J. Math. Fluid Mech. 19 (2017), 375 422 c 2016 Springer International Publishing 1422-6928/17/030375-48 DOI 10.1007/s00021-016-0289-y Journal of Mathematical Fluid Mechanics Solvability and Regularity for an Elliptic System Prescribing the Curl, Divergence, and Partial Trace of a Vector Field on Sobolev-Class Domains C. H. Arthur Cheng and Steve Shkoller Abstract. We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field u are prescribed in an open, bounded, Sobolev-class domain R n, and either the normal component u N or the tangential components of the vector field u N are prescribed on the boundary. For k > n/2, we prove that u is in the Sobolev space H k1 ()ifisanh k1 -domain, and the divergence, curl, and either the normal or tangential trace of u has sufficient regularity. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients, and with a rather general set of Dirichlet and Neumann boundary conditions. The resulting regularity theory for the vector u is fundamental in the analysis of free-boundary and moving interface problems in fluid dynamics. Mathematics Subject Classification. 35J57, 58A14. Keywords. Hodge theory, elliptic estimates, Sobolev-class coefficients, Sobolev-class domains, regularity. Contents 1. Introduction 376 1.1. Statements of the Main Results 376 1.2. Outline of the Paper 378 1.3. A Brief History of Prior Results 378 2. Notation and Preliminary Results 379 2.1. Definition of a C k -Domain 379 2.2. Definition of an H s -Domain 379 2.3. Basic Inequalities 381 2.4. Poincaré-Type Inequalities 383 2.5. Commutation with Mollifiers 383 2.6. The Piola Identity 384 3. Vector-Valued Elliptic Equations 384 3.1. The Case that the Coefficients a jk are of Class C k 385 3.2. The Case that the Coefficients a jk are of Sobolev Class 391 4. Regularity Theory: The Proof of Theorem 1.3 399 5. Existence and Uniqueness Theory: The Proof of Theorem 1.1 404 5.1. Uniqueness of the Solution 404 5.2. Existence of Solutions 405 5.3. Solutions with Prescribed Tangential Trace 413 6. Fractional-Order Regularity: The Proof of Theorem 1.6 417 Appendix A. Proofs of the Inequalities in Sect. 2.3 419 References 421
376 C. H. A. Cheng and S. Shkoller JMFM 1. Introduction 1.1. Statements of the Main Results Given a sufficiently smooth Sobolev-class bounded domain R n and forcing functions f and g in together with boundary data given by either h or h on, we establish the basic elliptic estimates for the vector elliptic system of Hodge-type: with boundary conditions given by either curl v = f in, div v = g in, v N = h or v N = h on, where N is the outward-pointing unit normal vector on. When the domain is of class C k1, elliptic estimates for solutions v in H k1 () are now classical. We extend this well-known theory to the case of domains of Sobolev class H k1. We first establish the following Theorem 1.1. Let R 3 be a bounded H k1 -domain with integer k > 3 2. Given f,g Hl 1 () with div f =0, consider the equations curl v = f in, (2a) div v = g in. (2b) (1) If f satisfies f N ds =0 for each connected component Γ of, (3) Γ and h H l 0.5 () satisfies hds = gdx, then, for 1 l k, there exists a solution v H l () to (2) with boundary condition such that v N = h on, (4) v Hl () C( H k0.5) f H l 1 () g H l 1 () h H l 0.5 (). The solution is unique if is the disjoint union of simply connected open sets. (2) If f satisfies (3) and f N =div h on (where div denotes the surface divergence operator defined in Definition 2.4), h H l 0.5 () satisfies h N =0as well as f n ds = (N h) dr if Σ has piecewise smooth boundary Σ Σ Σ with unit normal n, compatible with the orientation of Σ, (5) then, for 1 l k, there exists a solution v H l () to (2) with boundary condition such that v N = h on, (6) v Hl () C( H k0.5) f H l 1 () g H l 1 () h H l 0.5 (). The solution is unique if each connected component of has a connected boundary.
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 377 Remark 1.2. To explain condition (3), let be a connected bounded open set, and let {Γ i } I i=0 denote the connected components of in which Γ 0 is the boundary of the unbounded connected component of.foreachi =0,...,I,letq i be the solution to Δq i =0 in, (7a) q i = δ ij on Γ j. (7b) Then if u satisfies curlu = f in for some divergence-free vector f, applying the divergence theorem and then integrating by parts, shows that for i =0,...,I, f NdS = (f N)q i ds = (f N)q i ds = Γ i Γ i = curlu q i dx = (N u) q i ds = q i divfdx f q i dx (N u) q i ds =0, where is the tangential derivative defined below in Definition 2.4. Therefore, for curlu = f to be solvable, it is necessary that f N ds =0 i {0, 1,...,I}. Γ i In other words, (3) is a necessary condition for the solvability of (2a). We will show that it is also one of the sufficient conditions to solve (2) with the boundary conditions (4) or(6). The problem (2) with either boundary conditions (4)or(6) has been well studied. The characterization of the kernel of both problems, the solvability conditions, and the existence theory has been developed in a number of papers; see, for example, 1,2,5,6,13 15,17, and the references therein. The inequalities given in Theorem 1.1 are new for Sobolev-class domains. Motivated by the analysis of the free-boundary problems which arise in inviscid fluid dynamics, we next state a theorem which provides two fundamental elliptic estimates set on Sobolev-class domains: Theorem 1.3. Let R n, n=2or 3, be a bounded H k1 -domain with integer k > n. Then there exists 2 a generic constant C depending on H k0.5 such that for all u H k1 (), u H k1 () C u L2 () curlu H k () divu H k () u N H k 0.5 (), (8) u H k1 () C u L 2 () curlu Hk () divu Hk () u N H k 0.5 (), (9) where u is the tangential derivative on (defined in Definition 2.4). Remark 1.4. The inequalities (8) and (9) play a fundamental role in the regularity theory of the Euler equations with moving interfaces; see, for example, 10 for the incompressible setting and 11 for the compressible problem with vacuum. The use of the norm u N H k 0.5 () rather than u N H k0.5 () is crucial, as the regularity of the normal vector to field to is often worse than the regularity of the velocity vector u. On the other hand, if is at least of class H k2 then the inequalities (8) and (9) can be replaced, respectively, by u H k1 () C u L 2 () curlu Hk () divu Hk () u N H k0.5 () (10) u H k1 () C u L2 () curlu H k () divu H k () u N H k0.5 () (11) Remark 1.5. Recently, Amrouche and Seloula 5 established the inequalities (10) and (11) inthel p framework and for domains of class C k1 ; see Corollary 3.5 in 5. Of course, in the case of a C k1 - domain, the inequalities (8) and (9) follow immediately from (10) and (11), respectively. When is very close to a C -domain, we can obtain these inequalities for fractional-order Sobolev spaces, as in the following
378 C. H. A. Cheng and S. Shkoller JMFM Theorem 1.6. Let R n, n=2or 3, be a bounded H s1 -domain with s R such that s> n 2,andlet D denote a C -domain such that the distance between D and in the H s0.5 -norm is less than ɛ for 0 <ɛ 1. Then there exists a generic constant C depending only on D H s0.5, such that for all u H s1 (), u H s1 () C u L2 () curlu Hs () divu Hs () u N H s 0.5 (), (12) u H s1 () C u L 2 () curlu H s () divu H s () u N H s 0.5 (), (13) where u is the tangential derivative on (defined in Definition 2.4). The inequalities (12) and (13) set in fractional-order Sobolev spaces are fundamental to the analysis of Euler-type free-boundary problems. We remark that is assumed to be in a small tubular neighborhood of the normal bundle over D; hence, there is height function h(x, t) suchthateachpointon is given by x h(x)n(x), x D, where n is the outward-pointing unit normal to D. The assumption that the distance between D and intheh s0.5 -norm is less than ɛ 1 means that we assume that h H s0.5 ( D) <ɛ 1 1.2. Outline of the Paper In Sect. 2, we introduce our notation as well as a number of elementary technical lemmas, whose proofs we include (for completeness) in Appendix A. Section 3 is devoted to the analysis of the vector-valued elliptic system (31a) with mixed-type boundary conditions (31b) and (31c), which is fundamental to the proof of our two main theorems; in particular, we prove Theorem 3.6 which establishes the elliptic estimate for (31) when the coefficients are of Sobolev-class. As a corollary to this theorem, we state in Corollary 3.8 the basic elliptic estimates for both the classical Dirichlet and Neumann problems, again with Sobolev class regularity. Finally, for coefficients which are close to the identity, we give an improved estimate in Theorem 3.9 for solutions to (31), which is linear in the highest derivatives of the coefficient matrix. This latter theorem is essential for estimates in fractional-order Sobolev spaces via linear interpolation. In Sect. 4, we prove Theorem 1.3, using the elliptic regularity theory developed for the elliptic system (31). Then, in Sect. 5, we prove Theorem 1.1. Our proof relies on some basic geometric identities involving the mean curvature of, together with the elliptic regularity theory established in Sect. 3. Finally, in Sect. 6, we prove Theorem 1.6. 1.3. A Brief History of Prior Results In addition to the recent work of Amrouche and Seloula 5 noted above, there have been many other methods and results to study such elliptic systems on smooth domains. The elliptic system (2) can be viewed as a particular example of the systems studied by Agmon et al. 1, wherein both Schauder-type estimates and L p -estimates can be found. In 19, von Wahl proved that if the normal or the tangential trace of a vector field vanishes, and for bounded or unbounded, the inequality u L p () C ( divu L p () curlu L p ()) holds when the first Betti number or the second Betti number, respectively, is equal to zero. Vector potentials and the characterization of the kernel of problem (2) with boundary conditions (4) or (6) have been obtained by Foias and Temam 13, Georgescu 14, Bendali et al. 6, Amrouche et al. 2, and Amrouche et al. 3. Amrouche and Girault 4 derived the L p -regularity theory of the steady Stokes equation by establishing the equivalency between the Sobolev space W m,r and the direct sum of W m,r by divergence-free vector fields and the gradients of W m1,r functions. Schwarz 18, studied the Hodge decomposition on manifolds with boundaries and showed that a differential k-form can be written as the sum of an exact form, a coexact form, and a harmonic form.
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 379 Bolik and von Wahl 7 derived C α -estimates of the gradient of a vector field whose curl, divergence, and normal or tangential traces are prescribed. Mitrea et al. 16 studied the vector potential theory on non-smooth domains in R 3 with applications to electromagnetic scattering. Buffa and Ciarlet 8 and 9 established the Hodge decomposition of tangential vector fields defined on polyhedron domains, and studied the tangential trace and tangential components of vectors belonging to the space H(curl, ) := { u L 2 (; R 3 ) curlu L 2 (; R 3 ) }. In 15, Kozono and Yanagisawa proved the decomposition of a divergence-free vector-field as the sum of the curl of a vector-field and a vector-field which is solenoidal, irrotational and has zero normal trace. 2. Notation and Preliminary Results The Einstein summation convention is used throughout the paper. In particular, repeated Latin indices are summed from 1 to n, and repeated Greek indices are summed from 1 to n 1. For example, f i g i = n i=1 f ig i and f α g α = n 1 i=1 f αg α. The gradient operator is denoted by =( 1,..., n ). Below, we shall also define various tangential derivative operators. When it is not explicitly stated, k,l 0 denote integers, and s denotes a real number. 2.1. Definition of a C k -Domain We recall that a domain R n is said to be of class C k if isan(n 1)-dimensional C k -manifold; that is, there exists an open cover {U m } K m=1 R n of and a collection of C k -maps {φ m } K m=1 such that for each 1 m K, φ m : U m V m R n 1 is one-to-one, onto, and has a C k -inverse map for some open subset V m of R n 1. A domain is called a C -domain if it is a C k -domain for all k N. Equivalently, we shall use the following Proposition 2.1. Let R n be a C k -domain for some k N, andε>0 be given. Then there exists a collection of open sets {U m } K m=0 with each U m R n, a collection of C k -maps {ϑ m } K m=1 and positive numbers {r m } K m=1 such that K m=0 U m and for each 1 m K, 1. ϑ m : B(0,r m ) U m is a C k -diffeomorphism; 2. ϑ m : B(0,r m ) {y n =0} U m ; 3. ϑ m : B(0,r m ) {y n > 0} U m ; 4. det( ϑ m )=1; 5. ϑ m Id L (B(0,r m)) ε. and The proof of this proposition is given in Appendix A. K m=1 U m, 2.2. Definition of an H s -Domain In order to make our presentation self-contained, in this section, we collect a number of useful technical lemmas. These lemmas are well-known when the domains are smooth, but we shall need these basic results for Sobolev class domains. The proofs will be collected in Appendix A. We use the term domain to mean an open connected subset of R n.
380 C. H. A. Cheng and S. Shkoller JMFM Definition 2.2. Let R n be a bounded domain, and s> n 1 be a real number. is said to be an 2 H s -domain, or of class H s, if there exists a bounded C -domain O and a map ψ such that ψ : O is an H s -diffeomorphism; that is, 1. ψ : O is continuous; 2. ψ :O is one-to-one and onto, with differentiable inverse map ψ 1 : O; 3. ψ : O is one-to-one and onto, with differentiable inverse map ψ 1 : O; 4. ψ H s (O; ) and ψ 1 H s (; O). By the trace theorem, ψ O H s 0.5 ( O; ) and we shall often denote the value of this norm by H s 0.5. Definition 2.3. For s> n 1, given a local chart (U,ϑ) as defined in Proposition 2.1, the induced metric 2 in the local chart (U,ϑ)isthe(0, 2)-tensor g αβ given by g αβ = ϑ ϑ, y α y β and the induced second-fundamental form in a local chart (U,ϑ)isthe(0, 2)-tensor b αβ given by b αβ = 2 ϑ (N ϑ), y α y β where N is the outward-pointing unit normal to. Definition 2.4 (Tangential gradient and surface divergence operators). For s> n 2 1, let Rn be a bounded H s -domain. We let denote the tangential gradient of a function on. If ϕ : R is differentiable, then in local chart (U,ϑ), ϕ is given by ( ϕ) ϑ = g αβ (ϕ ϑ) ϑ, y α y β { } where g αβ ϑ 2 is the inverse matrix of the induced metric g αβ, and are tangent vectors to. y β β=1 We define the surface divergence operator div to be the formal adjoint of ;ifu is a tangent vector field on so that u N =0on, then u ϕds = ϕ div u ds ϕ H 1 (). In a local chart (U,ϑ), (div u) ϑ = 1 gg αβ ( (u ϑ) ϑ ), g y α y β where g = det(g) is the determinant of the induced metric g αβ. Definition 2.5 (Tangential projection of a vector field onto ). With N denoting the outward unit normal vector field to andv : R n, we define P N : R n R n to be the tangential projection operator given by P N (v) =v (v N) N = ( Id N N ) v. (14) We will also write v for P N (v). Definition 2.6 (Various tangential derivatives). We let u : R n and w : R n denote vectorvalued functions, and let w be given by (14). 1. w u denotes the directional derivatives of u in the direction w. In a local chart (U,ϑ), w u ϑ = g αβ (u ϑ) ϑ j (w j ϑ) =g αβ ϑ (u ϑ) (w ϑ). y α y β y β y α
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 381 2. u =( u 1,..., u n ), and u w = n w i ( u i ), so that u w is a vector in the i=1 tangent space of. In a local chart (U,ϑ), ( u w) ϑ = g αβ (u j ϑ) ϑ i (w j ϑ) =g αβ (u ϑ) ϑ i (w ϑ). y α y β y α y β The product rule holds: u w = (u w) w u. 3. u w is defined to be the linear map satisfying (at each point of ) ( u w)v =( v u) w for all v R n. In a local chart (U,ϑ), ( αβ ϑj (u r ϑ) u w) ij ϑ = ε irs g (w s ϑ), y β y α where ε irs is the permutation symbol, equaling 1 if (i, r, s) is an even permutation of (1, 2, 3), 1 if (i, r, s) is an odd permutation of (1, 2, 3), and 0 otherwise. 2.3. Basic Inequalities We now state some basic inequalities, that we use throughout the paper. Proposition 2.7. For k > n 2 and 0 l k, let Rn be a bounded C -domain. Then for all σ ( 0, 1 4), there exists a constant C σ depending on σ such that for all f H k () and g H l (), l j f l j g L 2 () C σ f Hk () g H l σ (). (15) j=1 Moreover, for some generic constant C>0, fg H l () C f H () g k H l () f H k (),g H l (). (16) Remark 2.8. Suppose that s> n and 0 r s for some real numbers r and s. Then there exists a 2 generic constant C s > 0 such that fg Hr (R n ) C s f Hs (R n ) g Hr (R n ) f H s (R n ),g H r (R n ). (17) By the Sobolev extension argument, we also conclude that fg Hr () C s f Hs () g Hr () f H s (),g H r () (18) if is a bounded C -domain. The following corollary is a direct consequence of Proposition 2.7 since by Leibniz s rule, l l ( l ),f g = j f l j g. j Corollary 2.9. Let R n be a bounded C -domain for some integer k > n 2. j=1 1. Suppose that spt(g). Then for 0 <σ< 1 and 1 l k1, 4 l,f g C L 2 () σ f H () g max{k,l} H l σ (), (19) where l,f g = l (fg) f l g. 2. Suppose that ζ is a smooth cut-off function such that (a) spt(ζ) U;
382 C. H. A. Cheng and S. Shkoller JMFM (b) there exists a C -diffeomorphism ϑ : B(0, 1) U satisfying (i) ϑ : B (0, 1) B(0, 1) {y n > 0} U ; (ii) ϑ : {y n =0}. Define F =(ζf) ϑ and G =(ζg) ϑ. Then for 0 <σ<1/4, 1 l k1, l,f G L2 (B (0,r)) C σ f H max{k,l} () g H l σ (), (20) where l,f G = l (FG) F l G and =( y1,..., yn 1 ) denotes the horizontal derivative. The following two corollaries are direct consequences of Proposition 2.7, and are the foundation of the study of inequalities on H s -domains. The proof of these two corollaries can also be found in Appendix A. Corollary 2.10. Let O R n be a bounded C -domain, and ψ : O R n be a H k1 -diffeomorphism for some integer k > n 2.IfJ = det( ψ) and A =( ψ) 1, then J H k (O) C ( ψ H (O)), (21a) k A Hk (O) C ( 1/J L (O), ψ Hk (O)). (21b) Corollary 2.11. Let O R n be a bounded C -domain, and ψ : O R n be an H k1 -diffeomorphism for some integer k > n. Then for all l k1, 2 f H l () C( ψ H (O)) f ψ k H l (O) f H l (), (22a) f ψ Hl (O) C( ψ Hk (O)) f Hl () f H l (). (22b) Remark 2.12. Note that Corollary 2.11 implies that the interpolation inequalities on a Sobolev class domain are still valid if the domain is bounded and has H k1 regularity for some integer k > n 2.For example, f H 0.5 () C( H k0.5) f ψ H 0.5 (O) C( H k0.5) f ψ 1 2 L2 (O) f ψ 1 2 H1 (O) C( H k0.5) f 1 2 L 2 () f 1 2 H 1 (). Similar arguments can be applied to prove the following theorem whose proof we omit. Theorem 2.13. Let R n be a bounded domain of class H k1 for an integer k > n. Then for all 2 σ ( 0, 1, there exists constant Cσ depending on 4) H k0.5 and σ such that for all 0 l k1, f H max{k,l} () and g H l (), l j f l j g L 2 () C σ f H max{k,l} () g H l σ (). (23) j=1 Moreover, for a generic constant C depending on H k0.5, fg Hl () C f H max{k,l} () g Hl () f H max{k,l} (),g H l (). (24) By using Theorem 2.13, we can easily establish the following Theorem 2.14. Let R n be a bounded H k1 -domain for an integer k > n. Then for each integers 2 l {0, 1} ( n 2, k1, there exists a generic constant C = C( H k0.5) such that fg H l () C f L () g H l () f H () g l L () f,g H l () L (). (25)
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 383 2.4. Poincaré-Type Inequalities We will make use of the following Poincaré-type inequalities, whose proofs are similar to the proof of the standard Poincaré inequality. Lemma 2.15. Let k > n 2 be an integer and R3 be a bounded H k1 -domain with outward-pointing unit normal N. Weset Hτ 1 () { u : R 3 u H 1 (), u N =0on }, Hn() 1 { u : R 3 u H 1 (), u N =0on }. Then and u L 2 () C u L 2 () u H 1 τ (), (26) u L2 () C u L2 () u H 1 n(). (27) 2.5. Commutation with Mollifiers Our proof of elliptic regularity relies on a mollification procedure (rather than the use of difference quotients). Definition 2.16 (Standard mollifiers). Let η(x) =C exp ( 1 ) for x < 1andη vanishes outside the x 2 1 unit ball, where C is chosen so that η L1 (R n ) = 1. The standard mollifier η ɛ is defined by We will make use of the following η ɛ (x) = 1 ɛ η( x). n ɛ Lemma 2.17. For f W 1, () and g L 2 () with compact support, there is a generic constant C independent of ɛ such that ( η ɛ,f g ) L2 () = η ɛ (fg) fη ɛ g L2 () C f W 1, () g L 2 () (28) for all 0 <ɛ<min { dist (, spt(f) ), dist (, spt(g) )}. Since we are dealing with problems on domains with boundaries, we make use of the horizontal convolution-by-layers operator, introduced in 10. We define the horizontal convolution-by-layers operator Λ ɛ as follows: Λ ɛ f(x h,x n )= ρ ɛ (x h y h )f(y h,x n )dy h for f(,x n ) L 1 (R n 1 ), R n 1 where ρ ɛ (x h )= 1 ɛ n 1 ρ( x h ɛ ),andρ C c (R n 1 ) is given by ρ(x) =C exp( 1 x 2 1 )if x < 1andρ(x) =0 if x h 1. The constant C is chosen so that R n 1 ρdx= 1. It follows that for ɛ>0, 0 ρ ɛ C c (R n 1 ) with spt(ρ ɛ ) B(0,ɛ). (Here, spt stands for support.) It should be clear that Λ ɛ smooths functions defined on R n along all horizontal subspaces, but does not smooth functions in the vertical x n -direction. On the other hand, we can restrict the operator Λ ɛ to act on functions f : R n 1 R as well, in which case Λ ɛ becomes the usual mollification operator. Associated to Λ ɛ, we need the following
384 C. H. A. Cheng and S. Shkoller JMFM Lemma 2.18. For f W 1, (R n ) and g L 2 (R n ), there is a generic constant C independent of ɛ such that ( Λɛ,f g ) = L 2 (R Λɛ (fg) fλ n ) ɛ g C L 2 (R f W g L (29) n ) 1, (R n ) 2 (R n ) for all ɛ>0. 2.6. The Piola Identity Lemma 2.19 (Piola identity). Letψ : R n R n be a diffeomorphism, and a ij n n be the cofactor matrix of ψ. Then a ji =0. (30) x j The proof can be found in 12. 3. Vector-Valued Elliptic Equations Let R n denote a bounded domain whose regularity will be specified below. In this section, we study a vector-valued elliptic equation (Lu) i = u i ( jk ) ui a = f i in, (31a) x j x k with special types of boundary conditions, where u =(u 1,...,u n )andf =(f 1,...,f n ) are vector-valued functions, and a jk is a two-tensor satisfying the positivity condition a jk ξ j ξ k λ ξ 2 ξ R n (32) for some λ>0. Since u R n, n boundary conditions are needed to solve the system uniquely. We consider a mixed-type boundary condition given by u w =0 on, (31b) ( P w a jk u ) N j g = 0 x k on, (31c) where w is a uniformly continuous vector field defined in a neighbourhood of which vanishes nowhere on, N is the outward-pointing unit normal to, g is a vector-valued function defined on, and P w : R n R n is the projection map given by P w (v) =v (v w) w 2 w = ( Id w w w 2 ) v. (33) The condition (31b) specifies the component of the vector u in the direction of w, while the condition jk ui (31c) specifies the (n 1) components of the Neumann derivative a N j. x k Integration by parts with respect to x j leads to the following identity: ( jk ) ui a ϕ i jk ui ϕ i jk ui dx = a dx a N j ϕ i dx x j x k x k x j x k jk ui ϕ i = a dx g i jk ur w r w i a N j x k x j x k w 2 ϕ i dx, which, in turn, motivates the following
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 385 Definition 3.1. Let V = { v H 1 () } v w =0on. A function u Vis called a weak solution to (31) if jk ui ϕ i (u, ϕ) L 2 () a dx =(f, ϕ) L x k x 2 () g, ϕ ϕ V, (34) j where, denotes the duality pairing between distributions in H 1 2 () and functions in H 1 2 (). With the help of the Lax-Milgram theorem it is easy to conclude the following Theorem 3.2 (Weak solutions). Suppose that a jk L () satisfies the positivity condition (32), andw is a uniformly continuous vector field defined in a neighborhood of which vanishes nowhere on. Then for all f L 2 () and g H 0.5 (), there exists a unique weak solution to (31) in V, and the weak solution u satisfies u H1 () C f L2 () g H 0.5 (). (35) Remark 3.3. Let u H 2 () V be a weak solution to (31). Integrating by parts with respect to x j,we find that ( u i ( a jk ui ) ) f i ϕ i ( dx a jk ui N j g ) ϕ i ds =0 ϕ V. x j x k x k Since ϕ w =0on, the Neumann-type boundary condition (31c) is thus shown to hold. We next establish the regularity theory for weak solutions satisfying (34). Definition 3.4 (Partition-of-unity subordinate to an open cover). For a given collection of open sets {U m } K m=1 R n, there exists a partition of unity {ζ m } K m=1 subordinate to {U m } K m=1 such that ζ m Cc (R n ) for all 1 m K. In fact, if {ξ m } K m=1 is a smooth partition-of-unity subordinate to {U m } K m=1, by defining {ζ m } K m=1 by ζ m = ξm 2 K, j=1 ξ2 j then 0 ζ m 1, spt(ζ m ) U m, ζ m C c (R n ) for all 1 m K, and that K m=1 ζ m =1. 3.1. The Case that the Coefficients a jk are of Class C k Now we study the regularity of the weak solution u to (31) when the coefficients a jk are of Sobolev class H k,k N, and the domain is C k1. To do so, we shall first establish this regularity result under the more restrictive assumption that the coefficients a jk are in C k (). Theorem 3.5 (Regularity for the case that a jk C k () and C k1 ). Suppose that for k N, R n is a bounded C k1 -domain, a jk C k () satisfies the positivity condition (32), w is C k1 in an open neighborhood U of, and w > 0 on. Then for all f H k 1 () and g H k 0.5 (), the weak solution u to (31) in fact belongs to H k1 (), and satisfies u H k1 () C f H k 1 () g H k 0.5 () (36) for a constant C depending on a C k (), w C k1 (U) and C k1. Proof. Our goal is to establish the regularity theory for weak solutions u V to (36). We prove this by induction and divide the proof into several steps as follows: Step 1: (Interior regularity) Suppose that u H l () for 1 l k. Let χ be a smooth function with spt(χ), and 0 <ɛ<dist(spt(χ), ). We define ϕ =( 1) l χ η ɛ 2l( η ɛ (χu) )
386 C. H. A. Cheng and S. Shkoller JMFM with no summation over the index l, and we let {η ɛ } ɛ>0 denote a sequence of standard mollifiers given in Definition 2.16. We note that this choice of test function ϕ is in V, and can hence be used in the variational formulation (34). First, we see that (u, ϕ) L2 () = l η ɛ (χu) 2 L 2 (). (37) Since convolution is self-adjoint, the product rule shows that jk ui ϕ i a dx = η ɛ (a jk l 1 (χu i ) ), k l η ɛ (χu i ), j dx x k x j l 2 ( l 1 ) η ɛ (( l 1 r a jk ) r (χu i ) ), k l η ɛ (χu i ), j dx r r=0 l η ɛ (a jk u i ) χ, k l η ɛ (χu i ), j dx l 1 η ɛ (a jk χ, j u i,k) l1 η ɛ (χu i ) dx. Using the commutator notation A, B f = A(Bf) B(Af), the first term on the right-hand side of the identity above can be rewritten as η ɛ (a jk l 1 (χu i ) ), k l η ɛ (χu i ), j dx = a jk η ɛ l 1 (χu i ), k l η ɛ (χu i ), j dx ηɛ,a jk l 1 (χu i ), k l η ɛ (χu i ), j dx = a jk l η ɛ (χu i ), k l η ɛ (χu i ), j dx ( a jk ) l 1 η ɛ (χu i ), k l η ɛ (χu i ), j dx ηɛ,a jk l 1 (χu i ), k l η ɛ (χu i ), j dx ; thus, after rearranging terms, the positivity condition (32) implies that λ l1 ( η ɛ (χu) ) 2 L 2 () jk ui ϕ i a dx ( a jk ) l 1 η ɛ (χu i ), k l η ɛ (χu i ), j dx x k x j ηɛ,a jk l 1 (χu i ), k l η ɛ (χu i ), j dx l 2 ( l 1 ) r r=0 η ɛ (( l 1 r a jk ) r (χu i ), k ) l η ɛ (χu i ), j dx l η ɛ (a jk u i χ, k ) l η ɛ (χu i ), j dx l 1 η ɛ (a jk χ, j u i,k) l1 η ɛ (χu i ) dx. The last five integrals on the right-hand side of the inequality above can be estimated using Hölder s inequality and the commutation estimate (28), and we obtain that λ l1( η ɛ (χu) ) 2 l1( η L 2 () ɛ (χu i ) ). (38) L2 () jk ui ϕ i a x k dx C a x C l () u H l () j
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 387 On the other hand, it is easy to see that f ϕ dx = l 1 η ɛ (χf i ) l1 η ɛ (χu i ) dx C f H l 1 () l1 ( η ɛ (χu) ) (39). L2 () Summing (37), (38) and (39), we find that l (η ɛ (χu) 2 λ l1( η L 2 () ɛ (χu) ) 2 L 2 () C f H l 1 () a C l () u l1 H () ( η l ɛ (χu) ) ; L 2 () therefore, by Young s inequality, l (η ɛ (χu) 2 λ L 2 () l1 ( η ɛ (χu) ) 2 L 2 () C f 2 H λ l 1 () a 2 C l () u 2 H l () λ l1 η ɛ (χu i ) 2 L 2 () which further implies that l1 ( η ɛ (χu) ) C L f 2 () λ H l 1 () a C l () u H l (). (40) Since f H k 1 () and a C k (), the assumption that u H l () implies that the right-hand side of (40) is bounded independent of the smoothing parameter ɛ. Therefore, we can pass to the limit as ɛ 0 in (40) and obtain that f H l 1 () a C l () u H l () l1 (χu) L 2 () C λ or χ l1 u L2 () C f λ H l 1 () ( a C l () λ) u H l (). (41) This implies that u H l1 loc (). So, we have shown that if u Hl () for 1 l k then, in fact, u H l1 loc () and u satisfies (41). Now, we note that by Theorem 3.2, u H 1 () and hence u Hloc 2 (). This allows us to integrate by parts in the variational formulation (34) and find that u The above identity implies that ( a jk u ) f x j x k u i ϕ dx =0 ϕ C c (). ( jk ) ui a = f i a.e. in. (42) x j x k Step 2: (Regularity for tangential derivatives of u in the w-direction near ) We now initiate the induction procedure. We assume that u H l () for 1 l k 1, and prove then that u H l1 (). Let {U m } K m=0, {ϑ m } K m=1, and{r m } K m=1 denote the system of local charts given in Proposition 2.1 with ε 1, and let 0 ζ m 1inCc (U m ) denote a partition-of-unity subordinate to the open covering U m as given in Definition 3.4. Wefixedm {1,...,K}, and work with the chart ϑ m : B(0,r m ) U m.on B(0,r m ), we define the new functions ζ = ζ m ϑ m, ũ = u ϑ m, w = w ɛ ϑ m, f = f ϑm, g = g ϑ m and ϕ = ϕ ϑ m. With A =( ϑ m ) 1, we define b rs =(a jk ϑ)a k sa j r. Then the matrix b is positive-define. In fact, since ϑ m Id L () 1, b rs ξ r ξ s =(a jk ϑ m )A s ka r jξ r ξ s λ A T ξ 2 λ 2 ξ 2. (43)
388 C. H. A. Cheng and S. Shkoller JMFM Setting x = ϑ m (y), the change-of-variables formula shows that the variational formulation (34) takes the form rs ũi ϕ i ũ ϕ dy b dy = f ϕ dy g ϕ ds ϕ y s y Ṽm, (44) r {y n=0} where Ṽm = { ϕ H 1 () } ϕ w =0on {y n =0}, ϕ =0onR n. With Δ 0 = n 1 2 α=1 denoting the horizontal Laplace operator, we define y 2 j ϕ i =( 1) l ζ w i Λ ɛ Δ l 0Λ ɛ ( ζũ w) ϑ 1 m, where Λ ɛ is the horizontal convolution-by-layers operator given by Λ ɛ φ(y h,y n )= ρ ɛ (y h z h )φ(z h,y n )dz h for φ(,y n ) L 1 (R n 1 ), R n 1 where y h =(y 1,...,y n 1 ). Recalling that = ( ),..., denotes the horizontal gradient, we note y 1 y n 1 that l v l w = l 1 v l1 w = n 1 α 1=1 n 1 α 1=1 n 1 α l =1 n 1 α l 1 =1 l v y α1 y αl l w y α1 y αl, l 1 v l 1 Δ 0 w, y α1 y αl 1 y α1 y αl 1 and so forth. Since ϕ w =0on, ϕ V and can be used as a test function. The use of ϕ as a test function in (34) implies that rs ũi ϕ i l (ũ, ϕ) L 2 () b dy C f y s y Hl () g H l 0.5 () Λɛ ( ζũ w) H1 (B r m). (45) Similar to (37), integrating by parts with respect to y h implies that (u, ϕ) L 2 () = l Λɛ ( ζũ w) 2 L 2 (B m). Now we focus on the second term on the left-hand side of (45). Integrating by parts in the horizontal direction (using ) yields B m rs ũi ϕ i b dy = y r y s = B m l 2 l Λ ɛ ( b rs ζũm,s w ) l Λ ɛ ( ζũ w), r dy Λ ɛ ( b rs l 1 ( ζũ w), s ) l Λɛ ( ζũ w), r dy ( l 1 ) ( l 1 k Λ ɛ b rs k ) l ( ζũ w), s Λɛ ( ζũ w), r dy k k=0 Bm l ( Λ ɛ b rsũ i ( ζ w i ) l ), s Λɛ ( ζũ w), r dy. (46) B m
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 389 For the first term on the right-hand side of (46), as in Step 1, we find that ( Λ ɛ b rs l 1 ) l ( ζũ w), s Λɛ ( ζũ w), r dy = b rs Λ ɛ l 1 l ( ζũ w), s Λɛ ( ζũ w), r dy ( Λɛ,b rs l 1 ) l ( ζũ w), s Λɛ ( ζũ w), r dy = b rs l Λ ɛ ( ζũ w), s l Λ ɛ ( ζũ w), r dy ( b rs )Λ ɛ l 1 ( ζũ w), s l Λ ɛ ( ζũ w), r dy ( Λɛ,b rs l 1 ) l ( ζũ w), s Λɛ ( ζũ w), r dy. (47) B m The positivity condition (43) and the commutation estimate (29) imply that ( Λ ɛ b rs l 1 ) l ( ζũ w), s Λɛ ( ζũ w), r dy λ l Λɛ ( ζũ w) 2 2 C u l L 2 () H () l Λɛ ( ζũ w) u H 1 () H l (). (48) For the remaining terms on the right-hand side of (46), we apply Hölder s inequality and find that l 2 ( l 1 k k=0 ) Bm B m ( l 1 k Λ ɛ b rs k ) l ( ζũ w), s Λɛ ( ζũ w), r dy l ( Λ ɛ b rsũ i ( ζ w i ) l ), s Λɛ ( ζũ w), r dy l C u Hl () Λɛ ( ζũ w) u L 2 () H l (), (49) where C depends on a C l (), w C l1 (U) and C (). As a consequence, Young s inequality together l1 with (45) (48) implies that l Λɛ ( ζũ w) 2 L 2 (B m) λ l Λɛ ( ζũ w) 2 L 2 (B m) C δ u 2 H l () f 2 H l 1 () g 2 H l 1.5 () δ l Λɛ ( ζũ w) 2 L 2 (B m), which, by choosing δ>0 sufficiently small, shows that l Λɛ ( ζũ w) C H f 1 () H l 1 () g H l 0.5 () u Hl () for a constant C = C( a C l (), w C (U), l1 C l1). Since the estimate above is independent of the smoothing parameter ɛ, by passing to the limit as ɛ 0 we conclude that l ζ (ũ w) H1 () C u Hl () f H l 1 () g H l 0.5 (). Since both w and ϑ m are C k1 in the support of ζ, it follows that lũ ζ w H C f 1 () H l 1 () g H l 0.5 () u Hl (). (50)
390 C. H. A. Cheng and S. Shkoller JMFM Step 3: (Regularity for tangential derivatives of u in the w -directions near ) Estimate (50) provides regularity for the vector ζ l ũ w. Next, we establish the regularity of ζ l ũ w. We define ϕ i =( 1) l ζλɛ Δ l 0Λ ɛ ( ζũ i ) ( ζ w Λɛ Δ l 0Λ ɛ ( ζũ) ) w i w 2 =( 1) l ζλɛ Δ l 0Λ ɛ ( ζũ i ) ( ζλɛ Δ l 0Λ ɛ ( ζũ j ) ) w j w i w 2. Note that ϕ is the projection of the vector ζλ ɛ Δ l 0Λ ɛ ( ζũ) onto the affine space with normal w, soϕ V and can be used as a test function in (44). Following the similar computation in Step 2 above, we have that l Λɛ ( ζũ) 2 λ l L 2 () Λɛ ( ζũ) 2 4 L 2 () C δ f 2 H l 1 () g 2 H l 1.5 () u 2 H l () ( 1) l1 b rs ũ i, s ( ζλɛ Δ l 0Λ ɛ ( ζũ j m) ) w j w i w,r 2 dy C δ f 2 H l 1 () g 2 H l 1.5 () u 2 H l () ( 1) l1 b rs w ζ ( w 2 ũ, s Λɛ Δ l 0Λ ɛ ( ζũ j, r ) ) w j dy C δ f 2 H l 1 () g 2 H l 1.5 () u 2 H l () b rs w 2 l ( ζũ m,s w) ( Λ ɛ l Λ ɛ ( ζũ j, r ) ) w j dy. B m Applying estimate (50) and Young s inequality, b rs w 2 l ( ζũ m,s w) ( Λ ɛ l Λ ɛ ( ζũ j m,r) ) w m j dy C l ( ζ w ũ) L2 l Λ () ɛ ( ζ ũ) L2 () C δ f 2 H l 1 () g 2 H l 1.5 () u 2 H l () thus by choosing δ>0 sufficiently small, we conclude that δ l Λɛ ( ζũ) 2 L 2 (B m) 2δ l Λɛ ( ζũ) 2 L 2 (B m) 3δ l Λɛ ( ζũ) 2 L 2 (B m) δ l Λɛ ( ζ ũ) 2 L 2 () ; l Λ ɛ ( ζũ) u 2 C 2 H 1 () H l () f 2 H l 1 () g 2 H l 1.5 () Again, due to the ɛ-independence of the right-hand side, we conclude that ζ l ũ H 1 (B u m) C Hl () f H l 1 () g H l 0.5 () for a constant C = C( a C l (), w C l1 (U), C l1). Step 4: (Regularity for normal derivatives of u near ) Multiplying (42) byζ m and then composing with ϑ m, by the Piola identity (30) we obtain that ζũ ζ ( b rs ) ũ, s,r = ζ(f ϑ m ) a.e. in. Letting l 1 j j act on the equation above, we find that ζb rs l 1 j j ũ, rs = F (l,j) a.e. in (52) for some F (l,j) L 2 () satisfying F (l,j) L 2 () C f H l 1 () u Hl (),. (51)
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 391 where the constant C depends on a C l (). Using (43), bnn > 0; thus (52) further implies that ζ l 1 j j ũ, nn = 1 b nn F (l,j) ζ b rs l 1 j ũ, rs. (r,s) (n,n) Now we argue by induction on 0 j l 1. When j =0,(51) shows that l 1ũ,nn ζ L2 C u () Hl () f H l 1 () g H l 0.5 () which, combined with (51), provides the estimate l 1 ζ 2ũ C L2 u () H l () f H l 1 () g H l 0.5 (). Repeating this process for j =1,...,l 1, we conclude that ζ l1 ũ L 2 (B m) C u Hl () f H l 1 () g H l 0.5 (). (53) The combination of (41) and (53), as well as the induction process, proves the theorem. 3.2. The Case that the Coefficients a jk are of Sobolev Class We are now in position to study the regularity of solution u to (31) when the coefficient a jk and the domain is of Sobolev class. We first prove the following rather technical Theorem 3.6 (Regularity for the case that a jk H k () and C ). Let R n be a bounded C -domain. Suppose that for an integer k > n 2 and 1 l k, ajk H k () satisfies the positivity condition a jk ξ j ξ k λ ξ 2 ξ R n, and w H max{k,l1} () (or w H max{k 1 2,l 1 2 } ()) such that w vanishes nowhere on. Then for all f H l 1 () and g H l 0.5 (), the weak solution u to (31) belongs to H l1 (), and satisfies u H l1 () C f H l 1 () g H l 0.5 () P ( a Hk ()) ( ) f L 2 () g H 0.5 () (54) for a constant C = C ( w H max{k,l1} ()) and a polynomial P. Proof. Let {U m,ϑ m } K m=1 be a collection of charts of given in Proposition 2.1, {ζ m } K m=1 a partitionof-unity subordinate to {U m } K m=1 given in Definition 3.4, and let E : H k1 () H k1 (R n ) denote a Sobolev extension operator. We define a ɛ = η ɛ (Ea), f ɛ = η ɛ (Ef), w ɛ = η ɛ (Ew). Finally, let g ɛ denote a smooth regularization of g defined by K ( ) g ɛ = ζm Λɛ ( ζm g) ϑ m ϑ 1 m. m=1 It follows that for ɛ 1 sufficiently small, a jk ɛ (x)ξ j ξ k λ 2 ξ 2 ξ R n,x. (55) Hence by Theorem 3.5, the solution u ɛ to the variational problem u ɛ ϕ dx a jk u ε ɛ ϕ dx = f x k x ɛ ϕ dx g ɛ ϕ ds ϕ V j satisfies u ɛ H k () for all k 1. We next establish an ɛ-independent upper bound for u ɛ H (). l1 Step 1: (Regularity for tangential derivatives of u in the w-direction near ) We fix m {1,...,K} and set ζ = ζ m ϑ m, ũ = u ɛ ϑ m, w = w ɛ ϑ m, f = fɛ ϑ m, g = g ɛ ϑ m and ϕ = ϕ ϑ m.
392 C. H. A. Cheng and S. Shkoller JMFM With A =( ϑ) 1, we define b rs ɛ positive-definite since using (55), =(a jk ɛ ϑ)a s k Ar j. Then, as ϑ Id L (B m) 1, the matrix b ɛ is b rs ɛ ξ r ξ s =(a jk ɛ ϑ)a s ka r jξ r ξ s λ 2 AT ξ 2 λ 8 ξ 2 ξ R n. (56) Setting x = ϑ m (y), and using the change-of-variables formula, we find that the variational formulation (34) can be written as ũ ϕ dy b rs ũ i ϕ i ɛ dy = f ϕ dy g ϕ ds ϕ y s y Ṽm, (57) r {y n=0} where Ṽm = { ϕ H 1 () } ϕ w =0on {y n =0}, ϕ =0onR n. With Δ0 denoting the horizontal Laplace operator and denoting the horizontal gradient defined in Step 2 in the proof of Theorem 3.5, we define ϕ i =( 1) l ζ w i Δ l 0( ζũ w), so that (ũ, ϕ) L2 () b rs ũ i ϕ i ɛ dy y s y r C C B m f H l 1 () g H l 0.5 ( {y n=0}) l ( ζũ w) H1 () l f H l 1 () g H l 0.5 () ( ζũ w) H 1 (), (58) where we have used Young s inequality for convolution to conclude the last inequality. We focus now on the left-hand side of (58). As in the proof of Theorem 3.5, we have that (ũ, ϕ) L 2 () = l ( ζũ w) 2. L 2 () Moreover, b rs ũ i ϕ i ɛ dy =( 1) l y s y r = B m B m b rs ɛ ũi, s ζ w i Δ l 0( ζũ w), r dy l b rs ɛ ( ζũ w), s l ( ζũ w),r dy B m B m l b rs ɛ ũi ( ζ w i l ), s ( ζũ w),r dy l 1 b rs ɛ ũi, s ( ζ w i ), r l1 ( ζũ w) dy. (59) For the first term on the right-hand side of (59), we make use of the positivity condition (56) and Young s inequality to conclude that l b rs l ɛ ( ζũ w), s ( ζũ w),r dy = b rs ɛ l ( ζũ w), s l l ( ζũ w), r dy,b rs l ɛ ( ζũ w), s ( ζũ w),r dy ( λ 8 δ) l ( ζũ w) 2 L 2 () C l δ,bɛ ( ζũ w) 2 L 2 (). (60) Then, Corollary 2.9 with σ = 1 shows that 8 l b rs l ɛ ( ζũ w), s ( ζũ w),r dy ( λ 8 δ) l ( ζũ w) 2 C L 2 () δ a 2 H k () uɛ 2. H l 7 8 () B m
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 393 For the second and the third terms on the right-hand side of (59), we use the inequality (16), and find that l b rs ɛ ũi ( ζ w i l ), s ( ζũ w),r dy l 1 b rs ɛ ũi, s ( ζ w i l1 ), r ( ζũ w) dy B m B m C δ a 2 H k () uɛ 2 H l () δ l ( ζũi w i ) 2 L 2 (B m) for a constant C δ depending on w H max{k,l1} (). Choosing δ>0 sufficiently small in (60) and (61), we conclude that l ( ζũ w) L2 l ( ζũ w) () L2 () (62) C f H l 1 () g H l 0.5 () a Hk () u ɛ H l 7 8 () for a constant C = C ( w H max{k,l1} ()). Step 2: (Regularity for tangential derivatives of u in the w -directions near ) Now we estimate u ɛ in the directions perpendicular to w. Similar to Step 3 in the proof of Theorem 3.5, weuse ϕ i = ζδ l 0( ζũ) ( ζ w Δ l 0 ( ζũ) ) w w 2 as a test function in (57) and find that l ( ζũ) 2 L 2 () ( λ 8 δ) l ( ζũ) 2 L 2 () C f 2 H l 1 () g 2 H l 0.5 () C δ a 2 H k () uɛ 2 H l 7 8 () ( 1) l w j w 2 ( ζũ w)δ l 0( ζũ j ) dy ( 1) l1 b rs ɛ ũi l, s ζδ 0 ( ζũ j ) wj w i, w 2 r dy. Integrating by parts in the horizontal direction (using ), by Corollary 2.9 and (62) we obtain that ( 1) l w j w 2 ( ζũ w) 2l ( ζũ j ) dy l 1( w j w ( ζũ L l1 w)) ( ζũ j ) 2 2 2 (B L 2 () m) (61) C δ u ɛ 2 H l 1 () δ l ( ζũ j ) 2 L 2 (B m) (63) and ( 1) l1 B m b rs ɛ ũi, s ζ 2l ( ζũ j ) wj w i, w 2 r dy C l (b ɛ ũ) L 2 (B m) b ɛ l ( ζũ w) L2 (B m) l,bɛ ( ζũ w) L2 (B m l ( ζũ) L 2 (B m) C δ a 2 H k () uɛ 2 H l 7 8 () δ l ( ζũ) 2 L 2 (B m), (64) in which the constant C δ also depends on w H max{k,l1} (). Therefore, choosing δ>0 sufficiently small in (63) and (64), we conclude that
394 C. H. A. Cheng and S. Shkoller JMFM lũi ζ l L2 ζ () ũ i L2 () C u ɛ H l 1 () f H l 1 () g H l 0.5 () a Hk () u ɛ H l 7 8 () for a constant C = C ( w H max{k,l1} ()). Step 3: (Regularity for normal derivatives of u near ) In this step, we follow the procedure of Step 4 in the proof of Theorem 3.5. Since u ɛ is a strong solution, it follows that u ɛ ( a jk u ɛ ) ɛ = f x j x ɛ in ; k thus the Piola identity (30) implies that ζ ( b rs ɛ ũ, ) s,r = ζ ( ũ (f ɛ ϑ) ) in. With ũ, n and ũ, nn denoting ũ y n ζb nn ɛ ũ, nn = ζ ũ (f ɛ ϑ) b nn and 2 ũ, respectively, we have that yn 2 ɛ,nũ, n (r,s) (n,n) Let G = ζ ũ (f ɛ ϑ) b nn ɛ,nũ, n (r,s) (n,n) brs ɛ,rũ, s b rs ɛ,rũ, s (r,s) (n,n) b rs ɛ ũ, sr,andfor0 j l 1 we define G (l,j) = l 1 j j G l 1 j j,b rs ɛ ũ, sr. Letting l 1 j j act on (66), we obtain that ζb nn ɛ l 1 j j ũ, nn = G (l,j) (r,s) (n,n) (65) in B m. (66) ζb rs ɛ l 1 j j ũ, rs (67) Now we estimate G (l,j) in L 2 (). First we note that l 1 j j ζ(ũ fɛ ϑ) C L u ɛ 2 () H l 1 () f H l 1 (). Moreover, since l k, by Proposition 2.7 with σ = 1 we find that 8 l 1 j j ( ζb nn,n ũ, n ) l 1 j L 2 () j (b rs ɛ,rũ, s ) L 2 () (r,s) (n,n) l 1 C j1 a l 1 j u ɛ L2 () = C j=0 C a Hk () u H l 7 8 (). Finally, by Corollary 2.9 with σ = 1 8, l 1 j j nn, ζb ɛ ũ, L2 nn () C ɛ a H k () u ɛ H l 7 8 (). (r,s) (n,n) l j a l j u ɛ L2 () j=1 l 1 j j rs, ζb ɛ ũ, L2 rs () Therefore, G (l,j) satisfies G (l,j) L2 () C u H l 1 () f H l 1 () a H () u ɛ k H l 7. 8 ()
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 395 Now we argue by induction on 0 j l 1. By the positivity condition (55), b nn ɛ λ so that when 4 j = 0, the inequalities (65) and (67) show that l 1ũ,nn ζ L 2 () G (l,j) L 2 () b rs ɛ l 1ũ,rs ζ L () L 2 () (r,s) (n,n) C u H l 1 () f H l 1 () g H l 0.5 () a H k () uɛ l 7 H 8 () which, combined with (65), provides the estimate l 1 ζ 2ũ L2 () C u H l 1 () f H l 1 () g H l 0.5 () a Hk () u ɛ H l 7 8 () Repeating this process for j =1,...,l 1, we conclude that ζ lũ i L2 ζ l1ũ i () L2 () (68) C u ɛ H l 1 () f H l 1 () g H l 0.5 () a Hk () u ɛ H l 7 8 () for a constant C = C ( w H max{k,l1} ()). Step 4: (Completing the regularity theory) Letχ 0 be a smooth cut-off function so that spt(χ). Arguing as in Step 1 of the proof of Theorem 3.5, we find that χ l u ɛ L2 () χ l1 u ɛ L2 () C f H l 1 () a H () u ɛ k H l 7. (69) 8 (). Combining (68) and (69) establishes the inequality u ɛ H l1 () C f H l 1 () g H l 0.5 () ( 1 a H ()) u ɛ k H l 7 8 () for a constant C = C ( w H ()). Since the interpolation inequality provides max{k,l1} (70) u ɛ H l 7 8 () C uɛ 1 1 8l H l1 () uɛ 1 8l H 1 (), Young s inequality further shows that u ɛ H l1 () C δ f H l 1 () g H l 0.5 () P ( a Hk ()) u ɛ H 1 () δ u ɛ H l1 () (71) for a polynomial function P. Finally, choosing δ>0sufficiently small and then passing to the limit as ɛ 0, by the fact that a jk ɛ a jk in H k (), w ɛ w in H max{k,l1} (), f ɛ f in H l 1 (), g ɛ g in H l 0.5 (), we find that u ɛ converges to the unique weak solution u to (31), and the inequality (36) is established by substitution of the H 1 -estimate (35) in the inequality (71). Having established the regularity theory for the case that a jk H k () and C, we can now prove the following Corollary 3.7 (Regularity for the case that a jk H k () and H k1 ). Let R n be a bounded H k1 -domain for an integer k > n 2. Suppose that ajk H k () satisfies the positivity condition a jk ξ j ξ k λ ξ 2 ξ R n,
396 C. H. A. Cheng and S. Shkoller JMFM and for 1 l k, w H max{k,l1} () (or w H max{k 1 2,l 1 2 } ()) such that w vanishes nowhere on. Then for all f H l 1 () and g H l 0.5 (), the weak solution u to (31) belongs to H l1 (), and satisfies u H l1 () C f H l 1 () g H l 0.5 () P ( a H ()) ( ) (72) f k L2 () g H 0.5 () for a constant C = C ( w H (), max{k,l1} H k0.5) and a polynomial P. Proof. Using Definition 2.2, weletψ : O beanh k1 -diffeomorphism, where O is a bounded C -domain. Making the change-of-variables x = ψ(y), with A denoting ( ψ) 1 we can rewrite (31) as ū ( ā jk A r y ja s ū ) k = r y f ā jk A s A r j ū k in O, s y r y s ū w =0 on O, ( P w ā jk A r ja s ū ) k Nr ḡ = 0 on O, y s where we use the bar notation to denote the variable defined on O through the composition with ψ: ā = a ψ, ū = u ψ, w = w ψ, f = f ψ, ḡ = g ψ, and N is the outward-pointing unit normal to O. By Proposition 2.7, Corollaries 2.10, and2.11, we find that ā jk A s ka r j H k (O) C( H k0.5) a H (), k w H max{k,l1} () C( H k0.5) w H (), max{k,l1} f H l 1 (O) ḡ H l 0.5 ( O) C( H k0.5) f H l 1 () g H l 0.5 (). Theorem 3.6 then implies that ū H l1 (O) C f H l 1 (O) ḡ H l 0.5 ( O) P ( AāA T H (O)) ( f k L2 (O) ḡ H 0.5 ( O) C f H l 1 (O) g H l 0.5 () P ( a Hk (), H k0.5) ( f L 2 () g H 0.5 () for a constant C = C ( w H max{k,l1} (), H k0.5). Estimate (72) then follows from Corollary 2.11. Corollary 3.8 (Regularity for the classical Dirichlet and Neumann problems). Let R n be a bounded H k1 -domain for an integer k > n 2,andajk H k () satisfies the positivity condition a jk ξ j ξ k λ ξ 2 ξ R n. Let l be an integer such that 1 l k. Then 1. For any f H l 1 (), the weak solution u H0 1 () to the Dirichlet problem ( jk u ) a = f in, x j x k u =0 on, belongs to H l1 (), and satisfies u H l1 () C f H l 1 () (73) for a constant C = C ( a Hk (), H k0.5). ) )
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 397 2. For any f H l 1 () and g H l 0.5 (), the weak solution v H 1 () to the Neumann problem v ( jk v ) a = f x j x k in, jk u a N j = g x k on, belongs to H l1 (), and satisfies v H l1 () C f H l 1 () g H l 0.5 () (74) for a constant C = C ( a H k (), H k0.5). Proof. It suffices to prove the case that u and v are both scalar functions. 1. Let w =(1, 0,...,0), and u be the solution to u ( a jk u ) =(f u, 0,...,0) in, x j x k u w =0 on, ( P w a jk u ) N j = 0 on. x k Then u = u 1 (in fact, u =(u, 0,...,0)); thus (72) implies that u H l1 () C f u H l 1 () C f H l 1 () u H l 1 () for a constant C = C ( a H k (), H k0.5). By interpolation and Young s inequality, u H l1 () C f H l 1 () C δ u H 1 () δ u H l1 () ; thus (73) follows from choosing δ>0 sufficiently small and the estimate for the weak solution. 2. Let w =(0, 1, 0,...,0), and v be the solution to v ( a jk v ) =(0,f,0,...,0) in, x j x k v w =0 on, ( P w a jk v ) N j =(0,g,0,...,0) on. x k Then v = v 2 (in fact, v =(0,v,0,...,0)); thus (74) follows from (72). In general, elliptic estimates with Sobolev class coefficients a jk have a nonlinear dependence on the Sobolev norm of a jk. There are, however, situations when the estimate becomes linear with respect to the Sobolev norm of a jk. Theorem 3.9 (Regularity estimate which is linear in the coefficient matrix a jk ). Suppose that the assumptions of Theorem 3.6 are satisfied with l =k, and that furthermore a Id L () ɛ 1. Then the solution u H k1 () to (31) satisfies u H k1 () C f H k 1 () g H k 0.5 () ( 1 a H ()) k u L () (75) for a constant C = C ( w H k1 ()). (Recall that w is an H k1 () vector field defined in a neighborhood of which vanishes nowhere on.)
398 C. H. A. Cheng and S. Shkoller JMFM Proof. By Theorem 3.6 we know that u H k1 () so Eq. (31) holds in the pointwise sense. We rewrite (31) as u Δu = f ( (a jk δ jk) u ) f x j x k in, u w =0 on, P w P w N j g k ( u ) = g N ( (δ jk a jk) u x ) on. We then conclude from Theorem 3.6 that u H k1 () C f H k 1 () g H k 0.5 () C f H k 1 () g H k 0.5 () ( (δ jk a jk) u ) H k 1 x j x k () Pw ( (δ jk a jk) u x k N j ) H k 0.5 () for a constant C = C ( w H ()). By Theorem 2.14, k1 ( (δ jk a jk) u ) H x j x k k 1 () C ( δ jk a jk) u x k H k () δ a L () u H k () δ a H k () u L () Cɛ u H k1 () C ( 1 a Hk ()) u L (). Similarly, by the trace estimate (and the fact that k 0.5 > n 1, where (n 1) is the dimension of ), 2 ( (δ P w jk a jk) u ) N j H C ( δ jk a jk) u H N j x k k 0.5 () x k k 0.5 () C δ a L () u H k 0.5 () δ a H k 0.5 () u L () Cɛ u H k1 () C ( 1 a Hk ()) u L () for a constant C = C ( ) n w H k1 (). The embedding H 2 δ () C 0,α () for some α>0 further suggests that u is uniformly Hölder continuous; thus u L () u L (). (75) then follows from the assumption that ɛ 1. Remark 3.10. As we noted, inequality (75) is linear with respect to the highest-order norms. This permits the use of linear interpolation to extend this inequality to fractional-order Sobolev spaces. In the same way that we proved Theorem 3.6, we can prove the following complimentary result: Theorem 3.11. Let R n be a bounded H k1 -domain for an integer k > n 2. Suppose that ajk H k () satisfies the positivity condition a jk ξ j ξ k λ ξ 2 ξ R n, and for 1 l k, w H max{k,l1} () (or w H max{k 1 2,l 1 2 } ()) such that w vanishes nowhere on. Then for all f H l 1 () and g H l 0.5 (), there exists a solution u to u i ( jk ) ui a = f i x j x k in, (76a) u w = 0 on, (76b) a jk ui x k N j w i = g on, (76c)
Vol. 19 (2017) Regularity Theory for Elliptic Equations of Hodge-Type 399 and satisfies u H l1 () C f H l 1 () g H l 0.5 () P ( a H ()) ( ) f k L2 () g H 0.5 () (77) for a constant C = C ( w H max{k,l1} (), H k0.5) and a polynomial P. 4. Regularity Theory: The Proof of Theorem 1.3 In this section, we prove our main regularity result given by Theorem 1.3. We first establish the following lemma which is also fundamental to the proof of Theorem 1.1. Lemma 4.1. Let R 3 be a bounded H k1 -domain with outward-pointing unit normal N. Then for every differentiable vector field w : R 3, the following identities hold: ( w ) P N = (curlw N) w N N on, (78a) divw = w N N 2H(w N) div (P N w) on, (78b) curlw N =div (w N) on, (78c) where H is the mean curvature of (in local chart (U,ϑ), H is given by H= 1 2 gαβ b αβ ). Proof. We define Θ(y) =ϑ(y 1,y 2, 0) y 3 (N ϑ)(y 1,y 2, 0), and G ij =Θ, i Θ, j with inverse G ij.letñ (N ϑ),and f y3=0 f Θiff N. Since Θ, 1,Θ, 2 Ñ, for every vector v R 3, ṽ can be expressed as the linear combination of Θ, 1,Θ, 2 and Ñ. In particular, we have ṽ i =(ṽ Ñ)Ñi (G αβ Θ j, β ṽ j )Θ i, α ṽ 3 Ñ i ṽ α Θ i, α (79a) and f, k Θ= f, 3 Ñ 3 G αβ f,β Θ k, α. (79b) To see (78a), we first note that w i N ϑ = w 3 Ñ i w α Θ i, α,3 = w 3,3 Ñ y3=0 i w α,3 Θ i, α w α Ñ i, α y3=0 ; (80) thus, since Ñ Θ, α = Ñ Ñ, α =0, ( w ) P N ϑ = w 3,3 Ñ i w α,3 Θ i, α w α Ñ i, α y3=0 N Moreover, by the identity w 3,3 Ñ k w α,3 Θ k, α w α Ñ k, α y3=0 Ñ k Ñ i = w α,3 ϑ i, α w α Ñ i, α. (81) (curlw N) i = ε ijk ε jrs w s, r N k =(δ is δ kr δ ir δ ks )w s, r N k =(w i, k w k, i )N k,