The in-the-large completion of Weyl's geodesic-field sufficiency theory for multiple integrals: global embedding of extremals and global solvability of the Legendre-Riccati system

For variational problems minimizing a multiple integral $J(\Sigma)=\int_G L(t^i, z^\alpha, z_i^\alpha)\,dt$ over graphs of vector-valued functions $z(t^1,\dots,t^r)$, $r\ge 2$, Hermann Weyl (1935) built a sufficiency theory modeled on the classical single-integral field theories: a minimum is certified by embedding the extremal in a geodesic field, a slope field $z_i^\alpha(t,z)$ with an elevation function $s^i(t,z)$ satisfying $L=D$ and $L_{z_i^\alpha}=D_{z_i^\alpha}$, where $D=\partial s^i/\partial t^i+(\partial s^i/\partial z^\alpha)z_i^\alpha$ is Weyl's trace (divergence) version of Carathéodory's determinant. The embedding reduces to a Hamilton-Jacobi equation of De Donder-Weyl type whose first approximation is a system of matrix Riccati (Legendre) equations, and Weyl's construction of it is local: analytic data, a Cauchy-Kowalevski-style majorant argument, and in places a cylinder-like specialization of the parameter domain. Weyl's construction left several limitations open. The invariant-formulation and analyticity limitations were substantially resolved (Lepage, Boerner, van Hove, Gawędzki, Goldschmidt-Sternberg, and the modern multisymplectic formalism), and the search for sharp simple local necessary conditions was transformed into the quasiconvexity program, where it received a definitive negative answer for $\nu\ge 3$ (Šverák) and survives independently as Morrey's conjecture in the $2\times 2$ case. The in-the-large core remains open: an intrinsic characterization of the parameter domains and data for which singularity-free embedding is achievable, and a global existence and uniqueness theory for the associated Legendre-Riccati and Hamilton-Jacobi field equations under general hypotheses — a conjugate-point theory in the large for multiple integrals. Only special classes have been settled — Klötzler's constructions of geodesic fields in the large for quadratic problems in codimension one, and the American school's conjugate-surface conditions on the second variation. The general core remains open.

Background

(1) The setting. A multiple-integral variational problem asks for the minimum of $J(\Sigma)=\int_G L(t^i, z^\alpha, z_i^\alpha)\,dt$ over surfaces $\Sigma$ that are graphs $z^\alpha=z^\alpha(t^1,\dots,t^r)$ of $\nu\ge 1$ functions of $r\ge 2$ variables over a parameter domain $G$, where $z_i^\alpha=\partial z^\alpha/\partial t^i$ denotes the gradient and $L$ is a smooth Lagrangian. Extremals satisfy the Euler-Lagrange equations $\partial L/\partial z^\alpha-\partial(\partial L/\partial z_i^\alpha)/\partial t^i=0$; a strong minimum is one taken against surfaces near $\Sigma_0$ in the supremum norm, a weak one against surfaces near in the $C^1$ norm. (2) The single-integral model. For $r=1$ there is a complete sufficiency theory in the large: an extremal curve embedded in a field of extremals with slope function $p(t,z)$ yields a minimum when Weierstrass's excess function $E=L(t,z,q)-L(t,z,p)-L_p(t,z,p)(q-p)$ is nonnegative, by Hilbert's independence integral, whose value depends only on endpoints; the global validity of the field is governed by Jacobi's conjugate-point condition — the requirement that no nontrivial solution of the linearized (Jacobi) equation vanish twice along the extremal, equivalent to global solvability of a Riccati equation for the second variation. For $r\ge 2$ no comparable in-the-large theory exists. (3) Carathéodory's royal road and Weyl's trace theory. Carathéodory (1929) extended the equivalent-field idea to multiple integrals using as 'elevation' the determinant $D=|\partial s^i/\partial t^k+(\partial s^i/\partial z^\alpha)z_k^\alpha|$ of the Jacobian of the map $(t,z)\mapsto(t+s(t,z))$; Weyl (1935) replaced the determinant by its first approximation in $\varepsilon$, the trace (divergence) $D=\partial s^i/\partial t^i+(\partial s^i/\partial z^\alpha)z_i^\alpha$ (summation over repeated indices), and showed the two theories are connected by the substitution $t^i\mapsto t^i+\varepsilon s^i$, $L\mapsto 1+\varepsilon L$ with $\varepsilon\to 0$, a device related to his 1934 observations on Hilbert's independence theorem. (4) Geodesic fields and sufficiency. A geodesic field is a slope field $z_i^\alpha(t,z)$ together with an elevation $s^i(t,z)$ such that on the field $L(t,z,z_i)=D$ and $L_{z_i^\alpha}(t,z,z_i)=D_{z_i^\alpha}$; then $D$ is a divergence, $\int_G L\,dt=\int_{\partial G}s\cdot n\,d\sigma$ for surfaces lying in the field, so the integral is boundary-determined (the 'complete figure'), and $J(\Sigma)-J(\Sigma_0)$ reduces to an integral of the excess function over $G$, which is nonnegative under McShane's rank-one inequalities. (5) Weyl's construction and its limitations. The elevation is found from the Hamilton-Jacobi (De Donder-Weyl) equation $\partial s^i/\partial t^i=H(t^k,z^\alpha,\partial s^k/\partial z^\alpha)$, where $H$ is obtained from $L$ by a Legendre transformation; Weyl solved it locally by the method of characteristics and by a Cauchy-Kowalevski-style majorant argument (hence analytic data), the first approximation along a given extremal being a system of matrix Riccati (Legendre) equations obtained by linearizing this Hamilton-Jacobi equation along the extremal. A singularity of the Riccati solution, equivalently vanishing of the characteristic Jacobian, is the multiple-integral analogue of a conjugate point and blocks the embedding; Weyl's global criterion is that the extremal is embeddable in a singularity-free geodesic field on all of $G$ exactly when the Riccati system is globally solvable, a criterion he could not implement beyond local, partially cylinder-specialized ($G\approx G_{r-1}\times$ interval) situations. (6) Necessary conditions and their modern transformation. McShane (1931) proved that a minimum forces $E(t,z,p,p+a\otimes b)\ge 0$ for all vectors $a\in\mathbb{R}^\nu$, $b\in\mathbb{R}^r$ — rank-one convexity of $L$ in the gradient variable. Morrey (1952) introduced quasiconvexity — $\int_Q f(A+D\varphi)\ge f(A)$ for all matrices $A$ and test functions $\varphi$ with compactly supported gradient — necessary for weak lower semicontinuity and strictly stronger than rank-one convexity; Ball (1977) added polyconvexity (convexity in the matrix minors) and made these notions the basis of existence theory in nonlinear elasticity; Šverák (1992) proved rank-one convexity does not imply quasiconvexity for $\nu\ge 3$, definitively closing the search for simple sharp local conditions in that range, and Kristensen (1999) proved quasiconvexity is non-local; the $2\times 2$ case is Morrey's conjecture and is open. (7) Geometric reformulations. Lepage (1941) unified field concepts via differential forms; Boerner (1940) recast the theory in the calculus of alternating differential forms; Goldschmidt and Sternberg (1973) gave the intrinsic Hamilton-Cartan formalism on fibered manifolds (spaces endowed with a projection onto the parameter domain, so that fields are their sections), the ancestor of the modern multisymplectic (De Donder-Weyl) formalism — the extension of Hamiltonian symplectic geometry from mechanics to field theory — used in classical field theory. (8) What was never done. The in-the-large questions — the intrinsic characterization of domains permitting singularity-free embedding, and global solvability of the Legendre-Riccati and Hamilton-Jacobi field equations — were pursued in the literature of the 1940s-1970s without a general global theory: Klötzler devoted a monograph (1961) to constructing geodesic fields in the large but carried out the worked global constructions only for structured classes (quadratic integrands in codimension one), and the American school treated the accessory second variation by spectral methods (Karush's semi-strong minima, Dennemeyer's conjugate surfaces), giving Jacobi-type conditions but not an intrinsic characterization of the domains and data nor a global Riccati theory. The field has since developed mainly through the multisymplectic formalism and numerical field integrators, with scattered results on necessary conditions. This problem is that unfinished in-the-large core.

Problem Statement

Complete the in-the-large sufficiency theory for multiple-integral variational problems that Weyl's 1935 geodesic-field construction began. Concretely: given a smooth Lagrangian $L(t^i,z^\alpha,z_i^\alpha)$ on a parameter domain $G\subset\mathbb{R}^r$ ($r\ge 2$) satisfying a Legendre-type positivity condition, and a regular extremal surface $\Sigma_0$ for $J(\Sigma)=\int_G L\,dt$, Weyl's method certifies a minimum by embedding $\Sigma_0$ in a geodesic field — an elevation $s^i(t,z)$ with slope field $z_i^\alpha(t,z)$ satisfying $L=D$ and $L_{z_i^\alpha}=D_{z_i^\alpha}$, $D=\partial s^i/\partial t^i+(\partial s^i/\partial z^\alpha)z_i^\alpha$ — constructed from the Hamilton-Jacobi equation $\partial s^i/\partial t^i=H(t^k,z^\alpha,\partial s^k/\partial z^\alpha)$, whose first approximation along $\Sigma_0$ is the Legendre-Riccati system. The construction is presently available only locally, for analytic data, and under a cylinder-like specialization of $G$. The in-the-large completion consists in establishing (a) an intrinsic characterization — topological and analytic — of the parameter domains $G$ and of the data $(L,\Sigma_0)$ for which a singularity-free geodesic-field embedding exists on all of $G$, resolving whether the cylinder hypothesis is necessary or removable; (b) a global existence and uniqueness theory for the Legendre-Riccati system and the associated Hamilton-Jacobi field equations under general (smooth, non-analytic) hypotheses, i.e. a conjugate-point/Jacobi theory in the large for multiple integrals analogous to the single-interval theory of the classical calculus of variations, of which only special-class fragments exist (Klötzler's quadratic codimension-one constructions; second-variation conjugate-surface theories for special problem classes); (c) conditions under which the global embedding yields sufficiency of a strong minimum on the whole domain, together with sharp obstructions — explicit counterexamples showing when the Riccati solution necessarily develops singularities and the embedding necessarily fails. The three components form a single question: a single counterexample, or a construction for a restricted class of Lagrangians and domains, leaves the stated in-the-large question open. The invariant-formulation and local-analyticity limitations of Weyl's construction have been substantially addressed by later literature and are not part of the open core; the necessary-conditions limitation survives only as the independent Morrey conjecture ($2\times 2$ quasiconvexity), which belongs to the direct-methods program rather than to field theory and is not part of this problem.

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