THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS

Transcription

1 MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages S XX THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS GERHARD DZIUK AND JOHN E. HUTCHINSON Abstract. We solve te problem of finding and justifying an optimal fully discrete finite element procedure for approximating minimal, including unstable, surfaces. In a previous paper we introduced te general framework and some preliminary estimates, developed te algoritm and give te numerical results. In tis paper we prove te convergence estimate. 1. Introduction We recall from [DH4] tat a minimal surface or solution of te Plateau Problem can be caracterised in a number of different ways. For our purposes it is convenient to begin wit te following formulation, wic we restate more precisely later, c.f. 3. Let D be te unit disc in IR 2 and Γ a smoot Jordan curve in IR n. Let F be te class of armonic maps u : D IR n suc tat u D : D Γ is monotone and satisfies a certain integral tree-point condition, c.f. 1. Te function u F is said to be a minimal surface if u is stationary in F for te Diriclet energy Du = 1 2 D u 2. Suc maps u provide an armonic conformal parametrisation of te corresponding minimal surface. Let D be a quasi-uniform triangulation of D wit grid size controlled by. Let F be te class of discrete armonic maps u : D IR n for wic u φ j Γ wenever φ j is a boundary node of D, and wic satisfy an analogue of te previous integral tree-point condition. Note tat we do not require monotonicity of u D. Te function u F is said to be a discrete minimal surface if u is stationary witin F for te Diriclet energy Du = 1 2 D u 2, c.f. 27 for te precise formulation. Te main result Teorem 5.5 is tat if u is a nondegenerate minimal surface spanning Γ ten tere exist discrete minimal surfaces u, unique in a ball of almost constant radius ɛ 0 log 1, suc tat u u H 1 D c, were c depends on Γ and te nondegeneracy constant λ for u but is independent of. Recall tat nondegeneracy corresponds to te absence of zero eigenvalues for te second variation of te Diriclet Energy at u, and generically corresponds to te absence of branc points. Te constant c blows up as λ 0, and tis is consistent wit te numerical results in [DH4]. We give an outline of te proof of te main result in Section 2.3. Te first autor would like to tank te Centre for Matematics and it Applications, and te second autor would like to tank te Institut für Angewandte Matematik, for teir ospitality during te course of tis work. Te second autor would also like to tank Gaven Martin for elpful discussions on trace teory 1991 Matematics Subject Classification. Primary 65N30; Secondary 49Q05, 53A10. Key words and prases. minimal surface, finite elements, order of convergence, Plateau Problem. 1 c 1997 American Matematical Society

2 2 GERHARD DZIUK AND JOHN E. HUTCHINSON and related matters. Tis researc as been partially supported by te Australian Researc Council. 2. Formulation of te problem 2.1. Te energy functional. We first recall some notation from [DH4] Section 3, to wic we refer for furter discussion. See Section 3 of te present paper for te properties of H 1/2. Let D be te open unit disc in IR 2, wit boundary D. It will be convenient to let S 1 denote anoter, distinct, copy of te unit circle. Let Γ be a Jordan curve in IR n wit regular C r -parametrisation γ :S 1 Γ were r 3. For f : D IR n we denote by Φf:D IR n its unique armonic extension to D specified by Φf = 0 in D, Φf = f on D. Ten Φ : H 1/2 D, IR n H 1 D, IR n is a bounded linear map wit bounded inverse. Harmonic maps are uniquely determined by te associated boundary maps. We will use te Hilbert space H of functions defined by H = {ξ : D IR ξ H 1/2 < and 1 is satisfied}, were 2π 1 ξφ dφ = 0, 0 2π 0 ξφ cos φ dφ = 0, 2π 0 ξφ sin φ dφ = 0. Te norm on H is te usual norm H 1/2. Te corresponding affine space of maps s : D S 1 suc tat sφ = φ + σφ for some σ H is denoted by H. We also need te Banac space T defined by T = H C 0 D, IR wit norm ξ T = ξ H 1/2 + ξ C 0. Te corresponding affine space T is defined by T = H C 0 D, S 1. Wit some abuse of standard notation, we write s = 1 + σ for various norms on σ. Te energy functional E is defined on H by Es = 1 2 Φγ s 2 = DΦγ s. 2 D Finiteness of E follows from 8. We say te armonic function u = Φγ s is a minimal surface spanning Γ, or a solution of te Plateau Problem for Γ, if and only if s is monotone and stationary for E, i.e. 3 E s, ξ = 0 ξ T. As discussed in [DH4], tis is equivalent to oter formulations of te notion of a minimal surface. For te proof of asymptotic convergence of te numerical metod we need te following regularity result, wic follows from standard regularity results c.f. [DH4] Teorem 3.2 and te Implicit Function Teorem used to write Γ locally as te grap of a C k,α function. Proposition 2.1. If γ C k,α were k 1 and 0 < α < 1, and s T is monotone and stationary for E, ten s C k,α c = c γ C k,α, γ 1 L. We next recall some properties of te energy functional from [DH4] Section 3.3. Using te notation 4 u = Φγ s, v = Φγ s ξ, w = Φγ s ξ 2

3 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 3 formal computation gives Es = 1 5 u 2, 2 D E s, ξ = d 6 dt Es + tξ = u v, t=0 D E sξ, ξ = d2 7 dt 2 Es + tξ = u w + v 2, t=0 D D wit an analogous expression for E sξ, η obtained by bilinearity in te case of distinct variations. If γ C 3 ten E C 2 T, IR and te Frécet derivatives are given by 6 and 7. Te functional E is not differentiable on H, but if γ and s are as smoot as is necessary for te following estimates, ten one as 8 Es c γ 2 C 1 s 2 H 1/2, 9 E s, ξ c γ 2 C 2 s 2 C 1 ξ H 1/2, 10 E sξ, η c γ 2 C 2 s 2 C 1 ξ H 1/2 η H 1/2. In particular, tese will be used in case s is stationary for E. It will be important to consider te beaviour of te second derivatives of E near a stationary point s T. Te second derivative E s can ten be interpreted as a self-adjoint bounded map 2 Es:H H. Let 11 H = H H 0 H +, ξ = ξ + ξ 0 + ξ + if ξ H, be te ortogonal decomposition generated by te eigenfunctions of 2 Es aving negative, zero and positive eigenvalues respectively. Proposition 2.2. Suppose γ C 3,α. Suppose s is monotone and stationary for E. Ten H and H 0 are finite dimensional. If ξ H H 0 ten ξ H 3/2 D and 12 wit ν = ν γ C 3,α, γ 1 L. ξ H 3/2 ν ξ H 1/2 Proof. Te finite dimensionality of H and H 0 follow from elliptic teory, see te proof of [St, Proposition II.5.6]. It is also sown tere, wit a different notation, tat Φγ s ξ H 2 D c γ C 3, γ 1 L, s C 3, ξ H 1/2 c γ C 3,α, γ 1 L, ξ H 1/2 by Proposition 2.1. But ξ H 3/2 γ s γ s 2 γ s ξ H 3/2 by 52 C 2 c γ C 3, γ 1 L Φγ s ξ H2 D from 52 and 40. Hence ξ H 3/2 c γ C 3,α, γ 1 L, ξ H 1/2, and so te required result follows by scaling. Higer regularity on γ implies iger regularity on ξ. In particular, γ C 4,α implies ξ H 5/2 D, see te proof of [St, Proposition II.5.6].

4 4 GERHARD DZIUK AND JOHN E. HUTCHINSON If s is monotone and stationary for E, we say s is nondegenerate if H 0 = {0}. Te corresponding minimal surface u = Φγ s is also said to be nondegenerate. If s is nondegenerate it follows tat tere exists a λ > 0 suc tat for ξ H, 13 E sξ, ξ + ξ = E sξ +, ξ + E sξ, ξ λ ξ 2 H 1/2. We call λ te nondegeneracy constant for s Te discrete energy functional. We recall te necessary notation from [DH4] Section 4 and prove some preliminary estimates. Let G be a quasi-uniform triangulation of D wit grid size comparable to. Let D = {G G G }, D = {E j 1 j M} were te E j are te boundary edges, B = {φ 1,..., φ M } be te set of boundary nodes. Te projection π : D D is defined by 14 π e i1 tφj+tφj+1 = 1 te iφj + te iφj+1 for 0 t 1, 1 j M. In order to ave a discrete analogue E of te functional E we define te following discrete analogues of H 1 D; IR n, H 1/2 D; IR n, H, T, H and T. X n = { u C 0 D ; IR n } 15 u P 1 G for G G, 16 x n = { f C 0 D ; IR n f P 1 E j for 1 j M }, 17 H = {ξ C 0 D; IR ξ P 1 π 1 E j if 1 j M, ξ satisfies 1}, 18 H = {s C 0 D; S 1 s φ = φ + σ φ for some σ H }. Tus H T H, H T H, and te space of variations at any s H is naturally identified wit H. Setting n = 1 we similarly define X and x. We will make frequent use of te following inverse-type estimates: Proposition 2.3. If ξ H ten ξ H 1 c 1/2 ξ H 1/2, ξ C 0 c log 1/2 ξ H 1/2. Proof. Te first estimate is standard. Te second is sown in [DH1] Proposition 5.3. Suppose f C 0 D; IR n. One defines te linear interpolants I f x n, I f 1 te iφj + te iφj+1 = 1 tfe iφj + tfe iφj+1, I D f C0 D; IR n, I f 1 te iφj + te iφj+1 = 1 tfe iφj + tfe iφj+1, were 0 t 1, 1 j M. Here and elsewere, φ M+1 = φ 1. Note te different domains of I f and I Df. Note also tat te image of I γ s is a polygonal approximation to Γ, and I γ sφ j = γ sφ j Γ for φ j B. Finally, 21 I D f = I f π. Anoter type of approximation operator we require is a map p : T T H H. Te usual interpolation operator does not preserve te normalisation conditions 1. However, if we first interpolate and ten project onto H, te resulting operator still satisfies all te usual estimates. Te proof of te following is essentially given in [DH1] Proposition 5.2.

5 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 5 Proposition 2.4. Tere is a bounded linear operator p : T H suc tat, in particular, 22 ξ p ξ H s c k s ξ H k for s = 0, 1 2, 1 and k = 1, 3 2, 2. Moreover, 23 ξ p ξ C 0,1 c ξ C 2, p ξ C 0,1 c ξ C 0,1, 24 ξ p ξ C 0 c 2 ξ C 2, ξ p ξ C 0 c ξ C 1. If s T and sφ = φ + σφ, ten p s is defined by p sφ = φ + p σφ, s p s := σ p σ, and ence p s satisfies similar estimates to tose for p ξ. 25 For f x te discrete armonic extension Φ f X is defined by Φ f = 0 in D, Φ f = f on D, were is te discrete Laplacian and te first equation in 25 is interpreted as D Φ f ψ = 0 for all ψ X suc tat ψ = 0 on D. If f x n te discrete armonic extension Φ f is defined componentwise. For s H, te discrete energy functional E is defined by E s = 1 26 Φ I γ s 2 = D Φ I γ s. 2 D Note tat E is of course not te restriction of E to H. Te discrete armonic function u = Φ I γ s is said to be a discrete minimal surface spanning Γ, or a solution of te discrete Plateau Problem for Γ, if and only if 27 E s, ξ = 0 ξ H. Note tat we do not require monotonicity of s, as is te case for s in 3. Te derivatives of E, c.f. 5 7, are given by E s = 1 28 u 2, 2 D 29 E s, ξ = u v, D 30 were E ξ, ξ = u w + D v 2, D u = Φ I γ s, v = Φ I γ s ξ, w = Φ I γ s ξ 2, 31 For ξ H and s stationary for E, we define te projection of te decomposition 11: 32 ξ = p ξ, ξ0 = p ξ 0, ξ + = p ξ +. Note tat ξ, ξ0 and ξ+ do not normally belong to H, in particular te first two are smoot functions. However, if H = H 0 = {0} ten ξ = ξ + = ξ+. Proposition 2.5. If ξ H ten 33 and ξ , ξ0, ξ+ ξ = ξ + ξ 0 + ξ+ H. Moreover, wit ν as in 12, ξ ξ H 1/2 cν ξ H 1/2, ξ ξ H 1 c1/2 ν ξ H 1/2, ξ 0 ξ 0 H 1/2 cν ξ H 1/2, ξ 0 ξ 0 H 1 c1/2 ν ξ H 1/2, ξ + ξ+ H 1/2 cν ξ H 1/2, ξ + ξ+ H 1 c1/2 ν ξ H 1/2.

6 6 GERHARD DZIUK AND JOHN E. HUTCHINSON Proof. Since p ξ = ξ, 33 follows from 11. Next, from 22 and 12, ξ ξ H 1/2 = ξ p ξ H 1/2 c ξ H 3/2 cν ξ H 1/2. Te proof of te oter inequality in 34, and of 35, is similar. For 36 just note tat, from 33 and 11, ξ + ξ+ and use te previous estimates. = ξ ξ ξ0 ξ0, We remark tat te decomposition 33, unlike 11, is not an ortogonal decomposition. We also note tat te powers of in can be increased by 1/2 if we assume ξ H 2 ν ξ H 1/2; tis latter olds if γ C 4,α, c.f. te remarks after Proposition Structure of te proof. Te main error estimates, Teorems 5.4 and 5.5, are proved in Section 5. To motivate our approac and te need for te various preliminary estimates in Sections 4 and 5, we outline in an informal manner te structure of te proof of Teorem 5.4. H ' slope λ/4 B grap of E ' c 1 A s 4c 1 /λ from A & B p s s c 3/2 Figure 1 Assume s is a nondegenerate stationary point for E wit nondegeneracy constant λ. One applies te Inverse Function Teorem to te derivative E :H H te dual space of H in a neigbourood of te point p s; remember tat H is te tangent space for H, i.e. te space of variations at any t H. Identify E wit te derivative E of E ; tus E t :H H for t H. Tere are tree estimates to be proved. A: E p s c 1 for sufficiently small. Tis follows immediately from te consistency estimate Proposition 4.2 for te first derivative. See 109. B 1 : E p s, ξ λ 2 ξ for sufficiently small; i.e. te slope of E at p s is bounded away from 0 by λ/2. Tis is a nondegeneracy estimate on E p s and is establised in Proposition 5.3, using te one-sided consistency estimate Proposition 4.3 for te second derivative, and te regularity of members of te negative eigenspaces to compensate for te one-sided nature of tat estimate. See 114. B 2 : E p s E p s + η λ/4 for sufficiently small and η sufficiently close to p s, more precisely for η ɛ 0 / log ; i.e. te slope of E near p s differs from tat at p s by at most λ/4. Tis is a consequence of Proposition 5.2. See 115. As indicated in te diagram, it follows

7 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 7 tere exists a unique stationary point s for E suc tat p s s H 1/2 4c 1 /λ, and ence suc tat s s H 1/2 c/λ. 3. Preliminary estimates One can define te H s D and H s D norms for any real s; but apart from non-negative integers s we will only need te following cases. For f : D IR te H 1/2 D seminorm is defined by 37 f 2 H 1/2 D = D D fφ fφ 2 φ φ 2 dφ dφ, and for u:d IR te H 1/2 D seminorm is defined by u 2 H 1/2 D = ux ux 2 38 x x 3 dx dx. In bot cases te corresponding norm is given by 39 Also, D D 2 H 1/2 = 2 L H 1/2. f H 3/2 D = f H 1/2 D, u H 3/2 D = u H 1/2 D, 2 H 3/2 = 2 L H 3/2. If u H s+1/2 D for s = 1/2, 1, 3/2 in fact for s > 0 ten u as a well-defined trace f on D and 40 f Hs D c u H s+1/2 D. More precisely, te previous estimate is true for u C D and te definition of trace is extended by continuity and density to te general case. Suc an estimate is not true for smoot u if s = 0, but in tis case if u L 2 D te following estimate olds and ten te trace operator is again defined by continuity and density to satisfy 41 f L2 D c u H 1/2 D + u L2 D. See [LM] Capter 1 Section 9.2, page 193 remarks before Teorem 8.1 and page 187 Teorem 7.3. Note tat by considering kernels it is clear tat in te case s = 1/2 one can replace norms in 40 by te corresponding seminorms. We will frequently use tis fact witout furter remark. Conversely, if f H s D for s = 0, 1/2, 1, 3/2 wit similar results for any real s ten tere is a unique armonic function Φf defined on D wit trace f as before, and in particular 42 Φf H s+1/2 D c f H s D. See [LM] Teorem 7.4 page 188. Note tat for s = 1/2, 1, 3/2 te norms can be replaced by seminorms as follows again by considering kernels; we will use tis fact witout furter comment. In te case of L 2 boundary data one also as, wit u = Φf, sup 0 r<1 lim urθ = fθ for a.e. θ D, r 1 ur c f L 2 D. L 2 D See [JK] Teorem 1.16 page 11. In particular, ur f in L 2 D as r 1. Te formulae 43 and 44 are establised using te Poisson integral representation of u from f. Te function u agrees wit Φf in 42 since it agrees for f

8 8 GERHARD DZIUK AND JOHN E. HUTCHINSON in te dense subset C D L 2 D by te maximum principle, and since te Poisson integral map and Φ are bot bounded as maps from L 2 D into L 2 D, for example. If f : D IR as te Fourier series expansion fφ = a 0 + a n cos nφ + b n sin nφ ten one can define n=1 f 2 L 2 D = a f 2 H s D = a 2 n + b 2 n, n=1 n 2s a 2 n + b2 n s > 0, n=1 f 2 H s D = f 2 L 2 D + f 2 H s D. Tese norms and seminorms are equivalent to te usual definitions in case s IN, and te previous definitions in case s = 1/2 or 3/2. Te armonic extension of f is given by Φfr, φ = a 0 + r n a n cos nφ + b n sin nφ. n=1 Proposition 3.1. Suppose f, g : D IR. Ten fg H 1/2 f C 0 g H 1/2 + f H 1/2 g C 0 fg H 1/2 c f C 0,1 g H 1/2 fg H 1 f C 0 g H 1 + f H 1 g C 0 fg H 1 c f C 0,1 g H 1 fg H 3/2 c f C 2 g H 3/2. Proof. Te first two inequalities follow from 37. Te next two inequalities are standard. Te last inequality follows from 49. Te next Proposition will be applied in case g is a component of γ, γ or γ, and s is eiter smoot or piecewise linear and continuous. In particular, g will be at least C 1 but s may be only C 0,1 in some cases. Proposition 3.2. Suppose s = id + σ : D S 1, g :S 1 IR. Ten g s C 0 g C 0 g C 0 s C 0 g s C 0,1 c g C 1 s C 0,1 g s C 1 c g C 1 s C 1 56 g s C 2 c g C 2 s 2 C 2 57 g s L2 c g C0 c g C0 s L g s H 1/2 c g C 1 s H 1/2 g s H 1 c g C 1 s H 1 Proof. Recall tat s C 0 = 1 + σ C 0 1. Ten 53 is immediate and similarly for 57. Also 54 is immediate. For 55, 56 and 59 note tat g s = g s s, g s = g s s 2 + g s s. Inequality 58 follows from 37.

9 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 9 Te following Proposition will typically be applied in case g is γ, γ or γ and in particular is C 1 ; and eiter s 1 = s 0 and s 2 = p s 0, or s 1 = p s 0 and s 2 = p s 0 + η for an arbitrary η H. Note tat in 61 only te C 0 norm of s 1 s 2 is required, and in case s 1 s 2 H tis will be estimated by te inverse estimate 20. Proposition 3.3. Suppose s i = id + σ i : D S 1 for i = 1, 2 and g : S 1 IR. Ten 60 g s 1 g s 2 L 2 c g C 1 s 1 s 2 L 2 61 g s 1 g s 2 H 1/2 c g C 2 s 1 C 0,1 + s 1 s 2 C 0 s 1 s 2 H 1/2 62 g s 1 g s 2 H 1 c g C 2 s 1 C 0,1 s 1 s 2 H 1 63 g s 1 g s 2 C 0,1 c g C 2 s 1 C 0,1 s 1 s 2 C 0,1 Proof. Te proof of 60 is immediate. For 61 write Ten gs 1 φ gs 1 φ + ηφ 1 = ηφ = s 2 φ s 1 φ. gs 1 φ gs 1 φ + ηφ 1 g s 1 φ + tηφdt ηφ + g s 1 φ + tηφdt ηφ g s 1 φ + tηφ dt ηφ ηφ 0 g s 1 φ + tηφ g s 1 φ + tηφ dt ηφ g C 1 ηφ ηφ + g C 2 It now follows from 37 tat g s 1 g s 1 + η H 1/2 0 s 1 φ s 1 φ + ηφ ηφ ηφ. g C 1 η H 1/2 + g C 2c s 1 C 0,1 η L 2 + η H 1/2 η C 0 c g C 2 s 1 C 0,1 + η C 0 η H 1/2, recalling tat s 1 C 0,1 1. Tis establises 61. For 62 we compute g s 1 g s 2 L 2 = g s 1 s 1 g s 2 s 2 L 2 g s 1 g s 2 L 2 s 1 L + g s 2 C 0 s 1 s 2 L 2 c g C 2 s 1 s 2 L 2 s 1 C 0,1 + g C 1 s 1 s 2 H 1 from 60 c g C 2 s 1 C 0,1 s 1 s 2 H 1. For 63 we similarly estimate g s 1 g s 2 L. Te next Proposition will be used repeatedly in te consistency and non-degeneracy estimates of Sections 4 and 5, particularly in case s = 1. In Corollary 3.5 we see tat te H 1 D seminorm of te discrete armonic extension of a discrete function f can be estimated by te H 1 D seminorm of te smoot armonic extension of te function. Tis is not true for general smoot f. Te Corollary is due to Bramble, Pasciak & Scatz [BPS].

10 10 GERHARD DZIUK AND JOHN E. HUTCHINSON Proposition 3.4. If f H s D, IR n were s = 1, 3/2 ten Φf Φ I f H 1 D c s 1/2 f H s D, Φ I f H1 D f H 1/2 D + c s 1/2 f Hs D. Proof. It is sufficient to take n = 1. Let u = Φf, u = Φ I f. Note tat I f is well-defined as f C 0 D. Since u = I u on D, it follows in te usual way from te weak form of Laplace s equation and 25 tat 66 0 = u u u I u, D and so 67 Hence u u 2 = u u u I u. D D u u H1 D u I u H1 D c s 1/2 u H s+1/2 D c s 1/2 u H s+1/2 D c s 1/2 u H s D, from a standard interpolation result and 42. Tis gives 64, and 65 ten follows from 42. Corollary 3.5. Suppose f H. Ten 68 Φ I f H 1 D c Φf H 1 D Proof. Tis follows from 65 wit s = 1, te inverse estimate 19, and 40. Proposition 3.7 will allow us to estimate various quantities involving an armonic function and te discrete disc D, in terms of te trace of te armonic function on D. But first we need an elementary lemma. Lemma 3.6. Suppose u H 1 D and π : D D is as in 14. Ten 69 u u π L 2 D c u H 1 D\D Proof. Let L θ be te straigt line segment joining θ D to πθ D. Ten 2 uθ u πθ 2 u c 2 u 2. L θ L θ Eac z D \ D can be written uniquely as θ, y D IR +, were z L θ and y is te distance of z from θ D. Te corresponding map θ, y z as Jacobian J satisfying 1 c 2 J 1. Hence u u π 2 c 2 dθ u 2 c 2 u 2. D D L θ D\D denote normal and tan- In te following Proposition and elsewere, u ν gential derivatives on te relevant curve. u and τ

11 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 11 Proposition 3.7. Suppose u is armonic in D wit trace u D H 1 D as appropriate. Ten 70 u L2 D\D c u L2 D, 71 u L2 D\D c u H1 D, u u π L 2 D c 2 u H 1 D, u c u H ν 1 D. L 2 D L 2 D or Proof. Let Dρ be te disc of radius ρ. From 44 u L 2 Dρ c u L 2 D. Integrating te square of tis inequality wit respect to ρ from 1 c 2 to 1 now establises 70. Since u is also armonic, it follows But 74 u L 2 D\D c u L 2 D. D u ν 2 = D u 2 τ, from a Fourier series expansion, and so 71 follows. Inequality 72 follows from 71 and te previous Lemma. For 73 we ave tat if u = 0 and u D H 1 D ten u = 0 and u H 1/2 D by 42. Hence u D is well-defined and u D L 2 D from 41. From 44 applied to u, u L2 D c u L2 D and in particular, using 74, u ν c u H 1 D. L 2 D Tis establises Consistency estimates for te energy In tis Section we compare E and its derivatives at s, wit E and its derivatives at p s. Apart from teir intrinsic significance, tese estimates will be needed in te next Section to establis te main convergence results. Remarks on te Proofs In te proof of eac of te tree Propositions in tis section, Proposition 3.4 is used to estimate te difference between an armonic function and te corresponding discrete armonic function, and also to estimate various discrete armonic error terms. Interpolation results in terms of te H 2 piecewise seminorm, togeter wit inverse estimates, are used to estimate te L 2 norm of te quantities γ p s I Dγ p s, γ p s ξ I Dγ p s ξ and γ p s ξ 2 I D γ p s ξ 2. Since piecewise second derivatives of p s and ξ vanis, tis enables us to gain an extra power of tan one migt at first expect. In eac of te tree proofs te term I 2 is estimated wit an integration by parts. In Proposition 4.1 tis improves te order of convergence from O to O 2. In te oter two Propositions, one could not oterwise expect any order of convergence; see also te remark preceding Proposition 4.3 concerning v v L2 D.

12 12 GERHARD DZIUK AND JOHN E. HUTCHINSON Proposition 4.1. Let s H C 2, γ C 2. Ten Proof. Let 75 Ten 76 E p s Es c 2 γ 2 C 2 s 4 C 2. u = Φγ s, u = Φ I γ p s. E p s Es = 1 u 2 1 u 2 2 D 2 D = 1 u u 2 u u u 1 u 2 2 D D 2 D\D = I 1 + I 2 + I 3. We will estimate tese terms separately. For I 1, 2I 1 1/2 = Φγ s Φ I γ p s H 1 D Φγ s Φ I γ s H 1 D + Φ I γ s γ p s H 1 D c γ s H 3/2 + γ s γ p s H 1/2 + c 1/2 γ s γ p s H 1 from 64 wit s = 3/2 and 65 wit s = 1. But also and Hence γ s H 3/2 c γ s C 2 c γ C 2 s 2 C2 from 56, γ s γ p s H 1/2 c γ C 2 s C 0,1 + p s C 0,1 s p s H 1/2 by 61 c 3/2 γ C 2 s C 1 s H 2 by 23 and 22 c 3/2 γ C 2 s 2 C 2, 1/2 γ s γ p s H 1 c 1/2 γ C 2 s C 1 s p s H 1 from 62 c 3/2 γ C 2 s 2 C2 from For I2 one as I 1 c 2 γ 2 C 2 s 4 C 2. But I 2 = u D ν u u u ν u u L2 D L2 D c u H1 D u u L2 D by 73 c γ C 1 s C 1 u π u π L2 D by 59 and 14 c γ C 1 s C 1 u π u L 2 D + u u π L 2 D. u π u L 2 D c 2 u H 1 D = c 2 γ s H 1 c 2 γ C 1 s C 1

13 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 13 by 72 and 59. Also u u π L2 D = γ s I D γ p s L 2 by 75 and 21 γ s γ p s L 2 + γ p s I D γ p s L 2 γ C 1 s p s L 2 + c 2 γ p s H 2, by 60 and a standard interpolation estimate, were te H 2 semi-norm is to be understood in a piecewise sense. On eac arc segment, γ p s = γ p s p s 2 since p s = 0, so we can continue te estimate by using 24 and 23. Hence 78 Finally, c γ C 1 2 s C 2 + c 2 γ C 2 s 2 C 1 c2 γ C 2 s 2 C 2, I 2 c 2 γ 2 C 2 s 3 C 2. I 3 = 1 2 u 2 L 2 D\D c 2 u 2 H 1 D by = c 2 γ s 2 H 1 c2 γ 2 C 1 s 2 C 1. Te Proposition now follows from 76, 77, 78 and 79. Te next Proposition sows tat E p s is a O approximation to E s H in te H norm for γ and s sufficiently smoot. Te actual power of is important in te proof of Teorem 5.4. Proposition 4.2. Let s H C 2, γ C 3. Ten for any ξ H E s, ξ E p s, ξ c γ 2 C 3 s 3 C 2 ξ H 1/2. Proof. Define u and u as in 75, and for ξ H define 80 Ten from 29, v = Φγ s ξ, v = Φ I γ p s ξ. E s, ξ E p s, ξ = u v u v D D = u v u v + u v D D\D = u u v + D u v v + D u v D\D 81 Now = I 1 + I 2 + I 3. Also I 1 u u L 2 D v L 2 D c γ C 2 s 2 C 2 v L 2 D from 77. v L2 D = Φ I γ p s ξ H1 D γ p s ξ H 1/2 + c 1/2 γ p s ξ H 1 by 65. Te first term is estimated by γ p s ξ H 1/2 c γ p s C 0,1 ξ H 1/2 from 49 c γ C 2 s C 1 ξ H 1/2 using 23;

14 14 GERHARD DZIUK AND JOHN E. HUTCHINSON te second by γ p s ξ H 1 c γ p s C 0,1 ξ H 1 by 51 c γ C 2 p s C 0,1 ξ H 1 by So altogeter, c 1/2 γ C 2 s C 1 ξ H 1/2 by 23 and I 1 c γ 2 C 2 s 3 C 1 ξ H 1/2. For I 2, as for I 2 in te proof of te previous Proposition, I 2 = u D ν v v c γ C 1 s C 1 v π v L 2 D + v v π L 2 D. From 72 and te argument for 82, Also, v π v L 2 D c 2 v H 1 D = c 2 γ s ξ H 1 c 3/2 γ C 2 s C 1 ξ H 1/2. v v π L 2 D Altogeter, = γ s ξ I D γ p s ξ L 2 γ s γ p s ξ L 2 + γ p s ξ I D γ p s ξ L 2 γ C 2 s p s C 0 ξ L 2 + c 2 γ p s ξ H 2 were te H 2 semi-norm is understood in te piecewise sense c 2 γ C 2 s C 2 ξ L 2 from 24 + c 2 γ C 3 s 2 C 1 ξ L 2 + γ C 2 s C 1 ξ H 1 since on eac arc segment, γ p s ξ = γ p sp s 2 ξ + 2γ p sp s ξ c 2 γ C 3 s 2 C 2 ξ H 1 c 3/2 γ C 3 s 2 C 2 ξ H 1/2 by 19. Finally, I 2 c 3/2 γ 2 C 3 s 3 C 2 ξ H 1/2. I 3 u L 2 D\D v L 2 D\D c γ C 1 s C 1 v L 2 D\D by 79 c 2 γ C 1 s C 1 v H1 D by 71 c 2 γ C 1 s C 1 γ s ξ H 1 c 2 γ C 1 s C 1 γ C 2 s C 1 ξ H 1 by 51 and 55 c 3/2 γ 2 C 2 s 2 C 1 ξ H 1/2 by 19. Te Proposition follows from 81, 83, 84 and 85. Remark Te following Proposition, and te related Proposition 5.3, are essential for te proof of te main Teorem. Assuming γ and s are sufficiently smoot, we see from 87 tat if E s is positive definite for example, if s is a strictly stable local minimum for E ten so is E p s for sufficiently small, wit te same ellipticity constant up to O 1/2 log 1/2. One migt ope to estimate te positive term 86 v v 2 L 2 D = Φγ s ξ Φ I γ p s ξ 2 H 1 D

15 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 15 in 87 by c α ξ 2 for some α > 0. But we cannot expect tis as we see by H 1/2 considering te model estimates: Φξ Φ I ξ H1 D c 1/2 ξ H 1 c ξ H 1/2 from 64 and 19 tese estimates cannot be improved. None-te-less, in Proposition 5.3 we do estimate 86 in case ξ = p ξ were ξ is in te negative eigenspace of E s. Te point is tat ξ is ten smoot by elliptic regularity teory altoug ξ is of course not smoot, and in fact we sow in 106 tat in tis case v v 2 L 2 D c ξ 2 H 1/2. As a consequence we deduce in Proposition 5.3 tat if E s is non-degenerate ten so is E p s for sufficiently small, wit te same nondegeneracy constant up to O 1/2 log 1/2. Proposition 4.3. Let s C 2 H and γ C 3. Ten for any ξ H 87 were and E p sξ, ξ E sξ, ξ = v v 2 L 2 D + R Proof. Let u and u be as in 75. Let 88 v = Φγ s ξ, v = Φ I γ p s ξ, R c 1/2 log 1/2 γ 2 C 3 s 3 C 2 ξ 2 H 1/2. w = Φγ s ξ 2, w = Φ I γ p s ξ 2. From 30, E s E p s ξ, ξ = v 2 + u w v 2 u w D D D D = v v v v v + u u w D D D + w w u + v 2 + u w D D\D D\D = I 1 + I 2 + I 3 + I 4 + I 5 + I 6. We estimate te terms oter tan I 1. For I 2 we ave I 2 = 2 v v v = 2 D 2 v ν v v L 2 D. L 2 D D v ν v v Te first factor is estimated wit te use of 73 and arguing as for 82; v ν c v H1 D = c γ s ξ H 1 c 1/2 γ C 2 s C 1 ξ H 1/2. L2 D Te second factor was estimated in te estimate for I 2 in te proof of Proposition 4.2: Altogeter for I 2, 89 v v L2 D c 3/2 γ C 3 s 2 C 2 ξ H 1/2. I 2 c γ 2 C 3 s 3 C 2 ξ 2 H 1/2.

16 16 GERHARD DZIUK AND JOHN E. HUTCHINSON For I 3, I 3 = u u w u u L2 D w L2 D D c γ C 2 s 2 C 2 w L 2 D from 77. From 65 we get But Similarly 90 w L 2 D = Φ I γ p s ξ 2 H 1 D γ p s ξ 2 H 1/2 + c1/2 γ p s ξ 2 H 1. γ p s ξ 2 H 1/2 c γ C 3 s C 1 ξ 2 H 1/2 from 49 and 54 c γ C 3 s C 1 ξ C 0 ξ H 1/2 from 48 c log 1/2 γ C 3 s C 1 ξ 2 H 1/2 from 20. γ p s ξ 2 H 1 c γ C 3 s C 1 ξ2 H1 using 51 c γ C 3 s C 1 ξ C 0 ξ H 1 from 50 from 20 and 19. Hence and so 91 For I 4 we ave c 1/2 log 1/2 γ C 3 s C 1 ξ 2 H 1/2 w L 2 D c log 1/2 γ C 3 s C 1 ξ 2 H 1/2, I 3 c log 1/2 γ 2 C 3 s 3 C 2 ξ 2 H 1/2. I 4 = w w u = w w u D D ν, and so as for I 2 in te proof of Proposition 4.1, 92 I 4 c γ C 1 s C 1 w π w L 2 D + w w π L 2 D. But w π w L 2 D c 2 w H 1 D by 72 Also c 2 γ s ξ 2 H 1 c 3/2 log 1/2 γ C 3 s C 1 ξ 2 H 1/2 as for 90. w w π L 2 D = γ s ξ 2 I D γ p s ξ 2 L2 from 88 and 21 γ s ξ 2 γ p s ξ 2 L 2 + γ p s ξ 2 I D γ p s ξ 2 L 2 γ s γ p s C 0 ξ 2 L + 4 c γ p s ξ 2 H 1 Hence 93 c γ C 3 s p s C 0 ξ 2 H + c 1/2 log 1/2 γ 1/2 C 3 s C 1 ξ 2 H 1/2 using a Sobolev embedding teorem and 90 c 1/2 log 1/2 γ C 3 s C 1 ξ 2 H from 24. 1/2 I 4 c 1/2 log 1/2 γ 2 C 3 s 2 C 1 ξ 2 H. 1/2

17 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 17 From te final part of te argument for I 3 in te proof of te previous Proposition, Finally, I 5 = v 2 L 2 D\D c γ 2 C 2 s 2 C 1 ξ 2 H 1/2. I 6 u L 2 D\D w L 2 D\D c 2 u H1 D w H1 D by 71 = c 2 γ s H 1 γ s ξ 2 H 1 c 2 γ C 1 s C 1 γ s ξ 2 H 1 by 59 c 3/2 log 1/2 γ 2 C 3 s 2 C 1 ξ 2 H 1/2 as for 90. Now collect te estimates 89, 91, 93, 94 and 95, and tis proves te Proposition. 5. Te main error estimates We will apply te following quantitative version of te Inverse Function Teorem wit X = H and Y = H in te proof of Teorem 5.4. Lemma 5.1. Let X be an affine Banac space wit Banac space X as tangent space, and let Y be a Banac space. Suppose x 0 X and f C 1 X, Y. Assume tere are positive constants α, β, δ and ɛ suc tat were 99 fx 0 Y δ, f x 0 1 LY,X α 1, f x f x 0 LX,Y β for all x B ɛ x 0, β < α, δ α βɛ. Ten tere exists a unique x B ɛ x 0 suc tat fx = 0. Proof. Tis follows from te proof of te Inverse Function Teorem in [Be; pp ]. Te modifications necessary since X is an affine space are trivial. Te next Proposition establises tat for any β > 0, E p s E p s + η H H β provided log η H 1/2 is sufficiently small depending on β but independent of. Tis will be used to establis te appropriate version of 98 in te proof of Proposition 5.4. Remark on te Proof Te precise form of estimate 61, and te inverse estimate 20, are used to control g s g s + η H 1/2 for g = γ, γ and γ. Proposition 5.2. Let s H C 1 and γ C 4. Ten for any η, ξ H E p s E p s + η ξ, ξ 2 c 1 + log 1/2 η H 1/2 log η H 1/2 γ 2 C 4 s 2 C 1 ξ 2 H. 1/2 Proof. For simplicity we write s for p s. From 30, E s E s + η ξ, ξ = v 2 + u w v 2 u w, D D D D

18 18 GERHARD DZIUK AND JOHN E. HUTCHINSON wit u = Φ I γ s, u = Φ I γ s + η, v = Φ I γ s ξ, v = Φ I γ s + η ξ, w = Φ I γ s ξ 2, w = Φ I γ s + η ξ 2. So 100 E s E s + η ξ, ξ v v v + v + u u w D D + u w w D u u L 2 D w L 2 D + v v L 2 D v L 2 D + v L 2 D + w w L 2 D u L 2 D = A 1 B 1 + A 2 B 21 + B 22 + A 3 B 3. Now estimate tese terms separately. By 65 A 1 = Φ I γ s γ s + η H1D γ s γ s + η H 1/2 + c 1/2 γ s γ s + η H 1 c γ C 2 s C 0,1 + η C 0 η H 1/2 + c 1/2 γ C 2 s C 0,1 η H 1 by 61 and 62 c γ C 2 s C 1 η H 1/2 1 + log 1/2 η H 1/2 + c γ C 2 s C 1 η H 1/2 by 20, 19 and 23 c 1 + log 1/2 η H 1/2 η H 1/2 γ C 2 s C 1. Again from 65 A 2 = Φ I γ s ξ γ H s + η ξ 1 D γ s γ s + η ξ H 1/2 + c 1/2 γ s γ s + η ξ H 1 γ s γ s + η H 1/2 ξ C 0 + γ s γ s + η C 0 ξ H 1/2 by 48 + c 1/2 γ s γ s + η C 0 ξ H 1 + c 1/2 γ s γ s + η H 1 ξ C 0 by 50 c 1 + log 1/2 η H 1/2 η H 1/2 γ C 3 s C 1 ξ C 0 + c γ C 2 η C 0 ξ H 1/2 + c 1/2 γ C 2 η C 0 ξ H 1 using te argument for A 1 on te first and last terms in te preceding inequality c 1 + log 1/2 η H 1/2 log 1/2 η H 1/2 γ C 3 s C 1 ξ H 1/2, again using te inverse estimates 20 and 19 on η and ξ.

19 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 19 In te same way we proceed for A 3 to first obtain te analogue of te second last inequality above: A 3 = Φ I γ s ξ 2 γ s + η ξ 2 H 1 D c 1 + log 1/2 η H 1/2 η 1/2 γ C 4 s C 1 ξ 2 C 0 + c γ C 3 η C 0 ξ 2 H 1/2 + c 1/2 γ C 3 η C 0 ξ 2 H 1 c 1 + log 1/2 η H 1/2 log η H 1/2 γ C 4 s C 1 ξ 2 H, 1/2 from te inverse estimates 19 and 20 applied to η and ξ, and since For B 3, ξ 2 H 1/2 2 ξ C 0 ξ H 1/2, ξ 2 H 1 2 ξ C 0 ξ H 1. B 3 = Φ I γ s + η H1 D For B 22, γ s + η H 1/2 + c 1/2 γ s + η H 1 by 65 γ C 1 s H 1/2 + η H 1/2 + c 1/2 γ C 1 s H 1 + η H 1 c γ C 1 s H 1/2 + η H 1/2 by 19 c1 + η H 1/2 γ C 1 s C 1. B 22 = Φ I γ s + η ξ H 1 D γ s + η ξ H 1/2 + c 1/2 γ s + η ξ H 1 by 65 γ s + η C 0 ξ H 1/2 + γ s + η H 1/2 ξ C 0 + c 1/2 γ s + η C 0 ξ H 1 + γ s + η H 1 ξ C 0 by 48 c γ C 1 ξ H 1/2 + c γ C 2 s + η H 1/2 ξ C 0 + c 1/2 γ C 1 ξ H 1 + γ C 2 s + η H 1 ξ C 0 c γ C 2 s + η H 1/2 log 1/2 ξ H 1/2 by 19 and 20 c1 + η H 1/2 log 1/2 γ C 2 s C 1 ξ H 1/2. For B 21 we ave te same estimate wit η set equal to 0; Finally, B 21 c log 1/2 γ C 2 s C 1 ξ H 1/2. B 1 = Φ I γ s ξ 2 H 1 D γ s ξ 2 H 1/2 + c 1/2 γ s ξ 2 H 1 by 65 γ s C 0 ξ 2 H 1/2 + γ s H 1/2 ξ 2 C 0 + c 1/2 γ s C 0 ξ 2 H 1 + γ s H 1 ξ 2 C 0 c γ C 2 ξ C 0 ξ H 1/2 + c γ C 3 s H 1/2 ξ 2 C 0 + c 1/2 γ C 2 ξ C 0 ξ H 1 + γ C 3 s H 1 ξ 2 C 0 c log γ C 3 s C 1 ξ 2 H 1/2.

20 20 GERHARD DZIUK AND JOHN E. HUTCHINSON Substituting te previous estimates in 100, E s E s + η ξ, ξ c 1 + log 1/2 η H 1/2 η H 1/2 γ C 2 s C 1 log γ C 3 s C 1 ξ 2 H 1/2 + c 1 + log 1/2 η H 1/2 log 1/2 η H 1/2 γ C 3 s C 1 ξ H 1/2 1 + η H 1/2 log 1/2 γ C 2 s C 1 ξ H 1/2 + c 1 + η H 1/2 γ C 1 s C log 1/2 η H 1/2 log η H 1/2 γ C 4 s C 1 ξ 2 H 1/2 2 c 1 + log 1/2 η H 1/2 log η H 1/2 γ 2 C 4 s 2 C 1 ξ 2 H. 1/2 It follows from te next result tat E p s is nondegenerate wit nondegeneracy constant arbitrarily close to λ, provided is sufficiently small. Tis will be used to establis te appropriate version of 97 in te proof of Teorem 5.4. Remarks on te Proof See also te Remark preceding Proposition 4.3. Nondegeneracy for interpolants of functions from te positive and negative spaces for E s must be treated separately see te separate treatment of I 1 and I 2 below. In te first case, nondegeneracy follows more or less directly from Proposition 4.3. In te second case, te term v v 2 L 2 D, were v = Φ γ s ξ and v = Φ I γ s ξ, must be estimated. Te main points are to apply te regularity results 34 and Proposition 2.2 to members of te negative space. Proposition 5.3. Let s H C 2 and γ C 3. Suppose E s is nondegenerate wit nondegeneracy constant λ, as in 13. Let ν be as in 12. Ten E p sξ, ξ + ξ λ c 1/2 log 1/2 γ 2 C 3ν2 s 4 C ξ 2 2 H 1/2 for every ξ H. Proof. We again denote p s by s and split 101 E s ξ, ξ + ξ = E = I 1 + I 2. s ξ +, ξ+ E s ξ, ξ It follows from 87, after discarding a positive term, tat I 1 E sξ +, ξ+ c1/2 log 1/2 γ 2 C 3 s 3 C 2 ξ+ = E sξ +, ξ+ + E s ξ +, ξ+ ξ+, ξ+ c 1/2 log 1/2 γ 2 C 3 s 3 C 2 ξ+ 2 H 1/2. Using te notation of 13 tere exists λ + > 0 suc tat E sξ +, ξ+ λ+ ξ + 2 H 1/2. 2 H 1/2

21 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 21 Also E s = E s ξ +, ξ+ ξ+, ξ+ ξ + ξ +, ξ+ + ξ + c γ 2 C 2 s 2 C 1 ξ+ ξ + H 1/2 ξ+ + ξ + H1/2 by 10 c γ 2 C 2ν2 s 2 C 1 ξ 2 H by 36. 1/2 Using 36 again, I 1 can ten be estimated from below by 102 I 1 λ + ξ + 2 H 1/2 c 1/2 log 1/2 γ 2 C 3ν2 s 3 C 2 ξ 2 H 1/2. Wit similar arguments, but now keeping all terms in 87 and using te notation of 13, 103 were ere I 2 λ ξ 2 H 1/2 c 1/2 log 1/2 γ 2 C 3ν2 s 3 C 2 ξ 2 H 1/2 v v 2 L 2 D, v = Φ γ s ξ, v = Φ I γ s ξ. From 101, 102, 103, and using 13, E s ξ, ξ + ξ λ c 1/2 log 1/2 γ 2 C 3ν2 s 3 C ξ 2 2 H 1/2 v v 2 L 2 D. It remains to estimate te last term. For tis it is important to notice tat ξ is a smoot function. But And Also v v L2 D = Φγ s ξ Φ I γ s ξ H 1 D Φγ s ξ ξ H 1 D + Φ I γ s ξ ξ H 1 D + Φγ s ξ Φ I γ s ξ H 1 D + Φ I γ s γ s ξ H 1 D = I 3 + I 4 + I 5 + I 6. I 3 γ s ξ ξ H 1/2 γ s C 1 ξ ξ H 1/2 c γ C 2 s C 1ν ξ H 1/2 by 34. I 4 γ s ξ ξ H + 1/2 c1/2 γ s ξ ξ H1 by 65 c γ s C 1 ξ ξ H + 1/2 1/2 ξ ξ H 1 c γ C 2 s C 1ν ξ H 1/2 from 34 and 34. I 5 c 1/2 γ s ξ H1 by 64 c 1/2 γ s C 1 ξ H 1 c 1/2 γ C 2 s C 1ν ξ H 1/2 by 12.

22 22 GERHARD DZIUK AND JOHN E. HUTCHINSON Finally, 106 I 6 c γ s γ s ξ H + 1/2 c1/2 γ s γ s ξ H1 by 65 c γ s γ s C 0,1 ξ H + 1/2 1/2 ξ H 1 by 49 and 51 c γ C 3 s C 1 s s C 0,1ν ξ H 1/2 by 63 and 12 c γ C 3 s 2 C 2ν ξ H 1/2 from 23. If we now put togeter te estimates for I 3, I 4, I 5 and I 6, we get v v L 2 D c 1/2 γ C 3ν s 2 C 2 ξ H 1/2. Tis togeter wit 105 proves te Proposition. Te next result gives te main error estimates for discrete maps s wic are stationary for E. As remarked at te beginning of tis Section, te proof uses Lemma 5.1. Te necessary estimates were establised in Propositions 4.2, 5.2 and 5.3. Teorem 5.4. Assume γ C 4. Let s be a monotone nondegenerate stationary point for E, wit nondegeneracy constant λ as in 13. Ten tere exist positive constants 0 and c 0 depending on γ C 4 and γ 1 L, and λ in te case of 0, suc tat if 0 < 0 ten tere exists s H wic is stationary for E and satisfies 107 s s H 1/2 c 0 λ 1. Moreover, tere exists ɛ 0 = ɛ 0 γ C 4, γ 1 L, λ > 0 suc tat s is te unique stationary point for E satisfying 108 s s H 1/2 ɛ 0 log 1. Proof. We will apply Lemma 5.1 wit X = H, X = H, Y = H te dual space of H, f = E and x 0 = p s. Note tat E :H H. From Propositions 4.2 and 2.1, since E s = 0, 109 E p s H c 1 were c 1 = c 1 γ C 3, γ 1 L. Te derivative E of E is a map E :H LH, H and is naturally identified in te usual way wit E via E t, ξ, η = E 110 t ξ, η for all t H and ξ, η H. From Propositions 5.3, 2.1 and 2.2, since 1/2 log 1/2 0 as 0, 111 E p sξ, ξ + ξ 3λ 4 ξ 2 H 1/2 for all ξ H, provided 0 < 0 = 0 γ C 4, γ 1 L, λ. But 112 ξ + ξ H 1/2 ξ+ ξ H + 1/2 ξ+ ξ+ H + 1/2 ξ ξ 1 + cν ξ H 1/2 H 1/2 from 36 and 34, and since ξ + ξ H 1/2 = ξ H 1/2. Hence from 110, 111 and 112, 113 E p s, ξ ξ + ξ λ 2 ξ H 1/2 ξ + ξ H 1/2

23 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 23 for all ξ H, provided 0 < 0 for a new 0 wit te same dependencies as before. Tus for eac ξ H wit ξ H 1/2 = 1, te map E p s, ξ H as norm λ/2. It follows, since H and H ave equal finite dimension, tat E p s is invertible and E p s 1 1 λ 114 LH.,H 2 Next note tat from 110 and Proposition 5.2, by te symmetry of te second derivative, E p s E p s + η LH,H c log 1/2 η H 1/2 2 log η H 1/2 for all η H, were c 2 = c 2 γ C 4, γ 1 L from Proposition 2.1. We can now coose ɛ 0 = ɛ 0 γ C 4, γ 1 L, λ so tat log η H 1/2 < ɛ 0 implies 115 E p s E p s + η LH,H λ 4. By furter restricting 0, again wit te same dependencies, we can ensure tat, see 109, < 0 implies c 1 λ 4 ɛ 0 log 1. From 109, 114, 115, 116 and Lemma 5.1 wit α = λ/2, β = λ/4, δ = c 1 and ɛ = ɛ 0 log 1, it follows tat for 0 < 0 tere is a unique s H wic is stationary for E and suc tat 117 s p s H 1/2 ɛ 0 log 1. Next apply Lemma 5.1 wit α = λ/2, β = λ/4, δ = c 1 and ɛ = 4c 1 /λ. Note tat δ = α βɛ = c 1. Moreover, if 0 < 0 ten ɛ ɛ 0 log 1 from 116. It follows tat te unique stationary s as in 117 satisfies 118 were c 0 = 4c 1. Since s p s H 1/2 c 0 λ, s p s H 1/2 c 3/2 s H 2 c 3/2 s C 2 c 3/2 γ C 3, we may replace p s by s in 117 and 118, after furter restricting 0, ɛ 0 and c 0 if necessary. Remark Suppose u = Φγ s:d IR n is armonic. Ten u H 1 D\D c u H 1 D from 70 and 71 c γ C 1 s C 1 from 57 and 59 c, were c = c γ C 2. Tus te contribution to u H1 D from te boundary strip D \ D is O. Tis is consistent wit te order of approximation in te following Teorem. Teorem 5.5. Assume γ C 4. Let u be a nondegenerate minimal surface spanning Γ wit nondegeneracy constant λ as in 13.

24 24 GERHARD DZIUK AND JOHN E. HUTCHINSON Ten tere exist positive constants 0 and c 0 depending on γ C 4 and γ 1 L, and λ in te case of 0, suc tat if 0 < 0 ten tere is a discrete minimal surface u satisfying 119 u u H 1 D c 0 λ 1. Moreover, tere exists ɛ 0 = ɛ 0 γ C 4, γ 1 L, λ > 0 suc tat if u = Φγ s and u = Φ I γ s ten u is te unique discrete minimal surface satisfying 120 s s H 1/2 ɛ 0 log 1. Proof. Let u = Φγ s, u = Φ I γ s, were s and s are as in Proposition 5.4. We begin wit 0, ɛ 0 and c 0 as in Proposition 5.4. Now 121 But 122 u u H1 D = Φγ s Φ I γ s H1 D Φγ s Φ I γ p s H1 D + Φ I γ p s γ s H1 D = A + B. A c γ C 2 s 2 C 2, by te estimate 77 for I 1 in te proof of Proposition 4.1. Also, B c 1 + log 1/2 s p s H 1/2 s p s H 1/2 γ C 2 s C 1 by te estimate for A 1 in te proof of Proposition 5.2, wit s and η tere replaced by p s and s p s respectively. Hence 123 B c 0 from 118 λ provided 0 < 0, for a new 0 and c 0 wit te same dependencies. It follows from 121, 122 and 123, possibly again wit new c 0 and 0 wit te same dependencies as before, tat 124 u u H 1 D c 0 λ if 0 < 0. Now 125 u u H1 D c u u H1 D + u u L2 D, as follows easily by integrating along rays c depends only on n. It is routine to estimate u u L 2 D. Let Ten ũ = Φ I γ p s. u u L 2 D u ũ L 2 D + ũ u L 2 D 126 c 2 γ C 2 s 2 C 2 + ũ u L2 D = c 2 γ C 2 s 2 C 2 + D

25 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS 25 from te argument used to estimate I 2 in te proof of Proposition 4.1, wit u tere replaced by ũ. Moreover, D = I γ p s γ s L 2 D I D γ p s γ s L2 D by 21 as π 1 γ p s γ s L 2 + c γ p s γ s H 1 c γ C 1 p s s L 2 + c γ C 2 s C 1 p s s H 1 by 60, 62 and 23 c γ C 2 s C 1 p s s L 2 by a standard inverse estimate 127 c 0 from 118 λ wit a new c 0 but wit te same dependencies, for 0 < 0. From 124, 125, 126 and 127 it follows tat if 0 < 0 ten u u H 1 D c 0 λ, were c 0 = c 0 γ C 3, γ 1 L and 0 = 0 γ C 3, γ 1 L, λ. References [Be] M. Berger, Nonlinearity and Functional Analysis, Academic Press [BPS] J.H. Bramble, J.E. Pasciak & A.H. Scatz, An Iterative Metod for Elliptic Problems on Regions Partitioned into Substructures, Mat. Comp , [DH0] G. Dziuk, J.E. Hutcinson, On te approximation of unstable parametric minimal surfaces, preprint No SFB 256, Bonn, or CMA Mat. Res. Rep , Australian National University. [DH1] G. Dziuk, J.E. Hutcinson, On te approximation of unstable parametric minimal surfaces, Calc. Var , [DH3] G. Dziuk, J.E. Hutcinson, A Finite Element Metod for te Computation of Parametric Minimal Surfaces, Tatra Mount. Mat. J , [DH4] G. Dziuk, J.E. Hutcinson, Te Discrete Plateau Problem: Algoritm and Numerics, to appear. [JK] D.S. Jerison & C.E. Kenig, Boundary Value Problems on Lipscitz Domains, Studies in Matematics 23, 1 68, ed. W. Littman, MAA Studies in Matematics. [LM] J.L. Lions & E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Grundleren der Matematiscen Wissenscaften 181, Springer-Verlag [St] M. Struwe, Plateau s Problem and te Calculus of Variations, Matematical Notes 35, Princeton University Press Institut für Angewandte Matematik, Universität Freiburg, Hermann Herder Str. 10, D Freiburg i. Br., GERMANY address: gerd@matematik.uni-freiburg.de Department of Matematics, Scool of Matematical Sciences, Australian National University, GPO Box 4, Canberra, ACT 0200, AUSTRALIA address: Jon.Hutcinson@anu.edu.au