Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs
aa r X i v : . [ m a t h . A P ] J a n SPECTRAL PROPERTIES OF NONASSOCIATIVEALGEBRAS AND BREAKING REGULARITY FORNONLINEAR ELLIPTIC TYPE PDES
VLADIMIR G. TKACHEV
Dedicated to V.G. Maz’ya on the occasion of his 80th birthday
Abstract.
In this paper, we address the following question: Why cer-tain nonassociative algebra structures emerge in the regularity theory ofelliptic type PDEs and also in constructing nonclassical and singular solu-tions? The aim of the paper is twofold. Firstly, to give a survey of diverseexamples on nonregular solutions to elliptic PDEs with emphasis on recentresults on nonclassical solutions to fully nonlinear equations. Secondly, todefine an appropriate algebraic formalism which makes the analytic partof the construction of nonclassical solutions more transparent. Introduction
The first examples of nonassociative algebras (including octonions and Liealgebras) appeared in the mid-19th century. Since then the theory has evolvedinto an independent branch of algebra, exhibiting many points of contact withother fields of mathematics, physics, mechanics, biology and other sciences.Algebras whose associativity is replaced by identities (as the famous Jacobiidentity in Lie algebras or the Jordan identity [ad( x ) , ad( x )] = 0 in Jordanalgebras) were a central topic in mathematics in the 20th century.Very recently, certain commutative nonassociative algebra structures emer-ged in a very different context: regularity of viscosity solutions to fully non-linear elliptic PDEs. More precisely, it appears that subalgebras of simplerank three formally real Jordan algebras can be employed for constructingtruly viscosity solutions to uniformly elliptic Hessian equations. To explainthis connection, we first review some relevant concepts and results.1.1. Linear and quasilinear elliptic type PDEs.
Starting with the pio-neering works of S.N. Bernstein, the maximum principle and a priori estimateshave provided decisive instruments in proving the existence of solutions of gen-eral linear and quasilinear elliptic type PDEs Lw = 0. Important progressin this direction was made in the late 1950s by E. De Giorgi, J. Moser andJ. Nash by establishing fundamental a priori regularity results. A fundamen-tal result of the regularity theory asserts that every solution w from a ‘natural’ Date : 12/NOV/2018.2000
Mathematics Subject Classification.
Primary 53A04; Secondary 52A40, 52A10.
Key words. viscosity solutions, elliptic type PDEs, compositions algebras.The author thanks the anonymous referee for several helpful comments. class W , of Lw = 0 must be H¨older continuous, i.e. w ∈ C ,α for a certain α > l ≥ C ∞ -coefficients whose gen-eralized solutions are not C ∞ regular. The strong ellipticity of an operator Lu = P D α ( a αβ D β u ) means that there exists c > R R n P a αβ D α φD β φ dx ≥ c R R n P ( D α φ ) dx holds for any φ ∈ C ∞ ( R n ) with acompact support, where summation is over all | α | = | β | = l . Then Maz’yanotes that the equation ν ∆ w + κ ∆ (cid:18) x i x j | x | w x i x j (cid:19) + κ ∆ (cid:18) x i x j | x | ∆ w (cid:19) x i x j + µ (cid:18) x i x j x k x l | x | w x i x j (cid:19) x k x l = 0is strongly elliptic for κ < µν and the radial symmetric function w = | x | a with a = 2 − n s n − ( n − κ n + µ ) ν + 2 κ + µ is a weak solution of the above equation in W , ( B ), where B is the openunit ball in R n . For κ = n ( n − µ = n and ν = ( n − + ε , with ε < , theabove strong ellipticity condition is satisfied and the corresponding solution w = | x | a is unbounded in B for all dimensions n ≥ ε .A very related problem is the regularity of p -harmonic functions. It is well-known that for p > p = 2, a weak (in the distributional sense) p -harmonicfunction is normally in the H¨older class C ,α [59], but need not to be a H¨oldercontinuous or even continuous in a closed domain with non-regular boundary,as it follows from examples constructed by Krol’ and Maz’ya in [26]. Onthe other hand, if ess sup | Du ( x ) | > u ( x )is in fact a real analytic function in E [30]. It is interesting whether theconverse non-vanishing property holds true. This question naturally leads tohomogeneous p -harmonic functions (also called quasi-radial solutions in [4]);see the paper of M. Akman, J. Lewis, A. Vogel [1] in the present volume for themodern status of the problem and further discussion. We only mention thatin certain dimensions n ≥ p -harmonic functions for some distinguished values p , 2 < p < n based onisoparametric forms and generalizing the angle solutions constructed by Krol’and Maz’ya, see [55].Another prominent example of breaking regularity is the existence of glob-ally minimal cones and entire solutions of the minimal surface equation indimensions n ≥ Bombieri, De Giorgio and Giusti [6] make an essential use of the quadraticisoparametric form(1) u ( x ) := x + x + x + x − x − x − x − x see also [50] and [49] for the general isoparametric case. The function u ( x )coincides with the norm in the split octonion algebra [5]. It is not clear,whether this is merely an coincidence or there is some natural explanation inthe context of singular and entire solutions.On the other hand, Lawson and Osserman [29] constructed non-parametricminimal cones of high codimensions providing examples of Lipschitz but non- C solutions to the minimal surface equations, thereby making sharp con-trast to the regularity theorem for minimal graphs of codimension one. TheLawson-Osserman examples based on Hopf foliations corresponding to theclassical division algebras A d : A = C , A = H and A = O . More explicitly,for d = 2 , , w ( x ) = 12 r d + 1 d − η ( x ) | x | : R d → R d +1 , where(3) η ( x ) = ( | z | − | z | , z ¯ z ) , x = ( z , z ) ∈ A d ∼ = R d . Then the map w ( x ) : R d → R d +1 provides a nontrivial Lipschitz solution tothe Bernstein problem in codimension d + 1.We finally mention some further classes of semilinear and quasilinear el-liptic PDEs with similar phenomena, for example, the recent entire solutionsof the Ginzburg-Landau system constructed by A. Farina in [16] by usingisoparametric forms; see also section 3 in [33] for a recent survey of furthercounterexamples as well as regularity results.1.2. Fully nonlinear elliptic type PDEs.
In the general, fully nonlinearcase, the regularity and existence issues become more involved and require twoprincipal ingredients: (i) the Harnack inequality for solutions of the 2nd ordernon-divergence elliptic equations with measurable coefficients established in1979 by N. Krylov and M.V. Safonov [27], [28], and (ii) a suitable extensionof the concept of a generalized solution, the so-called viscosity solution , animportant tool developed by Crandall-Lions, Evans, Jensen and Ishii. Werefer to [11] for exact definitions and self-contained exposition of the basictheory of viscosity solutions and briefly note that the latter concept generalizesweak solutions for divergence type elliptic equations by utilizing the maximumprinciple. Viscosity solutions play a central role in many problems from globalgeoemtry and analysis, and fit naturally into the contexts of optimal control,differential and stochastic differential games as well as mathematical finance.An important model example with a far-ranging set of applications in pureand applied mathematics is the Dirichlet problem for the uniformly elliptic The author is grateful to Seidon Alsaody for pointing out this.
VLADIMIR G. TKACHEV equation of the form(4) F ( D w ) = 0 in B w = φ on ∂B with a uniformly elliptic operator F . The latter means that there exist 0 <λ ≤ Λ such that the inequalities hold λ k N k ≤ F ( M + N ) − F ( M ) ≤ Λ k N k whenever M, N are symmetric matrices with N nonnegative semi-definite.For many geometrical applications, an important particular case of (4) isthe Hessian equations , i.e. when F ( X ) depends only on the eigenvalues ofthe (symmetric) matrix X . The Hessian equations include the classical ex-amples of Laplace’s equation, curvature Weingarten equation, Monge-Amp`ereequation and special Lagrange equation, and, more recently, the calibratedgeometries and Dirichlet duality theory [19], [20]. According to the generaltheory [11], a fully nonlinear elliptic equation (4) always has a unique con-tinuous viscosity solution for any continuous data φ . The question whether aviscosity solution is classical, i.e. twice differentiable, turned out to be verychallenging. For certain cases the regularity is known:(i) If n = 2 then w is classical ( C ,α ) solution (Nirenberg [44])(ii) If n ≥ ≤ ǫ ( n ) then w is C ,α (Cordes [10])(iii) w ∈ C ,α ( B / ), α = α (Λ , n ) (Trudinger [58], Caffarelli [7])(iv) F convex (concave) ⇒ w ∈ C ,α ( B / ) (Krylov, Evans [15])(v) w is C ,α ( B \ Σ), where the Hausdorff dimension dim H Σ < n − ǫ , ǫ = ǫ (Λ , n ) (Armstrong-Silvestere-Smart, [3]).Further regularity results including the modulus of continuity of viscositysolutions can be found in [51], [22]; see also [13] for the VMO-regularity andgradient estimates for viscosity solutions to nonhomogeneous fully nonlinearelliptic equations F ( D u ) = f with f ∈ L p .1.3. Truly viscosity solutions.
It follows from the above that in the generalcase and in dimensions n ≥ C ,α -regular viscosity can be expected.Until very recently the very existence of nonclassical solutions for n ≥ n ≥
12. The method of constructionmakes an essential use of an accumulation property of the spectrum of theHessian of a certain cubic form u in R . The latter is also well known as a triality form and comes from the Hamilton quaternions H [5]. Theorem 1.1 ([37], [41]) . Let (5) u ( x ) = Re( z z z ) , where x = ( z , z , z ) ∈ H ∼ = R and (6) w ,α ( x ) = u ( x ) | x | α . If ≤ α < then w ,α is a viscosity solution in R of a uniformly ellipticHessian equation (4) with a smooth F . In view of the appearance of the quaternions in (6), it is natural to expectthat the corresponding cubic form u based on the octonions also producessingular viscosity solutions. The paper [40] establishes that this is actuallythe case. Furthermore, in [35] a nonclassical viscosity solution w in R basedon the Cartan isoparametric cubic form u has been constructed and, shortlyafter, in [43] Nadirashvili and Vl˘adut¸ established that the corresponding w ,α is a singular viscosity solutions in R . More precisely, one has Theorem 1.2 ([35]) . There a cubic form u ( x ) such that w ,α ( x ) = u ( x ) | x | α is a viscosity solution in B ⊂ R . If ≤ α < then w ,α is a viscositysolution in R of a uniformly elliptic Hessian equation (4) with a smooth F .Furthermore, w := w , satisfies (4) with F ( D w ) = (∆ w ) + 2 (∆ w ) + 2 ∆ w + 2 det D ( w ) . This establishes the lowest possible dimension known so far where singularviscosity solutions to uniformly elliptic equations may exist.The algebraic part of the above construction relies heavily on certain ac-cumulation properties of the spectrum of the cubic forms u k involved. Itwas noticed in [39] that to obtain a nonclassical viscosity solution, the corre-sponding cubic form u n must be rather exceptional. Furthermore, numericalsimulations show that a random (‘generic’) cubic form does not produce anonclassical solution to (4). This makes it reasonable to ask why u , u and u are so exceptional and how to characterize the cubic forms producingsingular/nonclassical viscosity solutions?We already mentioned that u and u come from the triality concept andare intimately related to exceptional Lie groups and the classical divisionalgebras in dimensions 4 and 8 respectively. It follows from [39] that thecorresponding triality cubic form u over the field of complex numbers C alsoproduces a viscosity solution but only for a degenerated elliptic equation.The cubic form u is also rather special and has many important connec-tions in analysis and geometry. In the present context, at least the followingproperties are essential:(a) u is the generic norm on (the trace free subspace of) rank 3 formallyreal Jordan algebra H ( R ) of symmetric 3 ×
3, and matrices.(b) u is the simplest Cartan isoparametric cubic form.There exists a natural correspondence between the concepts in (a) and (b), see[52]. Explicitly, the cubic form u is given by the determinant representation(7) u ( x ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ x + x x x x − √ x x x x √ x − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Taking into account the above observations, it is highly desirable to finda conceptual explanation of the existence of nonclassical solutions and relate
VLADIMIR G. TKACHEV them to appropriate algebraic structures. In this paper, we discuss somepossible approaches to these questions.The appearance of Jordan algebras in (7) and (a) makes it natural to askwhether these algebras also relevant for the examples in section 1.1. As wealready mentioned, the isoparametric form u in (1) reminisces the norm ofsplit octonions. Moreover, the Hopf map (3) coincides with the multiplicationin Clifford type Jordan algebras (the so-called spin-factors), see [5]. Remark 1.3.
In all the above examples of non-regular solutions, includingthe p -harmonic homogenous examples, the regularity breaks at an isolatedpoint where (the blow-down of) a solution has a H¨older type singularity. Re-markably, the asymptotic cone at the singular point is always a minimal cone .It is also interesting to know whether the appearance of the minimality in thiscontext is mere coincidence or there is a general explanation of this phenom-enon? This question also naturally arises in connection to phase transitionsand the De Giorgi conjecture, see [46].The rest of the paper is structured as follows. In section 2 we discuss themain ideas underlying the construction of homogeneous nonclassical viscositysolutions of (4). There we also define an algebraic formalism which connectscubic forms with certain commutative nonassociative algebras. In this setting,the gradient of a cubic form determines the multiplicative algebra structure; inparticular, algebra idempotents correspond to stationary points of the cubicform. A cubic form is called special if it generates a nonclassical solution.We reformulate this definition in terms of the Peirce spectrum. We discussthe Peirce decomposition and generic cubic forms in sections 2.4 and 2.5.In section 2.6 we apply the algebraic approach to special forms. In the lastsection3 we revisit Cartan’s isoparametric cubics and determine their Peircespectrum. 2. Ingredients and definitions
Preliminaries.
Recall that according to the regularity result [3], if w isa nonclassical solution to (4) in the unit ball B then its singular set must becompactly contained in B . It is natural to consider with the simplest singlepoint singularity and look for candidates for nonclassical viscosity solutions inthe unit ball B ⊂ R n in the class of homogeneous solutions, i.e. the functions w satisfying w ( tx ) = t k w ( x ) , x ∈ R n , t > . The ambient dimension must be n ≥
5. Indeed, for n = 2 it follows from theNirenberg result (i) mentioned above that any (not necessarily homogeneous)viscosity solution is classical, and also if 3 ≤ n ≤ n = 3 follows from a theorem of A.D. Alexandroff in [2]). Furthermore, it is natural to specify w ( x ) and assume that it has the fol-lowing form:(8) w ( x ) = u ( x ) | x | α ,u ( x ) being a cubic form in the variable x ∈ R n and α ≥
1. Under themade assumptions, it follows from [35] that the class of nonclassical viscositysolutions with α = 1 is non-empty for any dimension n ≥ u ( x ) with α = 1 guarantees that (8) with 1 < α < singular viscosity solution to a certain F in (4), see Chapter 4 in [36] (it isbelieved that this observation holds in general, though no proof is known). Inview of this, we shall only consider nonclassical solutions.2.2. Nonclassical viscosity solutions.
We briefly recall the main ideas ofthe construction following to [39], [35]. For more information and the proofs,the reader is referred to chapters 4 and 5 in [36]. Let w be an arbitraryhomogeneous function of order 2 defined on R n and smooth in R n \ { } . Thenthe Hessian D w ( x ) is homogeneous of order 0 at any point x ∈ R n \ { } andit induces the Hessian map D w : R n \ { } → Sym n , where Sym n denotes the space of symmetric n × n matrices with real coeffi-cients. By the homogeneity assumption, we have D w ( tx ) = D w ( x ) for any t > D w is completely determined by its values on the unitsphere ∂B . We denote by H w = D w | ∂B the corresponding restriction map. By O ( n ) we denote the group of isometriesin R n .Following to [39] we say that a symmetric matrix A is M - hyperbolic forsome real M ≥ λ i = 0 or(9) 1 M ≤ − λ λ n ≤ M, where λ ≤ · · · ≤ λ n denote the eigenvalues of A .A subset A ⊂
Sym n is called hyperbolic if there exists a constant M suchthat for any two matrices A, B ∈ A , A − B is M -hyperbolic.The importance of the latter concepts is clear from the following result. In fact, one can also construct some counterexamples w as in (8) using isoparametrichomogeneous forms u of degree 3 , quadratic isoparametric forms, but they are non-appropriate for constructing thenonclassical solutions. In this paper we confine ourselves with the degree 3 case. VLADIMIR G. TKACHEV
Lemma 2.1 (Main Lemma in [37]) . Let function w be defined by (8) with α = 1 . Suppose that the restriction map H w : ∂B → Sym n is a smoothembedding. Then(i) if H w ( ∂B ) is hyperbolic then w is a viscosity solution of a uniformlyelliptic equation (4) in R n ;(ii) if additionally the (larger) subset { U AU − : A ∈ H w ( ∂B ) , U ∈ O ( n ) } is hyperbolic then w is a solution of a Hessian uniformly elliptic equa-tion (4) in R n . Thus, the above hyperbolicity condition is crucial for constructing viscositysolutions. Note also that all examples discussed in section 1.3 satisfy in factthe stronger condition (ii). This suggests the following definition.
Definition 2.2.
A cubic form u ( x ) is called special if w = u ( x ) / | x | satisfieshypotheses (ii) in Lemma 2.1.It is also interesting to characterize the cubic forms u which satisfy theweaker condition (i). For continuity reasons, if u satisfies (i) then all cubicforms ‘sufficiently close’ to u will also satisfy (i). Remark 2.3.
An example of such a form is the determinant u ( x ) = det X ,where X ∈ R × ∼ = R ( X is an arbitrary nonsymmetric matrix over R ).Then it is easy to see that u is the specialization of the triality form u onthe imaginary subspace (Im H ) , wee (5). On the other hand, it is known,see Remark 4.1.5 in [36], that u satisfies the ellipticity criterion (i) and thusproduces a nonclassical solution w = u/ | x | of a uniformly elliptic fully nonlin-ear equation in R . However, a more sophisticated analysis reveals that thisfunction fails to satisfy the sufficient conditions (ii) and thus may be not asolution of a Hessian uniformly elliptic fully nonlinear equation. Remarkably, u is one of the exceptional Hsiang cubic forms emerging in classification ofcubic minimal cones[36, Chapter 6], cf. also with the remarks in Section 4below.2.3. Algebras of cubic forms.
The main obstacle for identifying whichcubic forms u are special is an a priori hard problem to control the spectrumof the difference H w ( x ) − H w ( y ) for all possible pairs x, y ∈ ∂B . This problembecomes more tractable and conceptually more transparent if one pass to acertain natural nonassociative algebra attached to the cubic form u . Below,we recall some relevant concepts and results following [53], [54], [56]; see alsosection 6.3 in [36].By an algebra we always mean a (finite dimensional) vector space V overthe real numbers R with a multiplication on V , i.e. a bilinear map (denotedby juxtaposition) ( x, y ) → xy ∈ V. By an abuse of notation, we denote by V both the vector space and the corresponding algebra.In this paper, by an algebra we mean a commutative algebra. A symmetricbilinear form b : V × V → R on an algebra V is called associating if(10) b ( xy, z ) = b ( x, yz ) , ∀ x, y, z ∈ V. An algebra carrying an associating bilinear form is called metrised . The stan-dard example is the generic trace form on a Jordan algebra [32].The most striking corollary of the existence of an associating bilinear formis that the operator of left (=right) multiplication L x : y → xy = yx is self-adjoint, i.e.(11) h L x y, z i = h y, L x yz i , ∀ x, y, z ∈ V. Now we want to associate with an arbitrary cubic form an algebra. Recallthat a cubic form is a function u : V → R such that its full linearization u ( x, y, z ) = u ( x + y + z ) − u ( x + y ) − u ( x + z ) − u ( y + z ) + u ( x ) + u ( y ) + u ( z )is a trilinear form. The trilinear form u ( x, y, z ) is obviously symmetric andthe cubic form is recovered by(12) 6 u ( x ) = u ( x, x, x ) . A positive definite symmetric bilinear form h x, y i on V is called an innerproduct. Given an inner product and a cubic form u , its gradient ∇ u ( x ) at x is uniquely determined by the duality(13) h∇ u ( x ) , y i = 12 u ( x, x, y ) , ∀ y ∈ V. It follows from the above definitions, that thus defined, ∇ u coincides with thestandard gradient of a function u . Definition 2.4.
Let u be a cubic form on an inner product vector space( V, h , i ). Define the multiplication ( x, y ) → xy as the unique element satisfying(14) h xy, z i = u ( x, y, z ) for all z ∈ V The multiplication is commutative but maybe nonassociative. We call thusdefined algebra on V the algebra of the cubic form u and denote it by V ( u ).It is easy to see that V ( u ) is a zero algebra (i.e. V ( u ) V ( u ) = 0) if and onlyif the cubic form u is identically zero.An important corollary of the above definition and the symmetricity ofthe trilinear form u ( x, y, z ) is that the inner product on the algebra V ( u ) isassociating, i.e. V ( u ) is a metrised commutative algebra .Furthermore, it also follows from the definition that given an inner productvector space ( V, h , i ), there is a canonical bijection between the vector space C ( V ) of all cubic forms on V and commutative metrised algebra structures A ( V ) on V , where the correspondence C ( V ) → A ( V ) is given by (14), andthe converse correspondence A ( V ) → C ( V ) is defined by(15) u ( x ) := 16 h x , x i . With the above concepts in hand, we are able to identify the standardcalculus operations on V as appropriate algebraic concepts on V ( u ). First note that (13) combined with (14) immediately yields that the gradient of u ( x ) is essentially the square of the element x in V ( u ):(16) ∇ u ( x ) = 12 xx = 12 x , and, similarly, the multiplication in V ( u ) is explicitly recovered by(17) ( D u ( x )) y = xy, where D u ( x ) is the Hessian map of u at x . This implies that the Hessianmap of u at x is nothing else than the multiplication operator L x :(18) D u ( x ) = L x . Then it follows from (10) that L x is a self-adjoint operator with respect tothe inner product h , i .In summary, starting with a cubic form on an inner product vector space onecan construct an algebra structure which translates the standard calculus intoappropriate algebraic concepts. Then the Peirce decomposition relative to anidempotent is a significant tool in identifying of the underlying nonassociativealgebra structure, see [47], [32]. Therefore it is important to know whetherthe set of idempotents in V ( u ) is empty or not. The next proposition answersthis question in positive; its proof can be found in our papers [54], [56]. Proposition 2.5.
Let V ( u ) be the algebra of a cubic form u on an innerproduct vector space V . Denote by E the (nonempty) set of stationary pointsof the variational problem (19) h x, x i → max subject to a constraint h x, x i = 1 . Then for any x ∈ E , either x = 0 or c := x h x ,x i is an idempotent in V .In particular, the set of idempotents of V ( u ) is nonempty. Furthermore, if E ⊂ E denote the subset of local maxima in (19) then the correspondingidempotent c = x/ h x , x i satisfies the extremal property (20) spectrum( L c | c ⊥ ) ⊂ ( −∞ , ] , where c ⊥ = { x ∈ V : h x, c i = 0 } . In particular, the eigenvalue of L c issimple (i.e. c is a primitive idempotent). Jordan algebras.
Jordan algebras is an important class in the contextof cubic forms and having numerous applications in mathematics and math-ematical physics. Recall that a commutative algebra V is called Jordan if(21) [ L x , L x ] = 0 . It is well known that (21) implies that any Jordan algebra is power-associative,i.e. the powers x n do not depends on associations [23].In the most interesting for applications case when V is formally real, thebilinear form b ( x, y ) := tr L xy (the generic trace form) is positive definite andassociating, i.e. satisfies (10). In their famous work [24] Jordan, von Neumannand Wigner proved that the only simple formally real Jordan algebras arethe Jordan algebras of n × n self-adjoint matrices over A d , d = 1 , ,
3, the exceptional 27-dimensional Albert algebra of 3 × spectrum of an idempotent c in a commutativealgebra is called the spectrum of the corresponding multiplication operator L c . The algebra spectrum is the union of all idempotent spectra.An important structure property of any Jordan algebra V is that the spec-trum of any idempotent is a subset of { , , } . In particular, V admits theso-called Peirce decomposition :(22) V = V ( c ) ⊕ V ( c ) ⊕ V ( c ) , where V λ is the λ -subspace of L c . Then V c ( λ ) are subalgebras of V for λ ∈{ , } .The eigenvalue and the corresponding subspace is distinguished in manyways. For example, a Jordan algebra is simple if and only its -eigenspace isnontrivial for any idempotent. Moreover, V satisfies the Jordan fusion laws(23) V c ( λ ) V c ( ) ⊂ V c ( ) ∀ λ ∈ { , } , V c ( ) V c ( ) ⊂ V c (0) ⊕ V c (1) . As we shall see, plays an essential role in constructing nonclassical solutions.2.5. Generic cubic forms.
As we remarked in section 1.3 above, some nu-merical simulations support a believe that a randomly chosen cubic form isnot special. To give the latter observation a rigorous meaning we need toformalize what we mean by a generic cubic form. One natural way to do thisis to combine the correspondence given in the proceeding section with theconcept of a generic algebra introduced recently in [25].Recall that an algebra over C is called generic if it has exactly 2 n idempo-tents (counting x = 0). The idea comes back to the classical work of Segre[48]: given a fixed basis of an arbitrary nonassociative algebra V of dimension n ≥ C , one can identify the multiplication with adegree 2 homogeneous map on V . In this setting, the idempotent definingrelation x = x becomes a system of quadratic polynomial equations on C n .It is well known that a generic (in the Zariski sense) polynomial system hasthe B´ezout number of solutions equal to the product of the principal degreesof the system equations. In our case, the B´ezout number is 2 n .More precisely, let { e i } ≤ i ≤ n be an arbitrary fixed basis of a finite dimen-sional vector space V . Then any algebra structure on V is uniquely determinedby its multiplication table M := ( a ijk ) ≤ i,j,k ≤ n , where e i e j = P nk =1 a ijk e k . Inthis notation, the algebra product f ( x, y ) := xy in V is given component-wiseby(24) f k ( x, y ) := n X k =1 a ijk x i y j , k = 1 , , . . . , n. The idempotent defining relation becomes in this notation(25) f ( x, x ) = x. Then the algebra V is generic if (25) has exactly 2 n distinct solutions over C .In particular, the latter implies that each solution is a non-degenerate pointof (25). The latter condition is important because it can be reformulated interms of the spectrum of the algebra V .Now it is natural to call a cubic form u on V generic if the correspondingalgebra V ( u ) is so.Note that the above definitions work equally well in both analytic and alge-braic setting. However, it is more preferable to work with the latter becausethe genericity concept is easily translated to the well-developed nonassociativealgebra theory, including the Peirce decomposition.In [25], the following criterium has been established. Proposition 2.6.
A commutative algebra V over C is generic if and only ifthe spectrum of any idempotent in V does not contain . Because the essential role playing in the present context and also for thereader convenience, we give a sketch of the proof. Proof.
Note that ˜ f ( x ) := f ( x, x ) is a homogeneous of degree 2 endomorphismof V . Since V is commutative we have˜ f ( x + y ) − ˜ f ( x ) − ˜ f ( y ) = f ( x, y ) + f ( y, x ) = 2 f ( x, y ) . In particular, ∂ ˜ f k ∂x j ( x ) = 2 f k ( x, e j ), which yields D ˜ f ( x ) = 2 L x , where D ˜ f ( x ) is the Jacobi matrix of g at x . Rewriting the system (25) as g k ( x ) := ˜ f k ( x ) − x k = 0, k = 1 , , . . . , n, we see that Dg ( x ) = 2 L x − I. Thus, Dg ( x ) is non-degenerate if and only if 2 L x − I is so, i.e. is not in thespectrum of L x . In order to finish the proof it remains to note that the system(25) has the maximal finite number of solution if and only if each solution isa nondegenerate point of g , see for instance [12]. (cid:3) We have already seen in section 2.4 that any Jordan algebra has in itsspectrum and plays a distinguished role in the classification of formally realalgebras by Jordan, von Neumann and Wigner [24]. In general, the exception-ality of the Peirce eigenvalue λ = in the spectrum of many well establishednonassociative algebra structures (including Jordan algebras, Bernstein al-gebras and general genetic algebras) is a rather common phenomenon. Wemention its very recent appearance in the context of axial algebras (general-izing the Monster algebra of the largest sporadic finite simple group) [18]).Hsiang algebras emerging in the context of cubic minimal cones is another ex-ample of commutative nonassociative algebras where the presence of playsa distinguished role, see chapter 6 in [36]. In a wider context of algebras withidentities, the universality of has been recently discussed in [57]. Our next step is to explain why the Peirce eigenvalue is also distinguishedin the context nonclassical solutions.2.6. The spectral properties of the Hessian of w . It has been remarkedin Lemma 3.2 in [39] that the hyperbolicity condition (Lemma 2.1 above) fora cubic form u would be fulfilled if there would exist δ > d ∈ V the spectrum λ ( d ) ≤ . . . ≤ λ n ( d ) of the quadratic form ∂ d u ( x ) satisfies(26) ρ ( d ) := max { λ ( d ) λ ( d ) , λ n ( d ) λ n − ( d ) } < − δ. Unfortunately (see the remark after Lemma 3.2 in [39]), the above conditionis too strong and is failed for certain d for any cubic form. Indeed, using (16),the quadratic form ∂ d u ( x ) may be rewritten as ∂ d u ( x ) = h x , d i = h x, xd i = h x, L d x i , therefore its spectrum coincides with the spectrum of the scaled multiplicationoperator L d . Now, choosing d ∈ E in the notation of Proposition 2.5, itfollows from (20) and the fact that d is an idempotent that λ ( d ) = 1 and λ ( d ) ≤ , hence ρ ( d ) ≥ in the corresponding algebra V ( u ). Belowwe try to clarify its nature in more detail and relate to generic cubic forms.Let u V .Using (15) we rewrite (8) for α = 1 as w ( x ) = 6 u ( x ) | x | = h x , x i| x | . Then we have for the directional derivative(27) h∇ w ( x ) , y i = ∂ y w | x = 3 h x , y i| x | − h x , x ih x, y i| x | , implying by the duality the expression for the gradient ∇ w ( x ) = 3 x | x | − h x , x i x | x | . Arguing similarly, we find for the Hessian(28) H ( x ) := D w ( x ) = 6 | x | L x − h x , x i| x | − x b ⊗ x | x | + 3 x ⊗ x | x | h x , x i , where ( a ⊗ b )( x ) = a h b, x i , a b ⊗ b = a ⊗ b + b ⊗ a. These idempotents have interesting extremal properties and can be related to Cliffordtype Jordan algebras, see our recent paper [56]
Note that the Hessian is homogeneous degree 0 but it is an odd map. i.e. H ( − x ) = − H ( x ). Furthermore,(29) ∆ w ( x ) = tr D w ( x ) = 6 | x | tr L x − ( n + 3) h x , x i| x | . Recall that u is special if there exists M ≥ H ( x ) − H ( y ) is M -hyperbolic for any pair x, y ∈ V . The following elementary observationimmediately follows if one chooses y = − x in and the oddness of H ( x ). Lemma 2.7. If w is special then the set { H ( x ) } | x | =1 is hyperbolic. The particular case when x = c is an idempotent of V ( u ) is very specialbecause x and x coincide, implying considerable simplifications in (28), suchthat the spectrum of H ( c ) can be calculated explicitly. Lemma 2.8. If c is a nonzero idempotent of V ( u ) then (30) spectrum( H ( c )) = (cid:26) | c | , λ − | c | , . . . , λ n − − | c | (cid:27) where , λ , . . . , λ n − are the eigenvalues of L c counting the multiplicities. Inparticular, the characteristic polynomial of H ( c ) is given by (31) χ H ( c ) ( t ) = 6 n (6 | c | t − | c | n ( | c | t − · χ c (cid:18) | c | t (cid:19) where χ c ( z ) is the characteristic polynomial of L c .Proof. Indeed, since c = c we obtain from (28)(32) H ( c ) = 1 | c | (cid:18) L c − − | c | c ⊗ c (cid:19) . In particular, H ( c ) c = | c | c , i.e. c is an eigenvector of H ( c ) with eigenvalue | c | . Since H ( c ) is self-adjoint, the orthogonal complement c ⊥ is its invariantsubspace. We have H ( c ) = 1 | c | (6 L c −
1) on c ⊥ . This yields (30), and therefore (31). (cid:3)
Remark 2.9.
As an immediate corollary of (31) we obtain(33) χ c ( z ) = ( z − | c | n n · χ H ( c ) (cid:16) z − | c | (cid:17) z − . We point out a remarkable appearance of the eigenvalue in the denominator.This implies by virtue of Proposition 2.6 that if u is a generic cubic form thenthe eigenvalue λ = | c | must be a simple eigenvalue of the Hessian H ( c ). Algebras of Cartan’s isoparametric cubics
In this section we consider the principal model example which clarifies somealgebraic ingredients of the construction of nonclassical solutions. We estab-lish the corresponding Peirce decomposition and derive the finiteness of theeiconal algebras in Proposition 3.5.3.1.
Cartan-M¨unzner equations.
The only known so far example of non-classical solution in the lowest dimension n = 5 (see Theorem 1.2) is basedon the cubic form u that naturally appears in the context of isoparametrichypersurfaces in the Euclidean spheres.According to ´E. Cartan, a hypersurface of the unit sphere S n − ⊂ R n iscalled isoparametric if it has constant principal curvatures. Isoparametrichypersurfaces have been shown to be useful in various areas of mathematics,see [9] for the modern account of the isoparametric theory. Cartan himselfclassified all isoparametric hypersurfaces with g = 1 , , g ∈ { , , , , } , all fivepossibilities are realized, and each isopaprametric hypersurface with g distinctprincipal curvatures is obtained as a level set of a homogeneous degree g polynomial on the unit sphere.The case g = 3 is very distinguished in many way. In [8] Cartan proved thatany isoparametric hypersurface M d with g = 3 distinct principal curvatures,each principal curvature must have the same multiplicity, and the possiblemultiplicities are d = 1 , , , A d . More precisely, M d is a tube of constant radius over astandard Veronese embedding of a projective plane into the standard sphereover the division algebra A d . Equivalently, M d is a locus of a cubic form u ( x )in S d +1 ⊂ R d +2 with u satisfying the Cartan-M¨unzner system(34) |∇ u ( x ) | = 9 | x | , (35) ∆ u ( x ) = 0 . Cartan classified all cubic solutions of (34)–(35) and showed that the corre-sponding defining cubic polynomials are given explicitly by u = x + x ( | z | + | z | − | z | − x ) + √ x ( | z | − | z | )+ 3 √ z z ) z , (36)where x = ( x , x , z , z , z ) and z i ∈ R d ∼ = A d and d ∈ { , , , } . We referto (36) as to a Cartan isoparametric cubic . Each Cartan isoparametric cubicalso satisfies a determinantal representation like (7) above, where the latterdeterminant should be properly understood in an appropriate sense. Moreprecisely, u is the generic determinant in the Jordan algebra of 3 × A d . Algebras attached to (34) . Below, we apply the definitions given insection 2.3 to Cartan isoparametric cubics. Somewhat different approachusing the Freudenthal-Springer construction was suggested in [52].Our starting point is an arbitrary cubic homogeneous polynomial solution u ( x ) of the Cartan-M¨unzner equation (34) alone. By abusing of terminol-ogy, we call u an eiconal cubic . The harmonic eiconal cubics are exactly theCartan isoparametric cubics. Using (16), we introduce the commutative al-gebra structure V ( u ) on V = R n equipped with the standard Euclidean innerproduct. Then u ( x ) = h x , x i and (34) becomes h x , x i = 36 h x, x i , The exact value of the constant factor 36 is not essential and may be chosenarbitrarily (positive) by a suitable scaling of the inner product.Now we want to consider an arbitrary algebra satisfying the above identity.This motivates the following definition.
Definition 3.1.
A commutative, maybe nonassociative, algebra with a posi-tive definite associating form h , i satisfying(37) h x , x i = h x, x i is called an eiconal algebra.If V is an arbitrary eiconal algebra then the cubic form u ( x ) = h x, x i satisfies (scaled) eiconal equation (34). This translates the study of (34) intoa purely algebraic context.Note also that the sense of (37) becomes more clear if one introduces thenorm N ( x ) = h x, x i . Then (37) takes form of the composition algebra identity(38) N ( x ) = N ( x ) . Note, however, that (38) does not imply that N ( xy ) = N ( x ) N ( y )Our goal is the Peirce decomposition of V . To this end we need the standardlinearization technique which is an important tool in nonassociative algebra,see [32]. More precisely, we linearize (37) at x in the direction y to get4 h xy, x i = 4 | x | h x, y i . Since the inner product is associating, we have h xy, x i = h y, x x i = h y, x i . (Note that by virtue of the commutativity of V the third power in V is welldefined: x = x x = xx .) Therefore h x − h x, x i x, y i = 0 holds for all y ∈ V, implying by the nondegeneracy of the inner product that x − h x, x i x = 0 forall x . Conversely, if the latter identity holds, one easily gets (37). This proves Proposition 3.2.
An arbitrary commutative algebra with a positive definiteassociating form h , i is eiconal if and only if (39) x = h x, x i x, ∀ x ∈ V. Next note that the harmonicity condition (35) is equivalent (taking intoaccount (18)) to the trace free condition(40) tr L x = 0 , ∀ x ∈ V. A further linearization of (39) in direction y ∈ V yields x y + 2 x ( xy ) = h x, x i y + 2 h x, y i x, and eliminating y , we get(41) L x + 2 L x = h x, x i + 2 x ⊗ x. Remark 3.3.
The latter identity implies that eiconal algebras are ‘nearlyJordan’. Indeed, recall any Jordan algebra satisfies (21). On the other hand,we have from (39) and (41)[ L x , L x ] = 2[ x ⊗ x, L x ] = 2( x ⊗ x − x ⊗ x ) = 0 , i.e. L x commutes with L x . In fact, it follows from [52] that any eiconalalgebra has a natural structure of the trace free subspace in a rank 3 Jordanalgebra.3.3. The Peirce decomposition and fusion laws.
It follows from thedefinition of an eiconal algebra V that x = 0 for any x = 0, i.e. V isa nonzero algebra. Therefore Proposition 2.5 ensures that there are nonzeroidempotents in V . Let us denote by Idm( V ) the set of all nonzero idempotents.Note also that by (37) | c | = 1 for any c ∈ Idm( V ) . The multiplication operator L c is self-adjoint with respect to the inner prod-uct h , i , hence V decomposes into orthogonal sum of L c -invariant subspaces.Let us determine the spectrum of L c . To this end, note that c = c , hence c is an eigenvector of L c with eigenvalue 1, thus, R c is an invariant subspace of L c . Therefore, the orthogonal complement c ⊥ is an invariant subspace of L c too. Applying (41) we obtain(42) 2 L c + L c − c ⊗ c = 0 on c ⊥ , hence spectrum( L c | c ⊥ ) ⊂ {− , } . Note that it follows from the above inclu-sion that the eigenvalue 1 has multiplicity one, i.e. any idempotent in V is primitive . Remark 3.4.
We point out that the presence of the eigenvalue is crucialfor constructing the nonclassical solutions and closely related to the generacycondition discussed in sections 1.3 and 2.5 above, cf. with Lemma 3.2 in [39].Let V λ ( c ) be the λ -eigenspace of L c . Then the Peirce decomposition of V is V = R c ⊕ V − ( c ) ⊕ V ( c ) . In order to extract the multiplication table (the so-called fusion laws) be-tween the eigenspaces V λ ( c ), we linearize further (41). This yields(43) L yx + ( L x L y + L y L x ) = h x, y i + x b ⊗ y. Applying (43) to an arbitrary element z ∈ V yields the full linearization(44) x ( yz ) + y ( zx ) + z ( xy ) = h x, y i z + h y, z i x + h z, x i y. Specializing z = c ∈ Idm( V ) in (44) and setting x, y ∈ c ⊥ yields( cx ) y + x ( cy ) + c ( xy ) = h x, y i c. Taking the scalar product with z in the latter identity and assuming that x ∈ V λ ( c ), y ∈ V λ ( c ) and z ∈ V λ ( c ), where λ i ∈ { , − } , we obtain(45) h x x , x i ( λ + λ + λ ) = 0 . As a corollary we have V λ ( c ) V λ ( c ) ⊥ V λ ( c ) whenever λ + λ + λ = 0.For example, setting λ = λ = − V − ( c ) V − ( c ) isperpendicular to both V − ( c ) and V ( c ), hence(46) V − ( c ) V − ( c ) ⊂ R c. Similarly, for λ = − λ = one has V − ( c ) V ( c ) ⊂ R c ⊕ V ( c ). On theother hand, since eigenspaces V λ ( c ) are perpendicular for distinct λ we have h V − ( c ) V ( c ) , c i = h V − ( c ) , V ( c ) c i = h V − ( c ) , V ( c ) i = 0 , implying that V − ( c ) V ( c ) ⊂ V ( c ). Arguing similarly for λ = λ = , onearrives at the fusion (multiplication) laws shown in Table 1. V − V V − R c V V V R c ⊕ V − Table 1.
Fusion laws of an eiconal algebraRecall that a linear map A : X × Y → Y such that A ( x, · ) : Y → Y is self-adjoint for all x ∈ X and A ( x, · ) = h x, x i id Y is called a symmetricClifford system, cf. [9], [45]. It is well-known that in this case dim Y is evenand dim X ≤ ρ ( dim Y ), and the Hurwitz-Radon function ρ is defined by(47) ρ ( m ) = 8 a + 2 b , if m = 2 a + b · odd , b ≤ . Proposition 3.5. If dim V ≥ then (48) dim V − ( c ) − ≤ ρ ( dim V ) . If additionally V satisfies (40) then the possible dimensions of the Peirce sub-space V − ( c ) coincides with these of classical division algebras. In particular,there are only finitely many isomorphy classes of harmonic eiconal algebrasin dimensions dim V ∈ { , , , } . Proof.
First we show that V − ( c ) is nontrivial. Indeed, assume by contradic-tion that V − ( c ) = { } . Then V = R c ⊕ V ( c ), hence dim V ( c ) = n − ≥ V ( c ) V ( c ) ⊂ R c . Hence for any x ∈ V ( c ) we have x = h x , c i c = h x, xc i c = h x, x i c. Therefore x = h x, x i x = h x, x i x , a contradiction with (39) follows. Thusdim V − ( c ) ≥
1. Next, note that for any x ∈ V − ( c ) Table 1 yields that L x isan endomorphism of V ( c ). Since by (46) x = h x , c i c = h x, xc i c = −h x, x i c, we find from (41)(49) L x = h x, x i on V ( c ) . Thus, √ L x is a symmetric Clifford system. This imposes the dimensionalobstruction (48). If additionally V satisfies (40) then0 = tr L c = 1 + 12 dim V ( c ) − dim V − ( c ) , thus dim V ( c ) = 2 m , where m = dim V − ( c ) −
1, implying by (48) ρ ( m ) ≥ m .It is well known and also easily follows from (47) that the latter inequalityholds only if m = 1 , , ,
8, which implies that dim V = 3 m + 2. (cid:3) Concluding remarks and open questions
Minimal cones constitutes an important subclass of singular minimal hy-persurfaces in the Euclidean space R n for n ≥ u in Remark 2.3) are exceptional Hsiang cubic minimal cones (or radial eigencubics in the sense of chapter 6in [36]). It is known that there are only finitely many (congruence classesof) such cubics in some distinguished dimensions 5 ≤ n ≤
72, see Table 1 onp. 158 in [36]. From analytical point of veiw, any exceptional Hsiang cubicform can be characterized as a cubic polynomial solution of the followingHessian trace equations: tr D u ( x ) = 0 , tr( D u ( x )) = C | x | , tr( D u ( x )) = C u ( x ) , (50) Natural questions arise: Does any exceptional Hsiang cubic produce a nonclas-sical solution? Do there exist special cubics which are not Hsiang eigencubics?In general, it would be interesting to clarify the connections between minimalcones and construction of nonregular solutions.
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