Extension of Hilbert's inner-uniqueness sufficiency criterion to variational problems with non-prescribed boundary values

In the closing section of 'Zur Variationsrechnung' (Mathematische Annalen 62, 1906, pp. 351-370), David Hilbert stated a sufficiency criterion for variational problems in which the values of the sought functions are prescribed everywhere on the boundary: a solution system Z of the Lagrange (Euler-Lagrange) equations that is (i) of 'positive definite character' (the Weierstrass excess functions are positive along Z) and (ii) of 'innerly unique character' (innerlich eindeutiger Charakter: the boundary value problem for the Lagrange system with the problem's side conditions is uniquely solved by Z on the whole domain, and likewise on every subdomain with the boundary data induced there by Z) yields a minimum - and this against arbitrary variations, not merely neighboring ones, so the statement is a global sufficiency result. Applied to the second variation the criterion yields the Jacobi condition, just as the Weierstrass condition yields the Legendre condition, and it remains applicable in cases where the Jacobi criterion fails (for instance between conjugate points). The paper's final sentence explicitly leaves open to what extent the criterion also suffices when the boundary values are not prescribed, and how it must then be modified. While problems with free ends have a classical sufficiency theory in the field-theoretic formulation (fields of extremals with the transversality condition, from Kneser and Bolza to Bliss), Hilbert's question in his own uniqueness-based formulation remains open, in particular for the double-integral problems treated in the same paper. The problem is to settle this: give a precise free-boundary formulation of the two conditions, prove or disprove sufficiency, and establish the correct modification.

Background

The calculus of variations studies minima of integral functionals such as the single integral $J[y]=\int F(y',y,x)\,dx$ or the double integral $J[z]=\iint F(z_x,z_y,z;x,y)\,d\Omega$, possibly subject to side conditions, for example differential relations imposed on the unknown functions. Setting the first variation of $J$ to zero yields the Euler-Lagrange equations; in the presence of side conditions these become the Lagrange equations with multipliers. A curve or surface satisfying them is called an extremal. A central distinction is between a weak minimum (the comparison functions are close to the candidate together with their first derivatives) and a strong minimum (only the function values need be close): conditions for strong minima are harder because the derivatives of competitors are unconstrained. Weierstrass introduced the excess function $\mathcal E(x,y,p,q)=F(x,y,q)-F(x,y,p)-(q-p)F_{y'}(x,y,p)$, where $p$ is a reference slope and $q$ an arbitrary slope, and proved that a strong minimum requires $\mathcal E\ge 0$ along the extremal (the Weierstrass condition); the Legendre condition ($F_{y'y'}\ge 0$) follows from it by passing to the second variation. Jacobi's necessary condition - the absence of conjugate points, equivalently the triviality of the solutions of the linearized (Jacobi) equation that vanish at the endpoints - arises from the second variation in the analogous position.

Sufficiency is subtler. Weierstrass's construction was completed by Hilbert's independence theorem (1906): in a field of extremals (a family of extremals simply covering a region, so that through each point passes exactly one of them, with slope function $u$), the curvilinear integral $J^*=\int [F(x,y,u)+(y'-u)F_{y'}(x,y,u)]\,dx$ is path-independent - the Hilbert invariant integral - and comparing $J$ along a competitor with $J^*$ along the extremal exhibits the increment of $J$ as an integral of $\mathcal E$; hence an extremal embedded in a field with $\mathcal E>0$ yields a strong minimum. For problems with mobile (free) ends - the boundary values of the sought function not prescribed, the end point at most confined to a given manifold - the first variation yields, in place of the vanishing boundary terms, the transversality condition (a generalization of the orthogonality of geodesics to surfaces), and the classical sufficiency theorems for this case use fields transversal to the endpoint manifold. This variable-endpoint theory goes back to A. Kneser (whose 1900 Encyklopädie article devotes a section to sufficient conditions with variable limits) and was codified by the textbooks of O. Bolza ('Vorlesungen über Variationsrechnung', 1909) and G. A. Bliss ('Lectures on the Calculus of Variations', 1947).

Hilbert's 1906 paper 'Zur Variationsrechnung' treats the necessity of the Lagrange equations under side conditions, the independence theorem and its relation to the Jacobi-Hamilton integration theory, the extension of the invariant-integral method to double integrals, a mixed problem (minimum of the sum of a double and a single integral), and, in its final section (pp. 368-370), a 'general rule for the treatment of variational problems and a new criterion' for problems in which the values of the sought functions are prescribed everywhere on the boundary. Let $Z$ be a system of solutions of the Lagrange equations $L$ satisfying all side conditions $B$ of the problem. Call $Z$ of positive definite character if the Weierstrass $\mathcal E$-functions are positive along $Z$. Call $Z$ of innerly unique character if the boundary value problem for $L$ with conditions $B$ is solved uniquely by $Z$, and if the same holds on every subdomain $T$ of the domain of integration, where on the part of $\partial T$ lying on the original boundary the conditions $B$ remain in force and on the newly created part of $\partial T$ the boundary values are the values taken there by $Z$. Hilbert asserts that a solution system of positive definite and innerly unique character yields a minimum, valid for arbitrary variations, not merely close ones, so that the criterion is a global sufficiency statement; he derives from it, via the second variation, the Jacobi condition; and he notes that the criterion is still applicable where the Jacobi criterion fails, for instance for deciding the minimum of a curve between two conjugate points. The final sentence leaves the free-boundary case open: to what extent the criterion also suffices when the boundary values are not given, and how it must then be modified, requires investigation in each particular case.

Hilbert's inner-uniqueness criterion did not become the standard formulation. The modern sufficiency theorem (as in the Encyclopedia of Mathematics articles 'Extremal field' and 'Weierstrass conditions (for a variational extremum)') is field-based: a slope function defined in a region, with $\mathcal E\ge 0$ throughout, certifies a strong local minimum; the same encyclopedia's article 'Hilbert invariant integral' credits the 1906 paper for the integral. For double integrals a long twentieth-century line (Levi 1913, Lichtenstein 1917, Haar 1931, Miranda 1934, Morrey 1943, Hestenes 1948, Karush 1948, Serrin 1959) developed sufficient conditions under restrictive hypotheses, with uniqueness ingredients supplied by Haar-type lemmas (uniqueness theorems for elliptic boundary value problems of the kind introduced by Haar in 1931) rather than by Hilbert's subdomain condition. The relation between Hilbert's uniqueness formulation and field embeddability remains unsettled; the correct free-boundary modification of his criterion, especially for the double-integral problems of his own paper, is the object of the present problem.

Problem Statement

The problem is the closing question of Hilbert's 'Zur Variationsrechnung' (1906), p. 370: determine whether the sufficiency criterion stated there - positive definite character plus innerly unique character of a solution system $Z$ of the Lagrange equations - remains a sufficient condition for a minimum when the boundary values of the sought functions are not prescribed, and establish how the criterion must be modified in that case. Concretely, within Hilbert's framework (analytic integrands and side conditions; single integrals as well as the double integrals treated in the same paper):

1. Formulate precisely the free-boundary analogue of the two conditions - (i) positive definite character (positivity of the Weierstrass excess functions along $Z$) and (ii) inner uniqueness (unique solvability of the boundary value problem for the Lagrange system with the side conditions, on the whole domain and on every subdomain with the boundary data induced there by $Z$) - when the prescribed boundary values are replaced by the free-boundary conditions of the variable-ends setting described in the Background: the end points at most confined to a given manifold, with the transversality condition with respect to that manifold, completely free boundary values arising as a special case; including the subdomain condition with $Z$-induced data on the newly created boundary.
2. Settle sufficiency: either prove that a solution system satisfying the two conditions in the formulation of item 1 yields a minimum against all admissible comparison functions with free boundary values, not merely neighboring ones, matching the global character of Hilbert's fixed-boundary claim - or construct an explicit counterexample showing that the criterion, in this formulation, does not suffice.
3. Establish the correct modification of the criterion: if the free-boundary criterion formulated in item 1 fails for some admissible free-boundary problem, exhibit the modification that restores sufficiency, and if it does not fail, prove that no modification is required. Then address the double-integral setting of the same paper: either prove the free-boundary criterion formulated in item 1 (modified as in the preceding clause, if a modification is required) for the double integrals treated there under the paper's hypotheses, or refute that criterion by an explicit example.

References

1. David Hilbert, 'Zur Variationsrechnung', Mathematische Annalen 62, 351-370 (1906). DOI: 10.1007/BF01450516, URL: https://doi.org/10.1007/BF01450516. Reprinted in D. Hilbert, Gesammelte Abhandlungen, Band III, Springer, Berlin (1970), pp. 38-55. DOI: 10.1007/978-3-662-25726-5_5, URL: https://doi.org/10.1007/978-3-662-25726-5_5, ISBN 9783662257265.
2. Adolf Kneser, 'II A 8. Variationsrechnung', in Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Band II A, Teubner, Leipzig (article completed September 1900). Digitized volume: https://gdz.sub.uni-goettingen.de/id/PPN36050616X.
3. Georg Prange, 'Die Hamilton-Jacobische Theorie für Doppelintegrale', Dieterich'sche Verlagsbuchhandlung, Göttingen (1915). http://resolver.sub.uni-goettingen.de/purl?PPN313315329.
4. I. M. Belen'kii, 'On an application of Hilbert's independence theorem', Journal of Applied Mathematics and Mechanics 27, 1349-1353 (1963). DOI: 10.1016/0021-8928(63)90073-X, URL: https://doi.org/10.1016/0021-8928(63)90073-X.
5. James Serrin, 'On a fundamental theorem of the calculus of variations', Acta Mathematica 102, 1-22 (1959). DOI: 10.1007/BF02559565, URL: https://doi.org/10.1007/BF02559565.
6. H. A. Kastrup, 'Canonical theories of Lagrangian dynamical systems in physics', Physics Reports 101, 1-167 (1983). DOI: 10.1016/0370-1573(83)90037-6, URL: https://doi.org/10.1016/0370-1573(83)90037-6.
7. Ruediger Thiele, 'On Some Contributions to Field Theory in the Calculus of Variations from Beltrami to Carathéodory', Historia Mathematica 24(3), 281-300 (1997). DOI: 10.1006/hmat.1997.2158, URL: https://doi.org/10.1006/hmat.1997.2158.
8. Craig Fraser, 'Hamilton-Jacobi Methods and Weierstrassian Field Theory in the Calculus of Variations: A Study in the Interaction of Mathematics and Physics', in E. Grosholz, H. Breger (eds.), The Growth of Mathematical Knowledge, Synthese Library vol. 289, Springer, pp. 93-101 (2000). DOI: 10.1007/978-94-015-9558-2_7, URL: https://doi.org/10.1007/978-94-015-9558-2_7.
9. I. B. Vapnyarskii (originator), 'Extremal field', Encyclopedia of Mathematics (online edition). ISBN 1402006098. https://encyclopediaofmath.org/wiki/Extremal_field.
10. V. M. Tikhomirov (originator), 'Weierstrass conditions (for a variational extremum)', Encyclopedia of Mathematics (online edition). ISBN 1402006098. https://encyclopediaofmath.org/wiki/Weierstrass_conditions_(for_a_variational_extremum).
11. V. M. Tikhomirov (originator), 'Hilbert invariant integral', Encyclopedia of Mathematics (online edition). ISBN 1402006098. https://encyclopediaofmath.org/wiki/Hilbert_invariant_integral.
12. N. N. Moiseev (originator), 'Variational calculus', Encyclopedia of Mathematics (online edition). ISBN 1402006098. https://encyclopediaofmath.org/wiki/Variational_calculus.
13. Misir J. Mardanov, Telman K. Melikov, Samin T. Malik, 'On sufficient conditions in the classical problem of the calculus of variations', arXiv:2512.21918 (2025). https://arxiv.org/abs/2512.21918.