Optimal regularity for a Dirichlet-conormal problem in Reifenberg flat domain
aa r X i v : . [ m a t h . A P ] A p r OPTIMAL REGULARITY FOR A DIRICHLET-CONORMAL PROBLEM INREIFENBERG FLAT DOMAIN
JONGKEUN CHOI, HONGJIE DONG, AND ZONGYUAN LIA bstract . We study the divergence form second-order elliptic equations withmixed Dirichlet-conormal boundary conditions. The unique W , p solvability isobtained with p being in the optimal range (4 / , ffi cients areassumed to have small mean oscillations and the boundary of domain is Reifenbergflat. We also assume that the two boundary conditions are separated by someReifenberg flat set of co-dimension 2 on the boundary.
1. I ntroduction
In this paper, we discuss the mixed boundary value problem for second-orderelliptic operators: Lu = f + D i f i in Ω , Bu = f i n i on N , u = D , (1.1)where Ω is a domain (not necessarily bounded) in R d , d ≥ D and N . The di ff erential operator L isin divergence form acting on real valued functions u as follows: Lu = D i ( a ij ( x ) D j u + b i ( x ) u ) + ˆ b i ( x ) D i u + c ( x ) u . Here, all the coe ffi cients are assumed to be bounded measurable, and the lead-ing coe ffi cients a ij are symmetric and uniformly elliptic. We denote by Bu = ( a ij D j u + b i u ) n i the conormal derivative of u on N associated with the operator L . Dirichlet and conormal boundary conditions are prescribed on the portions D and N respectively, which are separated by their relative boundary Γ ⊂ ∂ Ω . Boththe equation and the boundary conditions are understood in the weak sense. Forprecise definition, see Definition 2.1.As is well known, solutions to purely Dirichlet / conormal boundary value prob-lems are smooth when coe ffi cients, data, and boundaries of domains are smooth.However, for mixed boundary value problems, such a regularity result does nothold near the interface Γ , and the regularity of solutions depends also on that of Γ and the way two boundary conditions meet (e.g., the meeting angle and certaincompatibility conditions). For instance, the best possible regularity of derivativesof solutions to (1.1) is Du ∈ L p for p < Mathematics Subject Classification.
Primary 35J25, 35B65; Secondary 35J15.
Key words and phrases.
Mixed boundary value problem, second-order elliptic equations of divergenceform, Reifenberg flat domains, W , p estimate and solvability.H. Dong and Z. Li were partially supported by the NSF under agreement DMS-1600593. when the two boundary portions meet tangentially (the angle between D and N is π ); see Example 2.6 for a classical counterexample. In this paper, we investigateminimal regularity assumptions of a ij , ∂ Ω , and Γ , which guarantee the aboveoptimal regularity as well as the solvability of the mixed problem (1.1).Regularity theory for mixed problems has been studied for a long time. For thecase when the two boundary portions D and N meet tangentially, we refer thereader to Shamir [22] and Savar´e [21]. In [22], the author proved W , − ε regularityfor non-divergence form elliptic equations with smooth coe ffi cients in half space.He also obtained W s , p regularity on a smooth bounded domain with the indices p > s < / + / p . At one end, the optimal C / − ε -H ¨older regularity can beobtained by passing p ր ∞ , which improved a general H ¨older regularity result ofDe Giorgi’s type by Stamppachia in [23]. It is also worth mentioning that in [21],the author proved optimal regularity in Besov space B / , ∞ for the divergence formelliptic equations with Lipschitz coe ffi cients on a C , domain.For the case when D and N do not meet tangentially, we refer the reader toI. Mitrea-M. Mitrea [17], where the authors studied the mixed problem (1.1) with Lu = ∆ u . They proved the W , p solvability with32 + ε < p < − ε for some ε = ε ( Ω , D , N ) ∈ (0 ,
1) (1.2)on the so-called creased domains in R d , d ≥
3, which means that D and N areseparated by a Lipschitz interface and the angle between D and N is less than π .This class of domains was introduced by Brown in [2] to answer a question raisedby Kenig in [14] regarding the non-tangential maximal function estimate k ( ∇ u ) ∗ k L ( ∂ Ω ) ≤ C (cid:16) k∇ tan u k L ( D ) + k ∂ u /∂ n k L ( N ) (cid:17) of harmonic functions. As mentioned in [14], the above regularity result can befalse when Ω is smooth so that D and N meet tangentially, whereas it holds forpurely Dirichlet / Neumann problem. For further work in this direction, see [3, 18]and the references therein.In this paper, we work on the so-called “Reifenberg flat” domain, which is,roughly speaking, at every small scale the boundary is close to certain hyperplane.A Reifenberg flat domain is much more general than a Lipschitz domain withsmall Lipschitz constant: locally it is not given by a graph, and typically it containsfractal structures. The Reifenberg flat domain was introduced by Reifenberg in[19] when he worked on the Plateau problem. Since then, there has been a lot ofwork on Reifenberg flat domains regarding minimal surfaces, harmonic measures,regularity of free boundaries, and divergence form elliptic / parabolic equations.An important fact for studying divergence form equation in such domains is thatany small Reifenberg flat domain is a W , p -extension domain for every p ∈ [1 , ∞ ].Hence we have all the Sobolev inequalities up to the first order. For this result andthe history of studying Reifenberg flat domains, one may refer to [16].Notice that although on Reifenberg flat domain, neither the outer normal northe trace operator of W , p is defined, the weak formulation in Definition 2.1 stillmakes sense due to the fact that no boundary integral term appears when Ω issmooth enough so that the outer normal and the trace operator are well defined.We prove the solvability in Sobolev spaces W , p and the L p -estimates with p being in the optimal range 4 / < p < IXED BOUNDARY VALUE PROBLEM 3 coe ffi cients on Reifenberg flat domains. The two boundary portions D and N areassumed to meet almost tangentially, which means D and N are separated by someReifenberg flat set of co-dimension 2 on the boundary. We note that our result holdsfor both bounded and unbounded domains. For the bounded domain case, wecan further relax the assumptions on the source term. As mentioned before, sinceLipschitz domains with small Lipschitz constant are Reifenberg flat, our resultscan be applied also on creased domains. Therefore, we see that in the restriction(1.2), the best possible range of ε is 0 < ε < / ffi cients and purely Dirichlet / conormal boundary con-ditions were studied. See also the series [4, 5] regarding second-order equationson bounded domains. Our proof is mainly based on a perturbation argumentsuggested in [6] by Ca ff arelli and Peral, by studying the level sets of maximal func-tions. The key step in our proof is to carefully design an approximation functionnear Γ , which combines the cut-o ff and reflection techniques in [8, 9]. Comparedto the purely Dirichlet or purely conormal problems, the approximation functionin our problem is less regular, which is only W , − ε , not Lipschitz. This situation issimilar to [11], where dedicated decay rates of the level sets are required.The paper is organized as follows. In Section 2, we introduce the basic notation,definitions, and assumptions. Our main results are given in Theorem 2.4 for bothbounded and unbounded domains and in Theorem 2.5 for bounded domains. InSection 3, we prove two useful tools for our problem: the local Sobolev-Poincar´einequality and the reverse H ¨older inequality. Then in Section 4, we study a modelproblem, which is the W , − ε regularity of harmonic functions on the upper halfspace with mixed boundary conditions. With all these preparation, the proof ofthe main theorem including the approximation via cut-o ff and reflection, and thelevel set argument is presented in Section 5. In Section 6, we relax the regularityassumptions on the source term for the bounded domain case, mainly by solvinga divergence form equation.2. N otation and M ain R esults Let d be the space dimension. We write a typical point x ∈ R d as x = ( x ′ , x ′′ ),where x ′ = ( x , x ) ∈ R , x ′′ = ( x , · · · , x d ) ∈ R d − . In the same spirit, for a domain Ω ⊂ R d and p , q ≥
1, we define the anisotropicspace L p , x ′ L q , x ′′ ( Ω ) as the set of all measurable functions u on Ω having a finite norm k u k L p , x ′ L q , x ′′ ( Ω ) = Z R Z R d − | u | q I Ω dx ′′ ! p / q dx ′ ! / p , where I is the usual indicator function. We abbreviate L p , x ′ L p , x ′′ ( Ω ) = L p ( Ω ). Wewill also use the notation R d + = { x = ( x , · · · , x d ) ∈ R d : x > } , B + R = R d + ∩ B R (0) , (2.1) B ′ R = { x ′ ∈ R : | x ′ | < R } , ( B ′ R ) + = R + ∩ B ′ R , and Ω R ( x ) = Ω ∩ B R ( x ) for all x ∈ R d and R > JONGKEUN CHOI, HONGJIE DONG, AND ZONGYUAN LI
Now we formulate our mixed boundary value problem. We consider domain Ω ⊂ R d with boundary divided into two non-intersecting portions, and Γ beingthe boundary of D relative to ∂ Ω : ∂ Ω =
D ∪ N , D ∩ N = ∅ , Γ = ∂ ∂ Ω D . We need the following notation for Sobolev spaces with boundary conditionsprescribed on the whole or part of the boundary. For 1 ≤ p ≤ ∞ , we denoteby W , p ( Ω ) the usual Sobolev space and by W , p ( Ω ) the completion of C ∞ ( Ω ) in W , p ( Ω ), where C ∞ ( Ω ) is the set of all smooth, compactly supported functions in Ω . Similarly, we let W , p D ( Ω ) be the completion of C ∞D ( Ω ) in W , p ( Ω ), where C ∞D ( Ω )is the set of all smooth functions on Ω which vanish in a neighborhood of D .Let L be a second-order elliptic operator in divergence form Lu = D i ( a ij ( x ) D j u + b i ( x ) u ) + ˆ b i ( x ) D i u + c ( x ) u , where the coe ffi cients A = ( a ij ) di , j = , b = ( b , . . . , b d ), ˆ b = (ˆ b , . . . , ˆ b d ), and c arebounded measurable functions defined on Ω : for some positive constants Λ and K , we have | A | ≤ Λ − , | b | + | ˆ b | + | c | ≤ K . Note that the summation convention is adopted throughout this paper. The leadingcoe ffi cients A = ( a ij ) are also assumed to be symmetric, satisfy the uniformlyellipticity condition: d X i , j = a ij ( x ) ξ j ξ i ≥ Λ | ξ | , ∀ ξ ∈ R d , ∀ x ∈ Ω . We denote by Bu = ( A Du + b u ) · n = ( a ij D j u + b i u ) n i the conormal derivative operator on the boundary of Ω associated with the operator L , where n = ( n , . . . , n d ) is the outward unit normal to ∂ Ω . We will see that in theweak formulation, this boundary condition is still well defined even when theouter unit normal is not defined point-wise. Now we give the formal definition ofweak solutions. Let p ∈ (1 , ∞ ). Definition 2.1 (Weak Solution) . For f , f i ∈ L p ( Ω ), i ∈ { , . . . , d } , we say that u ∈ W , p D ( Ω ) is a weak solution to the mixed boundary value problem Lu = f + D i f i in Ω , Bu = f i n i on N , u = D , (2.2)if Z Ω ( − a ij D j u − b i u ) D i φ + (ˆ b i D i u + cu ) φ dx = Z Ω f φ dx − Z Ω f i D i φ dx holds for any φ ∈ W , p / ( p − D ( Ω ).In this paper, we will work on the so-called Reifenberg flat domains, which isdefined below in ( i ). In ( ii ), we assume that locally the two types of boundaryconditions are almost separated: the relative boundary Γ is also Reifenberg flat. IXED BOUNDARY VALUE PROBLEM 5
Assumption 2.2 ( γ ) . There exists a positive constant R such that the followinghold.( i ) For any x ∈ ∂ Ω and R ∈ (0 , R ], there is a coordinate system depending on x and R such that in this new coordinate system (called the coordinate systemassociated with ( x , R )), we have { y : x + γ R < y } ∩ B R ( x ) ⊂ Ω R ( x ) ⊂ { y : x − γ R < y } ∩ B R ( x ) . ( ii ) Let Γ be the boundary (relative to ∂ Ω ) of D . If x ∈ Γ and R ∈ (0 , R ], we canfurther require that the coordinate system defined in ( i ) satisfy Γ ∩ B R ( x ) ⊂ { y : | y ′ − x ′ | < γ R } ∩ B R ( x ) , (cid:16) ∂ Ω ∩ B R ( x ) ∩ { y : y > x + γ R } (cid:17) ⊂ D , (cid:16) ∂ Ω ∩ B R ( x ) ∩ { y : y < x − γ R } (cid:17) ⊂ N . In this paper, we always assume that D , N , ∅ , since otherwise the boundarycondition becomes purely conormal or Dirichlet. Corresponding results have beenincluded in [8, 9]. See also [4, 5].We consider the equations with small “BMO” leading coe ffi cients with a smallparameter θ ∈ (0 ,
1) to be specified later.
Assumption 2.3 ( θ ) . There exists R ∈ (0 ,
1] such that for any x ∈ Ω and r ∈ (0 , R ],we have − Z Ω r ( x ) | a ij ( y ) − ( a ij ) Ω r ( x ) | dy < θ. In the following, we denote R : = min { R , R } .Now we can present our main result. First, in Ω (bounded or unbounded)we consider the existence and uniqueness of W , p D weak solution to the followingequation: Lu − λ u = f + D i f i in Ω , Bu = f i n i on N , u = D . (2.3)Compared to (2.2), here we introduce the − λ u term to create the required decay atinfinity for the unbounded domain case. For simplicity, we will use the followingnotation with λ > U : = | Du | + √ λ | u | , F : = d X i = | f i | + √ λ | f | . Theorem 2.4.
For any p ∈ (4 / , , we can find positive constants ( γ , θ ) = ( γ , θ )( d , p , Λ ) , λ = λ ( d , p , Λ , R , K ) , such that the following holds. If Assumptions 2.2 ( γ ) and 2.3 ( θ ) are satisfied, and λ > λ , then for any ( f i ) di = ∈ ( L p ( Ω )) d , f ∈ L p ( Ω ) there exists a unique weak solutionu ∈ W , p D ( Ω ) to (2.3) satisfying k U k L p ( Ω ) ≤ N k F k L p ( Ω ) , (2.4) where N = N ( d , p , Λ ) is a constant. JONGKEUN CHOI, HONGJIE DONG, AND ZONGYUAN LI
When Ω is bounded, we have better results: instead of taking large λ , we canassume the usual sign condition L ≤
0, which is understood in the weak sense: Z Ω ( − b i D i φ + c φ ) dx ≤ φ ∈ W , p / ( p − D ( Ω ) satisfying φ ≥
0. Also, the integrability of the non-divergence form source term f can be generalized to L p ∗ , where p ∗ = pd / ( p + d ) when p > d / ( d − , + ε when p ≤ d / ( d −
1) (2.5)for any ε > Theorem 2.5.
Let Ω be a bounded domain in R d . For any p ∈ (4 / , , we can find positiveconstants γ , θ depending on ( d , p , Λ ) , such that the following holds. If Assumptions 2.2 ( γ ) and 2.3 ( θ ) are satisfied, and L ≤ in the weak sense, then for any ( f i ) di = ∈ ( L p ( Ω )) d ,f ∈ L p ∗ ( Ω ) there exists a unique weak solution u ∈ W , p D ( Ω ) to (2.2) satisfying k u k W , p ( Ω ) ≤ N d X i = k f i k L p ( Ω ) + k f k L p ∗ ( Ω ) ! , (2.6) where N is a constant independent of u , f i and f . In the above theorems, we always assume that p ∈ (4 / , , A = ( a ij ) di , j = is symmetric . (2.7)Indeed, by the Lax-Milgram Lemma and the reverse H ¨older’s inequality, when p is close to 2, the symmetry of A is not needed. Otherwise, by the following twoexamples, we see that the restrictions in (2.7) are optimal for the solvability of mixedboundary value problems. Precisely, based on a duality argument, Example 2.6shows the restriction p ∈ (4 / ,
4) is optimal, and Example 2.7 shows the symmetryof A is required for the solvability in W , p ( Ω ) when p is away from 2. Here, for thereader’s convenience, we temporarily set R + = { x = ( x , x ) ∈ R : x > } , which is di ff erent from that in (2.1). Note that the examples below are applicableto higher dimensional cases by a trivial extension. Example 2.6. In R + , let u ( x , x ) = Im( x + ix ) / . One can simply check that ∆ u = R + , u = ∂ R + ∩ { x > } , ∂ u ∂ x = ∂ R + ∩ { x < } . Since Du is of order r − / , one could also check that near the origin Du ∈ L p for any p ∈ [1 , Du < L . Example 2.7. In R + , let u ( x , x ) = Im( x + ix ) s with s ∈ (0 , / D i ( a ij D j u ) = R + , u = ∂ R + ∩ { x > } , a ij D j un i = ∂ R + ∩ { x < } , where ( a ij ) i , j = = " π s ) − cot( π s ) 1 . IXED BOUNDARY VALUE PROBLEM 7
Since Du is of order r s − , near the origin we only have Du ∈ L p only if p < − s . Notethat − s < − s ց s ց ocal P oincar ´ e I nequality and R everse H ¨ older ’ s I nequality In this section, we introduce two useful tools for our problem. The first oneis the local Sobolev-Poincar´e inequality. Notice that a Reifenberg flat domainintersecting with a ball might no longer be Reifenberg flat. We cannot simplylocalize to obtain the required local version, although Sobolev inequalities of W , p hold for the Reifenberg flat domain since it is an extension domain. Theorem 3.1 (Local Sobolev-Poincar´e inequality) . Let γ ∈ [0 , / and Ω ⊂ R d be aReifenberg flat domain satisfying Assumption 2.2 ( γ ) ( i ) . Let x ∈ ∂ Ω and R ∈ (0 , R / .Then, for any p ∈ (1 , d ) and u ∈ W , p ( Ω R ( x )) , we have k u − ( u ) Ω R ( x ) k L dp / ( d − p ) ( Ω R ( x )) ≤ N k Du k L p ( Ω R ( x )) , where N = N ( d , p ) .Proof. See [7, Theorem 3.5]. (cid:4)
Corollary 3.2.
Let γ ∈ [0 , / and Ω ⊂ R d be a Reifenberg flat domain satisfyingAssumption 2.2 ( γ ) ( i ) . Let x ∈ Ω , R ∈ (0 , R / , and D ⊂ ∂ Ω with D ∩ B R ( x ) , ∅ . Ifthere exist z ∈ D ∩ B R ( x ) and α ∈ (0 , such thatB α R ( z ) ⊂ B R ( x ) , (cid:16) ∂ Ω ∩ B α R ( z ) (cid:17) ⊂ (cid:16) D ∩ B R ( x ) (cid:17) , (3.1) then the following hold. ( a ) For any p ∈ (1 , d ) and u ∈ W , p D ( Ω ) , we have k u k L dp / ( d − p ) ( Ω R ( x )) ≤ N k Du k L p ( Ω R ( x )) , where N = N ( d , p , α ) . ( b ) For any u ∈ W , D ( Ω ) , we have k u k L ( Ω R ( x )) ≤ NR k Du k L ( Ω R ( x )) , where N = N ( d , α ) .Proof. The assertion ( b ) is a simple consequence of the assertion ( a ). Indeed, bytaking p ∈ (cid:16) dd + , (cid:17) , and using H ¨older’s inequality and the assertion ( a ), we have k u k L ( Ω R ( x )) ≤ NR d / − d / p + k u k L dp / ( d − p ) ( Ω R ( x )) ≤ NR d / − d / p + k Du k L p ( Ω R ( x )) ≤ NR k Du k L ( Ω R ( x )) . To prove the assertion ( a ), we extend u by zero on B α R ( z ) \ Ω so that u ∈ W , p ( B α R ( z )).Since | B α R ( z ) \ Ω | ≥ N ( d )( α R ) d , by the boundary Poincar´e inequality, we have k u k L dp / ( d − p ) ( Ω α R ( z )) ≤ N ( d , q ) k Du k L p ( Ω α R ( z )) . (3.2)Notice from the triangle inequality and H ¨older’s inequality that k u k L dp / ( d − p ) ( Ω R ( x )) ≤ k u − ( u ) Ω R ( x ) k L dp / ( d − p ) ( Ω R ( x )) + k ( u ) Ω R ( x ) − ( u ) Ω α R ( z ) k L dp / ( d − p ) ( Ω R ( x )) + k ( u ) Ω α R ( z ) k L dp / ( d − p ) ( Ω R ( x )) ≤ N α − d / p (cid:16) k u − ( u ) Ω R ( x ) k L dp / ( d − p ) ( Ω R ( x )) + k u k L dp / ( d − p ) ( Ω α R ( z )) (cid:17) . JONGKEUN CHOI, HONGJIE DONG, AND ZONGYUAN LI
This combined with Theorem 3.1 and (3.2) gives the desired estimate. (cid:4)
In the rest of the section, we shall prove the reverse H ¨older’s inequality for thefollowing mixed boundary value problem without lower order terms D i ( a ij D j u ) − λ u = f + D i f i in Ω , a ij D j un i = f i n i on N , u = D . (3.3)Here, we do not impose any regularity assumption (including the symmetry con-dition) on the coe ffi cients a ij . Recall the notation that for λ > U : = | Du | + √ λ | u | , F : = d X i = | f i | + √ λ | f | . Lemma 3.3.
Let γ ∈ (0 , / , dd + < p < , and Ω ⊂ R d be a Reifenberg flat domainsatisfying Assumption 2.2 ( γ ) . Suppose that u ∈ W , D ( Ω ) satisfies (3.3) with f i , f ∈ L ( Ω ) .Let x ∈ Ω and R ∈ (0 , R ] , satisfying eitherB R / ( x ) ⊂ Ω or x ∈ ∂ Ω . Then, when λ > , we have Z Ω R / ( x ) U dx ≤ NR d (1 − / p ) Z Ω R ( x ) U p dx ! / p + N Z Ω R ( x ) F dx . When λ = and f ≡ , we have Z Ω R / ( x ) | Du | dx ≤ NR d (1 − / p ) Z Ω R ( x ) | Du | p dx ! / p + N Z Ω R ( x ) | f i | dx . In the above, the constant N depends only on d, p, and Λ .Proof. Here we only prove for the case λ >
0. When λ =
0, the proof still works ifwe replace U by | Du | and F by P i | f i | . Also, we prove only the case x ∈ ∂ Ω becausethe proof for the interior case is similar to the one in case (ii) for purely conormalboundary conditions. Without loss of generality, we assume that x =
0. Let us fix R ∈ (0 , R ]. We consider the following two cases: B R / ∩ Γ , ∅ , B R / ∩ Γ = ∅ . i ) B R / ∩ Γ , ∅ . We take y ∈ Γ such that dist(0 , Γ ) = | y | , and observe that B R / ⊂ B R / ( y ) ⊂ B R / ( y ) ⊂ B R . (3.4)Since u ∈ W , D ( Ω ), as a test function to (3.3), we can use η u ∈ W , D ( Ω ), where η is a smooth function on R d satisfying0 ≤ η ≤ , η ≡ B R / ( y ) , supp η ⊂ B R / ( y ) , |∇ η | ≤ NR − . Now, using H ¨older’s inequality and Young’s inequality, we have Z Ω R / ( y ) η U dx ≤ NR Z Ω R / ( y ) | u | dx + N Z Ω R / ( y ) F dx , (3.5) IXED BOUNDARY VALUE PROBLEM 9 where N = N ( d , Λ ). We fix a coordinate system associated with ( y , R / , Γ )satisfying the properties in Assumption 2.2 ( γ ) ( ii ). Since we have (cid:16) ∂ Ω ∩ B R / ( y ) ∩ { y : y > γ R / } (cid:17) ⊂ D , there exists z ∈ D satisfying B R / ( z ) ⊂ B R / ( y ) , (cid:16) ∂ Ω ∩ B R / ( z ) (cid:17) ⊂ (cid:16) D ∩ B R / ( y ) (cid:17) . Note that because dd + < p <
2, we have dpd − p >
2. Then by H ¨older’s inequalityand Corollary 3.2 ( a ), we see that1 R Z Ω R / ( y ) | u | dx ≤ NR d (1 − / p ) Z Ω R / ( y ) | u | dp / ( d − p ) dx ! d − p ) / dp ≤ NR d (1 − / p ) Z Ω R / ( y ) | Du | p dx ! / p , (3.6)where N = N ( d , p ). Combining this inequality and (3.5), and using (3.4), weobtain the desired estimate. ii ) B R / ∩ Γ = ∅ . Then ∂ Ω ∩ B R / is contained in either D or N . When it isin D , the proof for the previous case still works if we simply choose any y ∈ ∂ Ω ∩ B R / . When it is contained in N , as a test function to (2.2), we canuse ζ ( u − c ) ∈ W , D ( Ω ), where c = ( u ) Ω R / and ζ is a smooth function on R d satisfying0 ≤ ζ ≤ , ζ ≡ B R / , supp ζ ⊂ B R / , |∇ ζ | ≤ NR − . By testing (3.3) with ζ ( u − c ), we have Z Ω R / ( ζ U ) dx ≤ NR Z Ω R / (cid:12)(cid:12)(cid:12) u − ( u ) Ω R / (cid:12)(cid:12)(cid:12) dx + NR d Z Ω R / √ λ | u | dx ! + N Z Ω R / F dx , where N = N ( d , Λ ). Similar to (3.6), we get from Theorem 3.1 that1 R Z Ω R / (cid:12)(cid:12)(cid:12) u − ( u ) Ω R / (cid:12)(cid:12)(cid:12) dx ≤ NR d (1 − / p ) Z Ω R / | Du | p dx ! / p , where N = N ( d , p ). By H ¨older’s inequality, we also have1 R d Z Ω R / √ λ | u | dx ! ≤ NR d (1 − / p ) Z Ω R / (cid:16) √ λ | u | (cid:17) p dx ! / p . Combining these together, we obtain the desired estimate.The lemma is proved. (cid:4)
Based on Lemma 3.3 and Gehring’s lemma, we get the following reverseH ¨older’s inequality.
Lemma 3.4 (Reverse H ¨older’s inequality) . Let γ ∈ (0 , / , p > , and Ω ⊂ R d be aReifenberg flat domain satisfying Assumption 2.2 ( γ ) . Suppose that u ∈ W , D ( Ω ) satisfies (3.3) with f i , f ∈ L p ( Ω ) ∩ L ( Ω ) . Then there exist constants p ∈ (2 , p ) and N > , depending only on d, p, and Λ , such that for any x ∈ R d and R ∈ (0 , R ] , the followinghold. When λ > , we have (cid:16) U p (cid:17) / p B R / ( x ) ≤ N (cid:16) U (cid:17) / B R ( x ) + N (cid:16) F p (cid:17) / p B R ( x ) . When λ = and f ≡ , we have (cid:16) | Du | p (cid:17) / p B R / ( x ) ≤ N (cid:16) | Du | (cid:17) / B R ( x ) + N (cid:16) | f i | p (cid:17) / p B R ( x ) , where U, F, Du, and f i are the extensions of U, F, Du, and f i to R d so that they are zero on R d \ Ω .Proof. Again, we only prove for the case λ >
0. Let us fix a constant p ∈ (cid:16) dd + , (cid:17) ,and set Φ = U p , Ψ = F p . Then by Lemma 3.3, we have Z B R / ( x ) Φ / p dx ≤ NR d (1 − / p ) Z B R ( x ) Φ dx ! / p + N Z B R ( x ) Ψ / p dx (3.7)for any x ∈ R d and R ∈ (0 , R ], where N = N ( d , Λ , p ) = N ( d , Λ ). Indeed, if B R / ( x ) ⊂ Ω , then (3.7) follows from Lemma 3.3. In the case when B R / ( x ) ∩ ∂ Ω , ∅ , there exists y ∈ ∂ Ω such that | x − y | = dist( x , ∂ Ω ) and B R / ( x ) ⊂ B R / ( y ) ⊂ B R / ( y ) ⊂ B R ( x ) . Using this together with Lemma 3.3, we get (3.7). If B R / ( x ) ⊂ R d \ Ω , by thedefinition of U , (3.7) holds.By (3.7) and a covering argument, we have − Z B R / ( x ) Φ / p dx ≤ N − Z B R ( x ) Φ dx ! / p + N − Z B R ( x ) Ψ / p dx for any x ∈ R d and R ∈ (0 , R ], where N = N ( d , Λ ). Therefore, by Gehring’slemma (see, for instance, [12, Ch. V]), we get the desired estimate. The lemma isproved. (cid:4)
4. H armonic functions in half space with mixed boundary condition
In this section, we prove a regularity result for harmonic functions with mixedDirichlet-Neumann boundary conditions on half space. We denote B R = B R (0) , Γ + R : = B R ∩ { x = , x > } , Γ − R : = B R ∩ { x = , x < } . Theorem 4.1.
Suppose u ∈ W , Γ + R ( B + R ) is a weak solution to ∆ u − λ u = in B + R , ∂ u ∂ x = on Γ − R , u = on Γ + R , where λ > . Then for any p ∈ [2 , , we have u ∈ W , p ( B + R / ) with ( U p ) / pB + R / ≤ N ( d , p )( U ) / B + R . In the case when λ = , the same estimate holds with | Du | in place of U. IXED BOUNDARY VALUE PROBLEM 11
Remark . In Theorem 4.1, the boundary condition is only prescribed on the flatpart of the boundary. Hence the meaning of “weak solution” is slightly di ff erent.In the theorem and throughout the paper, a W , Γ + R ( B + R ) weak solution means: for any φ ∈ W , ( B + R ) satisfying φ = ∂ B + R \ Γ − R , Z B + R ( ∇ u · ∇ φ + λ u φ ) dx = . It is clear that, as a test function, one can use η u , where η ∈ C ∞ c ( B R ).For the proof of Theorem 4.1, we will use the following two dimensional regu-larity result. Lemma 4.3.
In the half ball B + R ⊂ R , consider u ∈ W , Γ + R ( B + R ) which solves ∆ u = f in B + R , ∂ u ∂ x = on Γ − R , u = on Γ + R , where f ∈ L ( B + R ) . Then for any p ∈ [2 , , we have u ∈ W , p ( B + R / ) with ( | Du | p ) / pB + R / ≤ N ( p ) (cid:16) ( | Du | ) / B + R + R ( | f | ) / B + R (cid:17) . (4.1)From the proof below, it is clear that in Lemma 4.3, R / r ∈ (0 , R ). In this case, the constant N also depends on r and R . Proof of Lemma 4.3.
By a scaling argument, we may assume R =
1. We consider thefollowing change of variables: ( y , y ) ∈ B ∩ { y > , y > } 7→ ( x , x ) ∈ B + : x = y y , x = y − y , or in complex variables: x + ix = ( y + iy ) . Write e u ( y , y ) = u ( x , x ) and e f ( y , y ) = f ( x , x ). Then we can rewrite the equationas ∆ y e u = | y | e f in B ++ , ∂ e u ∂ y = B ∩ { y > , y = } , e u = B ∩ { y = , y > } , where B ++ : = B ∩ { y > , y > } . Next, we take an even extension of e u and e f withrespect to y -variable. Still denote the extended functions on B + by e u and e f . Thenthe following equation is satisfied: ∆ y e u = | y | e f in B + , e u = B ∩ { y = } . Note that | D x u | ≤ N | y | | D y e u | , dx = | y | dy . By the Sobolev embedding theorem, the local W estimate for elliptic equations,and the boundary Poincar´e inequality, we obtain k D y e u k L q ( B + √ / ) ≤ k e u k W , ( B + √ / ) ≤ N (cid:16) k e u k L ( B + ) + k | y | e f k L ( B + ) (cid:17) ≤ N (cid:16) k D y e u k L ( B + ) + k | y | e f k L ( B + ) (cid:17) ≤ N (cid:16) k D y e u k L ( B ++ ) + k | y | e f k L ( B ++ ) (cid:17) , where N = N ( p ) > q = q ( p ) is a constant with q > p − p ≥ p . (4.2)Here we also used the fact that e u and e f are both even functions in y . Translatingback to x -variables, we obtain k| x | q − q D x u k L q ( B + / ) ≤ N (cid:16) k D x u k L ( B + ) + k| x | / f k L ( B + ) (cid:17) ≤ N (cid:16) k D x u k L ( B + ) + k f k L ( B + ) (cid:17) . (4.3)By H ¨older’s inequality and (4.2), we get k D x u k L p ( B + / ) ≤ k| x | q − q D x u k L q ( B + / ) k| x | − q − q k L qp / ( q − p ) ( B + / ) ≤ N k| x | q − q D x u k L q ( B + / ) . Combining this with (4.3), we obtain k D x u k L p ( B + / ) ≤ N (cid:16) k D x u k L ( B + ) + k f k L ( B + ) (cid:17) , which is exactly (4.1). The lemma is proved. (cid:4) We are now ready to present the proof of Theorem 4.1.
Proof of Theorem 4.1.
We first prove the theorem for λ =
0. By a scaling argumentand Lemma 4.3, we may assume R = d ≥
3. Noting that we can di ff eren-tiate both the equation and the boundary condition in x ′′ -direction, the followingCaccioppoli type inequality holds: k D x ( D kx ′′ u ) k L ( B + s ) ≤ N ( d , k ) | t − s | k k Du k L ( B + t ) for 0 < s < t ≤ k ∈ { , , , . . . } . Thus by anisotropic Sobolev embedding, wecan increase the integrability in x ′′ -variables so that D x ′ u ∈ L , x ′ L p , x ′′ ( B + r ) , D x ′′ u , D x ′′ u ∈ L p ( B + r ) ∀ r < , with the estimate k D x ′ u k L , x ′ L p , x ′′ ( B + r ) + k D x ′′ u k L p ( B + r ) + k D x ′′ u k L p ( B + r ) ≤ N ( d , p , r ) k Du k L ( B + ) . (4.4)It remains to estimate D x ′ u . From (4.4), for almost every | x ′′ | < /
2, we have D x ′′ u ( · , x ′′ ) ∈ L (cid:16) ( B ′ / ) + (cid:17) . IXED BOUNDARY VALUE PROBLEM 13
Now we rewrite the equation as a 2-dimension problem in x ′ -variables: ∆ x ′ u ( · , x ′′ ) = − ∆ x ′′ u ( · , x ′′ ) in ( B ′ / ) + , ∂ u ∂ x = B ′ / ∩ { x = , x < } , u = B ′ / ∩ { x = , x > } . We apply a properly rescaled version of Lemma 4.3 to see that for almost every | x ′′ | < / k D x ′ u ( · , x ′′ ) k L p (( B ′ / ) + ) ≤ N ( p ) (cid:16) k D x ′ u ( · , x ′′ ) k L (( B ′ / ) + ) + k ∆ x ′′ u ( · , x ′′ ) k L (( B ′ / ) + ) (cid:17) . Taking L p norm in { x ′′ ∈ R d − : | x ′′ | < / } for both sides, and using the Minkowskiiinequality and (4.4) with r = √ /
2, we obtain D x ′ u ∈ L p ( B + / ) and k D x ′ u k L p ( B + / ) ≤ N ( d , p ) k Du k L ( B + ) . This gives the desired estimate for λ = λ >
0, we use an idea by S. Agmon. We define v ( x , τ ) = u ( x ) cos( √ λτ + π/ , and observe that v satisfies ∆ ( x ,τ ) v = B + , ∂ v ∂ x = Γ − , v = Γ + . (4.5)where ˆ B = { ( x , τ ) ∈ R d + : | ( x , τ ) | < } , ˆ B + = ˆ B ∩ { x > } , ˆ Γ + = ˆ B ∩ { x = , x > } , ˆ Γ − = ˆ B ∩ { x = , x < } . By applying the result for λ = | D ( x ,τ ) v | p ) / p ˆ B + / ≤ N ( | D ( x ,τ ) v | ) / B + , (4.6)where N = N ( d , p ). Note that the function Φ given by Φ ( λ ) = Z / (cid:12)(cid:12)(cid:12) cos( √ λτ + π/ (cid:12)(cid:12)(cid:12) p d τ has a positive lower bound depending only on p . Thus by using (4.6) and the factthat (cid:12)(cid:12)(cid:12) Du ( x ) cos( √ λτ + π/ (cid:12)(cid:12)(cid:12) ≤ | D ( x ,τ ) v ( x , τ ) | ≤ U ( x ) , we have Z B + / | Du | p dx ≤ N Z / Z B + / | Du | p (cid:12)(cid:12)(cid:12) cos( √ λτ + π/ (cid:12)(cid:12)(cid:12) p dx d τ ≤ N Z ˆ B + / | D ( x ,τ ) v | p dx d τ ≤ N Z B + U dx ! p / , where N = N ( d , p ). Similarly, from the fact that (cid:12)(cid:12)(cid:12) √ λ u ( x ) sin( √ λτ + π/ (cid:12)(cid:12)(cid:12) ≤ | D ( x ,τ ) v ( x , τ ) | ≤ U ( x ) , we obtain Z B + / (cid:12)(cid:12)(cid:12) √ λ u (cid:12)(cid:12)(cid:12) p dx ≤ N Z B + U dx ! p / . Combining these together we get the desired estimate. The theorem is proved. (cid:4)
5. R egularity of W , D weak solutions The crucial step in proving unique W , p solvability is the following improvedregularity result. As in Theorem 2.4, we consider a domain Ω ⊂ R d which can bebounded or unbounded, together with nonempty boundary portions D , N . Again,recall the notation that for λ > U : = | Du | + √ λ | u | , F : = d X i = | f i | + √ λ | f | . Proposition 5.1 (Regularity of W , D weak solutions) . For any p ∈ (2 , , we can findpositive constants γ , θ depending on ( d , p , Λ ) , such that if Assumptions 2.2 ( γ ) and 2.3 ( θ ) are satisfied, the following holds. For any W , D ( Ω ) weak solution u to (3.3) with λ > and f i , f ∈ L p ( Ω ) ∩ L ( Ω ) , we have u ∈ W , p D ( Ω ) and k U k L p ( Ω ) ≤ N ( R d (1 / p − / k U k L ( Ω ) + k F k L p ( Ω ) ) . (5.1) Furthermore, if we also have f ≡ , then we can take λ = , and the following estimateholds: k Du k L p ( Ω ) ≤ N ( R d (1 / p − / k Du k L ( Ω ) + k f i k L p ( Ω ) ) . (5.2) In the above, the constant N only depends on d, p, and Λ . Based on Proposition 5.1, we obtain the following a priori estimate for theequations with lower order terms and large λ , which will be useful for the uniquesolvability in Theorem 2.4. Corollary 5.2.
Let p ∈ (2 , and γ , θ be the constants from Proposition 5.1. UnderAssumptions 2.2 ( γ ) and 2.3 ( θ ) , there exists a positive constant λ depending on ( d , p , Λ , R , K ) such that if u is a W , p D weak solution to the equation (2.3) with f i , f ∈ L p ( Ω ) and λ > λ , then we have k U k L p ( Ω ) ≤ N k F k L p ( Ω ) , (5.3) where N = N ( d , p , Λ ) . The rest of this section is devoted to the proofs of Proposition 5.1 and Corollary5.2.5.1.
Decomposition of Du . We will use an interpolation argument to prove Propo-sition 5.1. The key step is the following decomposition (approximation).
Proposition 5.3.
Suppose that u ∈ W , D ( Ω ) satisfies (3.3) with λ > and f i , f ∈ L p ( Ω ) ∩ L ( Ω ) , where p > . Then under Assumptions 2.2 ( γ ) and 2.3 ( θ ) with γ < / (32 √ d + and θ ∈ (0 , , for any x ∈ Ω and R < R , there exist positive functionsW , V ∈ L ( Ω R / ( x )) such thatU ≤ W + V in Ω R / ( x ) . IXED BOUNDARY VALUE PROBLEM 15
Moreover, we have for any q < , ( W ) / Ω R / ( x ) ≤ N (cid:16) ( θ µ ′ + γ µ ′ )( U ) / Ω R ( x ) + ( F µ ) µ Ω R ( x ) (cid:17) , (5.4)( V q ) / q Ω R / ( x ) ≤ N (cid:16) ( U ) / Ω R ( x ) + ( F µ ) µ Ω R ( x ) (cid:17) . (5.5) Here µ is a constant satisfying µ = p , where p = p ( d , p , Λ ) > comes from Lemma3.4, and µ ′ satisfies /µ + /µ ′ = . The constant N only depends on d, p, q, and Λ . The rest of Section 5.1 will be devoted to the proof of this proposition.
Proof.
According to the relative position of x to D , N , we will discuss the following3 cases. Case 1 : dist( x , ∂ Ω ) ≥ R / Ω . We cando the usual “freezing coe ffi cient” approximation. The existence of such W , V canbe found in [8, Lemma 8.3 (i)] or [9, Lemma 5.1 (i)].In the next two cases, we also need to approximate the Reifenberg flat boundaryby hyperplane and deal with corresponding boundary conditions. Case 2 : dist( x , ∂ Ω ) < R /
32, dist( x , Γ ) ≥ R / B R / ( x ) ∩ N = ∅ or B R / ( x ) ∩ D = ∅ . Correspond-ingly, we only deal with purely Dirichlet or purely conormal boundary condition.The functions W and V are constructed in Ω R / ( x ). Then for the estimates, weneed to shrink the radius to R /
32. Such construction and estimates can be foundin [8, Lemma 8.3 (ii)] and [9, Lemma 5.1 (ii)]. Briefly, we approximate the neigh-borhood of Ω R / ( x ) by a half ball thanks to the small Reifenberg flat assumption.Then we apply a cuto ff technique for the Dirichlet case, or a reflection techniquefor the conormal case. All these two techniques will be introduced in Case 3 below.In both Cases 1 and 2, actually we can take q = ∞ in (5.5). Case 3 : dist( x , ∂ Ω ) < R /
32, dist( x , Γ ) < R / y ∈ Γ withdist( y , x ) < R /
24. Consider the coordinate system associated with ( y , R /
4) as inAssumption 2.2 ( γ ). For simplicity, we shift the origin in x ′ = ( x , x ) − hyperplane,such that ∂ Ω ∩ B R / ( y ) ⊂ {− γ R / < x < } , Γ ∩ B R / ( y ) ⊂ {− γ R / < x < } . (5.6)In the following, we will omit the center when it is y . For example, Ω R / : = Ω ∩ B R / ( y ) , Ω + R / : = Ω R / ∩ R d + , Ω − R / : = Ω R / ∩ R d − , where R d − = { x ∈ R d : x < } . Note that this is slightly di ff erent from the usualconvention that we omit the center when it is the coordinate origin. The followinginclusion relation will be useful: Ω R / ( x ) ⊂ Ω R / ⊂ Ω R / ⊂ Ω R ( x ) . (5.7)Now we start to construct the decomposition. First, we introduce a cut-o ff function χ ∈ C ∞ ( R d ) with D χ supported in a “L-shaped” domain, satisfying χ = , on { x > − γ R } ∩ { x < γ R } ,χ = , on { x < − γ R } ∪ { x > γ R } , ≤ χ ≤ , | D χ | ≤ γ R . The following two lemmas should be read as parts of the proof of Proposition5.3. The first one is an important estimate of a typical term in our proof. Both theinequality itself and the decomposition technique in the proof will be used later.
Lemma 5.4.
We have ( | D χ u | ) / Ω R / ≤ N γ / (2 µ ′ ) (cid:16) ( U ) / Ω R ( x ) + ( F µ ) / (2 µ ) Ω R ( x ) (cid:17) , where N = N ( d , p , Λ ) .Proof. From the construction of χ , we have k D χ u k L ( Ω R / ) ≤ γ R k I supp { D χ } u k L ( Ω R / ) . Now we decompose the set supp { D χ } ∩ Ω R / to obtain the required smallness.Consider the following grid points on ∂ R d + : D grid : = { z ∈ R d : z = (0 , k γ R ) for k = ( k , . . . , k d ) ∈ Z d − , k ≥ − } ∩ Ω R / . Clearly S z ∈D grid Ω √ d + γ R ( z ) covers supp { D χ } ∩ Ω R / , and because γ < / (32 √ d + [ z ∈D grid Ω √ d + γ R ( z ) ⊂ Ω R / , with each point covered by at most N ( d ) of such neighborhoods. Due to (5.6), weknow that in each Ω √ d + γ R ( z ), (3.1) is satisfied with( z , α ) = ( z + ( c , ( − + √ d + γ R / , , · · · , , / , where c ∈ ( − γ R / ,
0) is chosen carefully to guarantee z ∈ ∂ Ω . Hence we can applythe Poincar´e inequality stated in Corollary 3.2 and H ¨older’s inequality to obtain k I supp { D χ } u k L ( Ω R / ) ≤ N X z ∈D grid k u k L ( Ω √ d + γ R ( z )) ≤ N ( γ R ) X z ∈D grid k Du k L ( Ω √ d + γ R ( z )) ≤ N ( γ R ) k I | x | < √ d + γ R Du k L ( Ω R / ) (5.8) ≤ N ( γ R ) · ( γ R d ) /µ ′ k Du k L µ ( Ω R / ) . (5.9)To obtain (5.9), we used H ¨older’s inequality. Now we rewrite this using thenotation of average and use a properly rescaled version of Lemma 3.4 as well as(5.7) to obtain ( | D χ u | ) / Ω R / ≤ N γ / (2 µ ′ ) ( | Du | µ ) / (2 µ ) Ω R / ≤ N γ / (2 µ ′ ) (cid:16) ( U ) / Ω R ( x ) + ( F µ ) / (2 µ ) Ω R ( x ) (cid:17) . The lemma is proved. (cid:4)
The second lemma shows how we “freeze” the boundary to be a hyperplaneusing a cut-o ff technique together with a reflection. IXED BOUNDARY VALUE PROBLEM 17
Lemma 5.5.
The function χ u ∈ W , ( Ω + R / ) satisfies the following equation in the weaksense D i ( a ij D j ( χ u )) − λχ u = D i g (1) i + D i g (2) i + g (3) i D i χ + g (4) i D i e χ + g (5) in Ω + R / , a ij D j ( χ u ) n i = g (1) i n i + g (2) i n i on Γ − ,χ u = on Γ + , (5.10) where g (1) i = a ij uD j χ + f i χ, g (2) i = ( − ε i ε j e a ij e χ D j e u + ε i e χ e f i ) I ( − x , x , x ′′ ) ∈ Ω − R / , g (3) i = a ij D j u − f i , g (4) i = ( ε i ε j e a ij D j e u − ε i e f i ) I ( − x , x , x ′′ ) ∈ Ω − R / , g (5) = χ f + e χ e f I ( − x , x , x ′′ ) ∈ Ω − R / + λ e χ e u I ( − x , x , x ′′ ) ∈ Ω − R / , Γ + = ∂ Ω + R / ∩ { x = , x > } , Γ − = ∂ Ω + R / ∩ { x = , x < } . Here we denote e f ( x , x , x ′′ ) : = f ( − x , x , x ′′ ) , and similarly for e a ij , e χ , and e u. We also usethe following notation ε i : = − if i = , if i , . Proof.
Take any test function ψ ∈ W , ∂ Ω + R / \ Γ − ( Ω + R / ). We extend ψ to R d + by setting ψ ≡ R d + \ Ω + R / , and then, we again extend evenly to R d . Denote this ex-tended function by E ψ , and note that χ E ψ ∈ W , D ( Ω ). Testing (3.3) with χ E ψ andrearranging terms will give us (5.10). (cid:4) We continue the proof of Proposition 5.3. Solve the following equation D i ( a ij D j ˆ w ) − λ ˆ w = D i (( a ij − a ij ) D j ( χ u )) + D i g (1) i + D i g (2) i + g (3) i D i χ + g (4) i D i e χ + g (5) in Ω + R / , a ij D j ˆ w · n i = ( a ij − a ij ) D j ( χ u ) n i + g (1) i n i + g (2) i n i on Γ − , ˆ w = ∂ Ω + R / \ Γ − , (5.11)for ˆ w ∈ W , ∂ Ω + R / \ Γ − ( Ω + R / ), where a ij = ( a ij ) Ω R / are constants. Due to the Lax-Milgramlemma, such ˆ w exists. For simplicity, we denoteˆ W : = | D ˆ w | + √ λ | ˆ w | . Testing (5.11) by ˆ w , and using the ellipticity and H ¨older’s inequality, we have k ˆ W k L ( Ω + R / ) ≤ k ( a ij − a ij ) D j ( χ u ) k L ( Ω + R / ) k D ˆ w k L ( Ω + R / ) (5.12) + (cid:13)(cid:13)(cid:13) g (1) i (cid:13)(cid:13)(cid:13) L ( Ω + R / ) k D ˆ w k L ( Ω + R / ) + (cid:13)(cid:13)(cid:13) g (2) i (cid:13)(cid:13)(cid:13) L ( Ω + R / ) k D ˆ w k L ( Ω + r / ) (5.13) + (cid:13)(cid:13)(cid:13) I supp { D χ } g (3) i (cid:13)(cid:13)(cid:13) L ( Ω + R / ) k D i χ ˆ w k L ( Ω + R / ) + (cid:13)(cid:13)(cid:13) g (4) i (cid:13)(cid:13)(cid:13) L ( Ω + R / ) (cid:13)(cid:13)(cid:13) D i e χ ˆ w I ( − x , x , x ′′ ) ∈ Ω − R / (cid:13)(cid:13)(cid:13) L ( Ω + R / ) (5.14) + (cid:13)(cid:13)(cid:13) λ − / g (5) (cid:13)(cid:13)(cid:13) L ( Ω + R / ) k λ / ˆ w k L ( Ω + R / ) . (5.15)For the term in (5.12), we use Assumption 2.3, H ¨older’s inequality, Lemma 3.4,and Lemma 5.4. Noting that | Ω + R / | , | Ω R / | and | Ω R ( x ) | are all comparable to R d , we have: (cid:16) | ( a ij − a ij ) D j ( χ u ) | (cid:17) / Ω + R / ≤ N (cid:16) | a ij − a ij | µ ′ (cid:17) / (2 µ ′ ) Ω R / (cid:16) | Du | µ (cid:17) / (2 µ ) Ω R / + N (cid:16) | D χ u | (cid:17) / Ω + R / ≤ N (cid:16) | a ij − a ij | (cid:17) / (2 µ ′ ) Ω R / (cid:16) | Du | µ (cid:17) / (2 µ ) Ω R / + N (cid:16) | D χ u | (cid:17) / Ω + R / ≤ N (cid:16) θ / (2 µ ′ ) + γ / (2 µ ′ ) (cid:17)(cid:16) ( U ) / Ω R ( x ) + ( F µ ) / (2 µ ) Ω R ( x ) (cid:17) , (5.16)where N = N ( d , p , Λ ) is a constant.For the terms in (5.13), we first estimate g (1) i . This is simply due to Lemma 5.4and H ¨older’s inequality: (cid:16)(cid:12)(cid:12)(cid:12) g (1) i (cid:12)(cid:12)(cid:12) (cid:17) / Ω + R / ≤ N ( | uD χ | ) / Ω R / + N ( | f i | ) / Ω R / ≤ N γ / (2 µ ′ ) ( U ) / Ω R ( x ) + N ( F µ ) µ Ω R ( x ) . (5.17)Now we estimate g (2) i as follows: (cid:16)(cid:12)(cid:12)(cid:12) g (2) i (cid:12)(cid:12)(cid:12) (cid:17) / Ω + R / ≤ (cid:16)(cid:12)(cid:12)(cid:12) e a ij e χ D j e u I ( − x , x , x ′′ ) ∈ Ω − R / (cid:12)(cid:12)(cid:12) (cid:17) / Ω + R / + (cid:16)(cid:12)(cid:12)(cid:12)e χ e f i I ( − x , x , x ′′ ) ∈ Ω − R / (cid:12)(cid:12)(cid:12) (cid:17) / Ω + R / ≤ N (cid:16)(cid:12)(cid:12)(cid:12) I Ω − R / Du (cid:12)(cid:12)(cid:12) (cid:17) / Ω R / + N ( | f i | ) / Ω R / ≤ N γ / (2 µ ′ ) ( | Du | µ ) / (2 µ ) Ω R / + N ( | f i | µ ) / (2 µ ) Ω R / ≤ N γ / (2 µ ′ ) (cid:16) ( U ) / Ω R ( x ) + ( F µ ) / (2 µ ) Ω R ( x ) (cid:17) + N ( | f i | µ ) / (2 µ ) Ω R / , (5.18)where in the last line, we used Lemma 3.4.For the terms in (5.14), we first use the same decomposition technique togetherwith Poincar´e’s inequality as in the proof of Lemma 5.4 (until the step (5.8)) toobtain: k D i χ ˆ w k L ( Ω + R / ) ≤ N k D ˆ w k L ( Ω + R / ) . To avoid the problem of increased integrating domain, here we need to modify thedecomposition to be [ z ∈D grid (cid:16) Ω √ d + γ R ( z ) ∩ Ω + R / (cid:17) , and the same proof still applies. Again, with the help of H ¨older’s inequality andLemma 3.4, we can estimate g (3) i as follows: (cid:13)(cid:13)(cid:13) I supp { D χ } g (3) i (cid:13)(cid:13)(cid:13) L ( Ω + R / ) ≤ N (cid:13)(cid:13)(cid:13) I supp { D χ } Du (cid:13)(cid:13)(cid:13) L ( Ω + R / ) + N (cid:13)(cid:13)(cid:13) I supp { D χ } f i (cid:13)(cid:13)(cid:13) L ( Ω + R / ) ≤ N γ / (2 µ ′ ) ( k U k L ( Ω R ( x )) + R d / (2 µ ′ ) k F k L µ ( Ω R ( x )) ) . Hence, (cid:13)(cid:13)(cid:13) I supp { D χ } g (3) i (cid:13)(cid:13)(cid:13) L ( Ω + R / ) k D i χ ˆ w k L ( Ω + R / ) ≤ N γ / (2 µ ′ ) ( k U k L ( Ω R ( x )) + R d / (2 µ ′ ) k F k L µ ( Ω R ( x )) ) k D ˆ w k L ( Ω + R / ) . (5.19)Using similar techniques as in (5.18), we can deduce that (cid:13)(cid:13)(cid:13) g (4) i (cid:13)(cid:13)(cid:13) L ( Ω + R / ) (cid:13)(cid:13)(cid:13) D i e χ ˆ w I ( − x , x , x ′′ ) ∈ Ω − R / (cid:13)(cid:13)(cid:13) L ( Ω + R / ) ≤ N γ / (2 µ ′ ) ( k U k L ( Ω R ( x )) + R d / (2 µ ′ ) k F k L µ ( Ω R ( x )) ) k D ˆ w k L ( Ω + R / ) . (5.20) IXED BOUNDARY VALUE PROBLEM 19
We are left to estimate the one last term in (5.15): (cid:13)(cid:13)(cid:13) λ − / g (5) (cid:13)(cid:13)(cid:13) L ( Ω + R / ) ≤ (cid:13)(cid:13)(cid:13) λ − / χ f (cid:13)(cid:13)(cid:13) L ( Ω + R / ) + (cid:13)(cid:13)(cid:13)(cid:13) λ − / e f e χ I ( − x , x , x ′′ ) ∈ Ω − R / (cid:13)(cid:13)(cid:13)(cid:13) L ( Ω + R / ) + (cid:13)(cid:13)(cid:13) λ / e χ e u I ( − x , x , x ′′ ) ∈ Ω − R / (cid:13)(cid:13)(cid:13) L ( Ω + R / ) ≤ k F k L ( Ω R / ) + N γ / (2 µ ′ ) ( k U k L ( Ω R ( x )) + k F k L ( Ω R ( x )) ) , (5.21)where for the last term, we applied similar techniques as we did to estimate g (2) i .Substituting (5.16)-(5.21) back, we obtain( ˆ W ) / Ω + R / ≤ N (cid:16) ( θ µ ′ + γ µ ′ )( U ) / Ω R ( x ) + ( F µ ) µ Ω R ( x ) (cid:17) . (5.22)Now we define W : = ˆ W + | D ((1 − χ ) u ) | + √ λ | (1 − χ ) u | in Ω R / ( x ) ∩ R d + , | Du | + √ λ | u | in Ω R / ( x ) ∩ R d − . Using (5.22), H ¨older’s inequality, Lemma 3.4, and Lemma 5.4, we can obtain (5.4).To construct V , we set v : = χ u − ˆ w . Clearly v ∈ W , Γ + ( Ω + R / ). Simple computation using Lemma 5.5 and (5.11) showsthat v satisfies D i ( a ij D j v ) − λ v = Ω + R / , a ij D j v · n i = Γ − , v = Γ + . Now we define V : = | Dv | + √ λ | v | in Ω + R / , Ω − R / . Then we have U ≤ W + V in Ω R / ( x )from the fact that u = v + ˆ w + (1 − χ ) u in Ω R / ( x ) ∩ R d + , W = U in Ω R / ( x ) ∩ R d − . Using (5.7), we can apply a properly rescaled version of Theorem 4.1 with a changeof variables to obtain that for any q ∈ [2 , V ∈ L q ( Ω R / ( x )) satisfying( V q ) / q Ω R / ( x ) ≤ ( V q ) / q Ω + R / ≤ N ( V ) / Ω + R / ≤ N (( | D ( χ u ) | ) / Ω + R / + √ λ ( | χ u | ) / Ω + R / + | ˆ W | / Ω + R / ) ≤ N ( U ) / Ω R ( x ) + N (cid:16) ( θ / (2 µ ′ ) + γ / (2 µ ′ ) )( U ) / Ω R ( x ) + ( F µ ) / (2 µ ) Ω R ( x ) (cid:17) ≤ N ( U ) / Ω R ( x ) + N ( F µ ) / (2 µ ) Ω R ( x ) . (5.23)Here we used the estimates for uD χ and ˆ W in previous steps. Clearly, from (5.23)we obtain (5.5). This finishes the proof of Proposition 5.3. (cid:4) Level Set Argument.
In previous steps, we treat the perturbation problemby decomposing U into two parts, with L and L q estimates respectively. Nowwe interpolate using a level set argument to obtain the required L p estimate forProposition 5.1. Such argument was suggested by Ca ff arelli in [6] for a “kernelfree” approach to W , p estimate of divergence form second-order elliptic equations.Note that our estimate is not an a priori estimate, i.e., we do not need to assume Du ∈ L p in advance.Define A ( s ) : = { x ∈ Ω : M Ω ( U ) / > s } , B ( s ) : = { x ∈ Ω : ( γ / (2 µ ′ ) + θ / (2 µ ′ ) ) − M Ω ( F µ ) / (2 µ ) + M Ω ( U ) / > s } , where µ, µ ′ ∈ (1 , ∞ ) are the constants from Proposition 5.3. Here we denote M Ω to be the Hardy-Littlewood maximal operator restricted on Ω , i.e., for f ∈ L , loc ( Ω )and x ∈ Ω : M Ω ( f )( x ) : = sup r > ? B r ( x ) | f | I Ω . By the Hardy-Littlewood theorem, for any f ∈ L q ( Ω ) with q ∈ [1 , ∞ ), we have |{ x ∈ Ω : M Ω ( f )( x ) > s }| ≤ N k f k qL q ( Ω ) s q , (5.24)where N = N ( d , q ).Proposition 5.3 leads to the following lemma. Lemma 5.6.
Under the same hypothesis of Proposition 5.3, for any q ∈ [2 , , there existsa constant N depending on ( d , p , q , Λ ) , such that for all κ > d / and s > , the followingholds: if for some R < R , x ∈ Ω , | Ω R / ( x ) ∩ A ( κ s ) | ≥ N (cid:16) κ − q + κ − ( γ /µ ′ + θ /µ ′ ) (cid:17) | Ω R / ( x ) | , (5.25) then Ω R / ( x ) ⊂ B ( s ) .Proof. Without loss of generality, we assume s =
1. We also extend U and F to bezero outside Ω . We will prove the contrapositive of the above statement.Suppose there exists a point z , with z ∈ Ω R / ( x ) , z < B (1) , then by the definition of B , we have( γ / (2 µ ′ ) + θ / (2 µ ′ ) ) − M Ω ( F µ ) / (2 µ ) ( z ) + M Ω ( U ) / ( z ) ≤ . In particular, for any r >
0, we have( γ / (2 µ ′ ) + θ / (2 µ ′ ) ) − ( F µ ) / (2 µ ) B r ( z ) + ( U ) / B r ( z ) ≤ . Using Proposition 5.3 with z in place of x , we can find W , V defined on Ω R / ( z ),such that for any q ∈ [2 , U ≤ V + W in Ω R / ( z ) , ( W ) / Ω R / ( z ) ≤ N ( γ / (2 µ ′ ) + θ / (2 µ ′ ) ) , ( V q ) / q Ω R / ( z ) ≤ N . (5.26)Notice that we have the following inclusion Ω R / ( x ) ⊂ Ω R / ( z ) ⊂ Ω R / ( z ) . (5.27) IXED BOUNDARY VALUE PROBLEM 21
Now for any y ∈ Ω R / ( x ) ∩ A ( κ ), by the definition of A , we can find some r > ? B r ( y ) U dx ! / > κ. We claim that r < R /
64. Otherwise noting y ∈ Ω R / ( z ), we have Ω r ( y ) ⊂ Ω r ( z ).Hence we can deduce that ? B r ( y ) U dx ! / ≤ d / ? B r ( z ) U dx ! / ≤ d / M Ω ( U ) / ( z ) ≤ d / < κ, which is a contradiction.Now, since r < R /
64, the decomposition U ≤ W + V is defined in Ω r ( y ) ⊂ Ω R / ( z ). Extending W and V to be zero outside Ω , we have ? B r ( y ) U dx ! / ≤ ? B r ( y ) W dx ! / + ? B r ( y ) V dx ! / ≤ M Ω ( W I Ω R / ( z ) ) / ( y ) + M Ω ( V I Ω R / ( z ) ) / ( y ) . Then by (5.24), (5.26) and (5.27), we obtain | Ω R / ( x ) ∩ A ( κ ) | ≤| Ω R / ( z ) ∩ A ( κ ) |≤ (cid:12)(cid:12)(cid:12) {M Ω ( W I Ω R / ( z ) ) / > κ/ } (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) {M Ω ( V I Ω R / ( z ) ) / > κ/ } (cid:12)(cid:12)(cid:12) ≤ N k W k L ( Ω R / ( z )) ( κ/ + N k V k qL q ( Ω R / ( z )) ( κ/ q ≤ N | Ω R / ( z ) | (cid:16) κ − ( γ /µ ′ + θ /µ ′ ) + κ − q (cid:17) ≤ N (cid:16) κ − ( γ /µ ′ + θ /µ ′ ) + κ − q (cid:17) | Ω R / ( x ) | . Here N = N ( d , p , q , Λ ) is exactly what we aim to find. (cid:4) Using a lemma in measure theory called “crawling of the ink spot” which wasfirst introduced by Krylov and Safonov in [15, 20], we obtain the following decayestimate from Lemma 5.6.
Corollary 5.7 (Decay of A ( s )) . Under the same hypothesis of Proposition 5.3, for anyq ∈ [2 , , there exists a constant N depending on ( d , p , q , Λ ) , such that for any κ > max { d / , κ } ands > s ( d , p , q , Λ , κ, R , k U k L ( Ω ) ) : = k U k L ( Ω ) N κ ( κ − q + κ − ( γ /µ ′ + θ /µ ′ )) | B R / | ! / , (5.28) we have |A ( κ s ) | ≤ N (cid:16) κ − q + κ − ( γ /µ ′ + θ /µ ′ ) (cid:17) |B ( s ) | , where κ is the constant satisfyingN (cid:16) κ − q + κ − ( γ /µ ′ + θ /µ ′ ) (cid:17) < / . (5.29)Here we only sketch the proof. The key idea is to use a stopping time argument(or the Calder ´on-Zygmund decomposition as in [6]). Di ff erent from Krylov andSafonov’s original version, we cover Ω by balls instead of dyadic cubes. For any x ∈ A ( κ s ), by (5.24), (5.28), and (5.29), we see that (5.25) does not hold with R in place of R . We shrink the “ball” Ω R / ( x ) from R = R until the first time (5.25)holds. Due to (5.28), (5.29) and the Lebesgue di ff erentiation theorem, such R existsand R ∈ (0 , R ). We are left to use the Vitali covering lemma to pick a “almostdisjoint” cover.5.3. Proof of Proposition 5.1 and Corollary 5.2.
Now we are ready to give theproof of Proposition 5.1.
Proof of Proposition 5.1.
Let us fix p ∈ (2 , γ , θ , and κ be positive constantsto be chosen later, such that γ < / (32 √ d + , θ < , κ > max { d / , κ } , where κ = κ ( d , p , Λ ) is a constant satisfying (5.29) with q = ( p + /
2. It su ffi ces toprove lim S →∞ Z S p |A ( s ) | s p − ds ≤ N (cid:16) R d (1 − p / k U k pL ( Ω ) + k F k pL p ( Ω ) (cid:17) (5.30)under Assumptions 2.2 ( γ ) and 2.3 ( θ ). The left-hand side becomeslim S →∞ Z S /κ p κ p |A ( κ s ) | s p − ds . We bound the integrand by using Corollary 5.7 with q = ( p + / s > s . For s ≤ s , we apply Chebyshev’s inequality. Then we have Z S /κ |A ( κ s ) | κ p s p − ds ≤ N Z s k U k L ( Ω ) ( κ s ) κ p s p − ds + N (cid:16) κ − q + κ − ( γ µ ′ + θ µ ′ ) (cid:17) Z S /κ |B ( s ) | κ p s p − ds ≤ N R d (1 − p / k U k pL ( Ω ) + N (cid:16) κ − q + κ − ( γ µ ′ + θ µ ′ ) (cid:17) κ p Z S /κ |A ( s / | s p − ds + N k F k pL p ( Ω ) , where N = N ( d , p , Λ ) and N depends also on κ . Here in the last line, we usedthe following relationship: B ( s ) ⊂ A ( s / ∪ { ( γ / (2 µ ′ ) + θ / (2 µ ′ ) ) − M Ω ( F µ ) / (2 µ ) > s / } and the Hardy-Littlewood inequality, noting that 2 µ < p . Now we choose κ su ffi cient large such that N κ p − q < − p − , and then θ and γ su ffi cient small suchthat N ( κ p − ( γ /µ ′ + θ /µ ′ )) < − p − . Then we have Z S p |A ( s ) | s p − ds ≤ NR d (1 − p / k U k pL ( Ω ) + N k F k pL p ( Ω ) + p Z S / (2 κ )0 |A ( s ) | s p − ds , where N = N ( d , p , Λ ). This yields (5.30). Hence, we have u ∈ W , p ( Ω ) satisfying(5.1). Note that all the previous proof including the reverse H ¨older inequality,the estimate for harmonic functions, the decomposition lemma, and the level setargument, also work when λ = U by | Du | and F by P i | f i | . Thuswe can also obtain (5.2) when λ = f = (cid:4) To end this section, we give the proof of Corollary 5.2.
IXED BOUNDARY VALUE PROBLEM 23
Proof of Corollary 5.2.
Consider the usual smooth cut-o ff function ζ ∈ C ∞ c ( B ε ) with ζ ∈ [0 , , | D ζ | ≤ N /ε . From (2.3), we can obtain the equation for ζ u : D i ( a ij D j ( u ζ )) − λ ( u ζ ) = D i ( f i ζ + h i ) + ( f ζ + h ) in Ω , a ij D j ( u ζ ) n i = ( f i ζ + h i ) n i on N , u ζ = D , where h i and h are given as follows: h i : = a ij uD j ζ − b i u ζ, h : = a ij D j uD i ζ + b i uD i ζ − ˆ b i D i u ζ − cu ζ − f i D i ζ. Hence by (5.1), for any λ >
0, we have k D ( u ζ ) k L p ( Ω ) + √ λ k u ζ k L p ( Ω ) ≤ N R d (1 / p − / (cid:16) k D ( u ζ ) k L ( Ω ) + √ λ k u ζ k L ( Ω ) (cid:17) + N k f i ζ k L p ( Ω ) + N √ λ (cid:16) k f i D ζ k L p ( Ω ) + k f ζ k L p ( Ω ) (cid:17) + N (cid:16) k uD ζ k L p ( Ω ) + k u ζ k L p ( Ω ) (cid:17) + N √ λ (cid:16) k Du · D ζ k L p ( Ω ) + k uD ζ k L p ( Ω ) + k Du ζ k L p ( Ω ) + k u ζ k L p ( Ω ) (cid:17) , (5.31)where N = N ( d , p , Λ ) and N depends also on K . Using H ¨older’s inequality, weobtain k D ( u ζ ) k L ( Ω ) + √ λ k u ζ k L ( Ω ) ≤ ε d / − d / p (cid:16) k D ( u ζ ) k L p ( Ω ) + √ λ k u ζ k L p ( Ω ) (cid:17) . Thus by taking ε = ε ( d , p , Λ , R ) > ffi ciently small such that N ( ε/ R ) d / − d / p < /
2, we can absorb the first two terms on the right-hand side of (5.31) to the left-hand side. Then by using the standard partition of unity technique and choosing λ large enough, we conclude (5.3). The corollary is proved. (cid:4)
6. S olvability and G eneral p With the regularity result in hand, we are now going to prove Theorem 2.4concerning the solvability. Note that in this section, we deal with more generalcases p ∈ (4 / , L well-posedness result, which is adirect consequence of the Lax-Milgram lemma. Lemma 6.1.
Let Ω be a domain with ∂ Ω =
D ∪ N . Consider the equation (2.3) withb i , ˆ b i , c ∈ L ∞ ( Ω ) . Then for anyf , f i ∈ L ( Ω ) , λ > λ : = (cid:16) k b i k L ∞ ( Ω ) / Λ + k ˆ b i k L ∞ ( Ω ) / Λ + k c k L ∞ ( Ω ) (cid:17) , there exists a unique W , D ( Ω ) weak solution u to (2.3) , satisfying k U k L ( Ω ) ≤ N k F k L ( Ω ) , where N = N ( Λ ) is a constant.Proof of Theorem 2.4. We prove by three cases under Assumptions 2.2 ( γ ) and 2.3( θ ), where γ , θ are the constants from Proposition 5.1. Assume that λ > λ ,where λ is a constant to be chosen below, which satisfies λ ≥ max { λ , λ } . Here, λ and λ are the constants from Corollary 5.2 and Lemma 6.1, respectively. Case 1 : p =
2. This is Lemma 6.1.
Case 2 : p ∈ (2 , ffi ces to prove the theorem when all the lower order coe ffi cientsare zero, i.e., b i ≡ ˆ b i ≡ c ≡ f , f i by f ( n ) , f ( n ) i strongly in L p ( Ω ), where f ( n ) , f ( n ) i ∈ L ( Ω ) ∩ L p ( Ω ). Then by Lemma 6.1, there exist W , D ( Ω ) weak solutions u ( n ) to(2.3) (without lower order terms) with f ( n ) , f ( n ) i in place of f , f i . Moreover, it followsfrom Proposition 5.1 and Corollary 5.2 that { u ( n ) } is a Cauchy sequence in W , p D ( Ω ).Denote its limit by u ∈ W , p D ( Ω ). Clearly u is a weak solution, and (2.4) is satisfied. Case 3 : p ∈ (4 / , u ∈ W , p D ( Ω ) is a solution to (2.3) with f , f i ∈ L p ( Ω ), we are to provethe estimate (2.4). For simplicity, we consider the following equivalent norm forthe space L p ( Ω ) × ( L p ( Ω )) d with λ > k ( f , ( f i ) di = ) k p ,λ : = λ − / k f k L p ( Ω ) + d X i = k f i k L p ( Ω ) , and its dual space L p ′ ( Ω ) × ( L p ′ ( Ω )) d : k ( f , ( f i ) di = ) k p ′ , /λ : = λ / k f k L p ′ ( Ω ) + d X i = k f i k L p ′ ( Ω ) , where p ′ ∈ (2 ,
4) satisfying 1 / p + / p ′ = ffi ces to provesup ϕ i ,ϕ ∈ C ∞ c ( Ω ) k ( ϕ, ( ϕ i ) di = ) k p ′ , /λ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω ( D i u ϕ i + λ u ϕ ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N (cid:16) k f i k L p ( Ω ) + λ − / k f k L p ( Ω ) (cid:17) . For this, we solve for v ∈ W , p ′ D ( Ω ) to the following adjoint problem: D i ( a ji D j v − ˆ b i v ) − b i D i v + cv − λ v = λϕ − D i ϕ i in Ω , a ji D j vn i − ˆ b i vn i = − ϕ i · n i on N , v = D . (6.1)Noting that u is a test function for (6.1), and v is a test function for (2.3), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω ( D i u ϕ i + λ u ϕ ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω ( − a ji D j vD i u + ˆ b i vD i u − b i D i vu + cvu − λ vu ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω ( − f i D i v + f v ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) k f i k L p ( Ω ) + λ − / k f k L p ( Ω ) (cid:17)(cid:16) k Dv k L p ′ ( Ω ) + λ / k v k L p ′ ( Ω ) (cid:17) = N (cid:16) k f i k L p ( Ω ) + λ − / k f k L p ( Ω ) (cid:17) . Here, we use the following W , p ′ estimate for the equation (6.1): k D i v k L p ′ ( Ω ) + √ λ k v k L p ′ ( Ω ) ≤ N (cid:16) k ϕ i k L p ′ ( Ω ) + λ − / k λϕ k L p ′ ( Ω ) (cid:17) = N . This gives us the W , p -a priori estimate, i.e., (2.4) when p ∈ (4 / , IXED BOUNDARY VALUE PROBLEM 25
To see the solvability, we approximate f , f i by f ( n ) , f ( n ) i ∈ C ∞ c ( ⊂ L p ) , f ( n ) → f , f ( n ) i → f i in L p . Let u ( n ) be the unique W , D weak solution associated with f ( n ) and f ( n ) i . Due to the W , p -a priori estimate that we just obtained, and the same argument as in case 2, itis enough to show u ( n ) ∈ W , p D ( Ω ). Due to H ¨older’s inequality, this can be furtherreduced to showing the following: X k k u k W , ( Ω k + \ Ω k ) · k ( d − / p − / < ∞ . (6.2)Here, we denoted Ω k : = Ω k (0). For this, we use a classical “hole-filling” technique.Take η ∈ C ∞ c ( B ck ), η = B ck + , | D η | ≤
2. Testing the equation by u η and rearrangingterms, we obtain that there exits some λ = λ ( d , p , Λ , R , k b i k ∞ , k ˆ b i k ∞ , k c k ∞ ), suchthat for λ > λ , Z Ω ck + ( | Du | + λ | u | ) dx ≤ N Z Ω k + \ Ω k | u | dx . Clearly, this leads to k Du k L ( Ω ck + ) + λ k u k L ( Ω ck + ) ≤ NN + λ ( k Du k L ( Ω ck ) + √ λ k u k L ( Ω ck ) ) . Hence, k u k W , ( Ω k + \ Ω k ) decays exponentially and in particular, (6.2) holds. Thisfinishes our proof. (cid:4)
7. B ounded D omain C ase In this section, we deal with the bounded domain case, i.e., Theorem 2.5. First,we reduce the problem to the case f = Definition 7.1 (John domain, [1]) . A bounded set Ω ⊂ R d is a John domain, if thereexist x ∈ Ω and λ > x ∈ Ω there exists a continuous rectifiablecurve γ : [0 , Ω , such that γ (0) = x , γ (1) = x , anddist( γ ( t ) , Ω c ) ≥ λ · | γ [0 , t ] | (7.1)for all t ∈ [0 , | γ [0 , t ] | represents the arc length. Lemma 7.2.
Assume Ω is a bounded Reifenberg flat domain with ∂ Ω =
D∪N , D , N , ∅ ,satisfying Assumption 2.2 ( γ ) , γ < / . Let p > and p ∗ be given as in (2.5) . Then forevery f ∈ L p ∗ ( Ω ) , there exists φ = ( φ , · · · , φ d ) ∈ ( W , p ∗ N ( Ω )) d ( ⊂ ( L p ( Ω )) d , by the Sobolevembedding), such that D i φ i = f in Ω , k φ i k L p ( Ω ) ≤ N k f k L p ∗ ( Ω ) , (7.2) where N = N ( d , p , diam( Ω ) , R ) is a constant.Proof. Noting that D , N , ∅ , we can choose a point x ∈ Γ . Taking the coordinatesystem in B R ( x ) from Assumption 2.2, we extend Ω beyond D as follows.We first take the Whitney decomposition of the open set e Ω R : = Ω R ( x ) ∩ { x + / R < y < x + / R } as in [24, Chapter IV], i.e., e Ω R = ∪ k Q k , where the disjoint cubes Q k satisfydiam( Q k ) ≤ dist( Q k , ( e Ω R ) c ) ≤ Q k ) . Denote the center of Q k to be x k . We extend Q k to ˆ Q k in the way thatˆ Q k − x k = Q k − x k ) . Let ˆ
Ω = Ω ∪ ( ∪ k ˆ Q k ). It is easy to see that N ⊂ ∂ ˆ Ω , C R d ≤ | ˆ Ω \ Ω | ≤ C R d , where C , C are constants only depending on the space dimension d . Next, wecheck that ˆ Ω is still a John domain, i.e., for any ˆ x ∈ ˆ Ω we construct the pathconnecting ˆ x and x , which satisfies the conditions in Definition 7.1. Case 1 : ˆ x ∈ Ω . Noting that any Reifenberg flat domain is also a John domain,we take the same path as in Definition 7.1. Noting that for any x ∈ Ω , dist( x , ˆ Ω c ) ≥ dist( x , Ω c ), (7.1) is satisfied with the same λ . Case 2 : ˆ x ∈ ˆ Ω \ Ω . We assume that ˆ x lies in the extended cube ˆ Q k with center x k and diam( ˆ Q k ) = r k . Let x be the point defined in Definition 7.1. If x ∈ Q k , we cantake the straight line path. In this case, (7.1) is satisfied with the constant 7 / (9 √ d ).Now, if x < Q k , we first consider the straight line path γ : [0 , / ˆ Q k , γ (0) = ˆ x , γ (1 / = x k . Since ˆ x < Q k , we have 12 r k ≤ | γ [0 , / | ≤ √ dr k . Noting that x k ∈ Ω , we consider the re-parametrized path coming from Definition7.1: γ : [0 , / Ω , γ (0) = x k , γ (1 / = x . Take γ = γ ◦ γ be the path connecting γ and γ . Now, when t ∈ [0 , / / √ d . When t ∈ [1 / , γ ( t ) ∈ Q k or γ ( t ) < Q k .If γ ( t ) ∈ Q k , we have | γ [0 , t ] | dist( γ ( t ) , ˆ Ω c ) ≤ | γ [0 , / | dist( γ ( t ) , ˆ Ω c ) + | γ [1 / , t ] | dist( γ ( t ) , Ω c ) ≤ √ dr k r k / + λ − = √ d + λ − . When γ ( t ) < Q k , we have | γ [1 / , t ] | ≥ r k /
2. Hence, | γ [0 , t ] | dist( γ ( t ) , ˆ Ω c ) = | γ [0 , t ] || γ [1 / , t ] | · | γ [1 / , t ] | dist( γ ( t ) , ˆ Ω c ) = (1 + | γ [0 , / || γ [1 / , t ] | ) · | γ [1 / , t ] | dist( γ ( t ) , ˆ Ω c ) ≤ (1 + √ dr k r k / · | γ [1 / , t ] | dist( γ ( t ) , ˆ Ω c ) ≤ (1 + √ d ) λ − . IXED BOUNDARY VALUE PROBLEM 27
With all above, we have proved that ˆ Ω is still a John domain. Now we extend f toˆ Ω as ˆ f : = f in Ω , ˆ f : = − | ˆ Ω \ Ω | R Ω f in ˆ Ω \ Ω . Then we have Z ˆ Ω ˆ f = , k ˆ f k L p ∗ ( ˆ Ω ) ≤ N ( R , | Ω | ) k f k L p ∗ ( Ω ) . Since ˆ Ω is a John domain, we apply the result in [1, Theorem 4.1] to find φ = ( φ , · · · , φ d ) ∈ ( W , p ∗ ( ˆ Ω )) d satisfying D i φ i = ˆ f in ˆ Ω , k φ i k W , p ∗ ( ˆ Ω ) ≤ N (diam( ˆ Ω ) , d , p ) k ˆ f k L p ∗ ( ˆ Ω ) . Now by Sobolev inequalities and our construction of ˆ Ω , ˆ f , we obtain that φ ∈ ( W , p ∗ N ( Ω )) d , and k φ i k L p ( Ω ) ≤ N (diam( Ω ) , R , d , p ) k ˆ f k L p ∗ ( ˆ Ω ) . The lemma is proved (cid:4)
Now we are ready to give the proof of Theorem 2.5.
Proof of Theorem 2.5.
Using Lemma 7.2, for every f ∈ L p ∗ ( Ω ), we can find( φ i ) di = ∈ ( W , p ∗ N ( Ω )) d ⊂ ( L p ( Ω )) d satisfying (7.2). Now we consider the following problem: Lu = D i ( f i + φ i ) in Ω , Bu = ( f i + φ i ) n i on N , u = D . (7.3)Since φ = N , one can easily check that any solution to (7.3) is also a solutionto (2.2). Hence, without loss of generality, we may assume f = λ , we can find a unique weaksolution u ∈ W , p D ( Ω ) to (2.3) satisfying (2.4), and hence (2.6).We write R ( λ, L ) as this solution operator, i.e., R ( λ, L ) : ( L p ( Ω )) d × L p ( Ω ) W , p D ( Ω ) , R ( λ, L )( f i , f ) = u . In particular, for any L p function f , we write R λ ( f ) : = R ( λ, L )(0 , f ) . From (2.6), R λ is a bounded linear operator from L p ( Ω ) to W , p D ( Ω ). Denote I as thecompact embedding from W , p D ( Ω ) to L p ( Ω ). Now, we write T : W , p D ( Ω ) → W , p D ( Ω ) , T ( u ) = R λ ◦ I ( u ) . From our construction, T is a compact operator. Noting that we have assumed that f =
0, applying the operator R ( λ, L ) to both sides of( L − λ ) u + λ u = D i f i , we can rewrite (2.2) as ( Id + λ T ) u = R ( λ, L )( f i , . (7.4) By the Fredholm alternative, (7.4) has a unique W , p D ( Ω ) solution satisfying k u k W , p ( Ω ) ≤ N k f i k L p ( Ω ) , if the following homogeneous equation only has zero solution Lv = Ω , Bv = N , v = D . (7.5)When v ∈ W , ( Ω ), this is true due to the weak maximum principle, noting thatthe proof in [13, Section 8.1] actually shows that sup Ω | v | has to be achieved at theDirichlet boundary. Hence the uniqueness of (7.5) is proved for the case p ≥ p <
2, we can use Theorem 2.4 and a bootstrap argument to improve theregularity. Suppose v ∈ W , p ( Ω ) is a solution to (7.5). Take λ large enough, notingthat v is also a W , p solution to ( L − λ ) v = − λ v in Ω , Bv = N , v = D . By the Sobolev embedding, − λ v ∈ L pd / ( d − p ) ( Ω ). Take p ∗ = min { pd / ( d − p ) , } . By theuniqueness of W , p ∗ solutions in Theorem 2.4, we obtain v ∈ W , p ∗ . Repeating thisprocess if needed, in finite steps, we can reach v ∈ W , . Hence we can use theweak maximum principle to deduce v = (cid:4) R eferences [1] Gabriel Acosta, Ricardo G. Dur´an, and Mar´ıa A. Muschietti. Solutions of the divergence operatoron John domains. Adv. Math. , 206(2):373–401, 2006.[2] Russell Brown. The mixed problem for Laplace’s equation in a class of Lipschitz domains.
Comm.Partial Di ff erential Equations , 19(7-8):1217–1233, 1994.[3] Russell Brown and Irina Mitrea. The mixed problem for the Lam´e system in a class of Lipschitzdomains. J. Di ff erential Equations , 246(7):2577–2589, 2009.[4] Sun-Sig Byun and Lihe Wang. Elliptic equations with BMO coe ffi cients in Reifenberg domains. Comm. Pure Appl. Math. , 57(10):1283–1310, 2004.[5] Sun-Sig Byun and Lihe Wang. The conormal derivative problem for elliptic equations with BMOcoe ffi cients on Reifenberg flat domains. Proc. London Math. Soc. (3) , 90(1):245–272, 2005.[6] L. A. Ca ff arelli and I. Peral. On W , p estimates for elliptic equations in divergence form. Comm.Pure Appl. Math. , 51(1):1–21, 1998.[7] Jongkeun Choi, Hongjie Dong, and Doyoon Kim. Conormal derivative problems for stationaryStokes system in Sobolev spaces.
Discrete Contin. Dyn. Syst. , 38(5):2349–2374, 2018.[8] Hongjie Dong and Doyoon Kim. Higher order elliptic and parabolic systems with variably partiallyBMO coe ffi cients in regular and irregular domains. J. Funct. Anal. , 261(11):3279–3327, 2011.[9] Hongjie Dong and Doyoon Kim. The conormal derivative problem for higher order elliptic systemswith irregular coe ffi cients. In Recent advances in harmonic analysis and partial di ff erential equations ,volume 581 of Contemp. Math. , pages 69–97. Amer. Math. Soc., Providence, RI, 2012.[10] Hongjie Dong and Doyoon Kim. Weighted L q -estimates for stationary Stokes system with partiallyBMO coe ffi cients. J. Di ff erential Equations , 264(7):4603–4649, 2018.[11] Hongjie Dong and Doyoon Kim. L p -estimates for time fractional parabolic equations with coe ffi -cients measurable in time. Adv. Math , 345:289–345, 2019.[12] Mariano Giaquinta.
Multiple integrals in the calculus of variations and nonlinear elliptic systems , volume105 of
Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 1983.[13] David Gilbarg and Neil S. Trudinger.
Elliptic partial di ff erential equations of second order . Classics inMathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. IXED BOUNDARY VALUE PROBLEM 29 [14] Carlos E. Kenig.
Harmonic analysis techniques for second order elliptic boundary value problems , vol-ume 83 of
CBMS Regional Conference Series in Mathematics . Published for the Conference Board ofthe Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence,RI, 1994.[15] N. V. Krylov and M. V. Safonov. A property of the solutions of parabolic equations with measurablecoe ffi cients. Izv. Akad. Nauk SSSR Ser. Mat. , 44(1):161–175, 239, 1980.[16] Antoine Lemenant, Emmanouil Milakis, and Laura V. Spinolo. On the extension property ofReifenberg-flat domains.
Ann. Acad. Sci. Fenn. Math. , 39(1):51–71, 2014.[17] Irina Mitrea and Marius Mitrea. The Poisson problem with mixed boundary conditions in Sobolevand Besov spaces in non-smooth domains.
Trans. Amer. Math. Soc. , 359(9):4143–4182, 2007.[18] Katharine A. Ott and Russell M. Brown. The mixed problem for the Laplacian in Lipschitz domains.
Potential Anal. , 38(4):1333–1364, 2013.[19] E. R. Reifenberg. Solution of the Plateau Problem for m -dimensional surfaces of varying topologicaltype. Acta Math. , 104:1–92, 1960.[20] M. V. Safonov. Harnack’s inequality for elliptic equations and H¨older property of their solutions.
Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) , 96:272–287, 312, 1980. Boundaryvalue problems of mathematical physics and related questions in the theory of functions, 12.[21] Giuseppe Savar´e. Regularity and perturbation results for mixed second order elliptic problems.
Comm. Partial Di ff erential Equations , 22(5-6):869–899, 1997.[22] Eliahu Shamir. Regularization of mixed second-order elliptic problems. Israel J. Math. , 6:150–168,1968.[23] Guido Stampacchia. Problemi al contorno ellitici, con dati discontinui, dotati di soluzionieh¨olderiane.
Ann. Mat. Pura Appl. (4) , 51:1–37, 1960.[24] Elias M. Stein.
Singular integrals and di ff erentiability properties of functions . Princeton MathematicalSeries, No. 30. Princeton University Press, Princeton, N.J., 1970.S chool of M athematics , K orea I nstitute for A dvanced S tudy , 85 H oegiro , D ongdaemun - gu ,S eoul epublic of K orea E-mail address : [email protected] D ivision of A pplied M athematics , B rown U niversity , 182 G eorge S treet , P rovidence , RI 02912,USA E-mail address : hongjie [email protected] D ivision of A pplied M athematics , B rown U niversity , 182 G eorge S treet , P rovidence , RI 02912,USA E-mail address ::