Isometric Immersions and Compensated Compactness
aa r X i v : . [ m a t h . A P ] M a y ISOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS
GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG
Abstract.
A fundamental problem in differential geometry is to characterize intrinsicmetrics on a two-dimensional Riemannian manifold M which can be realized as iso-metric immersions into R . This problem can be formulated as initial and/or boundaryvalue problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, wedevelop a general approach, which combines a fluid dynamic formulation of balance lawsfor the Gauss-Codazzi system with a compensated compactness framework, to deal withthe initial and/or boundary value problems for isometric immersions in R . The com-pensated compactness framework formed here is a natural formulation to ensure theweak continuity of the Gauss-Codazzi system for approximate solutions, which yieldsthe isometric realization of two-dimensional surfaces in R .As a first application of this approach, we study the isometric immersion problemfor two-dimensional Riemannian manifolds with strictly negative Gauss curvature. Weprove that there exists a C , isometric immersion of the two-dimensional manifold in R satisfying our prescribed initial conditions. To achieve this, we introduce a vanishingviscosity method depending on the features of initial value problems for isometric im-mersions and present a technique to make the apriori estimates including the L ∞ controland H − –compactness for the viscous approximate solutions. This yields the weak con-vergence of the vanishing viscosity approximate solutions and the weak continuity of theGauss-Codazzi system for the approximate solutions, hence the existence of an isometricimmersion of the manifold into R satisfying our initial conditions. Introduction
A fundamental problem in differential geometry is to characterize intrinsic metrics on atwo-dimensional Riemannian manifold M which can be realized as isometric immersionsinto R (cf. Yau [39]; also see [20, 33, 35]). Important results have been achieved forthe embedding of surfaces with positive Gauss curvature which can be formulated as anelliptic boundary value problem (cf. [20]). For the case of surfaces of negative Gausscurvature where the underlying partial differential equations are hyperbolic, the compli-mentary problem would be an initial or initial-boundary value problem. Hong in [22] firstproved that complete negatively curved surfaces can be isometrically immersed in R if theGauss curvature decays at certain rate in the time-like direction. In fact, a crucial lemmain Hong [22] (also see Lemma 10.2.9 in [20]) shows that, for such a decay rate of the nega-tive Gauss curvature, there exists a unique global smooth, small solution forward in timefor prescribed smooth, small initial data. Our main theorem, Theorem 5.1(i), indicatesthat in fact we can solve the corresponding problem for a class of large non-smooth initialdata. Possible implication of our approach may be in existence theorems for equilibrium Date : November 4, 2018.2000
Mathematics Subject Classification.
Primary: 53C42, 53C21, 53C45, 58J32, 35L65, 35M10, 35B35;Secondary: 53C24, 57R40, 57R42, 76H05, 76N10. configurations of a catenoidal shell as detailed in Vaziri-Mahedevan [38]. When the Gausscurvature changes sign, the immersion problem then becomes an initial-boundary valueproblem of mixed elliptic-hyperbolic type, which is still under investigation.The purpose of this paper is to introduce a general approach, which combines a fluiddynamic formulation of balance laws with a compensated compactness framework, to dealwith the isometric immersion problem in R (even when the Gauss curvature changessign). In Section 2, we formulate the isometric immersion problem for two-dimensionalRiemannian manifolds in R via solvability of the Gauss-Codazzi system. In Section 3, weintroduce a fluid dynamic formulation of balance laws for the Gauss-Codazzi system forisometric immersions. Then, in Section 4, we form a compensated compactness frameworkand present one of our main observations that this framework is a natural formulation toensure the weak continuity of the Gauss-Codazzi system for approximate solutions, whichyields the isometric realization of two-dimensional surfaces in R .As a first application of this approach, in Section 5, we focus on the isometric immersionproblem of two-dimensional Riemannian manifolds with strictly negative Gauss curvature.Since the local existence of smooth solutions follows from the standard hyperbolic theory,we are concerned here with the global existence of solutions of the initial value problemwith large initial data. The metrics g ij we study have special structures and forms usuallyassociated with(i) the catenoid of revolution when g = g = cosh ( x ) and g = 0;(ii) the helicoid when g = λ + y , g = 1, and g = 0.For these cases, while Hong’s theorem [22] applies to obtain the existence of a solutionfor small smooth initial data, our result yields a large-data existence theorem for a C , isometric immersion.To achieve this, we introduce a vanishing viscosity method depending on the featuresof the initial value problem for isometric immersions and present a technique to make theapriori estimates including the L ∞ control and H − –compactness for the viscous approx-imate solutions. This yields the weak convergence of the vanishing viscosity approximatesolutions and the weak continuity of the Gauss-Codazzi system for the approximate so-lutions, hence the existence of a C , –isometric immersion of the manifold into R withprescribed initial conditions.We remark in passing that, for the fundamental ideas and early applications of compen-sated compactness, see the classical papers by Tartar [37] and Murat [31]. For applicationsto the theory of hyperbolic conservation laws, see for example [4, 9, 12, 17, 36]. In particu-lar, the compensated compactness approach has been applied in [3, 6, 10, 11, 24, 25] to theone-dimensional Euler equations for unsteady isentropic flow, allowing for cavitation, inMorawetz [28, 29] and Chen-Slemrod-Wang [7] for two-dimensional steady transonic flowaway from stagnation points, and in Chen-Dafermos-Slemrod-Wang [5] for subsonic-sonicflows.2. The Isometric Immersion Problem for Two-Dimensional RiemannianManifolds in R In this section, we formulate the isometric immersion problem for two-dimensional Rie-mannian manifolds in R via solvability of the Gauss-Codazzi system. SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 3
Let Ω ⊂ R be an open set. Consider a map r : Ω → R so that, for ( x, y ) ∈ Ω, the twovectors { ∂ x r , ∂ y r } in R span the tangent plane at r ( x, y ) of the surface r (Ω) ⊂ R . Then n = ∂ x r × ∂ y r | ∂ x r × ∂ y r | is the unit normal of the surface r (Ω) ⊂ R . The metric on the surface in R is ds = d r · d r (2.1)or, in local ( x, y )–coordinates, ds = ( ∂ x r · ∂ x r ) ( dx ) + 2( ∂ x r · ∂ y r ) dxdy + ( ∂ y r · ∂ y r ) ( dy ) . (2.2)Let g ij , i, j = 1 , , be the given metric of a two-dimensional Riemannian manifold M parameterized on Ω. The first fundamental form I for M on Ω is I := g ( dx ) + 2 g dxdy + g ( dy ) . (2.3)Then the isometric immersion problem is to seek a map r : Ω → R such that d r · d r = I, that is, ∂ x r · ∂ x r = g , ∂ x r · ∂ y r = g , ∂ y r · ∂ y r = g , (2.4) so that { ∂ x r , ∂ y r } in R are linearly independent .The equations in (2.4) are three nonlinear partial differential equations for the threecomponents of r ( x, y ).The corresponding second fundamental form is II := − d n · d r = h ( dx ) + 2 h dxdy + h ( dy ) , (2.5)and ( h ij ) ≤ i,j ≤ is the orthogonality of n to the tangent plane. Since n · d r = 0, then d ( n · d r ) = 0 implies − II + n · d r = 0 , i.e., II = ( n · ∂ x r ) ( dx ) + 2( n · ∂ xy r ) dxdy + ( n · ∂ y r ) ( dy ) . The fundamental theorem of surface theory (cf. [13, 20]) indicates that there exists asurface in R whose first and second fundamental forms are I and II if the coefficients ( g ij ) and ( h ij ) of the two given quadratic forms I and II with ( g ij ) > satisfy the Gauss-Codazzi system . It is indicated in Mardare [27] (Theorem 9; also see [26]) that this theoremholds even when ( h ij ) is only in L ∞ for given ( g ij ) in C , , for which the immersion surfaceis C , . This shows that, for the realization of a two-dimensional Riemannian manifoldin R with given metric ( g ij ) >
0, it suffices to solve ( h ij ) ∈ L ∞ determined by theGauss-Codazzi system to recover r a posteriori.The simplest way to write the Gauss-Codazzi system (cf. [13, 20]) is as ∂ x M − ∂ y L = Γ (2)22 L − (2)12 M + Γ (2)11 N,∂ x N − ∂ y M = − Γ (1)22 L + 2Γ (1)12 M − Γ (1)11 N, (2.6)with LN − M = κ. (2.7)Here L = h p | g | , M = h p | g | , N = h p | g | , GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG | g | = det ( g ij ) = g g − g , κ ( x, y ) is the Gauss curvature that is determined by therelation: κ ( x, y ) = R | g | , R ijkl = g lm (cid:16) ∂ k Γ ( m ) ij − ∂ j Γ ( m ) ik + Γ ( n ) ij Γ ( m ) nk − Γ ( n ) ik Γ ( m ) nj (cid:17) ,R ijkl is the curvature tensor and depends on ( g ij ) and its first and second derivatives, andΓ ( k ) ij = 12 g kl ( ∂ j g il + ∂ i g jl − ∂ l g ij )is the Christoffel symbol and depends on the first derivatives of ( g ij ), where the summationconvention is used, ( g kl ) denotes the inverse of ( g ij ), and ( ∂ , ∂ ) = ( ∂ x , ∂ y ).Therefore, given a positive definite metric ( g ij ) ∈ C , , the Gauss-Codazzi system givesus three equations for the three unknowns ( L, M, N ) determining the second fundamentalform II . Note that, although ( g ij ) is positive definite, R may change sign and sodoes the Gauss curvature κ . Thus, as we will discuss in Section 3, the Gauss-Codazzisystem (2.6)–(2.7) generically is of mixed hyperbolic-elliptic type, as in transonic flow (cf.[2, 7, 8, 30]). In § § §
4. As an example of direct applications of this approach, in §
5, we show how this approach can be applied to establish an isometric immersion of atwo-dimensional Riemannian manifold with negative Gauss curvature in R .3. Fluid Dynamic Formulation for the Gauss-Codazzi System
From the viewpoint of geometry, the constraint condition (2.7) is a Monge-Amp`ereequation and the equations in (2.6) are integrability relations. However, our goal hereis to put the problem into a fluid dynamic formulation so that the isometric immersionproblem may be solved via the approaches that have shown to be useful in fluid dynamicsfor solving nonlinear systems of balance laws. To achieve this, we formulate the isometricimmersion problem via solvability of the Gauss-Codazzi system (2.6) under constraint(2.7), that is, solving first for h ij , i, j = 1 , , via (2.6) with constraint (2.7) and then recovering r a posteriori.To do this, we set L = ρv + p, M = − ρuv, N = ρu + p, and set q = u + v as usual. Then the equations in (2.6) become the familiar balancelaws of momentum: ∂ x ( ρuv ) + ∂ y ( ρv + p ) = − ( ρv + p )Γ (2)22 − ρuv Γ (2)12 − ( ρu + p )Γ (2)11 ,∂ x ( ρu + p ) + ∂ y ( ρuv ) = − ( ρv + p )Γ (1)22 − ρuv Γ (1)12 − ( ρu + p )Γ (1)11 , (3.1)and the Monge-Amp`ere constraint (2.7) becomes ρpq + p = κ. (3.2)From this, we can see that, if the Gauss curvature κ is allowed to be both positive andnegative, the “pressure” p cannot be restricted to be positive. Our simple choice for p is SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 5 the Chaplygin-type gas: p = − ρ . Then, from (3.2), we find − q + 1 ρ = κ, and hence we have the “Bernoulli” relation: ρ = 1 p q + κ . (3.3)This yields p = − p q + κ, (3.4)and the formulas for u and v : u = p ( p − M ) , v = p ( p − L ) , M = ( N − p )( L − p ) . The last relation for M gives the relation for p in terms of ( L, M, N ), and then the firsttwo give the relations for ( u, v ) in terms of (
L, M, N ).We rewrite (3.1) as ∂ x ( ρuv ) + ∂ y ( ρv + p ) = R ,∂ x ( ρu + p ) + ∂ y ( ρuv ) = R , (3.5)where R and R denote the right-hand sides of (3.1).We now find the corresponding “geometric rotationality–continuity equations”. Multi-plying the first equation of (3.5) by v and the second by u , and setting ∂ x v − ∂ y u = − σ, we see vρ div( ρu, ρv ) − ∂ y κ = R ρ + σu,uρ div( ρu, ρv ) − ∂ x κ = R ρ − σv, and hence div( ρu, ρv ) = 12 ρv ∂ y κ + R v + ρuσv , div( ρu, ρv ) = 12 ρu ∂ x κ + R u − ρvσu . (3.6)Thus, the right hand sides of (3.6) are equal, which gives a formula for σ : σ = 1 ρq (cid:16) v (cid:0) ρ∂ x κ + R (cid:1) − u (cid:0) ρ∂ y κ + R (cid:1)(cid:17) . (3.7) GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG
If we substitute this formula for σ into (3.6), we can write down our “rotationality-continuity equations” as ∂ x v − ∂ y u = 1 ρq (cid:16) u (cid:0) ρ∂ y κ + R (cid:1) − v (cid:0) ρ∂ x κ + R (cid:1)(cid:17) , (3.8) ∂ x ( ρu ) + ∂ y ( ρv ) = 12 ρuq ∂ x κ + 12 ρvq ∂ y κ + vq R + uq R . (3.9)In summary, the Gauss-Codazzi system (2.6)–(2.7), the momentum equations (3.1)–(3.4),and the rotationality-continuity equations (3.3) and (3.8)–(3.9) are all formally equivalent.However, for weak solutions, we know from our experience with gas dynamics that thisequivalence breaks down. In Chen-Dafermos-Slemrod-Wang [5], the decision was made(as is standard in gas dynamics) to solve the rotationality-continuity equations and viewthe momentum equations as “entropy” equalities which may become inequalities for weaksolutions. In geometry, this situation is just the reverse. It is the Gauss-Codazzi systemthat must be solved exactly and hence the rotationality-continuity equations will become“entropy” inequalities for weak solutions.The above issue becomes apparent when we set up “viscous” regularization that pre-serves the “divergence” form of the equations, which will be introduced in § c = p ′ ( ρ ) , (3.10)which in our case gives c = 1 ρ . (3.11)Since our “Bernoulli” relation is (3.3), we see c = q + κ. (3.12)Hence, under this formulation,(i) when κ >
0, the “flow” is subsonic, i.e., q < c , and system (3.1)–(3.2) is elliptic;(ii) when κ <
0, the “flow” is supersonic, i.e., q > c , and system (3.1)–(3.2) is hyperbolic;(iii) when κ = 0, the “flow” is sonic, i.e., q = c , and system (3.1)–(3.2) is degenerate.In general, system (3.1)–(3.2) is of mixed hyperbolic-elliptic type. Thus, the isomet-ric immersion problem involves the existence of solutions to nonlinear partial differentialequations of mixed hyperbolic-elliptic type.4. Compensated Compactness Framework for Isometric Immersions
In this section, we form a compensated compactness framework and present our newobservation that this framework is a natural formulation to ensure the weak continuity ofthe Gauss-Codazzi system for approximate solutions, which yields the isometric realizationof two-dimensional Riemannian manifolds in R .Let a sequence of functions ( L ε , M ε , N ε )( x, y ), defined on an open subset Ω ⊂ R ,satisfy the following Framework (A):(A.1) | ( L ε , M ε , N ε )( x, y ) | ≤ C a.e. ( x, y ) ∈ Ω, for some
C > ε ;(A.2) ∂ x M ε − ∂ y L ε and ∂ x N ε − ∂ y M ε are confined in a compact set in H − loc (Ω); SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 7 (A.3) There exist o εj (1) , j = 1 , ,
3, with o εj (1) → ε → ∂ x M ε − ∂ y L ε = Γ (2)22 L ε − (2)12 M ε + Γ (2)11 N ε + o ε (1) ,∂ x N ε − ∂ y M ε = − Γ (1)22 L ε + 2Γ (1)12 M ε − Γ (1)11 N ε + o ε (1) , (4.1)and L ε N ε − ( M ε ) = κ + o ε (1) . (4.2)Then we have Theorem 4.1 (Compensated compactness framework) . Let a sequence of functions ( L ε , M ε , N ε )( x, y ) satisfy Framework (A) . Then there exists a subsequence (still labeled) ( L ε , M ε , N ε )( x, y ) that converges weak-star in L ∞ (Ω) to ( ¯ L, ¯ M , ¯ N ) as ε → such that (i) | ( ¯ L, ¯ M , ¯ N )( x, y ) | ≤ C a.e. ( x, y ) ∈ Ω ; (ii) the Monge-Amp´ere constraint (2.7) is weakly continuous with respect to the sub-sequence ( L ε , M ε , N ε )( x, y ) that converges weak-star in L ∞ (Ω) to ( ¯ L, ¯ M , ¯ N ) as ε → ; (iii) the Gauss-Codazzi equations in (2.6) hold.That is, the limit ( ¯ L, ¯ M , ¯ N ) is a bounded weak solution to the Gauss-Codazzi system (2.6) – (2.7) , which yields an isometric realization of the corresponding two-dimensional Riemann-ian manifold in R .Proof. By the div-curl lemma of Tartar-Murat [37, 31] and the Young measure repre-sentation theorem for a uniformly bounded sequence of functions (cf. Tartar [37]), weemploy (A.1)–(A.2) to conclude that there exist a family of Young measures { ν x,y } ( x,y ) ∈ Ω and a subsequence (still labeled) ( L ε , M ε , N ε )( x, y ) that converges weak-star in L ∞ (Ω) to( ¯ L, ¯ M , ¯ N ) as ε → L, ¯ M , ¯ N )( x, y ) = ( h ν x,y , L i , h ν x,y , M i , h ν x,y , N i ) a . e . ( x, y ) ∈ Ω;(b) | ( ¯ L, ¯ M , ¯ N )( x, y ) | ≤ C a.e. ( x, y ) ∈ Ω;(c) the following commutation identity holds: h ν x,y , M − LN i = h ν x,y , M i − h ν x,y , L ih ν x,y , N i = ( ¯ M ) − ¯ L ¯ N . (4.3)Since the equations in (4.1) are linear in ( L ε , M ε , N ε ), then the limit ( ¯ L, ¯ M , ¯ N ) alsosatisfies the equations in (2.6) in the sense of distributions.Furthermore, condition (4.2) yields that h ν x,y , LN − M i = κ ( x, y ) a . e . ( x, y ) ∈ Ω . (4.4)The combination (4.3) with (4.4) yields the weak continuity of the Monge-Amp´ere con-straint with respect to the sequence ( L ε , M ε , N ε ) that converges weak-star in L ∞ (Ω) to( ¯ L, ¯ M , ¯ N ) as ε →
0: ¯ L ¯ N − ( ¯ M ) = κ. Therefore, ( ¯ L, ¯ M , ¯ N ) is a bounded weak solution of the Gauss-Codazzi system (2.6)–(2.7).Then the fundamental theorem of surface theory implies an isometric realization of thecorresponding two-dimensional Riemannian manifold in R . This completes the proof. (cid:3) GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG
Remark . In the compensated compactness framework, Condition (A.1) can be relaxedto the following condition:(A.1)’ k ( L ε , M ε , N ε ) k L p (Ω) ≤ C, p >
2, for some
C > ε .Then all the arguments for Theorem 4.1 follow only with the weak convergence in L p (Ω) , p >
2, replacing the weak-star convergence in L ∞ (Ω), with the aid of the Young measure rep-resentation theorem for a uniformly L p bounded sequence of functions (cf. Ball [1]).There are various ways to construct approximate solutions by either analytical methods,such as vanishing viscosity methods and relaxation methods, or numerical methods, suchas finite difference schemes and finite element methods. Even though the solution tothe Gauss-Codazzi system may eventually turn out to be more regular, especially in theregion of positive Gauss curvature κ >
0, the point of considering weak solutions here is todemonstrate that such solutions may be constructed by merely using very crude estimates.Such estimates are available in a variety of approximating methods through basic energy-type estimates, besides the L ∞ estimate. On the other hand, in the region of negativeGauss curvature κ <
0, discontinuous solutions are expected so that the estimates can beimproved at most up to BV in general.The compensated compactness framework (Theorem 4.1) indicates that, in order tofind an isometric immersion, it suffices to construct a sequence of approximate solutions( L ε , M ε , N ε )( x, y ) satisfying Framework (A), which yields its weak limit ( ¯ L, ¯ M , ¯ N ) to bean isometric immersion. To achieve this through the fluid dynamic formulation (3.1) and(3.3) (or (3.4)), it requires a uniform L ∞ estimate of ( u ε , v ε ) such that the sequence( L ε , M ε , N ε ) = ( ρ ε ( v ε ) + p ε , − ρ ε u ε v ε , ρ ε ( u ε ) + p ε )with p ε = − ρ ε = p ( u ε ) + ( v ε ) + κ satisfies Framework (A).The fluid dynamic formulation, (3.1) and (3.3) (or (3.4)), and the compensated compact-ness framework (Theorem 4.1) provide a unified approach to deal with various isometricimmersion problems even for the case when the Gauss curvature changes sign, that is, forthe equations of mixed elliptic-hyperbolic type.5. Isometric Immersions of Two-Dimensional Riemannian Manifolds withNegative Gauss Curvature
As a first example, in this section, we show how this approach can be applied to establishan isometric immersion of a two-dimensional Riemannian manifold with negative Gausscurvature in R .5.1. Reformulation.
In this case, κ < κ = − γ , γ > . For convenience, we assume γ ∈ C in this section and rescale ( L, M, N ) in this case as˜ L = Lγ , ˜ M = Mγ , ˜ N = Nγ ,
SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 9 so that (2.7) becomes ˜ L ˜ N − ˜ M = − . Then, without ambiguity, we redefine the “fluid variables” via˜ L = ρv + p, ˜ M = − ρuv, ˜ N = ρu + p, and set q = u + v , where we have still used ( u, v, p, ρ ) as the scaled variables and willuse them hereafter (although they are different from those in § § ∂ x ( ρuv ) + ∂ y ( ρv + p ) = R ,∂ x ( ρu + p ) + ∂ y ( ρuv ) = R , (5.1)where R := − ( ρv + p )˜Γ (2)22 − ρuv ˜Γ (2)12 − ( ρu + p )˜Γ (2)11 , (5.2) R := − ( ρv + p )˜Γ (1)22 − ρuv ˜Γ (1)12 − ( ρu + p )˜Γ (1)11 , (5.3)˜Γ (1)11 = Γ (1)11 + γ x γ , ˜Γ (1)12 = Γ (1)12 + γ y γ , ˜Γ (1)22 = Γ (1)22 , ˜Γ (2)11 = Γ (2)11 , ˜Γ (2)12 = Γ (2)12 + γ x γ , ˜Γ (2)22 = Γ (2)22 + γ y γ . Furthermore, the constraint ˜ L ˜ N − ˜ M = − ρpq + p = − . (5.4)From p = − ρ and (5.4), we have the “Bernoulli” relation: ρ = 1 p q − p = − p q − , (5.5)which yields u = p ( p − ˜ N ) , v = p ( p − ˜ L ) , ( ˜ M ) = ( ˜ N − p )( ˜ L − p ) . (5.6)Then the last relation in (5.6) gives the relation for p in terms of ( ˜ L, ˜ M , ˜ N ), and the firsttwo give the relations for ( u, v ) in terms of ( ˜ L, ˜ M , ˜ N ).Similarly to the calculation in §
3, we can write down our “rotationality–continuityequations” as ∂ x v − ∂ y u = − ρq ( vR − uR ) =: S , (5.7) ∂ x ( ρu ) + ∂ y ( ρv ) = vq R + uq R =: S . (5.8)Under the new scaling, the “sound” speed is c = p ′ ( ρ ) = 1 ρ > . (5.9)Then the “Bernoulli” relation (3.3) yields c = q − . (5.10)Therefore, q > c , and the “flow” is always supersonic, i.e., the system is purely hyperbolic. Riemann invariants.
In polar coordinates ( u, v ) = ( q cos θ, q sin θ ), we have R = ρq cos θ ˜Γ (2)22 − ρq sin θ cos θ ˜Γ (2)12 + ρq sin θ ˜Γ (2)11 − ρ (cid:0) ˜Γ (2)22 + ˜Γ (2)11 (cid:1) ,R = ρq cos θ ˜Γ (1)22 − ρq sin θ cos θ ˜Γ (1)12 + ρq sin θ ˜Γ (1)11 − ρ (cid:0) ˜Γ (1)22 + ˜Γ (1)11 (cid:1) , and then (5.7) and (5.8) becomesin θ∂ x q + q cos θ ∂ x θ − cos θ ∂ y q + q sin θ ∂ y θ = S , (5.11)cos θq ( q − ∂ x q + sin θ ∂ x θ + sin θq ( q − ∂ y q − cos θ ∂ y θ = − p q − q S . (5.12)That is, as a first-order system, (5.7) and (5.8) can be written as (cid:20) sin θ q cos θ q ( q − cos θ sin θ (cid:21) ∂ x (cid:20) qθ (cid:21) + (cid:20) − cos θ q sin θ q ( q − sin θ − cos θ (cid:21) ∂ y (cid:20) qθ (cid:21) = " S − √ q − q S . (5.13)One of our main observations is that, under this reformation, the two coefficient matricesin (5.13) actually commute, which guarantees that they have common eigenvectors. Theeigenvalues of the first and second matrices are λ ± = sin θ ± cos θ p q − , µ ± = − cos θ ± sin θ p q − , and the common left eigenvectors of the two coefficient matrices are( ± q p q − , . Thus, we may define the Riemann invariants W ± = W ± ( θ, q ) as ∂ θ W ± = 1 , ∂ q W ± = ± q p q − , (5.14)which yields W ± = θ ± arccos (cid:0) q (cid:1) . (5.15)Now multiplication (5.13) by ( ∂ q W ± , ∂ θ W ± ) from the left yields λ + (cid:0) ∂ q W + ∂ x q + ∂ θ W + ∂ x θ (cid:1) + µ + (cid:0) ∂ q W + ∂ y q + ∂ θ W + ∂ y θ (cid:1) = S ∂ q W + − p q − q S , (5.16) λ − (cid:0) ∂ q W − ∂ x q + ∂ θ W − ∂ x θ (cid:1) + µ − (cid:0) ∂ q W − ∂ y q + ∂ θ W − ∂ y θ (cid:1) = S ∂ q W − − p q − q S . (5.17)From (5.15), ∂ x W ± = ∂ q W + ∂ x q + ∂ θ W + ∂ x θ, ∂ y W ± = ∂ q W + ∂ y q + ∂ θ W + ∂ y θ, SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 11 then we can write (5.16) and (5.17) as λ + ∂ x W + + µ + ∂ y W + = 1 q p q − S − p q − q S , (5.18) λ − ∂ x W − + µ − ∂ y W − = − q p q − S − p q − q S . (5.19)5.3. Vanishing viscosity method via parabolic regularization.
Now we introduce avanishing viscosity method via parabolic regularization to obtain the uniform L ∞ estimateby identifying invariant regions for the approximate solutions.First, if ˜ R and ˜ R denote the additional terms that should be added to the right-handside of the Gauss-Codazzi system (5.1), our first choice is˜ R = ε∂ y ( ρv ) , ˜ R = ε∂ y ( ρu ) , (5.20)which gives us the system of “viscous” parabolic regularization: ∂ x ( ρuv ) + ∂ y ( ρv + p ) = R + ε∂ y ( ρv ) = R + ˜ R ,∂ x ( ρu + p ) + ∂ y ( ρuv ) = R + ε∂ y ( ρu ) = R + ˜ R . (5.21)From equations (3.8) and (3.9), we see˜ S = − ρq (cid:16) v ˜ R − u ˜ R (cid:17) , ˜ S = 1 q (cid:16) v ˜ R + u ˜ R (cid:17) (5.22)should be added to S and S on the right-hand side of (3.8) and (3.9). In polar coordinates( u, v ) = ( q cos θ, q sin θ ), (5.22) becomes˜ S = ε ρq ∂ y θ∂ y ( ρq ) + ε∂ y θ, ˜ S = ε q ∂ y ( ρq ) − ερ ( ∂ y θ ) . (5.23)Note the identity ε∂ y (cid:16) arccos (cid:0) q (cid:1)(cid:17) = ε∂ y (arccsc( ρq ))= − ε∂ y (cid:16) ρq p ρ q − (cid:17) ∂ y ( ρq ) − ερ q ∂ y ( ρq )= − ε∂ y (cid:16) ρq p ρ q − (cid:17) ∂ y ( ρq ) − ˜ S ρ − ερ ( ∂ y θ ) . Then ˜ S ρ = − ε∂ y (cid:16) arccos (cid:0) q (cid:1)(cid:17) − ε∂ y (cid:16) ρq p ρ q − (cid:17) ∂ y ( ρq ) − ερ ( ∂ y θ ) , and thus˜ S − ( q −
1) ˜ S = ˜ S − ˜ S ρ = 2 ερq ∂ y θ ∂ y ( ρq ) + ε∂ y θ + ε∂ y (cid:16) arccos (cid:0) q (cid:1)(cid:17) + ε∂ y (cid:16) ρq p ρ q − (cid:17) ∂ y ( ρq ) + ερ ( ∂ y θ ) . Since ∂ y θ = ∂ y W + + ∂ y ( ρq ) ρq p ρ q − , then ˜ S − ( q −
1) ˜ S = ε∂ y W + + 2 εqρ ∂ y W + ∂ y ( ρq ) + ε ( ∂ y W + ) . Similarly, using ∂ y θ = ∂ y W − − ∂ y ( ρq ) ρq p ρ q − , we have − ˜ S − ( q −
1) ˜ S = − ε∂ y W − − εqρ ∂ y W − ∂ y ( ρq ) + ε ( ∂ y W − ) . Thus, if we add the above ˜ S and ˜ S to the original S and S , (5.18) and (5.19) become q p q − (cid:16) λ + ∂W + ∂x + µ + ∂W + ∂y (cid:17) , = ε∂ y W + + 2 εqρ ∂ y W + ∂ y ( ρq ) + ε ( ∂ y W + ) + S − (cid:0) q − (cid:1) S (5.24) q p q − (cid:16) λ − ∂W − ∂x + µ − ∂W − ∂y (cid:17) = − ε∂ y W − − εqρ ∂ y W − ∂ y ( ρq ) + ε ( ∂ y W − ) − S − (cid:0) q − (cid:1) S . (5.25)Plugging R and R into S and S yields S ± (cid:0) q − (cid:1) S = − q sin θ (cid:16) ˜Γ (1)22 cos θ − (1)12 sin θ cos θ + ˜Γ (1)11 sin θ − q (cid:0) ˜Γ (1)22 + ˜Γ (1)11 (cid:1)(cid:17) − q cos θ (cid:16) − ˜Γ (2)22 cos θ + 2˜Γ (2)12 sin θ cos θ − ˜Γ (2)11 sin θ + 1 q (cid:0) ˜Γ (2)22 + ˜Γ (2)11 (cid:1)(cid:17) ± ρ (cid:26) q cos θ (cid:16) ˜Γ (1)22 cos θ − (1)12 sin θ cos θ + ˜Γ (1)11 sin θ − q (cid:0) ˜Γ (1)22 + ˜Γ (1)11 (cid:1)(cid:17) + q sin θ (cid:16) ˜Γ (2)22 cos θ − (2)12 sin θ cos θ + ˜Γ (2)11 sin θ − q (cid:0) ˜Γ (2)22 + ˜Γ (2)11 (cid:1)(cid:17)(cid:27) . (5.26)Then system (5.24)–(5.25) is parabolic when λ + > λ − < E, F, G ) = ( g , g , g ), we recall the following classical identi-ties: Γ (1)11 = GE x − F F x + F E y EG − F ) , Γ (1)22 = 2 GF y − GG x − F G x EG − F ) , Γ (2)11 = EF x − EE y − F E x EG − F ) , Γ (2)22 = EG y − F F y + F G x EG − F ) , Γ (1)12 = GE y − F G x EG − F ) , Γ (2)12 = EG x − F E y EG − F ) , SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 13 ( EG − F ) κ = det − E yy + F xy − G xx E x F x − F y F x − G x E F G y F G − det E y G x E y E F G x F G , and γ = − κ .5.4. L ∞ –estimate for the viscous approximate solutions. Based on the calculationabove for the Riemann invariants, we now introduce an approach to make the L ∞ estimate.First we need to sketch the graphs of the level sets of W ± . If W ± = θ ± arccos (cid:0) q (cid:1) = C ± for constants C ± , then dθdq = ∓ ddq (cid:16) arccos (cid:0) q (cid:1)(cid:17) = ∓ q p q − W ± = C ± , and, as q → ∞ , dθdq → , θ → C ± ∓ arccos(0) = C ± ∓ π W ± = C ± . See Fig. 1 for the graphs of the level sets W ± = C ± . uv uvC + C − W + = C + W − = C − W − ≥ C − W + ≤ C + (a) (b) q = 1 q = 1 Figure 1.
Level setsNext we examine the meaning of inequality W + ≤ C + , i.e., θ + arccos (cid:0) q (cid:1) ≤ C + . For example, if q = 1, then θ ≤ C + . This indicates the region of W + ≤ C + as sketched inFig. 1(a). Similarly, W − ≥ C − means θ − arccos (cid:0) q (cid:1) ≥ C − , and, if q = 1, then θ ≥ C − , and the region of W − ≥ C − is sketched in Fig. 1(b). Thus wesee that W + ≤ C + means the region below W + = C + ,W − ≥ C − means the region below W − = C − ;and W + ≥ C + means the region above W + = C + ,W − ≤ C − means the region above W − = C − . As an example, we now focus on the case that F = 0 , E ( x ) = G ( x ) . (5.27)Then Γ (1)11 = E ′ E , Γ (1)12 = 0 , Γ (1)22 = − E ′ E ; Γ (2)11 = 0 , Γ (2)12 = E ′ E , Γ (2)22 = 0 . Therefore, we have˜Γ (1)11 = E ′ E + γ ′ γ , ˜Γ (1)12 = 0 , ˜Γ (1)22 = − E ′ E ; ˜Γ (2)11 = 0 , ˜Γ (2)12 = E ′ E + γ ′ γ , ˜Γ (2)22 = 0 , and the right-hand side of (5.26) is equal to12 γ (cid:18) κ ′ ρ q − γ q E ′ E (cid:19) sin θ ± γ ρ (cid:18) − κ ′ q + qγ E ′ E (cid:19) cos θ. Thus, the two solutions θ ± ( q ) that make the right-hand side of (5.26) equal to zero satisfytan θ = ± ρ (cid:16) κ ′ q − γ q E ′ E (cid:17) κ ′ ρ q − γ q E ′ E . (5.28)If we fix the intersection point of θ ± ( q ) at θ = 0 , q = q = β, where β > β κ ′ ( x ) κ ( x ) + E ′ ( x ) E ( x ) = 0 , (5.29)i.e., ddx ln (cid:16) | κ ( x ) | β E ( x ) (cid:17) = 0 . Thus, | κ ( x ) | β E ( x ) = const. > . Since κ ( x ) <
0, then κ ( x ) = − κ E ( x ) − β , where κ > θ becomestan θ = ± p q − β − q ) β − ( β − q . Assume that we have a solution E ( x ) to (5.29). Fix another constant 1 < α < β . Thenthe curves θ ± ( q ) are independent of ( x, y ) and look like the sketch in Fig. 2. SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 15 α βv uθ − ( q ) θ + ( q ) Figure 2.
Graphs of θ ± α W − = W − (0 , α ) W + = W + (0 , β ) W − = W − (0 , β ) W + = W + (0 , α ) βv uθ + ( q ) θ − ( q ) Figure 3.
Invariant regionsNext, note that W ± , when evaluated at θ = 0 , q = α , and θ = 0 , q = β , take on theconstant values, i.e., independent of ( x, y ). Equations (5.24) and (5.25) imply that, at anypoint where ∇ W + = 0 (respectively ∇ W − = 0), εγ∂ y W + ( > θ > θ + ( q ) ,< θ < θ + ( q ) ,εγ∂ y W − ( > θ > θ − ( q ) ,< , for θ < θ − ( q ) . Hence, when λ + > λ − <
0, by the maximum principle (cf. [18, 34]), W + has no internal maximum for θ > θ + ( q ) ,W + has no internal minimum for θ < θ + ( q ) , W − has no internal maximum for θ > θ − ( q ) ,W − has no internal minimum for θ < θ − ( q ) . Define W ± (0 , β ) = ± cos − (cid:0) β (cid:1) , W ± (0 , α ) = ± cos − (cid:0) α (cid:1) . Therefore, if the data is such that W + ≤ W + (0 , β ), then W + can have no internal maximumgreater than W + (0 , β ) for θ > θ + ( q ). Similarly, if the data is such that W − ≥ W − (0 , β )(= W + (0 , β )), then W − can have no internal minimum less than W − (0 , β ) for θ < θ − ( q ).Furthermore, if the data is such that W + ≥ W + (0 , α ), then W + can have no internalminimum less than W + (0 , α ) for θ < θ + ( q ); if that data is such that W − ≤ W − (0 , α ),then W − can have no internal maximum greater than W − (0 , α ) for θ > θ − ( q ). Thus, thediamond-shaped region in Fig. 3 provides the upper and lower bounds for W ± .From the definition of λ ± : λ ± = sin θ ± cos θ p q − , we easily see that the lines λ ± = 0 are as sketched in Fig. 4. λ − = 0 λ + = 0 v u Figure 4.
Graphs of λ ± = 0Notice that λ + > λ − < λ + = 0 and below λ − = 0. If wenow super-impose Fig. 3 on top of Fig. 4 and choose α sufficiently close to β , we see thatthere is a region where the four-sided region of Fig. 3 is entirely confined in the regionabove λ + = 0 and below λ − = 0 in Fig. 4. Hence, the parabolic maximum/minimumprinciples apply and the four-sided region is an invariant region. For example, in thehalf-plane: Ω := { ( x, y ) : x ≥ , y ∈ R } (5.30)with periodic initial data ( q (0 , y ) , θ (0 , y )) prescribed in the four-sided region, the maxi-mum/minimum principles yield the invariant region for the periodic solution.There is an alternative symmetry about θ = π . If we set θ = ψ + π , then the right-handside of (5.26) becomes12 γ (cid:18) − κ ′ ρ q + γ q E ′ E (cid:19) cos ψ ± γ ρ (cid:18) q ∂κ∂y − γ q E ′ E (cid:19) sin ψ = 0 . SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 17 If ψ ± satisfy the above equations, then ψ + and ψ − are symmetric about ψ = 0, i.e., θ = π .Look for the crossing on ψ = 0 so that12 ρ (cid:18) − κ ′ ρ q + γ q E ′ E (cid:19) = 0 . With the crossing at q = β , this gives β − β κ ′ κ + E ′ E = 0 , (5.31)that is, | κ ( x ) | β − β E ( x ) = const. To exploit the symmetry, we now take x as a space-like variable and y as a time-like variableand replace ∂ y by ∂ x . Now it is the interior between the lines µ − = 0 and µ + = 0 that givesthe preferred signs µ + > µ − <
0, which keep (5.24) and (5.25) parabolic. Hence,similar to (i), in the case that the initial data and the metric E ( x ) = G ( x ) , F ( x ) = 0 , is periodic in x , then the periodic solution will stay in the four-sided invariant region inwhich the initial data lies.All these arguments yield the uniform L ∞ bounds for ( u ε , v ε , p ε , ρ ε ), which implies | ( L ε , M ε , N ε ) | ≤ C, for some constant C > k γ k L ∞ .Finally, let us examine the following examples: Example 5.1.
Catenoid: E ( x ) = ( cosh ( cx )) β − , κ ( x ) = − κ E ( x ) − β , where c = 0 and κ > are two constants. Substitution them into (5.29) (where x is time-like and y isspace-like) yields β > . Of course, (5.29) is satisfied when E ( x ) = ( cosh ( cx )) β − , κ ( x ) = − κ E ( x ) − β β − , with β > (where x is space-like and y is time-like). Example 5.2.
Helicoid: The metric associated with the helicoid is ds = E ( dX ) + ( dY ) with E ( Y ) = λ + Y and the Gauss curvature κ = − λ ( λ + Y ) , where λ > is a constant. To apply our previous result, for the special case of isothermalcoordinates given in (5.27) , we first make a change of variables to rewrite the helicoidmetric in isothermal coordinates, i.e., allow X, Y to depend on x, y so that ds = ( EX x + Y x ) dx + 2( EX x X y + Y x Y y ) dxdy + ( EX y + Y y ) dy . Hence, if we set Y x = −√ EX y , Y y = √ EX x , then ds = E ( X x + X y )( dx + dy ) , which gives the metric in isothermal coordinates. The above equations for X and Y maybe rewritten as − Y x √ E = X y , Y y √ E = X x , and with φ ( Y ) = Z dY √ λ + Y = ln( Y + p λ + Y ) , we have − φ x = X y , φ y = X x , i.e., the Cauchy-Riemann equations. A convenient solution is given by φ = − x, X = y, which yields − x = ln( Y + p λ + Y ) , that is, Y = −
12 ( λ e x − e − x ) . Thus, in the new ( x, y ) -coordinates, E = λ + Y = 12 λ + 14 ( λ e x + e − x ) , κ = − λ ( λ + Y ) = − λ (cid:0) λ + ( λ e x + e − x ) (cid:1) , and ds = (cid:0) λ + 14 ( λ e x + e − x ) (cid:1) ( dx + dy ) . Hence, we have − E ′ ( x ) E ( x ) = κ ′ ( x ) κ ( x ) and so relation (5.29) is satisfied with β = √ . Example 5.3.
Torus: The metric for the torus is usually written as ds = EdX + b dY , with E = ( a + b cos Y ) , κ ( Y ) = cos Yb ( a + b cos Y ) , where a > b > are constants. The same argument as given in Example above yieldsthe metric in isothermal coordinates as ds = E ( dx + dy ) with E = ( a + b cos Y ) = ( a + b cos( φ − ( x )) ) , where φ ( Y ) = b √ a − b arctan √ a − b sin Yb + a cos Y ! , SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 19 κ ( x ) = cos Yb ( a + b cos Y ) = cos( φ − ( x )) b (cid:0) a + b cos( φ − ( x )) (cid:1) . A direct computation yields κ ′ ( x ) κ ( x ) = − a ( φ − ( x )) ′ tan( φ − ( x )) a + b cos( φ − ( x )) , E ′ ( x ) E ( x ) = − b ( φ − ( x )) ′ sin( φ − ( x )) a + b cos( φ − ( x )) , and the ratio κ ′ ( x ) κ ( x ) (cid:30) E ′ ( x ) E ( x ) = ab cos( φ − ( x )) is not a constant. So (5.29) does not hold. Hence our Proposition will not directlyapply to that piece of the torus possessing negative Gauss curvature. H − loc –compactness. We now show how the H − loc –compactness can be achieved forthe viscous periodic approximate solutions via parabolic regularization.In § y with period P , we have a uniform L ∞ estimate on ( u ε , v ε , p ε , ρ ε ) as the periodic solution to the viscousequations (5.21). From the equations in (5.23), we have ∂ x ( ρu ) + ∂ y ( ρv ) = vq R + uq R + ε q ∂ ∂y ( ρq ) − ερ ( ∂ y θ ) = B ( x, y ) + ε q ∂ y ( ρq ) − ερ ( ∂ y θ ) , (5.32)where B ( x, y ) = ρq sin θ (cid:16) − (cid:0) ρq sin θ − ρ (cid:1) ˜Γ (2)22 − ρq sin(2 θ )˜Γ (2)12 − (cid:0) ρq cos θ − ρ (cid:1) ˜Γ (2)11 (cid:17) + ρq cos θ (cid:16) − (cid:0) ρq sin θ − ρ (cid:1) ˜Γ (1)22 − ρq sin(2 θ )˜Γ (1)12 − (cid:0) ρq cos θ − ρ (cid:1) ˜Γ (1)11 (cid:17) . Our L ∞ estimate in § B ( x, y ) is uniformly bounded with respect to ε .Using the periodicity, we have Z x Z P q ∂ y ( ρq ) dydx = Z x Z P ∂ y qq ∂ y ( ρq ) dydx = Z x Z P ∂ y qq (cid:0) − ρ ∂ y q (cid:1) dydx = − Z x Z P ρ ( ∂ y q ) q dydx. Now integrating both sides of (5.32) over { ( x, y ) : 0 ≤ x ≤ x , ≤ y ≤ P } , we find ε Z x Z P (cid:16) ρ ( ∂ y q ) q + ρ ( ∂ y θ ) (cid:17) dydx = Z x Z P B ( x, y ) dydx − Z P (cid:0) ( ρu )( x , y ) − ( ρu )( x , y ) (cid:1) dy ≤ C, where C > ε , but may depend on x and P . This implies that √ ε∂ y θ, √ ε∂ y q are in L loc (Ω) uniformly in ε. Therefore, we have
Proposition 5.1. (i)
Consider the viscous system (5.21) in Ω = { ( x, y ) : x ≥ , y ∈ R } with periodic initial data ( q, θ ) | x =0 = ( q ( y ) , θ ( y )) , then √ ε∂ y q, √ εθ y are in L loc (Ω) uniformly in ε ;(ii) If we replace ∂ y in system (5.21) by ∂ x , the initial data ( q, θ ) | x =0 = ( q ( y ) , θ ( y )) by ( q, θ ) | y =0 = ( q ( x ) , θ ( x )) (5.33) which is periodic in x with period P and, in addition, we assume that the metric E ( x ) = G ( x ) is also periodic with period P , then the periodic solution with period P satisfies that √ ε∂ x q, √ ε∂ x θ are in L loc (Ω) uniformly in ε, where Ω = { ( x, y ) : x ≤ x < x , y > } . Using Proposition 5.1 and the viscous system (5.21), we conclude that ∂ x ˜ M ε − ∂ y ˜ L ε , ∂ x ˜ N ε − ∂ y ˜ M ε are compact in H − loc (Ω) . Since γ ∈ C , we conclude that ( L ε , M ε , N ε ) = γ ( ˜ L ε , ˜ M ε , ˜ N ε ) satisfies Framework (A)in §
4. Then the compensated compactness framework (Theorem 4.1) implies that there isa subsequence (still labeled) ( L ε , M ε , N ε )( x, y ) that converges weak-star to ( ¯ L, ¯ M , ¯ N ) as ε → L, ¯ M , ¯ N ) is a bounded, periodic weak solution to the Gauss-Codazzi system (2.6)–(2.7). Therefore, ( ¯ L, ¯ M , ¯ N ) is a weak solution of (2.6)–(2.7). Wesummarize this as Proposition 5.2. Proposition 5.2.
For either initial value problem (i) or (ii) of Proposition , ( L ε , M ε , N ε ) possesses a weak-star convergent subsequence which converges to a periodic weak solution ofthe associated initial value problem for the Gauss-Codazzi system (2.6) – (2.7) when ε → . Existence of isometric immersions: Main theorem and examples.
We nowfocus on the case (5.27): F = 0 , and E = G depends only on x. to state an existence result for isometric immersions and analyze examples for this case.Let us look for a special solution:( θ, q ) = (0 , β ) (constant state) , for the Gauss-Codazzi system for the case (5.27). In this case,˜Γ (1)11 = E ′ E + γ ′ γ , ˜Γ (1)12 = 0 , ˜Γ (1)22 = − E ′ E ; ˜Γ (2)11 = 0 , ˜Γ (2)12 = E ′ E + γ ′ γ , ˜Γ (2)22 = 0 , and the Gauss-Codazzi system (5.1) becomes ∂ x (cid:0) ρ (cid:1) = − p E ′ E +( ρq + p )( E ′ E + γ ′ γ ) = ρq E ′ E +( ρq + p ) γ ′ γ = ρq E ′ E + ρ γ ′ γ , ρ = 1 p q − . When q ( x ) ≡ β , this reduces to 1 β γ ′ γ = − E ′ E , or 1 β κ ′ ( x ) κ ( x ) = − E ′ E . (5.34)
SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 21
Hence, q ( x ) = β becomes an exact solution precisely in this special case. Our theoremgiven below shows that, in fact, we can satisfy the prescribed initial conditions in thisspecial case and that, for this choice of E, F, G , there exists a weak solution for arbitrarybounded data in our diamond-shaped region when α ∈ (1 , β ) (see Fig. 3).Consider the initial value problem for the Gauss-Codazzi system (2.6)–(2.7) with initialdata ( q, θ ) | x =0 = ( q ( y ) , θ ( y )) , y ∈ R , (5.35)or ( q, θ ) | y =0 = ( q ( x ) , θ ( x )) , x ∈ R . (5.36)Our next result shows that, for this choice of E, F, G , there exists a weak solution forarbitrary bounded initial data in our diamond-shaped region when α ∈ (1 , β ) (see Fig. 3). Theorem 5.1.
Assume that the initial data (5.35) , or (5.36) , is L ∞ and lies in thediamond-shaped region of Figs. . Then (i) The Gauss-Codazzi system (2.6) – (2.7) has a weak solution with the initial data ( q, θ ) | x =0 = ( q ( y ) , θ ( y )) . This case includes Example for the catenoid with the metric E ( x ) = G ( x ) = ( cosh ( cx )) β − , F ( x ) = 0 , c = 0 , and β > , and Example for thehelicoid (in isothermal coordinates) with E ( x ) = G ( x ) = λ + ( λ e x + e − x ) , F ( x ) = 0 ,and β = √ . (ii) The Gauss-Codazzi system (2.6) – (2.7) has a weak solution with the initial data ( q, θ ) | y =0 = ( q ( x ) , θ ( x )) . This case includes the catenoid with the metric E ( x ) = G ( x ) =( cosh ( cx )) β − , F ( x ) = 0 , c = 0 , and β > , and the helicoid (in isothermal coordinates)with E ( x ) = G ( x ) = λ + ( λ e x + e − x ) , F ( x ) = 0 , and β = √ .Proof. We start with case (i).
Step 1.
For the initial data ( q ( y ) , θ ( y )), we can find ( q P ( y ) , θ P ( y )) for P > q P , θ P ∈ C ( R ) , q P , θ P are periodic with period P ;(ii) q P → q , θ P → θ a.e. in R and weakly in L ∞ ( R ) as P → ∞ .In particular, the functions q P and θ P are bounded in L ∞ ( R ), and ( q P , θ P ) converges to( q , θ ) in L ploc ( R ), p ∈ [1 , ∞ ), as P → ∞ .This can be achieved by the standard symmetric mollification procedure: First truncatethe initial data ( W − , , W + , ) = ( W − ( q , θ ) , W + ( q , θ )) in the interval − P ≤ y ≤ P and make the periodic extension to the whole space y ∈ R , and then take the standardsymmetric mollification approximation to get the C ∞ approximate sequence ( W P − , , W P + , )that yields the corresponding C ∞ approximate sequence( q P , θ P ) = ((cos( W P + , − W P − , − , W P + , + W P − , q , θ ) = ((cos( W + , − W − , )) − , W + , + W − , ) a.e. in R as P → ∞ . Since thestandard symmetric mollification is an average-smoothing operator, then the approximatesequence ( q P , θ P ) still lies in our diamond-shaped region of Figs. 3–4. Step 2 . Following the arguments in Section 5.4, we can established the uniform L ∞ apriori estimates for the corresponding viscous solutions in two parameters ǫ > P >
0. For fixed P , then we can show that there exists a unique periodic viscous solution with period P to the parabolic system (5.21), which can be achieved by combining thestandard local existence theorem with the L ∞ estimates. The H − loc –compactness followsfrom the argument in § P , letting the viscous coefficient ǫ tend 0, we employ Proposition 5.2 to obtainthe global periodic weak solution ( L P , M P , N P ) of the Gauss-Codazzi system (2.6)–(2.7),periodic in y with period P , in the half plane { ( x, y ) : x ≥ , y ∈ R } . Step 3 . Since the sequence of periodic solutions ( L P , M P , N P ) still stays in the invariantregion, which yields the uniform L ∞ bound in P as P → ∞ . This uniform bound alsoyields the H − loc –compactness of ∂ x ˜ M P − ∂ y ˜ L P , ∂ x ˜ N P − ∂ y ˜ M P . Using the compensated compactness framework (Theorem 4.1) again and letting P →∞ , we obtain a global weak solution ( L, M, N ) of the Gauss-Codazzi system (2.6)–(2.7)in the half plane { ( x, y ) : x ≥ , y ∈ R } . Step 4 . For (ii), it also requires the periodic approximation for the metric E ( x ) = G ( x )by E P ( x ) = G P ( x ) with period P in x , besides the period approximation for the initialdata. This can be achieved as follows: First truncate the function a ( x ) := E ′ ( x ) E ( x ) in theinterval − P ≤ x ≤ P and make odd extension along the lines x = − P , P respectively.Then make the periodic extension from the interval − P ≤ x ≤ P to the whole space x ∈ R with period P and take the standard symmetric mollification approximation to get the C ∞ approximate sequence a P ( x ) with zero mean over each period (i.e. R P − P a P ( x ) dx = 0)that yields the corresponding C ∞ approximate sequence E P ( x ) with period P uniquelydetermined by the differential equation: Y ′ ( x ) = a P ( x ) Y ( x ) , Y | x =0 = E (0) , and then take the C ∞ approximate sequence κ P ( x ) with period P determined by thedifferential equation: K ′ ( x ) = − β β − a P ( x ) K ( x ) , K | x =0 = κ (0)for the corresponding β > E P = G P and κ P into the viscous Gauss-Codazzi system so that the equa-tions in the system is periodic in x with period P . For fixed P , the same argument as thatfor the first case yields the global period weak solution ( L P , M P , N P ) of the Gauss-Codazzisystem (2.6)–(2.7) (with E ( x ) replaced by E P ( x ), periodic in x with period P , in the halfplane { ( x, y ) : x ∈ R , y ≥ } . Then, using the same argument, noting the strong conver-gence of E P to E , and letting P → ∞ , we again obtain a global weak solution ( L, M, N )of the Gauss-Codazzi system (2.6)–(2.7) in the half plane { ( x, y ) : x ∈ R , y ≥ } . (cid:3) If we repeat the similar argument for Theorem 5.1 through the corresponding vanishingviscosity method on the domain Ω = { ( x, y ) : x < , y ∈ R } for problem (5.35) and thedomain Ω = { ( x, y ) : x ∈ R , y < } for problem (5.36), we obtain again a weak solution.Together, they form a weak solution in R . As before, the associated immersion is in C , . SOMETRIC IMMERSIONS AND COMPENSATED COMPACTNESS 23
Theorem 5.2.
Assume that the initial data (5.35) , or (5.36) , is L ∞ and lies in thediamond-shaped region of Figs. for the case of the catenoid or helicoid metric (asgiven in Theorem ), then the initial value problem (2.6) – (2.7) and (5.35) , or (5.36) ,has a weak solution in L ∞ ( R ) . This yields a C , ( R ) immersion of the Riemannianmanifold into R .Remark . The catenoid with circular cross-section is sketched in Fig. 5. Our theoremasserts the existence of a C , -surface for the associated metric for a class of non-circularcross-sections prescribed at x = 0. Similarly, the C , -helicoid in the ( x, y, z )–coordinatesis sketched in Fig. 6. y zx Figure 5. C , -catenoid in the ( x, y, z )–coordinates Figure 6. C , -helicoid in the ( x, y, z )–coordinates Acknowledgments.
This paper was completed when the authors attended the “Work-shop on Nonlinear PDEs of Mixed Type Arising in Mechanics and Geometry”, which washeld at the American Institute of Mathematics, Palo Alto, California, March 17–21, 2008.
Gui-Qiang Chen’s research was supported in part by the National Science Foundation un-der Grants DMS-0807551, DMS-0720925, DMS-0505473, and an Alexander von HumboldtFoundation Fellowship. Marshall Slemrod’s research was supported in part by the NationalScience Foundation under Grant DMS-0243722. Dehua Wang’s research was supported inpart by the National Science Foundation under Grants DMS-0244487, DMS-0604362, andthe Office of Naval Research Grant N00014-01-1-0446.
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G.-Q. Chen, Department of Mathematics, Northwestern University, Evanston, IL 60208.
E-mail address : [email protected] M. Slemrod, Department of Mathematics, University of Wisconsin, Madison, WI 53706.
E-mail address : [email protected] D. Wang, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260.
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