FINITE ELEMENT APPROXIMATIONS AND THE DIRICHLET PROBLEM FOR SURFACES OF PRESCRIBED MEAN CURVATURE

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1 FINITE ELEMENT APPROXIMATIONS AND THE DIRICHLET PROBLEM FOR SURFACES OF PRESCRIBED MEAN CURVATURE GERHARD DZIUK AND JOHN E. HUTCHINSON Abstract. We give a finite element procedure for te Diriclet Problem corresponding to surfaces of prescribed mean curvature and prove an optimal convergence estimate in te H 1 -norm. 1. H-Harmonic Maps Te numerical solution of te classical H-Plateau Problem consists of approximating disc-like surfaces wit prescribed boundary curve and prescribed mean curvature H. For a detailed discussion of te algoritms and teory see [6] for te case of zero mean curvature, and [7] for te constant mean curvature case. In tis paper we consider te associated H-Diriclet problem. Estimates for finite element approximations to solutions of general nonlinear elliptic systems are obtained in [4], using a continuity metod involving L estimates for te discrete problem. Here we give a muc sorter proof of te H 1 estimate, avoiding te need for L estimates and only assuming te discrete and smoot data are close in te H 1 sense. Our tecniques apply to a wide class of nonlinear systems. We treat te case of a non-polygonal and non-convex boundary and give te explicit dependence on te non-degeneracy constant of te smoot solution being approximated. Te arguments are prototypes of tose used in [7] for treating te more difficult case of te (free boundary) H-Plateau Problem. Te main tool for avoiding L norms in te present borderline case is te isoperimetric inequality due to Rado, see Remark 3.1. Tis work was partially supported by te Australian Researc Council. Trougout, ( R 2 ) is a bounded domain wit C 2 boundary. Function spaces will consist of functions defined over wit values in R 3 unless oterwise clear from context. Constants will depend on and oter quantities as indicated Matematics Subject Classification. Primary: 65N30; Secondary: 49Q05, 53A10. Key words and prases. finite elements, mean curvature, H-surface, H-armonic. 1

2 2 GERHARD DZIUK AND JOHN E. HUTCHINSON By H 1 is meant te H 1 seminorm, and by H 1 te full norm. Note tat by Poincaré s inequality, H 1 () is a norm on H 1 0(). For vectors a, b, c R 3, te triple product is defined by [a, b, c] =a b c. Tis is invariant under cyclic permutations of a, b and c, and antisymmetric wit respect to intercanging any two. It is te volume of te parallelopiped spanned by a, b and c. Definition 1.1. Suppose H is a real number. A function u H 2 (; R 3 ) is H-armonic wit boundary data u 0 H 2 (; R 3 )if (1.1) (1.2) u=2hu x u y u = u 0 a.e. in on Example 1.1. Let D be te closed unit disc in R 2. Let u 0 (x, y) =(x, y, 0): D R 3 wit 0 <H<1. Tere are two solutions of (1.1) and (1.2) obtained by mapping te unit disc D conformally, i.e. stereograpically projecting from a suitable point, onto te lower sperical caps obtained from eac of te two speres of radius 1/H (mean curvature H) wic contain te image of u 0. Tese solutions are called small or large depending on weter teir images do not, or do, contain a emispere. (We use tis example for test computations, see Tables 1 and 2.) If 1 <H< 0 ten one similarly obtains two solutions from te upper sperical caps. If H = 0 ten tere is exactly one solution, te map u(x, y) = (x, y, 0): D R 3. If H = 1 ten one obtains a solution by mapping onto te lower emispere of a spere of radius 1, and onto te upper emispere if H = 1. Equation (1.1) is te Euler-Lagrange system associated to te H- Diriclet integral D H (u) =D H (u;)= 1 (1.3) u 2 +2HV (u), 2 were V (u) =V(u;):= 1 (1.4) [u, u x,u y ] 3 can be tougt of as te signed volume of te cone over te origin obtained from te image of u. In fact, direct computation and integration by parts easily gives D H (u),ϕ = D H (u;),ϕ := d D dt H (u + tϕ) (1.5) t=0 = u ϕ +2H [ϕ, u x,u y ]

3 PRESCRIBED MEAN CURVATURE DIRICHLET PROBLEM 3 for u C 2 ( ; R 3 ) and ϕ C0( ; 2 R 3 ), and ence for u H 1 L (; R 3 ) and ϕ H0 1 L (; R 3 ) by a limit argument, for example see [10, Remark III.1.1]. If u H 1 L is stationary for D H, i.e. (1.6) u ϕ +2H [ϕ, u x,u y ]=0 for all ϕ H0 1 L, ten u is said to be a weak solution of (1.1). Example 1.1 is fairly typical. Arguing euristically, te energy functional D H (u) is cubic in u and tus one expects (generically) eiter two or no stationary points. In te former case one expects te smaller solution to be a local minimum and te larger solution to be unstable. Indeed, one as te following result due to te combined work of Heinz, Werner, Hildebrandt, Jäger, Wente, Brezis Coron, Struwe and Steffen. For detailed references see Struwe [10, 11]. Teorem 1.1. Assume u 0 H 1 L (; R 3 ) and H R satisfy u 0 L H 1. Ten tere exists u u 0 + H0 1 L suc tat D H (u) = min { D H (v) :v u 0 +H 1 0, v L H 1 }. Moreover, u L u 0 L ( ) and u is a weak solution to (1.1) and (1.2). If furtermore u 0 L H < 1 ( ) ten u is te unique local minimum of D H in u 0 + H0 1 L. Moreover, u is te unique weak solution of (1.1) and (1.2) wic satisfies ( ). Te function u is called te small solution of (1.1) and (1.2). Under te same assumption ( ) if H 0and u 0 is not constant, tere is also a second weak solution ū to (1.1) and (1.2) wic satisfies ū L > u 0 L. Any suc solution is called a large solution to (1.1) and (1.2). If u 0 H 2 (, R 3 ) ten any weak solution to (1.1) and (1.2) belongs to H 2 (, R 3 ). Remark Te large solution need not be unique, altoug one would expect tat tis is te generic situation. An example of Wente, [11, Example IV.3.7], gives a continuum of solutions for te unit disc and boundary data u 0 (x, y) =(x, 0, 0). See Fig. 1 for te image of te trivial small solution and of one of te large solutions on a relatively coarse grid. Rotation of u() around te u 1 -axis gives a continuum of solutions.

4 4 GERHARD DZIUK AND JOHN E. HUTCHINSON Figure 1. Wente s example (discrete approximations); small and one of a continuum of large solutions 2. Te existence of a large solution is obtained by a mountain pass type argument, see [2] and [11, Teorem III.4.8] Remark 1.2 (Nondegeneracy). We will be interested in approximating functions u H 2 (; R 3 ) wic are H-armonic and nondegenerate in te sense tat te second variation D H (u) as no zero eigenvalues. Tis is always true for small solutions, see [11, Lemma III.4.7]. More precisely, for u H 2 (; R 3 ) and ϕ, ψ H0(; 2 R 3 ) one first easily cecks by direct computation and integration by parts tat D H (u)(ϕ, ψ) =D H (u; )(ϕ, ψ) := 2 D s t H (u + tϕ + sψ) s=t=0 (1.7) = ϕ ψ +2H [u, ϕ x,ψ y ]+[u, ψ x,ϕ y ] (1.8) = ϕ ψ +2H [ψ, u x,ϕ y ]+[ψ, ϕ x,u y ], see [10, Remark III.1.1] and te paragrap following (3.2). From (1.8) D H (u) extends to a bounded symmetric bilinear functional on H 1 0, since u ϕ ϕ u L 4 ϕ L 4 ϕ L 2 c u H 2 ϕ 2 L 2 It follows tat te inner product H 1 induces a bounded self-adjoint linear operator 2 D H (u): H0 1 H0. 1 Te eigenvalues of 2 D H (u) are real, bounded below by λ 0 (say) and ave no accumulation point. Moreover, λ 0 = λ 0 (,H, u H 2), as follows from using (1.8) to estimate te Raleig-Ritz quotient. Te nondegeneracy constant λ of D H (u) is defined by λ = min { γ : γ is an eigenvalue of 2 D H (u) }

5 PRESCRIBED MEAN CURVATURE DIRICHLET PROBLEM 5 Ten 2 D H (u) is one-one and onto iff λ>0. Let (1.9) ϕ = ϕ + + ϕ denote te H 1 ortogonal decomposition of ϕ H 1 0 into members of te positive and negative spaces H + and H corresponding to te eigenvalues and eigenfunctions of 2 D H (u). Ten (1.10) D H (u)(ϕ, ϕ + ϕ ) λ ϕ 2 H 1 for all suc ϕ and λ is te largest suc real. From te eigenfunction equation, c.f. (1.8), togeter wit te estimate for λ 0, one obtains ϕ H 2 and (1.11) ϕ H 2 ν ϕ H 1 were ν = ν(,h, u H 2,d) wit d te dimension of H. 2. Discrete H-Harmonic Maps For >0 let T be a triangulation of by triangles T wose side lengts are bounded above by c for some c independent of and wose interior angles are bounded away from zero uniformly and independently of. Te intersection of any two different triangles is eiter empty, a common vertex, or a common edge. Let = T. T T Let X = { } u C 0 ( ; R 3 ):u T P 1 (T) T T, X 0 ={ϕ X :ϕ =0}, were P 1 (T ) is te set of polynomials over T of degree at most one. For some δ>0 and all sufficiently small, := { x R 2 : d(x, ) <δ}. If u H 2 () ten by te C 2 regularity of tere exists an extension of u to, also denoted by u, suc tat (2.1) u H 2 ( ) c u H 2 (). Definition 2.1. Te discrete H-Diriclet integral is defined by D H (u ; )= 1 u 2 +2HV (u ; ) 2 for u X.

6 6 GERHARD DZIUK AND JOHN E. HUTCHINSON It follows from (1.5) and (1.7) wit replaced by and a limit argument, or by direct computation and noting tat boundary integrals on internal edges cancel, tat (2.2) D H(u ; ),ϕ = u ϕ +2H [ϕ,u x,u y ], D H (u ; )(ϕ,ψ )= (2.3) ϕ ψ +2H [u,ϕ x,ψ y ]+[u,ψ x,ϕ y ], for u X and ϕ,ψ X 0. Motivated by (1.6), one as Definition 2.2. A function u X is discrete H-armonic if (2.4) u ϕ +2H [u x,u y,ϕ ]=0, for all ϕ X 0. We will prove te following. Teorem 2.1. Let u H 2 (; R 3 ) be H-armonic and u = u 0 on were u 0 H 2 (; R 3 ). Assume u is nondegenerate wit nondegeneracy constant λ. Let u 0 X and assume u 0 u 0 H 1 ( ) α. Ten tere exist constants 0 = 0 ( u H 2, u 0 H 2,α,,d,H,λ),ε 0 = ε 0 (H, λ), and c 0 = c 0 ( u H 2, u 0 H 2,α,H) suc tat if 0 < 0 ten: 1. Tere exists a unique discrete H-armonic function u suc tat u = u 0 on and 2. Moreover, u u H 1 ( ) ε 0 ; u u H 1 ( ) c 0 λ Proof of Main Teorem Wit u, u 0 and u 0 as in te main teorem define (3.1) J u = u 0 + I (u u 0 ) u 0 + X 0, were I is te standard nodal interpolation operator. Te proof of te main teorem will use te following quantitative version of te Inverse Function Teorem wit X = u 0 +X 0, X = X 0, Y = X 0 (te dual space of X 0 ), x 0 = J u, f = D H( ; ). Te proof of te lemma follows from tat in [1] pp Lemma 3.1. 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

7 PRESCRIBED MEAN CURVATURE DIRICHLET PROBLEM 7 f C 1 (X,Y). Assume tere are positive constants α, β, δ and ε suc tat were f(x 0 ) Y δ, f (x 0 ) 1 L(Y,X) α 1, f (x) f (x 0 ) L(X,Y ) β for all x B ε (x 0 ), β<α, δ (α β)ε. Ten tere exists a unique x B ε (x 0 ) suc tat f(x )=0. Remark 3.1 (Te Volume Functional). A fundamental result wic is due to Wente [12] states tat for any u 0 H 1 L (; R 3 ) te functional V (and ence D H ) extends to an analytic functional on te affine space u 0 + H0(; 1 R 3 ). Tis is peraps surprising, since from (1.5) and (1.7) one migt expect bounds for te relevant integrals to also involve ϕ L and u L respectively. More generally, one as te following. For u, v, w H 1 L (; R 3 ) define te trilinear functional V (u, v, w) =V(u, v, w;)= 1 (3.2) [u, v x,w y ]+[u, w x,v y ]. 6 Note tat V (u) =V(u, u, u). Assume now tat at least one of u, v, w also belongs to H0(; 1 R 3 ). Ten V is invariant under cyclic permutations of its arguments, as follows from integration by parts in te C 2 case and in general by a limit argument, see [10, Remark III.1.1.iii]. Since V is invariant under permutation of its first two arguments, it ten follows it is invariant under any permutation of its arguments. Moreover under te same assumptions, from an argument similar to tat in [10, proof of Teorem III.2.3] wic uses an isoperimetric inequality due to Radó [8], one also as (3.3) V (u, v, w) c u H 1 () v H 1 () w H 1 (). Similar remarks and estimates apply if is everywere replaced by. Assume now u, v H 1 L (; R 3 ) and ϕ H 1 0 L (; R 3 ). It follows tat (3.4) (3.5) (3.6) V (u),ϕ =3V(u, u, ϕ) c u 2 H () ϕ 1 H 1 (), V (u)(v, ϕ) =6V(u, v, ϕ) c u H 1 () v H 1 () ϕ H 1 (), V ( )(u, v, ϕ) =6V(u, v, ϕ) c u H 1 () v H 1 () ϕ H 1 (). (In particular, if u H 1 L (; R 3 ) tese estimates allow one to define te integrals in (1.5) and (1.7) for arbitrary ϕ, ψ H 1 0(; R 3 ).) Similar results also old if is replaced by.

8 8 GERHARD DZIUK AND JOHN E. HUTCHINSON For te remainder of tis section, u is as in te Main Teorem. Extend u to as in (2.1) and restrict to as necessary. Bot te extension and restriction will also be denoted by u. Lemma 3.2. u J u H 1 ( ) c 1 were c 1 = c 1 ( u H 2 (), u 0 H 2 (),α). Proof. u J u H 1 ( ) = (u 0 u 0 ) (I (u u 0 ) (u u 0 )) H 1 ( ) u 0 u 0 H 1 ( ) + c u u 0 H 2 () c, were c = c( u H 2, u 0 H 2,α). Since u H 1 ( ) c u H 2 ( ) c u H 2 () by elementary estimates and (2.1), te result follows. Lemma 3.3. If ϕ X 0 ten D H(J u; ),ϕ c 2 ϕ H 1 ( ), were c 2 = c 2 ( u H 2 (), u 0 H 2 (),α,h). Proof. D H(J u; ),ϕ = ( D H(J u; ),ϕ D H(u; ),ϕ ) + D H(u; ),ϕ =: A + B From te Taylor series expansion for V ( ; ) and Remark 3.1 A = (J u u) ϕ +2H V (J u; ),ϕ V (u; ),ϕ J u u H 1 ( ) ϕ H 1 ( ) +2 H V (u; )(J u u, ϕ ) + H V (u; )(J u u, J u u, ϕ ) J u u H 1 ( ) ϕ H 1 ( ) + c H u H 1 ( ) J u u H 1 ( ) ϕ H 1 ( ) + c H J u u 2 H 1 ( ) ϕ H 1 ( ) c ϕ H 1 ( ), from Lemma 3.2 and (2.1), were c = c( u H 2 (), u 0 H 2 (),α,h). Also, B = u ϕ +2H [ϕ,u x,u y ] = ( u +2Hu x u y ) ϕ = ( u +2Hu x u y ) ϕ c ϕ L 2 ( ) c ϕ H 1 ( ) were c = c( u H 2,H), as follows from (2.1), a Sobolev imbedding teorem, and elementary calculus. Te required result follows.

9 PRESCRIBED MEAN CURVATURE DIRICHLET PROBLEM 9 Remark 3.2 (A Discrete Eigenspace Decomposition). If ϕ X 0 let ϕ also denote te zero extension to. Note tat ϕ / H0() 1 unless is convex. For tis reason define P : X 0 H0() 1 to be te H 1 projection, i.e. (Pϕ ) ϕ = ϕ ϕ for all ϕ H0(). 1 One as (3.7) ϕ H 1 ( ) c 1/2 ϕ H 1 ( ) (3.8) Pϕ H 1 () ϕ H 1 ( ) (3.9) ϕ Pϕ H 1 () c 1/2 ϕ H 1 ( ) To see (3.7) note tat ϕ is constant on any triangle T T and tat T ( ) c T. Inequality (3.8) is immediate, since P is just H 1 ortogonal projection onto H0(). 1 For (3.9) first note tat ϕ Pϕ H 1 () ϕ ϕ H 1 () for any ϕ H0(), 1 by ortogonality. Now coose ϕ by suitably deforming ϕ in a boundary strip. Let (Pϕ ) +,(Pϕ ) H0() 1 be te components of Pϕ as in (1.9). Note tat (Pϕ ) is smoot, and in particular (3.10) (Pϕ ) H 2 () ν ϕ H 1 ( ) since (Pϕ ) H 2 () ν (Pϕ ) H 1 () ν Pϕ H 1 () ν ϕ H 1 ( ) from (1.11), (1.9) and te H 1-ortogonality of (Pϕ ) + and (Pϕ ), and (3.8). Define a discrete analogue of (1.9) by ϕ ( ) = I (Pϕ ) X 0, ϕ (+) = ϕ, (3.11) ϕ = ϕ (+) + ϕ ( ). Taking te zero extension of (Pϕ ) and (Pϕ ) + to, and of ϕ ( ) and ϕ (+) to, we claim (Pϕ ) ϕ ( ) H 1 ( ) c ϕ H 1 ( ), (3.12) (Pϕ ) + ϕ (+) H 1 ( ) c 1/2 ϕ H 1 ( ), were c = c(ν) Proof of claim. (Pϕ ) ϕ ( ) H 1 ( ) c (Pϕ ) H 2 () cν ϕ H 1 ( ) from (3.10). Also (Pϕ ) H 1 ( ) c (Pϕ ) H 2 () cν ϕ H 1 ( ). Tis gives te first result.

10 10 GERHARD DZIUK AND JOHN E. HUTCHINSON For te second, (Pϕ ) + ϕ (+) H 1 ( ) Pϕ ϕ H 1 ( ) + (Pϕ ) ϕ ( ) H 1 ( ) c( 1/2 + ν) ϕ H 1 ( ) from te first result and (3.9). On, ϕ = ϕ (+) required estimate now follows from (3.9). = 0 and so te We also ave (3.13) ϕ ( ) H 1 ( ) (1 + c) ϕ H 1 ( ) ϕ (+) H 1 ( ) (1 + c 1/2 ) ϕ H 1 ( ) from (3.12), te ortogonal decomposition Pϕ =(Pϕ ) +(Pϕ ) + and (3.8). Tus (3.11) is an almost ortogonal decomposition for small. Lemma 3.4. If ϕ X 0 ten D H(J u; )(ϕ,ϕ (+) ) 3λ 4 ϕ 2 H 1 ( ) provided 1 were 1 = 1 ( u H 2 (), u 0 H 2 (),α,,d,h,λ). Proof. Since V ( ; ) is cubic, from (2.3) D H(J u; )(ϕ,ϕ (+) = D H (u; )(ϕ,ϕ (+) But ) 2HV (u; )(J u u, ϕ,ϕ (+) )+2HV (u; )(J u u, ϕ,ϕ (+) ) c J u u H 1 ( ) ϕ H 1 ( ) ϕ (+) H 1 ( ) c ϕ 2 H 1 ( ) ). were c = c( u H 2 (), u 0 H 2 (),α,ν,h), from Remark 3.1, and also from Lemma 3.2 and (3.13). Now were D H (u; )(ϕ,ϕ (+) E 1 c(1 + u L ) ϕ H 1 ( ) ϕ (+) )=D H (u; )(ϕ,ϕ (+) )+E 1 H 1 ( ) c ϕ 2 H 1 ( ) wit c = c( u H 2 (),ν,h), from (3.7) and (3.12), since (Pϕ ) + = (Pϕ ) = 0 in.

11 PRESCRIBED MEAN CURVATURE DIRICHLET PROBLEM 11 Also D H (u; )(ϕ,ϕ (+) ) = D H (u; )(Pϕ,(Pϕ ) + (Pϕ ) ) +D H (u; )(ϕ Pϕ,(Pϕ ) + (Pϕ ) ) from (1.10). But +D H (u; )(ϕ, (ϕ (+) λ Pϕ 2 H 1 () + E 2 + E 3 (Pϕ ) + ) (ϕ ( ) (Pϕ ) )) E 2, E 3 c(1 + u L ) 1/2 ϕ 2 H 1 ( ) from (3.9) and (3.8), and (3.12) respectively. It follows tat D H(J u; )(ϕ,ϕ (+) ) λ Pϕ 2 H 1 () c 1/2 ϕ 2 H 1 ( ) 3λ 4 ϕ 2 H 1 ( ) from (3.9), for 1 = 1 ( u H 2 (), u 0 H 2 (),α,,d,h,λ). Lemma 3.5. If v u 0 + X 0 and ϕ,ψ X 0 ten D H(v ; )(ϕ,ψ ) D H(J u; )(ϕ,ψ ) λ 4 ϕ H 1 ( ) ψ H 1 ( ) provided v J u H 1 ( ) ε 1 were ε 1 = ε 1 (H, λ). Proof. Tis follows from D H(v ; )(ϕ,ψ ) D H(J u; )(ϕ,ψ ) =2HV (J u; )(v J u, ϕ,ψ ), (3.6) and Lemma 3.2. Completion of proof of Main Teorem. We use Lemma 3.1 wit X = u 0 + X 0, X = X 0, Y = X 0, x 0 = J u, f = D H( ; ). Te norm on X 0 is H 1 ( ) and on X 0 is te corresponding dual norm. Note tat D H( ; ): u 0 +X 0 X0 wit derivative D H( ; ): u 0 +X 0 L(X 0,X0) using standard identifications. From Lemma 3.3 (3.14) D H(J u; ) c 2. From Lemma 3.4, D H(J u; ) is invertible and ( [D H(J u; )] 1 / 3λ 4 ϕ H 1 ( ) ϕ (+) H 1 ( ) ) 1

12 12 GERHARD DZIUK AND JOHN E. HUTCHINSON provided 1. But ϕ (+) H 1 ( ) (Pϕ ) + (Pϕ ) H 1 ( ) + ϕ (+) (Pϕ ) + H 1 ( ) + ϕ ( ) (Pϕ ) H 1 ( ) (1 + c 1/2 ) ϕ H 1 ( ) were c = c(ν), from (3.8) and (3.12). Hence (3.15) ( ) 1 [D H(J u; )] 1 λ 2 if 3 were 3 = 3 ( u H 2 (), u 0 H 2 (),α,,d,h,λ). Finally, from Lemma 3.5 (3.16) D H(v ; ) D H(J u; ) λ 4 if v J u H 1 ( ) ε 1 were ε 1 = ε 1 (H, λ). Take δ = c 2, α = λ/2, β = λ/4 and ε = ε 1. Ten from (3.14) (3.16) te ypoteses of Lemma 3.1 are satisfied provided 3, c 2 λ ε 4 1. Tis establises te first (uniqueness) part of te main teorem wit ε 0 = ε 1 and 0 = 0 ( 3,ε 1,λ,c 2 )= 0 ( u H 2, u 0 H 2,α,ν,H,λ). Taking δ = c 2, α = λ/2, β = λ/4 and ε = λ 1 c 0 te ypoteses of Lemma 3.1 are again satisfied from (3.14) (3.16) provided 3, c 2 1c 4 0. Tis establises te second (O() convergence) part of te main teorem wit 0 = 3 and c 0 =4c Numerical Results In Tables 1 and 2 we present te results of test computations for te explicitly known sperical solutions described in Example 1.1 wit H =0.5 and = B 1 (0). Denote by e te error between te continuous solution and te discrete solution in te cosen norm. For two successive grids wit grid sizes 1 and 2 te experimental order of convergence is eoc =ln e / 1 ln 1. e 2 2 Te test computations confirm te order 1 for te H 1 ()-norm and additionally sow te order 2 for te L 2 ()-norm. Figures 2 and 3 sow computational results wit = B 1 (0), H = 0.5 and boundary values u(e iφ ) = (cos(φ), sin(φ), (2 + 3) cos(2φ) 0.5 cos(6φ)) on a grid wit 8192 triangles. For better visibility te resulting surfaces are scaled, but te boundaries of te solution surfaces are te same. Figure 4 sows a solution for te annular domain = { x 1 < x < 2 } and boundary data wic give knotted boundary curves.

13 PRESCRIBED MEAN CURVATURE DIRICHLET PROBLEM 13 nodes level L 2 -error L 2 -eoc H 1 -error H 1 -eoc e e e e e e e e e e e e Table 1. Small solution, H =0.5 nodes level L 2 -error L 2 -eoc H 1 -error H 1 -eoc Table 2. Large solution, H =0.5 Figure 2. Small solution, H =0.5 References [1] M. Berger, Nonlinearity and Functional Analysis, Academic Press, [2] H. Brezis, J.-M. Coron, Multiple solutions of H systems and Rellic s conjecture, Comm. Pure Appl. Mat. 37, 1984,

14 14 GERHARD DZIUK AND JOHN E. HUTCHINSON Figure 3. Large solution, H =0.5 Figure 4. Annulus, H =0.5 [3] P. G. Ciarlet, Te Finite Element Metods for Elliptic Problems, Nort Holland, [4] M. Dobrowolski, R. Rannacer Finite element metods for nonlinear elliptic systems of second order, Mat. Nacr. 94, 1974, [5] G. Dziuk, J.E. Hutcinson, On te approximation of unstable parametric minimal surfaces, Calc. Var. 4, 1996, [6] G. Dziuk, J.E. Hutcinson, A finite element metod for approximating minimal surfaces, Preprint 4 Matematisce Fakultät Freiburg, 1996 CMA Mat. Res. Rep. 5, Australian National University, [7] G. Dziuk, J.E. Hutcinson, A finite element metod for approximating surfaces of prescribed mean curvature, in preparation.

15 PRESCRIBED MEAN CURVATURE DIRICHLET PROBLEM 15 [8] T. Radó, Te isoperimetric inequality and te Lebesgue definition of surface area, Trans. Amer. Mat. Soc. 61, 1947, [9] M. Rumpf, A. Scmidt et al, GRAPE, Grapics Programming Environment, Report 8, SFB 256, Bonn (1990). [10] M. Struwe, Plateau s Problem and te Calculus of Variations, Princeton University Press, [11] M. Struwe, Variational Metods, Springer Verlag, [12] H. C. Wente, An existence teorem for surfaces of constant mean curvature, J. Mat. Anal. Appl. 26, 1969, Institut für Angewandte Matematik, Universität Freiburg, Hermann Herder Str. 10, D Freiburg i. Br., GERMANY address: gerd@matematik.uni-freiburg.de Scool of Matematical Sciences, Australian National University, GPO Box 4, Canberra, ACT 0200, AUSTRALIA address: Jon.Hutcinson@anu.edu.au