ∈ C ( Ω ′ ) for any Ω ′ ⊂⊂ Ω .Proof. Since the proof is very similar to that of [4, Proposition 2.6], we will onlygive an outline of the proof. Since the coe ffi cients a ij are continuous, the standard W , p theory yields that u ∈ W , p loc ( Ω ) for any p ∈ (1 , ∞ ). To see that u ∈ W , p up tothe boundary, we locally flatten the boundary so that ν = − e n and the boundarycondition becomes − n X j = a nj D j u − a n u = − g n + g − a u on Γ ⊂ { x n = } . Note that if we set˜ g n ( x ) = ˜ g n ( x ′ , x n ) : = g n ( x ′ , x n ) − g ( x ′ , + a ( x ′ , u ( x ′ , , then we have − D n ˜ g n = − D n g n . Therefore, by replacing g n with ˜ g n , the aboveboundary condition reduces to the standard conormal boundary condition (seee.g., [5, Theorem 5]). Then we can apply the boundary W , p theory and a bootstrapargument to conclude that u ∈ W , p ( Ω ) for any p ∈ (1 , ∞ ).By the Morrey-Sobolev embedding, we have u ∈ C ,µ ( Ω ) for any µ ∈ (0 , D i ( a ij D j u ) = f − b i D i u − cu + D i ( g i − a i u ) . Let g ′ = g − a u . By Lemma 2.9, we see that g ′ is of Dini mean oscillation. Also, bytaking a su ffi ciently large p and using H ¨older’s inequality, we have f − b i D i u − cu ∈ L r ( Ω ) for some r ∈ ( n , q ). We set g ′′ = ∇ v , where v solves ∆ v = f − b i D i u − cu in Ω , ∂ v /∂ν = ∂ Ω . Then, we have g ′′ ∈ C ,δ ( Ω ) with δ = − nr . Therefore, we see that g ′ and g ′′ are ofDini mean oscillation and D i ( a ij D j u ) = div( g ′ + g ′′ ) in Ω . By [6, Theorem 1.5], we conclude that u ∈ C ( Ω ′ ) for any Ω ′ ⊂⊂ Ω . (cid:4) Next, we prove C estimate near the boundary. Note that g ′′ introduced in theproof of Proposition 3.1 satisfies g ′′ · ν = ∂ Ω . By the same reasoning explainedjust before [4, Proposition 2.7] and replacing g n by ˜ g n after locally flattening theboundary so that ν = − e n (note that ˜ g n is of Dini mean oscillation), we are reducedto prove the following. Proposition 3.2.
If u ∈ W , ( B + ) is a weak solution ofD i ( a ij D j u ) = div g in B + , A Du · e n = g · e n on T (0 , , then u ∈ C ( B + ) . The rest of this section is devoted to the proof of Proposition 3.2. We shallassume u ∈ C ( B + ) and derive an a priori estimate of the modulus of continuity of Du . We fix some p ∈ (0 ,
1) and introduce φ ( x , r ) : = inf q ∈ R n ? B ( x , r ) ∩ B + | Du − q | p p . We shall derive an estimate for φ ( ¯ x , r ) for ¯ x ∈ T (0 ,
3) and 0 < r ≤ . Recall thenotation D ( ¯ x , r ) introduced at the beginning of this section. We split u = v + w ,where w ∈ W , ( D ( ¯ x , r )) is a weak solution of the problemdiv( ¯ A ∇ w ) = − div(( A − ¯ A ) ∇ u ) + div( g − ¯ g ) in D ( ¯ x , r ) , ¯ A ∇ w · ν = − ( A − ¯ A ) ∇ u · ν + ( g − ¯ g ) · ν on ∂ D ( ¯ x , r ) , where ¯ A = ¯ A B + (¯ x , r ) and ¯ g = ¯ g B + (¯ x , r ) . By Lemma 2.12 with scaling, we see that (cid:12)(cid:12)(cid:12) { x ∈ B + ( ¯ x , r ) : | Dw ( x ) | > t } (cid:12)(cid:12)(cid:12) ≤ Ct k Du k L ∞ ( B + (¯ x , r )) Z B + (¯ x , r ) | A − ¯ A | + Z B + (¯ x , r ) | g − ¯ g | ! . Then, we have (see [6, (2.11)]) ? B + (¯ x , r ) | Dw | p ! p ≤ C ω A (2 r ) k Du k L ∞ ( B + (¯ x , r )) + C ω g (2 r ) . (3.3)Note that v = u − w satisfiesdiv( ¯ A ∇ v ) = div ¯ g = B + ( ¯ x , r ) , ¯ A ∇ v · e n = ¯ g · e n on T ( ¯ x , r ) . Then for any c ∈ R and k = , , . . . , n −
1, ˜ v = D k v − c satisfiesdiv( ¯ A ∇ ˜ v ) = B + ( ¯ x , r ) , ¯ A ∇ ˜ v · e n = T ( ¯ x , r ) . N OBLIQUE DERIVATIVE PROBLEM 11
By the standard elliptic estimates for the constant coe ffi cients equations with zeroconormal boundary data, we have k DD k v k L ∞ ( B + (¯ x , r )) ≤ Cr − ? B + (¯ x , r ) | D k v − c | p ! p , ∀ k ∈ { , . . . , n − } , ∀ c ∈ R . Then by using D nn v = − a nn X ( i , j ) , ( n , n ) ¯ a ij D ij v , we obtain k D v k L ∞ ( B + (¯ x , r )) ≤ C k DD x ′ v k L ∞ ( B + (¯ x , r )) ≤ Cr − ? B + (¯ x , r ) | D x ′ v − c | p ! p , ∀ c ∈ R n − , where we used the notation D x ′ v = ( D v , . . . , D n − v ). Therefore, we have k D v k L ∞ ( B + (¯ x , r )) ≤ Cr − ? B + (¯ x , r ) | Dv − q | p ! p , ∀ q ∈ R n . Let 0 < κ < be a number to be fixed later. Since ? B + (¯ x ,κ r ) (cid:12)(cid:12)(cid:12) Dv − Dv B + (¯ x ,κ r ) (cid:12)(cid:12)(cid:12) p ! p ≤ κ r k D v k L ∞ ( B + (¯ x ,κ r )) and κ < , we see that there is a constant C = C ( n , λ, Λ , p ) > ? B + (¯ x ,κ r ) (cid:12)(cid:12)(cid:12) Dv − Dv B + (¯ x ,κ r ) (cid:12)(cid:12)(cid:12) p ! p ≤ C κ ? B + (¯ x , r ) | Dv − q | p ! p , ∀ q ∈ R n . By using the decomposition u = v + w , we obtain from the above that ? B + (¯ x ,κ r ) (cid:12)(cid:12)(cid:12) Du − Dv B + (¯ x ,κ r ) (cid:12)(cid:12)(cid:12) p ! p ≤ − pp ? B + (¯ x ,κ r ) (cid:12)(cid:12)(cid:12) Dv − Dv B + (¯ x ,κ r ) (cid:12)(cid:12)(cid:12) p ! p + − pp ? B + (¯ x ,κ r ) | Dw | p ! p ≤ − pp C κ ? B + (¯ x , r ) | Du − q | p ! p + C ( κ − np + ? B + (¯ x , r ) | Dw | p ! p . Since q ∈ R n is arbitrary, by using (3.3), we obtain φ ( ¯ x , κ r ) ≤ − pp C κ φ ( ¯ x , r ) + C ( κ − np + (cid:16) ω A (2 r ) k Du k L ∞ ( B + (¯ x , r )) + ω g (2 r ) (cid:17) . (3.4)Therefore, we see that φ ( ¯ x , r ) enjoys the same estimates for the auxiliary quantity ϕ ( ¯ x , r ) defined in [4, (2.10)] for Dirichlet boundary problem, and thus [4, Lemma2.8] is valid with φ ( ¯ x , ρ ) in place of ϕ ( ¯ x , ρ ). Also, we note that if B ( x , ρ ) ⊂ B + ( ¯ x , R )and ρ ≃ R , then we have (see (2.26) and (2.28) in [4]) φ ( x , ρ ) . φ ( ¯ x , R ) . (3.5)Consequently, we have Lemmas 2.9, 2.11, and 2.12 in [4] also available in oursetting. For any given β ∈ (0 , κ ∈ (0 , ) su ffi ciently small such that − pp C κ ≤ κ β , we thus have the following estimate | Du ( x ) − Du ( y ) | ≤ C k Du k L ( B + ) | x − y | β + C k Du k L ( B + ) + Z ˆ ω g ( t ) t dt ! ω ∗ A ( | x − y | ) + C ω ∗ g ( | x − y | ) , ∀ x , y ∈ B + . (3.6)Here, ˆ ω g is a function determined by ω g satisfying the Dini condition and ω ∗ A ( t ),and ω ∗ g ( t ) are as defined by the formula [4, (2.34)]. More precisely, in [4] we defined˜ ω • ( t ) : = ∞ X i = κ i β (cid:16) ω • ( κ − i t ) [ κ − i t ≤ + ω • (1) [ κ − i t > (cid:17) ,ω ♯ • ( t ) : = sup s ∈ [ t , (cid:18) ts (cid:19) β ˜ ω • ( s ) for 0 < t ≤ , ˆ ω • ( t ) : = ˜ ω • ( t ) + ˜ ω • (4 t ) + ω ♯ • (4 t ) ( . ω ♯ • (4 t ) ) for 0 < t ≤ / ,ω ∗• ( t ) : = ˆ ω • ( t ) + Z t ˜ ω • ( s ) s ds + ˜ ω • (4 t ) + Z t ˜ ω • (4 s ) s ds for 0 < t ≤ / . In particular, they satisfy lim t → + ω ∗• ( t ) = . This completes the proof of Proposition 3.2 and that of Theorem 1.7. (cid:4)
Remark . In the case when A and g are H ¨older continuous with an exponent α ∈ (0 , β ∈ ( α,
1) in (3.6), one can show that ω ∗ A ( t ) . t α and ω ∗ g ( t ) . t α . Therefore, Du is H ¨older continuous with the same exponent α , whichrecovers the classical result.4. P roof of T heorem n , ellipticity constants λ , Λ , the obliqueness constant µ , and the domain Ω ; the mean oscillation functions for the coe ffi cients ω A , ω b , and ω c , all of whichsatisfy the Dini condition; C , Dini characteristics of the coe ffi cients β , β in theoblique derivative operator and those of γ (or equivalently ψ in Remark 2.6),which (locally) represents the boundary.The following proposition provides key estimates, the proof of which is deferredto the end of this section. Proposition 4.1.
If u ∈ W , ( B + ) is a strong solution ofa ij D ij u = f in B + , D n u = on T (0 , , then we have [ u ] B + ≤ C k D u k L ( B + ) + C Z ˆ ω f ( t ) t dt , (4.2) where as in the previous section ˆ ω f ( t ) is a nonnegative function derived from ω f ( t ) satis-fying the Dini condition. Moreover, for any x , y ∈ B + , we have | D u ( x ) − D u ( y ) | ≤ C n k D u k L ( B + ) | x − y | µ + [ u ] B + ω ∗ A ( | x − y | ) + ω ∗ f ( | x − y | ) o , (4.3) where µ ∈ (0 , is any number, C = C ( n , λ, Λ , ω A , µ ) , and ω ∗• ( t ) is a modulus of continuitydetermined by ω • ( t ) and µ , which goes to zero as t → . N OBLIQUE DERIVATIVE PROBLEM 13
We take the proposition for now. Our first goal is to establish the followingestimate under a qualitative assumption that u ∈ C ( Ω ):[ u ] Ω ≤ C k u k W , ( Ω ) + Z r ˜ ω f ( t ) t dt + | g | Ω + Z r ˜ ̺ Dg ( t ) ln t t dt , (4.4)where r > C are constants depending on the prescribed data; ˜ ω f ( t ) satisfiesthe Dini condition and is determined by ω f ( t ) and the prescribed data; ˜ ̺ Dg ( t )satisfies the double Dini condition and is determined by ̺ Dg ( t ) and the prescribeddata.With the estimate (4.4) at hand, we then show that for any x , y ∈ Ω , we have | D u ( x ) − D u ( y ) | ≤ C k D u k L ( Ω ) | x − y | µ + C [ u ] Ω ω ∗ A ( | x − y | ) + C ω ∗ f ( | x − y | ) + C | g | Ω + Z r ˜ ̺ Dg ( t ) ln t t dt ω ∗ ( | x − y | ) + C Z | x − y | ˜ ̺ Dg ( t ) ln t t dt + C ω ∗ ( | x − y | ) . (4.5)Here, µ ∈ (0 ,
1) is an arbitrary constant, C is a constant depending on µ and theprescribed data; ω ∗ A ( t ) and ω ∗ f ( t ) are nonnegative functions determined by ω A ( t )and ω f ( t ), respectively, as well as µ and the prescribed data; ω ∗ ( t ) and ω ∗ ( t ) arenonnegative functions determined by µ and the prescribed data. Moreover, all thefunction ω ∗• ( t ) in (4.5) satisfy lim t → + ω ∗• ( t ) = . Once the estimates (4.4) and (4.5) are available, we can drop the assumptionthat u ∈ C ( Ω ) by the usual bootstrap and approximation argument. We break theproof of the estimates into several steps.Unlike Dirichlet or conormal derivative boundary condition cases, we could notfind a global W , p estimate suitable to us in the existing literature. For this reason,we provide a proof which dispenses with a global L p estimates, which also worksfor other boundary conditions. Step 1.
We first establish interior estimates. Let us rewrite the equation as a ij D ij u = f : = f − b i D i u − cu . For B ( x , r ) ⊂ Ω , the proof of [6, Theorem 1.10] with scaling (c.f. [6, (2,17)]) yieldsthe estimate k D u k L ∞ ( B ( x , r )) ≤ Cr − n k D u k L ( B ( x , r )) + C Z r ˜ ω f ( t ) t dt . (4.6)Here, we adopted an abuse of notation˜ ω f ( t ) = ∞ X i = κ i µ (cid:16) ω f ( κ − i t )[ κ − i t ≤ r ] + ω f ( r )[ κ − i t > r ] (cid:17) , where µ ∈ (0 ,
1) is an arbitrary number, κ = κ ( n , λ, Λ , µ ) ∈ (0 , ) is a constant, and ω f ( t ) : = sup x ∈ B ( x , r ) ω f ; x ( t ) , where ω f ; x ( t ) : = ? B ( x , t ) (cid:12)(cid:12)(cid:12) f − ¯ f B ( x , t ) (cid:12)(cid:12)(cid:12) . It should be noted that ˜ ω f satisfies the Dini condition provided ω f satisfies theDini condition (see [3, Lemma 1]). In particular, if ω f ( t ) . t a with 0 < a < µ , then˜ ω f ( t ) . t a as well.By the proof of Lemma 2.9, for B ( x , t ) ⊂ Ω , we have ω f ; x ( t ) ≤ ω f ; x ( t ) + C ( n ) (cid:16) [ u ] B ( x , t ) ω b ; x ( t ) + t µ [ u ] ,µ ; B ( x , t ) k b k L ∞ ( B ( x , t )) (cid:17) + [ u ] B ( x , t ) ω c ; x ( t ) + t µ [ u ] ,µ ; B ( x , t ) k c k L ∞ ( B ( x , t )) . (4.7)Then, by the estimate (4.6), we obtain[ u ] B ( x , r ) ≤ Cr − n k D u k L ( B ( x , r )) + C Z r ˜ ω f ( t ) t dt + C [ u ] B ( x , r ) Z r ˜ ω b ( t ) t dt + r µ [ u ] ,µ ; B ( x , r ) k b k L ∞ ( B ( x , r )) ! + C [ u ] B ( x , r ) Z r ˜ ω c ( t ) t dt + r µ [ u ] ,µ ; B ( x , r ) k c k L ∞ ( B ( x , r )) ! . (4.8)Recall the interpolation inequalities[ u ] k ; B r + [ u ] k ,µ ; B r ≤ [ u ] B r + C r k u k L ( B r ) ( k = , . (4.9)where C r is a constant depending on r (and n , k , and µ ). By setting θ ( r ) : = Z r ˜ ω b ( t ) t dt + Z r ˜ ω c ( t ) t dt + r µ k b k L ∞ ( Ω ) + r µ k c k L ∞ ( Ω ) and applying (4.9) to (4.8), we obtain[ u ] B ( x , r ) ≤ C θ ( r )[ u ] Ω + C k u k L ( Ω ) + C k D u k L ( Ω ) + C Z r ˜ ω f ( t ) t dt . Therefore, by choosing r small, for Ω ′ : = { x ∈ Ω : dist( x , ∂ Ω ) ≥ r } , we have[ u ] Ω ′ ≤
12 [ u ] Ω + C k u k L ( Ω ) + C k D u k L ( Ω ) + C Z r ˜ ω f ( t ) t dt , (4.10)where C is a constant depending on the prescribed data and r . Step 2.
We turn to estimates near the boundary by closely following the idea ofSafonov [18]. In this step, we shall temporarily assume that β ≡
0. First, wemodify [18, Theorem 2.1] to C , Dini setting (see Lemma 5.21 in the Appendix), sothat via a local C , Dini di ff eomorphism, the boundary condition becomes β ( x ) · ∇ u ( x ) = D n ˆ u ( y ) = ˆ g ( y ) , where ˆ g ( y ) = g ( x ) is still of C , Dini . Moreover, ˆ u satisfies the equationˆ a ij D ij ˆ u + ˆ b i D i ˆ u + ˆ c ˆ u = ˆ f , where the coe ffi cients ˆ a ij ( y ), ˆ b i ( y ), ˆ c ( y ), and the data ˆ f ( y ) are of Dini mean oscillationby Lemma 2.9. Preserving the same notation for the transformed objects, we seethat the proof is reduced to the case D n u = g on ∂ Ω ∩ B ( x , r ) , r = const. > . N OBLIQUE DERIVATIVE PROBLEM 15
A slight modification of [18, Theorem 2.2] (see Lemma 5.29 in the Appendix),gives us r > v ∈ C , Dini ( Ω ) satisfying D n v = g on ∂ Ω ∩ B ( x , r )with its C , Dini characteristic determined by g and other prescribed data. Setting u = u − v , we have L u = f − L v = f in Ω , D n u = ∂ Ω ∩ B ( x , r ) . By Lemma 5.29, we also have | v | Ω ≤ C | g | Ω + C Z r ̺ Dg ( ct ) t + C Z r ̺ D γ ( ct ) t dt and for x , y ∈ Ω , we have | D v ( x ) − D v ( y ) | ≤ C Z | x − y | ̺ Dg ( ct ) t dt + C Z | x − y | ̺ D γ ( ct ) t dt . By Lemma 2.9, we see that f is of Dini mean oscillation in Ω and ω f ( t ) ≤ ω f ( t ) + C Z t ̺ Dg ( cs ) s ds + C | g | Ω + Z r ̺ Dg ( ct ) t dt ! ω ( t ) + C ω ( t ) , where C , c > ω ( t ), ω ( t )are nonnegative functions determined by the prescribed data satisfying the Dinicondition. As a matter of fact, we have ω ( t ) = ω A ( t ) + ω b ( t ) + t k b k ∞ + ω c ( t ) + t k c k ∞ ,ω ( t ) = Z r ̺ D γ ( ct ) t dt · ( ω A ( t ) + t k b k ∞ ) + Z t ̺ g ( cs ) + ̺ D γ ( cs ) s ds , where γ is a C , Dini function that represents ∂ Ω ∩ B ( x , r ); see Remark 4.25 below.Therefore, in light of (4.4) and (4.5), by considering u − v instead of u , it remains toprove the theorem under the assumption D n u = ∂ Ω ∩ B ( x , r ) , r = const. > . (4.11)Next, we flatten the boundary by using a “regularized distance” function ψ described in Lemma 5.1 in Appendix, which is originally introduced by Lieberman[11]. We note that D ψ , ∂ Ω ∩ B ( x , r ). Therefore, C , Dini di ff eomorphism x ∈ Ω s ( x ) ←→ z = z ( x ) ∈ ˜ Ω s : = z ( Ω ( x , s )) , where z i = x i − x i ( i = , . . . , n − , z n = ψ ( x ) , (4.12)is well defined for some s ∈ (0 , r / x ∈ Ω ( x , s ), z = z ( x ), let us define˜ u ( z ) = u ( x ). Then, we have D i u ( x ) = D k ˜ u ( z ) D i z k ( x ) , D ij u ( x ) = D kl ˜ u ( z ) D i z k ( x ) D j z l ( x ) + h ij ( x ) , (4.13)where h ij ( x ) = D n ˜ u ( z ) D ij ψ ( x ) . (4.14)Therefore, the equation is turned into˜ a ij D ij ˜ u + ˜ b i D i ˜ u + ˜ c ˜ u = ˜ f in ˜ Ω s = z ( Ω ( x , s )) ⊂ R n + . where ˜ a ij = ˜ a ij ( z ) = a kl ( x ) D k z i ( x ) D l z j ( x ) , ˜ b i = ˜ b i ( z ) = b k ( x ) D k z i ( x ) , ˜ c = ˜ c ( z ) = c ( x ) , ˜ f = ˜ f ( z ) = f ( x ) − a kl ( x ) h kl ( x ) , and the boundary condition (4.11) yields D n ˜ u = z ( ∂ Ω ∩ B ( x , s )) ⊂ ∂ R n + = { z n = } . (4.15)Now, let us choose s ≃ s such that B + (0 , s ) ⊂ ˜ Ω s . Since z = z ( x ) is of C , Dini ,we see from Lemma 2.9 that ˜ a ij , ˜ b i and ˜ c are of Dini mean oscillation in B + (0 , s ).Moreover, the next lemma shows that ˜ h ij ( z ) = h ij ( x ) are Dini continuous in B + (0 , s ). Lemma 4.16.
Denote B + s = B + (0 , s ) and let ˜ h ij ( z ) = h ij ( x ) . We have | ˜ h ij ( z ) | ≤ C k D ˜ u k L ∞ ( B + s ) ϑ ( z n ) , | ˜ h ij ( z ) − ˜ h ij ( z ) | ≤ C k D ˜ u k L ∞ ( B + s ) ϑ ( | z − z | ) , where C is a constant and ϑ ( t ) = ̺ D ψ ( t ) is a nonnegative function satisfying the Dinicondition; see Lemma 5.1.Proof. By Lemma 2.10, we may assume that ϑ ( t ) / t is decreasing for t ∈ (0 , s ). Forany z ∈ B + , by (4.15) and the mean value theorem, we get | D n ˜ u ( z ) | = | D n ˜ u ( z ) − D n ˜ u (¯ z ) | ≤ z n k D ˜ u k L ∞ ( B + s ) , where ¯ z = ( z , . . . , z n − , . Then, by (4.14), (4.12), and Lemma 5.1, we have | ˜ h ij ( z ) | = | h ij ( x ) | ≤ C k D ˜ u k L ∞ ( B + s ) ϑ ( z n ) , ϑ ( t ) = ̺ D ψ ( t ) . Also, since D k ˜ h ij ( z ) = D nm ˜ u ( z ) D k z m ( x ) D ij ψ ( x ) + D n ˜ u ( z ) D ijk ψ ( x ) , we also get | D ˜ h ij ( z ) | ≤ C k D ˜ u k L ∞ ( B + s ) ϑ ( z n ) / z n . (4.17)Consider any two points z , z ∈ B + s with z n ≥ z n . In the case when | z − z | > z n ,we have | ˜ h ij ( z ) − ˜ h ij ( z ) | ≤ | ˜ h ij ( z ) | + | ˜ h ij ( z ) | ≤ C k D ˜ u k L ∞ ( B + s ) ϑ ( z n ) + C k D ˜ u k L ∞ ( B + s ) ϑ ( z n ) ≤ C k D ˜ u k L ∞ ( B + s ) ϑ ( | z − z | ) , where we used ϑ ( at ) & ϑ ( t ) for a ≥ /
2. On the other hand, in the case when | z − z | ≤ z n , we have z n ≤ | z − z | + z n ≤ z n + z n , and thus, we have | z − z | ≤ z n ≤ z n . By the mean value theorem, there is z inthe line segment [ z , z ] satisfying | ˜ h ij ( z ) − ˜ h ij ( z ) | ≤ | D ˜ h ij ( z ) | | z − z | . Note that we have | z − z | ≤ z n ≤ z n . Hence, by using (4.17), we obtain | ˜ h ij ( z ) − ˜ h ij ( z ) | ≤ | D ˜ h ij ( z ) | | z − z | ≤ C k D ˜ u k L ∞ ( B + s ) ϑ ( z n ) z n | z − z |≤ C k D ˜ u k L ∞ ( B + s ) ϑ ( | z − z | ) , N OBLIQUE DERIVATIVE PROBLEM 17 where we used that ϑ ( t ) / t is decreasing. This completes the proof. (cid:4) By Lemmas 4.16 and 2.9, we find that ˜ f = ˜ f ( z ) = f ( x ) − a kl ( x ) h kl ( x ) is of Dini meanoscillation in B + = B + (0 , s ) and there is a constant a > ω ˜ f ( t ) ≤ C (cid:18) ω f ( at ) + ω A ( at )[ ˜ u ] B + s + ϑ ( at )[ ˜ u ] B + s (cid:19) , < ∀ t < s . (4.18)Now we set ˜ f : = ˜ f − ˜ b i D i ˜ u − ˜ c ˜ u . Note that by Lemma 2.9, we have (c.f. (4.7) and (4.18) above) ω ˜ f ( t ) ≤ ω ˜ f ( t ) + C (cid:18) [ ˜ u ] B + s ω ˜ b ( t ) + t µ [ ˜ u ] ,µ ; B + s k ˜ b k L ∞ ( B + s ) (cid:19) + C (cid:18) [ ˜ u ] B + s ω ˜ c ( t ) + t µ [ ˜ u ] ,µ ; B + s k ˜ c k L ∞ ( B + s ) (cid:19) , < ∀ t < s . Also, by the interpolation inequalities (c.f. (4.9) above) we have[ ˜ u ] B + s + [ ˜ u ] ,µ ; B + s + [ ˜ u ] B + s + [ ˜ u ] ,µ ; B + s ≤ [ ˜ u ] B + s + C k ˜ u k L ( B + s ) . Then by (4.18), for any 0 < t < s , we have ω ˜ f ( t ) ≤ C ω f ( at ) + C (cid:16) ϑ ( t ) + ϑ ( t ) (cid:17) [ ˜ u ] B + s + C ϑ ( t ) k ˜ u k L ( B + s ) . (4.19)where we set ϑ ( t ) : = ω A ( at ) + ϑ ( at ) ,ϑ ( t ) : = ω ˜ b ( t ) + ω ˜ c ( t ) + k ˜ b k L ∞ ( B + s ) t µ + k ˜ c k L ∞ ( B + s ) t µ . Note that ϑ ( t ) and ϑ ( t ) both satisfy the Dini condition.Therefore, we are reduced to˜ a ij D ij ˜ u = ˜ f in B + (0 , s ) , D n ˜ u = T (0 , s ) , where ˜ f is of Dini mean oscillation. By Proposition 4.1 and (4.19), we have[ ˜ u ] B + s ≤ C k D ˜ u k L ( B + s ) + C Z s ˆ ϑ ( t ) t dt + Z s ˆ ϑ ( t ) t dt ! [ ˜ u ] B + s + C Z s ˆ ω f ( at ) t dt + C Z s ˆ ϑ ( t ) t dt ! k ˜ u k L ( B + s ) . (4.20)Note that the equalities (4.13) and Lemma 4.16 imply[ u ] Ω ( x ,δ s ) ≤ C [ ˜ u ] B + s for some constant 0 < δ < . We also have[ ˜ u ] B + s ≤ C [ u ] Ω , because the mapping x = x ( z ) has the same properties as z = z ( x ).By requiring s so small that we have C Z s ˆ ϑ ( t ) t dt + Z s ˆ ϑ ( t ) t dt ! ≤ . Therefore, we get from (4.20) that[ u ] Ω ( x ,δ s ) ≤ C k u k W , ( Ω ) +
12 [ u ] Ω + C Z s ˆ ω f ( at ) t dt + C Z s ˆ ϑ ( t ) t dt ! k u k L ( Ω ) . (4.21) By combining (4.10) and (4.21), we get (4.4). Then, (4.5) is obtained by combining(4.3) and the interior estimate appears in the proof of [6, Theorem 1.6].
Step 3.
Finally, we drop the temporary assumption that β ≡
0. We rewrite theboundary condition as β · ∇ u = g : = g − β u on ∂ Ω . Recall Definition 2.4 and observe that ̺ D ( β u ) ( t ) ≤ ̺ D β ( t )[ u ] Ω + [ β ] Ω [ u ] ,µ ; Ω t µ + [ β ] Ω [ u ] Ω t + [ β ] Ω [ u ] ,µ ; Ω t µ . Therefore, by the interpolation inequalities[ u ] Ω + [ u ] ,µ ; Ω + [ u ] Ω + [ u ] ,µ ; Ω ≤ ε [ u ] Ω + C ε k u k L ( Ω ) , we find that β u ∈ C , Dini ( Ω ) and its C , Dini characteristic is determined by that β and the right-hand side of the above inequality. By choosing ε small, we can hide[ u ] Ω contribution. This completes the proof of Theorem 1.10. (cid:4) Proof of Proposition 4.1.
Once again, we derive an a priori estimate of the modulusof continuity of D u by assuming that u is in C ( B + ). As before, we fix some p ∈ (0 , φ ( x , r ) : = inf q ∈ S ( n ) ? B ( x , r ) ∩ B + | D u − q | p p , where S ( n ) is the set of all n × n symmetric real matrices.We shall derive an estimate for φ ( ¯ x , r ) for ¯ x ∈ T (0 ,
3) and 0 < r ≤
1. We split u = v + w , where w ∈ W , ( B + ( ¯ x , r )) is a strong solution of the mixed problem¯ a ij D ij w = − tr(( A − ¯ A ) D u ) + f − ¯ f in B + ( ¯ x , r ) , u = ∂ B ( ¯ x , r ) ∩ R n + , D n u = T ( ¯ x , r ) , where ¯ A = ¯ A B + (¯ x , r ) and ¯ f = ¯ f B + (¯ x , r ) . By Lemma 2.13 with scaling, we see that (cid:12)(cid:12)(cid:12) { x ∈ B + ( ¯ x , r ) : | D w ( x ) | > t } (cid:12)(cid:12)(cid:12) ≤ Ct Z B + (¯ x , r ) | f − ¯ f | + [ u ] B + (¯ x , r ) Z B + (¯ x , r ) | A − ¯ A | ! . Then similar to [6, (2.11)], we get ? B + (¯ x , r ) | D w | p ! p ≤ C [ u ] B + (¯ x , r ) ω A ( r ) + C ω f ( r ) . (4.22)Next v : = u − w solves¯ a ij D ij v = ¯ f in B + ( ¯ x , r ) , D n v = T ( ¯ x , r ) . Hence, for any k , l ∈ { , . . . , n − } and c ∈ R , the function V : = D kl v − c satisfies¯ a ij D ij V = B + ( ¯ x , r ) , D n V = T ( ¯ x , r ) . By applying Lemma 2.20 with scaling, we see that k DD kl v k L ∞ ( B + (¯ x , r )) ≤ Cr − ? B + (¯ x , r ) | D kl v − c | p ! p . Therefore, by setting D x ′ v : = { D ij v : 1 ≤ i , j ≤ n − } , N OBLIQUE DERIVATIVE PROBLEM 19 we find that k DD x ′ v k L ∞ ( B + (¯ x , r )) ≤ Cr − ? B + (¯ x , r ) | D v − q | p ! p , ∀ q ∈ S ( n ) . (4.23)Since D nn v = − a nn X ( i , j ) , ( n , n ) ¯ a ij D ij v + ¯ f , by taking the partial derivative with respect to x m , we obtain D nnm v = − a nn X ( i , j ) , ( n , n ) ¯ a ij D ijm v , (4.24)and thus it follows from (4.23) that for any m ∈ { , . . . , n − } , we have k D D m v k L ∞ ( B + (¯ x , r )) ≤ Cr − ? B + (¯ x , r ) | D v − q | p ! p , ∀ q ∈ S ( n ) . Then, by taking m = n in (4.24) we get k D v k L ∞ ( B + (¯ x , r )) ≤ Cr − ? B + (¯ x , r ) | D v − q | p ! p , ∀ q ∈ S ( n ) . Let 0 < κ ≤ be a constant to be fixed later. By the mean value theorem, we have ? B + (¯ x ,κ r ) | D v − ( D v ) κ r | p ! p ≤ κ r k D v k L ∞ ( B + (¯ x , r )) , where ( D v ) κ r : = ? B + (¯ x ,κ r ) D v . Hence, we see that there is some constant C = C ( n , λ, Λ ) such that ? B + (¯ x ,κ r ) | D v − ( D v ) κ r | p ! p ≤ C κ ? B + (¯ x , r ) | D v − q | p ! p , ∀ q ∈ S ( n ) . By using the decomposition u = v + w , similar to (3.4), we obtain ? B + (¯ x ,κ r ) (cid:12)(cid:12)(cid:12) D u − ( D v ) κ r (cid:12)(cid:12)(cid:12) p ! p ≤ − pp ? B + (¯ x ,κ r ) (cid:12)(cid:12)(cid:12) D v − ( D v ) κ r (cid:12)(cid:12)(cid:12) p ! p + − pp ? B + (¯ x ,κ r ) | D w | p ! p ≤ − pp C κ ? B + (¯ x , r ) | D u − q | p ! p + C ( κ − np + ? B + (¯ x , r ) | D w | p ! p . Since q ∈ S ( n ) is arbitrary, by using (4.22), we obtain φ ( ¯ x , κ r ) ≤ − pp C κ φ ( ¯ x , r ) + C ( κ − np + (cid:16) ω A ( r ) k Du k L ∞ ( B + (¯ x , r )) + ω f ( r ) (cid:17) , which is analogous to (3.4). Also, we note that (3.5) is available whenever B ( x , ρ ) ⊂ B + ( ¯ x , R ) and ρ ≃ R .Consequently, we have Lemmas 2.16, 2.17, 2.18, and 2.19 in [4] available in oursetting with φ ( ¯ x , r ). In particular, by [4, Lemma 2.18], we obtain (4.2) . Also, theestimate (4.3) follows by a similar argument employed in deriving [4, (2.36)]. (cid:4) Remark . Observe that by Lemmas 2.9 and 5.29, we have ω A D v ( t ) . [ v ] Ω ω A ( t ) + k A k ∞ ̺ D v ( t ) . | g | Ω + Z b ̺ Dg ( ct ) t + ̺ D γ ( ct ) t dt ! ω A ( t ) + Z t ̺ Dg ( cs ) s ds + Z t ̺ D γ ( cs ) s ds ,ω b Dv ( t ) . [ v ] Ω ω b ( t ) + k b k ∞ ̺ Dv ( t ) . | g | Ω ω b ( t ) + t k b k ∞ [ v ] Ω . | g | Ω ω b ( t ) + t | g | Ω + Z b ̺ Dg ( ct ) t + ̺ D γ ( ct ) t dt ! k b k ∞ ,ω cv ( t ) . [ v ] Ω ω c ( t ) + k c k ∞ ̺ v ( t ) . | g | Ω ω c ( t ) + t k c k ∞ [ v ] Ω . | g | Ω ω c ( t ) + t | g | Ω k c k ∞ . Therefore, we have ω f ( t ) ≤ ω f ( t ) + ω L v ( t ) . ω f ( t ) + ω A D v ( t ) + ω b Dv ( t ) + ω cv ( t ) . ω f ( t ) + | g | Ω + Z b ̺ Dg ( ct ) t dt ! · ( ω A ( t ) + ω b ( t ) + t k b k ∞ + ω c ( t ) + t k c k ∞ ) + Z b ̺ D γ ( ct ) t dt · ( ω A ( t ) + t k b k ∞ ) + Z t ̺ Dg ( cs ) s ds + Z t ̺ D γ ( cs ) s ds .
5. A ppendix
In the Appendix, we provide the proofs for some technical lemmas used beforeby slightly modifying those in Safonov’s paper [18].
Lemma 5.1.
Let Ω be a C , Dini domain with a defining C , Dini function ψ (see Remark2.6). Then there exists a function ψ ∈ C , Dini ( R n ) ∩ C ∞ ( Ω ) such that δψ ( x ) ≤ d x : = dist( x , ∂ Ω ) ≤ δ − ψ ( x ) , ∀ x ∈ Ω , [ ψ ] Ω ≤ , ̺ D ψ ( t ) ≤ C ̺ D ψ ( ct ) , where [ · ] k ; Ω and ̺ • are as defined in (2.2) and Definition 2.4, respectively, δ = δ ( n , ψ ) ∈ (0 , , C = C ( n , ψ ) , and c = c ( n , ψ ) > . Moreover, for any multi-index l with | l | = m ≥ , we have | D l ψ ( x ) | ≤ C ψ ( x ) − m ̺ D ψ ( ψ ( x )) , ∀ x ∈ Ω , where C = C ( n , m , ψ ) .Proof. We modify the proof of [18, Lemma 2.4]. Since ψ is Lipschitz and | D ψ | ≥ ∂ Ω , there exist constants K > δ ∈ (0 , n and ψ , such that | ψ ( x ) − ψ ( y ) | ≤ K | x − y | , ∀ x , y ∈ R n and δψ ( x ) ≤ dist( x , ∂ Ω ) ≤ δ − ψ ( x ) , ∀ x ∈ Ω . (5.2)Let us consider a function Ψ ( t , x ) = ψ ( t )0 ( x ) = Z ψ ( x − ty ) ζ ( y ) dy on R n + , (5.3)where ζ is a standard mollifier, that is a C ∞ function supported in the unit ball B (0 ,
1) satisfying 0 ≤ ζ ≤ R ζ =
1. Then it follows | Ψ ( t , x ) − Ψ ( t , x ) | ≤ K | t − t | . N OBLIQUE DERIVATIVE PROBLEM 21
Therefore, we can define the implicit function ψ ( x ) = t = (2 K ) − Ψ ( t , x ) on R n . (5.4)We have | K ψ ( x ) − ψ ( x ) | = | Ψ ( t , x ) − Ψ (0 , x ) | ≤ K | t | = K | ψ ( x ) | . This inequality implies 13 ≤ K ψ ( x ) ψ ( x ) ≤ R n \ ∂ Ω . Hence, by (5.2), we have δ Kt = δ K ψ ( x ) ≤ dist( x , ∂ Ω ) ≤ K δ − ψ ( x ) = K δ − t , ∀ x ∈ Ω . Now, it follows from (5.3) that Ψ ∈ C , Dini ( R n + ) andsup R n + | D Ψ | ≤ sup R n | D ψ | ≤ K . (5.5)Also, since D t Ψ ( t , x ) − D t Ψ ( t , x ) = Z B (0 , (cid:0) D ψ ( x − t y ) − D ψ ( x − t y ) (cid:1) · ( − y ) ζ ( y ) dy and D x i Ψ ( t , x ) − D x i Ψ ( t , x ) = Z B (0 , (cid:0) D i ψ ( x − t y ) − D i ψ ( x − t y ) (cid:1) ζ ( y ) dy , we have ̺ D Ψ ( r ) ≤ C ̺ D ψ ( cr ) , (5.6)where C = C ( n ) and c = c ( n ) >
0. Therefore, we have ψ ∈ C , Dini ( R n ). Moreover, by(5.4), we find that D i ψ ( x ) = (2 K ) − D t Ψ ( ψ ( x ) , x ) D i ψ ( x ) + (2 K ) − D x i Ψ ( ψ ( x ) , x ) , and thus by (5.5), we obtain sup R n | D ψ | ≤ . (5.7)Moreover, since | D i ψ ( x ) − D i ψ ( y ) | ≤ (2 K ) − | D t Ψ ( ψ ( x ) , x ) | | D i ψ ( x ) − D i ψ ( y ) | + (2 K ) − | D t Ψ ( ψ ( x ) , x ) − D t Ψ ( ψ ( y ) , y ) | | D i ψ ( y ) | + (2 K ) − | D i Ψ ( ψ ( x ) , x ) − D i Ψ ( ψ ( y ) , y ) | , we get | D i ψ ( x ) − D i ψ ( y ) | ≤ | D i ψ ( x ) − D i ψ ( y ) | + (cid:18) + K (cid:19) ̺ D Ψ ( a | x − y | ) , a = √ + K , and thus by (5.6) ̺ D ψ ( τ ) ≤ C ̺ D ψ ( c τ ) , where C = C ( n , K ) and c = c ( K ) . (5.8)Furthermore, for any multi-index l ∈ Z n + + with | l | = m ≥ t ,
0, using (5.3), weobtain similar to Lemma 2.3 of [18] that | D l Ψ ( t , x ) | ≤ C ( n , m ) t − m ̺ D ψ ( t ) . (5.9) Indeed, we have D t Ψ ( t , x ) = Z D ψ ( x − ty ) · ( − y ) ζ ( y ) dy = −| t | − n Z D k ψ ( z ) ζ k (cid:18) x − zt (cid:19) dz , where we set ζ k ( x ) : = x k ζ ( x ), and D x i Ψ ( t , x ) = Z D i ψ ( x − ty ) ζ ( y ) dy = | t | − n Z D i ψ ( z ) ζ (cid:18) x − zt (cid:19) dz . Therefore, for t >
0, we have D tt Ψ ( t , x ) = nt − n − Z D k ψ ( z ) ζ k (cid:18) x − zt (cid:19) dz + t − n − Z D k ψ ( z ) ˜ ζ k (cid:18) x − zt (cid:19) dz , where we set ˜ ζ k ( x ) : = x · D ζ ( x ). Since R ζ k = R ˜ ζ k = − n R ζ k =
0, we have D tt Ψ ( t , x ) = nt − n − Z (cid:0) D k ψ ( z ) − D k ψ ( x ) (cid:1) ζ k (cid:18) x − zt (cid:19) dz + t − n − Z (cid:0) D k ψ ( z ) − D k ψ ( x ) (cid:1) ˜ ζ k (cid:18) x − zt (cid:19) dz . Since the above integrals are actually taken over B ( x , t ), we have | D tt Ψ ( t , x ) | ≤ C ( n ) t − n − ̺ D ψ ( t ) | B ( x , t ) | (cid:16) k ζ k k ∞ + k ˜ ζ k k ∞ (cid:17) ≤ C ( n ) t − ̺ D ψ ( t ) . By a similar computation, we get | D tx i Ψ ( t , x ) | ≤ C ( n ) t − ̺ D ψ ( t ) , | D x i x j Ψ ( t , x ) | ≤ C ( n ) t − ̺ D ψ ( t ) . We have thus shown (5.9) for m = t >
0. The general cases can be deducedin the same fashion.For a multi-index l ∈ Z n + with | l | = m ≥
2, by the chain rule and a directcomputation, we obtain from (5.4), (5.5), (5.7), and (5.9) that | D l ψ ( x ) | ≤ C ( ψ ( x )) − m ̺ D ψ ( ψ ( x )) , ∀ x ∈ Ω , The lemma is proved. (cid:4)
Corollary 5.10.
Assume the same hypothesis as in Lemma 5.1. Then, for any functionu ∈ C , Dini ( Ω ) , there exists a function ˜ u ∈ C , Dini ( Ω ) ∩ C ∞ ( Ω ) such that ˜ u = u on ∂ Ω , | ˜ u | Ω ≤ C | u | Ω and ̺ D ˜ u ( t ) ≤ C (cid:16) k Du k L ∞ ( Ω ) ̺ D ψ ( ct ) + ̺ Du ( ct ) (cid:17) , where |·| k ; Ω and ̺ • are as defined in (2.3) and Definition 2.4, respectively, C = C ( n , ψ ) ,and c = c ( n , ψ ) > . Moreover, for any multi-index l with | l | = m ≥ , we have | D l ˜ u ( x ) | ≤ Cd − mx (cid:16) ̺ Du ( d x ) + ̺ D ψ ( cd x ) (cid:17) , ∀ x ∈ Ω , d x : = dist( x , ∂ Ω ) , where C = C ( n , m , ψ ) and c = c ( ψ ) > .Proof. We modify the proof of [18, Corollary 2.1]. Let ψ be from Lemma 5.1. Similarto (5.3), define U ( t , x ) = u ( t ) ( x ) = Z u ( x − ty ) ζ ( y ) dy on R n + . Then, the function ˜ u ( x ) = U ( δψ ( x ) , x ) (5.11) N OBLIQUE DERIVATIVE PROBLEM 23 is well defined in Ω , and ˜ u = u on ∂ Ω . It is clear that[ ˜ u ] Ω ≤ [ U ] R × Ω ≤ [ u ] Ω , where [ · ] k ; Ω is as defined in (2.2). Moreover, since D i ˜ u ( x ) = δ D t U ( δψ ( x ) , x ) D i ψ ( x ) + D i U ( δψ ( x ) , x ) , we have [ ˜ u ] Ω ≤ C [ U ] R × Ω ≤ C [ u ] Ω , (5.12)where C = C ( n , ψ ). As in (5.6), we also have ̺ DU ( t ) ≤ C ̺ Du ( ct ) , (5.13)where C = C ( n ) and c = c ( n ). Furthermore, since | D i ˜ u ( x ) − D i ˜ u ( y ) | ≤ δ | D t U ( δψ ( x ) , x ) | | D i ψ ( x ) − D i ψ ( y ) | + δ | D t U ( δψ ( x ) , x ) − D t U ( δψ ( y ) , y ) | | D i ψ ( y ) | + | D i U ( δψ ( x ) , x ) − D i U ( δψ ( y ) , y ) | , we have (recall k D ψ k ∞ ≤ ̺ D ˜ u ( τ ) ≤ C k DU k ∞ ̺ D ψ ( τ ) + C ̺ DU ( a τ ) , where C = C ( n , ψ ) and a = √ + δ . Then, by (5.12), (5.13), and (5.8), we have ̺ D ˜ u ( τ ) ≤ C k Du k ∞ ̺ D ψ ( c τ ) + C ̺ Du ( c τ ) , where C = C ( n , ψ ) and c = c ( n , ψ ) > l ∈ Z n + + with | l | = m ≥
2, we get | D l U ( t , x ) | ≤ C ( n , m ) t − m ̺ Du ( t ) , t > . Also, by [18, (2.20)], for any multi-index l ∈ Z n + + with | l | =
1, we have | D l U ( t , x ) | ≤ C ( n ) k Du k ∞ , t > . Then, by using the above two inequalities, for any multi-index l ∈ Z n + with | l | = m ≥
2, we derive from (5.11) that | D l ˜ u ( x ) | ≤ C ψ ( x ) − m (cid:16) ̺ Du ( δψ ( x )) + k Du k ∞ ̺ D ψ ( ψ ( x )) (cid:17) . (cid:4) Lemma 5.14.
Let τ > , n ∈ N , and let a n × n matrix function A ( t ) = [ A ij ( t )] and a vector valued function B ( t ) with values in R n be defined and continuous on [0 , τ ) .Suppose that | A ( t ) | ≤ K , | B ( t ) | ≤ K e K t ( τ − t ) − ̺ ( τ − t ) (5.15) on [0 , τ ) for some constants K , K ≥ , and a function ̺ on [0 , τ ) satisfying ̺ ( t ) > and ddt ( t − µ ̺ ( t )) ≤ for some µ ∈ (0 , . Then every solution X ( t ) of the systemd X dt = A X + B , ≤ t ≤ τ, satisfies the estimate | X ( t ) | ≤ N e K t ( τ − t ) − ̺ ( τ − t ) , ≤ t < τ, (5.16) where N = max n τ̺ ( τ ) − | X (0) | , K / (1 − µ ) o . Proof.
We modify the proof of [18, Lemma 2.5]. Consider the function f ( t ) = e − K t ( τ − t ) | X ( t ) | /̺ ( τ − t ) , ≤ t < τ. (5.17)Obviously the inequality (5.16) is equivalent to f ( t ) ≤ N . By the choice of N , wehave f (0) ≤ N . Suppose that (5.16) fails for some t ∈ (0 , τ ). Then there exist ε > t ∈ (0 , τ )such that f ( t ) < N + ε on [0 , t ) , f ( t ) = N + ε. (5.18)Moreover, since K ≤ (1 − µ ) N , by (5.15) we have | B ( t ) | ≤ (1 − µ ) N e K t ( τ − t ) − ̺ ( τ − t ) < (1 − µ )( τ − t ) − | X ( t ) | . Therefore, for t = t , ddt | X | = X · d X dt = X · ( A X + B ) < K + − µτ − t ! | X | . (5.19)Also, by the assumption that ddt ( t − µ ̺ ( t )) ≤
0, we obtain ̺ ′ ( t ) ≤ µ t − ̺ ( t ) . (5.20)By di ff erentiating the equation (5.17) and using the above two inequalities, we get f ′ ( t ) = − K e − K t ( τ − t ) | X ( t ) | /̺ ( τ − t ) − e − K t τ − t ) | X ( t ) | /̺ ( τ − t ) + e − K t ( τ − t ) ddt | X ( t ) | /̺ ( τ − t ) + e − K t ( τ − t ) | X ( t ) | ̺ ′ ( τ − t ) /̺ ( τ − t ) . Then, by using inequalities (5.19) and (5.20) we get f ′ ( t ) < e − K t ( τ − t ) | X ( t ) | ̺ ( τ − t ) (cid:0) − K ( τ − t ) − + K ( τ − t ) + − µ ) + µ (cid:1) = . On the other hand, (5.18) yields f ′ ( t ) ≥
0. This contradiction proves the estimate(5.16). (cid:4)
Lemma 5.21.
Let Ω ⊂ R n be a C , Dini domain with a defining function ψ and β = ( β , . . . , β n ) ∈ C , Dini ( Ω ) satisfy the condition (1.5) . Then there exists a constant r > ,and for every x ∈ ∂ Ω , there exists a one-to-one C , Dini mapping Φ : B ( x , r ) → R n suchthat upon writing y = Φ ( x ) and x = Ψ ( y ) , we have β i = ∂ x i ∂ y n = ∂ Ψ i ∂ y n on ∂ Ω ∩ B ( x , r ) , i = , . . . , n . The C , Dini characteristics of Φ and Ψ are determined only by the given data, namely, µ and C , Dini characteristics of ∂ Ω and β .Proof. We slightly modify the proof of [18, Theorem 2.1] using Corollary 5.10 andLemma 5.14 instead of [18, Corollary 2.1] and [18, Lemma 2.5], respectively.We follow exactly the same proof of Theorem 2.1 in [18] up to the beginning ofthe evaluation of second and third derivatives of x = x ( y ). In particular, we use thesame symbolic notation there so that ddt ∂ x i ∂ y j = X k ∂β i ∂ x k ∂ x k ∂ y j , i.e., ddt ∂ x ∂ y = ∂ β ∂ x ∂ x ∂ y (5.22) N OBLIQUE DERIVATIVE PROBLEM 25 turns into the form dx ′ / dt = P β ′ x ′ . Then, we have the estimates | β ′ | ≤ N , | x ′ | ≤ N with di ff erent constants N >
0. Di ff erentiating (5.22) twice, we obtain that x ′′ = ∂ x /∂ y i ∂ y j and x ′′′ = ∂ x /∂ y i ∂ y j ∂ y k satisfy the systems ddt x ′′ = X β ′ x ′′ + X β ′′ x ′ x ′ , (5.23) ddt x ′′′ = X β ′ x ′′′ + X β ′′ x ′′ x ′ + X β ′′′ x ′ x ′ x ′ , which correspond to [18, (2.53)] and [18, (2.54)] . Here, by using Corollary 5.10,we modified β in such a manner that β ∈ C , Dini ( Ω ( x , r )) ∩ C ∞ ( Ω ( x , r )), where Ω ( x , r ) = Ω ∩ B ( x , r ) for some r >
0, and for | l | = m ≥ x ∈ Ω ( x , r ), | D l β ( x ) | ≤ Nd − mx (cid:16) ̺ D β ( d x ) + ̺ D ψ ( d x ) (cid:17) , d x = dist( x , ∂ Ω ) . (5.24)Then by (5.24) and [18, (2.47)], we get (by replacing ̺ • ( t ) = ̺ • ( Nt ) if necessary) | β ′′ ( x ) | ≤ N ( τ − t ) − (cid:16) ̺ D β ( τ − t ) + ̺ D ψ ( τ − t ) (cid:17) , | β ′′′ ( x ) | ≤ N ( τ − t ) − (cid:16) ̺ D β ( τ − t ) + ̺ D ψ ( τ − t ) (cid:17) . (5.25)Let us apply Lemma 5.14 to the system (5.23), where X = { x ′′ } , A X = { P β ′ x ′′ } , B = { P β ′′ x ′ x ′ } , to get (note that ( τ − t ) − ≤ N ( τ − t ) − ) | x ′′ | ≤ N ( τ − t ) − (cid:16) ̺ D β ( τ − t ) + ̺ D ψ ( τ − t ) (cid:17) ≤ N ( τ − t ) − . (5.26)Now, let us apply Lemma 5.14 to (5.26), where X = { x ′′′ } , A X = { P β ′ x ′′′ } , B = { P β ′′ x ′′ x ′ + P β ′′′ x ′ x ′ x ′ } . The estimate (5.25), (5.26) provide us | B ( t ) | ≤ N ( τ − t ) − (cid:16) ̺ D β ( τ − t ) + ̺ D ψ ( τ − t ) (cid:17) , hence (by replacing ̺ • ( t ) with ̺ • ( ct ) if necessary) | x ′′′ | ≤ N ( τ − t ) − (cid:16) ̺ D β ( τ − t ) + ̺ D ψ ( τ − t ) (cid:17) ≤ Nd − x (cid:16) ̺ D β ( d x ) + ̺ D ψ ( d x ) (cid:17) . (5.27)Since x = x ( y ) is the C di ff eomorphism, the inverse mapping y = y ( x ) ∈ C ( Ω r ),and ˆ d y = dist( y , ∂ ˆ Ω r ) ≤ Nd x for x ∈ Ω r , y = y ( x ) ∈ ˆ Ω r = y ( Ω r ) . Therefore, from (5.27) it follows | x ′′′ ( y ) | ≤ N ˆ d − y (cid:16) ̺ D β ( ˆ d y ) + ̺ D ψ ( ˆ d y ) (cid:17) . (5.28)Finally, we estimate the modulus of continuity of x ′′ by modifying the proof of[18, Lemma 2.1]. Let us fix y , y ∈ ˆ Ω r , and set r = | y − y | . One can choose y ∈ ˆ Ω r such that B ( y , r / N ) ∈ ˆ Ω r , | y k − y | ≤ Nr for k = , , for some N >
0. Furthermore, we can connect y k with y by means of a smoothpath in ˆ Ω r , { y = h k ( s ) : 0 ≤ s ≤ s k } , s k ≤ Nr , h k (0) = y k , h k ( s k ) = y , parameterized by the arc length s in such a manner that s / N ≤ ˆ d h k ( s ) ≤ N , ≤ s ≤ s k . By the mean value theorem and (5.28), we get | x ′′ ( y ) − x ′′ ( y k ) | ≤ Z s k | x ′′′ ( h k ( s )) | ds ≤ C Z s k ̺ D β ( ˆ d h k ( s ) )ˆ d h k ( s ) + ̺ D ψ ( ˆ d h k ( s ) )ˆ d h k ( s ) ds . Again, by Lemma 2.10, we may assume without loss of generality that the functions ̺ D β ( t ) / t and ̺ D ψ ( t ) / t are decreasing. Then, we have | x ′′ ( y ) − x ′′ ( y ) | ≤ X k = | x ′′ ( y ) − x ′′ ( y k ) |≤ CN X k = Z s k ̺ D β ( s / N ) s + ̺ D ψ ( s / N ) s ds ≤ ˜ C Z r ̺ Dg ( s ) s + ̺ D ψ ( s ) s ds , where ˜ C is a constant depending only on n and γ . (cid:4) Lemma 5.29.
Let Ω ⊂ R n be a C , Dini domain with a defining function ψ , and γ : R n − → R be in Definition 2.5. Fix a small b > so that | D γ ( x ′ ) | < for any x ′ ∈ R n − with | x ′ | < b. Denote U b = { x = ( x ′ , x n ) ∈ R n : γ ( x ′ ) < x n < b , | x ′ | < b } . Then for any function g ∈ C , Dini ( Ω ) , there exists a function v ∈ C , Dini (U b ) such thatD n v = g on ∂ Ω . Moreover, we have | v | b ≤ C | g | Ω , [ v ] b ≤ C | g | Ω + C Z b ̺ Dg ( ct ) t + ̺ D γ ( ct ) t dt , and for a multi-index l with | l | = m ≥ , we have | D l v ( x ) | ≤ Cd − mx (cid:16) ̺ Dg ( cd x ) + ̺ D γ ( cd x ) (cid:17) , ∀ x ∈ U b , d x = dist( x , ∂ Ω ) . Furthermore, we have ̺ D v ( t ) ≤ C Z t ̺ Dg ( cs ) s + ̺ D γ ( cs ) s ds . In the above, C and c are constants, which vary from line to line, depending only on n, γ ,and b.Proof. We modify the proof of Theorem 2.2 in [18]. For x = ( x ′ , x n ) ∈ U b , we set¯ d x = x n − γ ( x ′ ) . Note that we have ¯ d x ≃ d x = dist( x , ∂ Ω ) for x ∈ U b . By Corollary 5.10, there exists afunction ˜ g ∈ C , Dini ( Ω ) ∩ C ∞ ( Ω ) such that ˜ g = g on ∂ Ω and | ˜ g | Ω ≤ C | g | Ω , (5.30)and for any multi-index l with | l | = m ≥ | D l ˜ g ( x ) | ≤ C ¯ d − mx (cid:16) ̺ Dg ( c ¯ d x ) + ̺ D ψ ( c ¯ d x ) (cid:17) , ∀ x ∈ U b . (5.31) N OBLIQUE DERIVATIVE PROBLEM 27
Now, we define v ( x ) = v ( x ′ , x n ) = − Z bx n ˜ g ( x ′ , t ) dt , x ∈ U b . Then, it is clear that D n v = ˜ g = g on ∂ Ω . Moreover, by (5.30), for x ∈ U b , we have | v ( x ) | ≤ b [ ˜ g ] Ω ≤ C | g | Ω , | D i v ( x ) | ≤ b [ ˜ g ] Ω ≤ C | g | Ω , for i = , . . . , n −
1, and also | D n v ( x ) | = | ˜ g ( x ) | ≤ C | g | Ω . Now, for l = ( l ′ , l n ), | l | = | l ′ | + l n = m ≥
2, and x ∈ U b , we consider separately thecases l n ≥ l n =
0. If l n ≥
1, then D l v ( x ) = D l ′ D l n n v ( x ) = D l ′ D l n − n ˜ g ( x ) , and thus, when m =
2, we have | D l v ( x ) | ≤ [ ˜ g ] Ω ≤ C | g | Ω , and when m ≥
3, by (5.31), we have | D l v ( x ) | ≤ C ¯ d − mx (cid:16) ̺ Dg ( c ¯ d x ) + ̺ D γ ( c ¯ d x ) (cid:17) . If l n =
0, by using (5.31) and noting ¯ d ( x ′ , t ) = t − γ ( x ′ ), we obtain | D l v ( x ) | ≤ Z bx n | D l ˜ g ( x ′ , t ) | dt ≤ C Z b − γ ( x ′ )¯ d x ̺ Dg ( c t ) t m − + ̺ D γ ( c t ) t m − dt . (5.32)In particular, when m =
2, we derive from (5.32) that | D l v ( x ) | ≤ C Z b ̺ Dg ( c t ) t + ̺ D γ ( c t ) t dt . Let us fix µ ∈ (0 , ̺ Dg ( c t ) / t µ and ̺ D γ ( c t ) / t µ are decreasing on (0 , b ]. Hence, when m ≥ | D l v ( x ) | ≤ C ̺ Dg ( c ¯ d x )¯ d µ x + ̺ D γ ( c ¯ d x )¯ d µ x ! Z b − γ ( x ′ )¯ d x t µ + − m dt ≤ C ̺ Dg ( c ¯ d x )¯ d µ x + ̺ D γ ( c ¯ d x )¯ d µ x ! ( m − − µ ) ¯ d µ + − mx . Therefore, in conclusion, for | l | =
2, we have | D l v ( x ) | ≤ C | g | Ω + C Z b ̺ Dg ( c t ) t + ̺ D γ ( c t ) t dt , ∀ x ∈ U b , and for | l | = m ≥
3, we have | D l v ( x ) | ≤ C ¯ d − mx ̺ Dg ( c ¯ d x ) + C ¯ d − mx ̺ D γ ( c ¯ d x ) , ∀ x ∈ U b . (5.33)Finally, we estimate the modulus of continuity of D v . Let us fix x , x ∈ U b , andset r = | x − x | . One can choose x ∈ U b such that B ( x , r / N ) ∈ U b , | x k − x | ≤ Nr for k = , , for some N >
0. Furthermore, we can connect x k with x by means of a smoothpath in U b , { x = h k ( s ) : 0 ≤ s ≤ s k } , s k ≤ Nr , h k (0) = x k , h k ( s k ) = x , parameterized by the arc length s in such a manner that s / N ≤ ¯ d h k ( s ) ≤ N , ≤ s ≤ s k . By the mean value theorem and (5.33) with m =
3, we get | D ij v ( x ) − D ij v ( x k ) | ≤ Z s k | DD ij v ( h k ( s )) | ds ≤ C Z s k ̺ Dg ( c ¯ d h k ( s ) )¯ d h k ( s ) + ̺ D γ ( c ¯ d h k ( s ) )¯ d h k ( s ) ds . Again, by Lemma 2.10, we may assume without loss of generality that the functions ̺ Dg ( c t ) / t and ̺ D γ ( c t ) / t are decreasing on (0 , b ]. Then, we have | D ij v ( x ) − D ij v ( x ) | ≤ X k = | D ij v ( x ) − D ij v ( x k ) |≤ CN X k = Z s k ̺ Dg ( c s / N ) s + ̺ D γ ( c s / N ) s ds ≤ ˜ C Z r ̺ Dg ( ˜ cs ) s + ̺ D γ ( ˜ cs ) s ds , where ˜ C , ˜ c are constants that depend only on n and γ . (cid:4) acknowledgment The authors would like to thank the anonymous referee for his careful readingand valuable comments. R eferences [1] Acquistapace, P.
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