The Gauss-Green Formula (And Elliptic Boundary Problems On Rough Domains) Joint Work with Steve Hofmann and Marius Mitrea
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1 The Gauss-Green Formula And Elliptic Boundary Problems On Rough Domains) Joint Work with Steve Hofmann and Marius Mitrea Dirichlet Problem on open in a compact Riemannian manifold M), dimension n: ) Lu = 0 on, u = f on. L = V, V L M), V 0, V > 0 on a set of positive measure, on each connected component of M \. Assume is rough. Examples: C, Lipschitz, bmo, vmo, vanishing chord-arc Semmes-Kenig-Toro domains ). Layer potential method: Look for solution as 2) u = Dg, where D is double layer potential: 3) Dgx) = νy Ex, y)gy) dσy), Ex, y) integral kernel for L : H M) H M). 4) gx) /2 Ex, y) = e 0 x, x y) + e x, y), in local coordinates, where ) n 2)/2, 5) e 0 x, z) = C n gjk x)z j z k e x, y) somewhat tamer. Layer Potential Estimates for various classes of. 6) K g L p ) C g L p ), < p <.
2 2 K gx) = sup K ε gx), 0<ε< K ε gx) = νy Ex, y)gy) dσy). y, x y <ε Nontangential maximal estimate: 7) N Dg L p ) C g L p ), < p <. Boundary behavior, ) 8) lim Dgy) = y x 2 I + K gx), a.e. on, as y x nontangentially, from within, where 9) Kgx) = lim ε 0 K ε gx). Validity of these estimates and limits Elementary if is smoother than C. C case: Fabes-Jodeit-Riviere following Calderon) Lipschitz case: 6) 7) Coifman-McIntosh-Meyer, 8) Verchota [MT] for variable coefficients) More general case: Uniformly rectifiable boundary UR domain). One assumes is Ahlfors regular, i.e., p, C r n H n B r p) ) C 2 r n, and one assumes contains big pieces of Lipschitz surfaces uniformly, on all scales). For UR domains, 6) is due to G. David, 7) 8) to [HMT]. Solving Dirichlet Problem done by inverting 2I + K. To show: 0) Fredholm of index zero on L p ), ) Adjoint injective on L p ). Done by Fabes-Jodeit-Riviere for < p <, for C domains. Done by Verchota for 2 ε < p <, for Lipschitz domains. [MT] for variable coefficients) Done by [HMT] for < p <, for regular Semmes-Kenig-Toro domains.
3 3 Definition. is a regular SKT domain vanishing chord-arc domain) provided it is Ahlfors regular, δ-reifenberg flat for small δ, and 2) lim r 0 lim r 0 inf H n B r p) ) p ω n r n = sup H n B r p) ) p ω n r n =. Example. Any VMO domain. Two ingredients to get 0) ). Theorem [HMT]). If is a regular SKT domain, 3) K : L p ) L p ) is compact, p, ). Theorem 2 [HMT]). If is a UR domain, 4) 2 I + K : L p ) L p ) is injective, p 2. Corollary. If is a regular SKT domain, 5) 2 I + K : Lp ) L p ) is bijective, p, ). Proof. We have 3) and 4) 5) for p, 2] 2 I + K injective on Lp ) for p, ) 5) again invoking 3)). Proof of Theorem 2. Assume f L 2 ) in Ker 2 I + K, and set 6) u = Sfx) = Ex, y)fy) dσy), single layer potential. Counterparts to 7) 8): 7) N Sf L p ) C f L p ),
4 4 8) ν Sf ± = 2 I + K ) f, + =, = M \. Take v = u u, so 9) div v = u 2 + u u = u 2 + V u 2 on. Apply Gauss-Green formula 20) div v dv = ν v dσ, with replaced by, to get 2) u 2 + V u 2) dv = u 2 I + K ) f dσ = 0, hence u 0 on, hence u 0 on. Hence u = 0 on, so by a second application of 20) on ), 22) u 2 + V u 2) dv = u 2 I + K ) f dσ = 0. So u = const on. Also u, given by 6), does not jump across, so u 0 on M. Then 8) gives, for both choices of sign, 23) ± 2 I + K ) f = 0, hence f = 0. The proof is done, modulo: Need to justify applying the Gauss-Green formula 20). What is known: 24) f L 2 ) N u L 2 ) N v L p ), some p >, whenever is a UR domain, by 7), and as shown in [HMT], 25) f L 2 ) u 2 L q ) div v L q ), some q >, whenever is Ahlfors regular. Also u and v are continuous on the interior regions and.
5 5 Applicable Gauss-Green Theorem For p [, ), set 26) L p = {v C, T M) : N v L p ), and nontangential limit v b, σ-a.e.}. Theorem 3 [HMT]). Assume is Ahlfors regular and 27) H n \ ) = 0. Assume that, for some p >, 28) v L p, and div v L ). Then 29) div v dx = ν v b dσ. The proof of Theorem 3 uses the results of De Giorgi and Federer that 29) holds for v Lip). See below.) In addition, it looks at χ δ v when χ δ is a family of cutoffs, and it uses the following results: 30) where v dx C N v ), v L, δ L O δ 3) O δ = {x : distx, ) δ}, and, for p, ), 32) If v L p, w L, w b = v b, and w k Lip), N w w k ) L ) 0. The proof of 30) uses a covering argument. The proof of 32) uses an explicit operator resembling a Poisson kernel, acting on v b. Hardy-Littlewood maximal function estimates make an appearance which is partly why we need p > ). Remark. For Lipschitz, 29) is due to Verchota.
6 6 Finite Perimeter Domains. DeGiorgi-Federer Results Definition. Open R n has locally finite perimeter provided 33) χ = µ, a locally finite R n -valued measure. Radon-Nikodym 34) µ = ν σ, σ locally finite positive measure, ν L, σ), νx) =, σ-a.e. Besicovitch 35) lim r 0 σb r x)) B r x) ν dσ = νx), for σ-a.e. x. Distribution theory given v C 0 R n, R n ) vector field), 36) div v, χ = v, χ, so 33) 34) equivalent to 37) DeGiorgi-Federer results: div v dx = ν v dσ. 38) σ = H n, where reduced boundary) consists of x where 35) holds, with νx) =. Also is countably rectifiable: 39) = k M k N, M k compact subset of C hypersurface, H n N) = 0. Measure-theoretic boundary : 40) x lim sup r 0 r n L n B r x) ± ) > 0, where + =, = R n \. Federer proved that and 42) H n \ ) = 0,
7 7 so 37) can be written 43) div v dx = ν v dh n. So far, we have 37) and 43) for v C 0 R n, R n ). Want such identities for more general v. Easy extension to v C 0R n, R n ), even to Lipschitz v. Better extension still easy). If has locally finite perimeter, then 37) holds for v in 44) D = {v C 0 0R n, R n ) : div v L R n )}. Proof. Apply 37) to mollifications v k = ϕ k v C 0 R n, R n ) and pass to the limit. We have div v k div v in L R n ) and ν v k ν v uniformly on. Drawback. Want to treat functions defined on, not on a neighborhood of. Still better extension though not implying Theorem 3) We say has a tame interior approximation { k : k N} provided open k k k+ with 45) χ k TVBR ) CR) <, k. For such, 37) holds for v in 46) D = {v C 0 0, R n ) : div v L )}. Proof. Use previous extension to get 47) k div v dx = v, χ k, and examine limit as k, using bounds 45). Result of Federer 952). If is bounded and H n ) <, then 37) holds provided v C) and each term j v j in div v belongs to L ). Examples of locally finite perimeter sets. Assume 48) A CR n ), A L locr n ).
8 8 With x = x, x n ), set 49) = {x R n : x n > Ax )}. Proposition A. Such has locally finite perimeter. Given this, we can consider { 50) x R n : x n > Ax ) + } k and deduce: Corollary B. Such has a tame interior approximation. Proof of Prop. A. Use a mollifier to produce A k C R n ) such that 5) A k A in CR n ), A k A in L locr n ). Set 52) k = {x R n : x n > A k x )}. Then 53) χ k χ in L locr n ), so χ k χ in D R n ). For k, Gauss-Green formula is elementary: 54) χ k = ν k σ k, σ k surface area on graph of A k, given in x -coordinates by 55) dσ k x ) = + A k x ) 2 dx. Hypothesis 48) {ν k σ k : k N} bounded set of R n -valued measures, on each set B R = {x R n : x R}. So passing to limit in 53) gives 55) χ = µ, locally finite R n -valued measure. Hence 37) holds, i.e., for v C0 R n, R n ), 56) div v dx = ν v dσ.
9 9 Going further, the Gauss-Green formula for k gives 57) div v dx = A k x ), ) vx, A k x )) dx, k R n valid for v C 0 R n, R n ). As k, 58) Hence 59) with 60) LHS 57) LHS 56) RHS 57) Ax ), ) vx, Ax )) dx. R n ν v dσ = R n νx ) vx, Ax )) dσx ), νx ) = Ax ), ) + Ax ) 2, dσx ) = + Ax ) 2 dx. Remark. If A is Lipschitz, one easily has =. More generally: Proposition C. For given by 48) 49), 6) H n \ ) = 0. Proof. Uses results of Tompson 954) and Federer 960). Given K R n, set 62) Σ K = {x, Ax )) : x K}. Tompson proved the first identity in I n Σ K ) = 63) K = σσ K ), + Ax ) 2 dx the second identity holding by 59) 60). Federer proved 64) H n Σ K ) = I n Σ K ). So 65) H n Σ K ) = σσ K ) = H n Σ K ), the last identity by 38). This gives 6).
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