In this paper we are concerned with the construction of a special class of appropriately defined vectorial minimisers in calculus of variations in , that is, for supremal functionals of the form
In the above, , is an open set and is a continuous function. The calculus of variations in has been pioneered by Aronsson in the 1960s who studied the scalar case quite systematically ([2, 3, 4, 5, 6, 7]). A major difficult associated to the study of (1.1) is that the standard minimality notion used for the respective more classical integral functional
is not appropriate in the case due to the lack of “locality” of (1.1). The remedy is to require minimality on all subdomains , a notion now known as absolute minimality (hence the emergence of the domain as second argument of the functional). The field has seen an explosion of interest especially after the 1990s when the development of viscosity solutions ([13, 11]) allowed the rigorous study of the non-divergence equation arising as the analogue of the Euler–Lagrange for (1.1) (for a pedagogical introduction to the scalar case and numerous references, we refer to ). Inthe special case of being the Euclidean norm on , the respective PDE is the -Laplace equation
Despite the importance for applications and the deep analytical interest of the area, the vectorial case of remained largely unexplored until the early 2010s. In particular, not even the correct form of the respective PDE systems associated to the variational problem was known. A notable exception is the early vectorial contributions [9, 8], wherein (among other deep results) versions of lower semi-continuity and quasiconvexity were introduced and studied, and the existence of absolute minimisers was established in some generality with H depending on u itself but for .
The author in a series of recent papers (see [18, 20, 21, 19, 26, 25, 30, 29, 28, 24]) has laid the foundations of the vectorial case and, in particular, has derived and studied the analogues of (1.2) associated to general functionals (see also the joint contributions with Croce, Pisante, Pryer and Abugirda [1, 14, 31, 27]). In the model case of H being the Euclidean norm on and independent of x, i.e.,
the respective equation is called the -Laplace PDE system and when applied to smooth maps reads
Here is the orthogonal projection on the orthogonal complement of the range of gradient matrix :
In index form, (1.3) reads
In the full vector case of (1.3), even more intriguing phenomena occur since a further difficulty, which is not present in the scalar case, is that the coefficient involving is discontinuous even for maps; for instance, is a smooth -harmonic map near the origin and the rank of gradient is 1 on the diagonal, but it is 2 otherwise. The emergence of discontinuities is a genuine vectorial phenomenon which does not arise if (see [18, 21, 19]). For , the scalar version (1.2) has continuous coefficients, whilst for , (1.3) reduces to
A problem associated to the discontinuities is that Aronsson’s notion of absolute minimisers is not appropriate in the vectorial case of rank . By the perpendicularity of and , actually consists of two independent systems and each one is characterised in terms of the norm of the gradient via different sets of variations. In  we proved the following variational characterisation in the class of classical solutions. A map is a solution to
if and only if it is a rank-one absolute minimiser on Ω, namely, when for all , all scalar functions vanishing on and all directions , u is a minimiser on D with respect to variations of the form (see Figure 1):
Further, if const., u is a solution to
if and only if has -minimal area, namely, when for all , all scalar functions (not vanishing on ) and all vector fields which are normal to , u is a minimiser on D with respect to normal free variations of the form (see Figure 2):
We called a map -minimal with respect to functional when it is a rank-one absolute minimiser on Ω and has -minimal area.
Perhaps the greatest difficulty associated to (1.1) and (1.3) is how to define and study generalised solutions since for the highly nonlinear non-divergence model system (1.3) all standard arguments based on the maximum principle or on integration by parts seem to fail. In the very recent work , the author proposed the theory of so-called -solutions, which applies to general fully nonlinear PDE systems of any order, namely,
and allows for merely measurable solutions u to be rigorously interpreted and studied. This notion is duality-free and is based on the probabilistic representation of derivatives which do not exist classically. -solutions have already borne substantial fruit and in [30, 29, 28, 24, 23] we have derived several existence-uniqueness, variational and regularity results. In particular, in , we have obtained a variational characterisation of (1.3) in the setting of general -solutions for appropriately defined minimisers which are relevant to (1.4)–(1.7) but different.
In this paper we consider the obvious generalisation of the rank-one minimality notion of (1.5) adapted to the functional (1.1). To this end, we identify a large class of rank-one absolute minimisers: for any , every solution to the vectorial Hamilton–Jacobi equation
actually is a rank-one absolute minimiser. Namely, for any , any and any , we have
For the above implication to be true we need the solutions to be in and not just in . This is not a technical difficulty, since it is well known, even in the scalar case, that if we allow only for one non-differentiability point, then the strong solutions of the eikonal equation are not absolutely minimising for the norm of the gradient (e.g., the cone function ). However, due to regularity results which available in the scalar case, it suffices to assume everywhere differentiability (see [12, 10]).
Our only hypothesis imposed on H is that for any , the partial function is rank-one level-convex. This means that for any , the sublevel sets are rank-one convex sets in . A set is called rank-one convex when for any matrices with , the convex combination is in for any . An equivalent way to phrase the rank-one level-convexity of is via the inequality
This convexity assumption is substantially weaker than the versions of quasiconvexity which we call “BJW-quasiconvexity”, named after Barron, Jensen and Wang who introduced it in .
We note that Hamilton–Jacobi equations are very important for variational problems and their equations. In the scalar case, solutions are viscosity solutions to the respective second order single equations in (see, e.g., ). Heuristically, for the case of the -Laplace equation this can be seen by rewriting (1.2) as and this reveals that solutions of are -harmonic. In the vectorial case, Hamilton–Jacobi equations give rise to certain first-order differential inclusions of the form
for which the Dacorogna–Marcellini Baire category (the analytic counterpart of Gromov’s convex integration) method can be utilised to establish existence of -solutions to the systems of PDE with extra geometric properties (see  and [16, 15]).
The main result of the present paper is the following.
Let be an open set, , and a continuous function such that for all , is rank-one level-convex, that is, is a rank-one convex in for all , . Let be a solution to the vectorial Hamilton–Jacobi PDE
for some . Then, u is a rank-one absolute minimiser of the functional
In addition, the following marginally stronger result holds true: for any , any and any , we have
where is the set of open balls centred at local extrema (maxima or minima) of ϕ inside :
An immediate consequence of Theorem 1 is the following result.
In the setting of Theorem 1, we additionally have
for any at which ϕ achieves a local maximum or a local minimum.
The quantity of the right-hand side above is known as the local functional at x and in the scalar case it has been used as a substitute of the pointwise values due to its upper semi-continuity regularity properties.
2 The proof of Theorem 1
Let be as in the statement and fix and a unit vector . We introduce the following notation for the projections on and the orthogonal hyperplane :
Let be such that , that is, the projections of ψ and u on the hyperplane coincide. Then and, because the scalar function vanishes on , there exist at least one local extremum of in , whence the set , given by (1.8), is non-empty. Fix a ball centred at such an extremal point of .
We illustrate the idea by assuming first in addition that . In this case, the point x is a critical point of and we have . Hence,
because on . Thus,
for any , whence the conclusion ensues.
Now we return to the general case of . We extend ψ by u on and consider the sets
where is a constant small enough so that . We set
and consider a partition of unity over so that
(Such a partition of unity can be easily constructed explicitly by mollifying the characteristic functions and rescaling them appropriately.) Let be the standard mollifier (as, e.g., in ) and set
whilst, for , we similarly have
By the standard properties of mollifiers, we have that the function
Since the regularity of is obvious (because u, by assumption, is such and ), the claim has been established.
Note now that since , the set given by (1.8) is non-empty because the scalar function which vanishes on necessarily attains an interior extremum. Fix a ball
Since in as , by a standard stability argument of maxima/minima of scalar-valued function under uniform convergence (see, e.g., ), there exists a local extremum of such that as . By the differentiability of u and by choosing ε small enough, we may arrange
Then, by arguing as in (2.1), we have
our continuity assumption and the regularity of ψ imply that there exists a positive increasing modulus of continuity with such that on the ball we have
By further restricting , we may arrange
This implies that for any ,
forming a convex combination. We now recall for immediate use right below the following Jensen-like inequality for level-convex functions (see, e.g., [9, 8]): for any probability measure μ on an open set and any μ-measurable function , we have
when is any continuous level-convex function. Further, by our rank-one level-convexity assumption on H and if ψ is as above, for any and with , the function
is level-convex. Indeed, given and with , we set
Then, , and hence . Moreover, and , which gives
for any , as desired.
By the continuity of H and , there exists a positive increasing modulus of continuity with such that
for all and . By using the fact that on , (2.15) and the above, we have
and since , we get
and by letting , the conclusion follows.
H. Abugirda and N. Katzourakis, Existence of vectorial absolute minimisers in under minimal assumptions, Proc. Amer. Math. Soc. 145 (2017), 2567–2575. Google Scholar
G. Aronsson, Minimization problems for the functional , Ark. Mat. 6 (1965), 33–53. Google Scholar
G. Aronsson, Minimization problems for the functional II, Ark. Mat. 6 (1966), 409–431. Google Scholar
G. Aronsson, On the partial differential equation , Ark. Mat. 7 (1968), 395–425. Google Scholar
G. Aronsson, Minimization problems for the functional III, Ark. Mat. 7 (1969), 509–512. Google Scholar
G. Aronsson, On certain singular solutions of the partial differential equation , Manuscripta Math. 47 (1984), no. 1–3, 133–151. Google Scholar
E. N. Barron, R. Jensen and C. Wang, Lower semicontinuity of functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), no. 4, 495–517. Google Scholar
E. N. Barron, R. Jensen and C. Wang, The Euler equation and absolute minimizers of functionals, Arch. Ration. Mech. Anal. 157 (2001), 255–283. Google Scholar
M. G. Crandall, A visit with the -Laplacian, Calculus of Variations and Non-Linear Partial Differential Equations, Lecture Notes in Math. 1927, Springer, Berlin (2008), 75–122. Google Scholar
M. G. Crandall, L. C. Evans and R. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations 13 (2001), 123–139. Google Scholar
G. Croce, N. Katzourakis and G. Pisante, -solutions to the system of vectorial calculus of variations in via the Baire category method for the singular values, preprint (2016), http://arxiv.org/pdf/1604.04385.pdf.
B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd ed., Appl. Math. Sci. 78, Springer, Berlin, 2008. Google Scholar
B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations, Progr. Nonlinear Differential Equations Appl. 37, Birkhäuser, Boston, 1999. Google Scholar
L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 1998. Google Scholar
N. Katzourakis, -variational problems for maps and the Aronsson PDE system, J. Differential Equations 253 (2012), no. 7, 2123–2139. Google Scholar
N. Katzourakis, Explicit -harmonic maps whose interfaces have junctions and corners, C. R. Math. Acad. Sci. Paris 351 (2013), 677–680. Google Scholar
N. Katzourakis, An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in , Springer Briefs Math., Springer, Cham, 2015. Google Scholar
N. Katzourakis, Equivalence between weak and -solutions for symmetric hyperbolic first order PDE systems, preprint (2015), http://arxiv.org/pdf/1507.03042.pdf.
N. Katzourakis, Mollification of -solutions to fully nonlinear PDE systems, preprint (2015), http://arxiv.org/pdf/1508.05519.pdf.
N. Katzourakis, Nonuniqueness in vector-valued calculus of variations in and some linear elliptic systems, Comm. Pure Appl. Anal. 14 (2015), no. 1, 313–327. Google Scholar
N. Katzourakis and T. Pryer, On the numerical approximation of -harmonic mappings, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 6, 1–23. Google Scholar
N. Katzourakis, A new characterisation of -harmonic and p-harmonic maps via affine variations in , Electron. J. Differential Equations 2017 (2017), no. 29, 1–19. Google Scholar
N. Katzourakis and T. Pryer, Second order variational problems and the -polylaplacian, preprint (2016), http://arxiv.org/pdf/1605.07880.pdf.
About the article
Published Online: 2017-06-04
Funding Source: Engineering and Physical Sciences Research Council
Award identifier / Grant number: EP/N017412/1
The author has been partially supported by an EPSRC grant EP/N017412/1.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 508–516, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0164.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0