Institut für Angewandte Analysis,Universit?t Ulm,89069 Ulm,Germany

E-mail:ibrokhimbek.akramov@uni-ulm.de

Marcel OLIVER

School of Engineering and Science,Jacobs University,28759 Bremen,Germany

E-mail:m.oliver@jacobs-university.de

Abstract In this article,we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampère operators acting in different two-dimensional coordinate sections.This equation is elliptic,for example,in the class of convex functions.We show that the notion of Monge-Ampère measures and Aleksandrov generalized solutions extends to this equation,subject to a weaker notion of convexity which we call bi-planar convexity.While the equation is also elliptic in the class of bi-planar convex functions,the contrary is not necessarily true.This is a substantial difference compared to the classical Monge-Ampère equation where ellipticity and convexity coincide.We provide explicit counter-examples:classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced.We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.

Key words Fully nonlinear elliptic equations;generalized solution;bi-planar convexity

1 Introduction

We study the fully nonlinear second order equation

on a three-dimensional domain ??R3.Setting

for j=1,2,we can write(1.1)in the form

Thus,the operator on the left is the sum of two planar Monge-Ampère operators on perpendicular sections.For this reason,we shall refer to(1.3)as the bi-planar Monge-Ampère equation.

The characteristic matrix(see[3])for(1.3)reads

where?denotes the Laplacian in the(x1,x2)-plane.Equation(1.3)is elliptic(in the sense of linearization)when Λ is positive definite.This is the case if and only if

In particular,(1.3)is elliptic in the class of convex functions.

The study of the bi-planar Monge-Ampère equation is motivated by a recent article[8]on variational balance models for rapidly rotating stratified fluid flow.For a class of models that includes the so-called L1-model first proposed by R.Salmon[2,10],the vertically integrated potential temperature Θ is related to the potential vorticity of the fluid via

where,up to rescaling,ω = ω(x1,x2)is the vorticity of the horizontal mean flow and ε is the Rossby number.Setting

where??1denotes the inverse Laplacian on the two-dimensional horizontal domain U with homogeneous Dirichlet boundary conditions on?U,we see that(1.6)can be written in the form of the bi-planar Monge-Ampère equation(1.3)on the cylindrical domain ?=U×(0,1).In particular,(1.6)is elliptic if and only ifand q>0.

In this article,we ask the question in which sense the well-established theory of generalized solution of the classical Monge-Ampère equation[3–5]carries over to the bi-planar Monge-Ampère equation.We find that it is possible to construct a bi-planar analog of the Monge-Ampère measure which can be used to define generalized solutions and assert their uniqueness[4,5,11].For this construction,it is necessary to require that the solution is convex on the respective coordinate sections.This notion,which we term bi-planar convexity,is more general than convexity.However,it is also more restrictive than the ellipticity condition(1.5).Indeed,we show by example that there exist classical solutions to the Dirichlet problem for the bi-planar Monge-Ampère equation such that the equation is elliptic in the vicinity of these solutions;yet,these solutions are not bi-planar convex.This is in contrast to the situation for the classical Monge-Ampère equation equation,where the notions of convexity and ellipticity coincide.

It should be noted that the classical Monge-Ampère equation is closely related to the geometric notion of convexity.However,bi-planar Monge-Ampère equation is related to the property of convexity for the two planar sections.Surely,if a function is convex,so are its planar sections.The converse,however,is not true(see Remark 2.2 below).This illustrates that bi-planar convexity does not have an intrinsic geometric meaning in three dimensions.Correspondingly,convex analysis does not lead to a natural notion of solution for the bi-planar Monge-Ampère equation.

The remainder of this article is structured as follows.In Section 2,we develop the convexity based theory:We introduce the notion of bi-planar convexity,define the bi-planar Monge-Ampère measure,prove monotonicity and a comparison principle,and finally use these notions to define the bi-planar analog of Aleksandrov generalized solutions.Section 3 is devoted to counter-examples which show that there is a gap between the concept of convexity,or even bi-planar convexity,and ellipticity for associated Dirichlet problem.This article concludes with a brief discussion.

2 Bi-Planar Monge-Ampère Measure

2.1 Construction

We define a measure on the Borel σ-algebra of R3,the bi-planar Monge-Ampère measure,by using planar Monge-Ampère measures on sections,then integrating over the remaining dimension.We begin the construction by defining a weaker notion of convexity adapted to the bi-planar structure of our equation.

Definition 2.1Let φ be a continuous function defined on the set ? ? R3.The function φ is bi-planar convex if for any fixed x1and x2,φx1(x2,z) ≡ φ(x1,x2,z)and φx2(x1,z)≡φ(x1,x2,z)are convex functions on the respective sections

and

whenever these are nonempty.

Remark 2.2A convex function is bi-planar convex,but the converse is not necessarily true.For example,

is bi-planar convex but not convex.

For the classical Monge-Ampère equation on a domain ??Rn,

where ν is a given Borel measure on ?,an Aleksandrov generalized solution is a convex function φ∈C(?)such that Mφ=ν,where Mφ denotes the Monge-Ampère measure

for every Borel set E ? U.Here,?φ is the normal map or sub differential defined at a point x∈? by

and for a Borel set E?? by

The Monge-Ampère measure(2.5)relates to the Monge-Ampère equation(2.4)via the identity

for all Borel sets E ? ?,which holds true whenever φ ∈ C2(?);see,for example,[4]for details.Derivatives of generalized solutions exist generally only in the sense of sub differentials but,being convex,generalized solutions have classical derivatives of second order a.e.[1].

In the following,we mimic this correspondence for the bi-planar Monge-Ampère equation.Suppose that ? ? R3is open andis bi-planar convex.For every x1∈ R,we define the measureon ?x1as the planar Monge-Ampère measure associated with the convex continuous function φx1;when ?x1is empty,we take this measure to be zero.Likewise,for every x2∈ R,we define the measure M13φx2on ?x2as the planar Monge-Ampère measure associated with φx2.

When E=E1×E23?? is compact with E1?R and E23?R2,thenμx1(E23)≡M23φx1(E23)is continuous on E1by[4,Lemma 1.2.3].We can thus Lebesgue-integrate over E1and define

is increasing,and

is Lebesgue measurable on E1.Hence,by the monotone convergence theorem,

is well-defined.Because of the countable additivity of the Monge-Ampère measure and countable additivity of the Lebesgue integral,μ1defines a pre-measure which can be extended to a Borel measure on ?.

Analogously,we define a measure μ2corresponding to the function φx2.Finally,the biplanar Monge-Ampère measure is defined as

2.2 Basic properties

We now provide the basic characterization of the bi-planar measure associated with smooth functions and prove monotonicity of the measure.

Lemma 2.3Ifis bi-planar convex,then μφis absolutely continuous with respect to the Lebesgue measure and

for all compact E??.

ProofWe will show that

Because of the properties of the Lebesgue measure,it suffices to prove this relation for cylindrical sets of the form E=E1×E23? ?;an arbitrary compact E ? ? can be approximated by a union of such sets.

By[4,Example 1.1.14],we have

Hence,by definition(2.12),we have

Similar arguments show that

Combining(2.15)and(2.18),we complete the proof.

Lemma 2.4Let ? ? R3be open and bounded,and letbe bi-planar convex.If φ = ψ on ?? andin ?,then for any fixed y1∈ R and y2∈ R,

ProofThe strategy of proof closely follows[4,pp.10–11].We write out the argument for(2.19a)explicitly;the proof of(2.19b)is analogous.

for all(x2,z) ∈ ?y1.Subtracting φy1(x2,z)from(2.20),taking the supremum on the right hand side,and using that ψy1≥ φy1,we find that

As ? and ?y1are bounded and φy1is continuous,the supremum in(2.21)is attained at someso that,by the definition of a,

for all(x2,z)∈ ?y1.Clearly,the right hand side of(2.22)defines a supporting hyperplane forWhenwe conclude thatand we are done.Otherwise,when,then,by assumption,.Furthermore,by continuity of ψ,we may letin(2.20),so that

As a≥ 0,this implies a=0.Therefore,By assumption,the reverse inequality is also true,so thatWith these provisions,the first line in(2.22)reads

Clearly,the right hand side defines a supporting hyperplane for φy1at(y2,z0),so that(p2,p3) ∈?φy1(?y1)in this case,too.

Lemma 2.5Let ? ? R3be open and bounded,and let φ,be bi-planar convex.If φ ≤ ψ in ? and φ = ψ on ??,then

ProofLemma 2.4 implies,for any fixed x1,the inclusionHence,

Integrating this inequality with respect to x1,we obtainwhereandare the measures μ1corresponding to φ and ψ,respectively.

An analogous argument yieldswhereandare the measures μ2corresponding to φ,and ψ,respectively.Asthe proof is complete.

Finally,superadditivity of the Monge-Ampère measure(for example[4,p.17])directly implies the following inequality.

Lemma 2.6Let ? ? R3be open and bounded and let φ,ψ ∈ C(?)be bi-planar convex functions.Then

for any Borel set E??.

2.3 Comparison principle

The first central result which carries over from the classical Monge-Ampère measure to the bi-planar case is the comparison principle.The proof is a close adaptation of[4,pp.16–17].

Then

ProofSuppose that(2.29)does not hold,that is,

and consider the set

On one hand,x0∈G.Indeed,using(2.31)and b?a>0,we find that

On the other hand,for x ∈ ??,we have φ(x)? ψ(x)≥ b so that,by(2.31),

Hence,G∩ ?? = ?and consequently?G={x∈ ?:?(x)= φ(x)}.Hence,using Lemma 2.5 and Lemma 2.6,we conclude

which contradicts(2.28).This completes the proof.

2.4 Generalized solutions

The bi-planar Monge-Ampère measure can be used to define the analog of Aleksandrov generalized solutions for the bi-planar Monge-Ampère equation.

Definition 2.8Let ? ? R3be open and let ν be a Borel measure on ?.Then,the bi-planar convex function φ ∈ C(?)is a generalized solution of the bi-planar Monge-Ampère equation

if the bi-planar Monge-Ampère measure associated with φ equals ν.

The following statement is then a direct consequence of Lemma 2.3.

Proposition 2.9Let ? ? R3be open,and suppose that the Borel measure ν is absolutely continuous with respect to the Lebesgue measure with non-negative density function f ∈ C(?).Then,φ is a generalized solution of(2.36)if and only if

Finally,the comparison principle implies the uniqueness of generalized solutions.

Theorem 2.10Let ? ? R3be open and bounded, ν a Borel measure on ?,and g ∈C(??).If φ1and φ2are generalized solutions to the Dirichlet problem

then u1=u2.

3 Non-Existence Results

In this section,we present main results of this article with corresponding examples:there is a domain and boundary data such that no generalized solution to the Dirichlet problem with zero or constant right hand side exist.

3.1 Non-existence of convex generalized solutions

We begin the discussion with a weaker result,namely,there is no generalized solution in the class of convex functions.This construction illustrates in a particularly transparent way how convexity over-constrains the system.We begin with a simple observation.

Lemma 3.1Let A be a symmetric positive semi-definite matrix,written as

Then detS=0 implies detA=0.

ProofBy assumption,the submatrix S must also be symmetric positive semi-definite.Moreover,as S is singular,we can take a nonzero v∈KerS and set wT=(0,vT).Then,wTAw=vTSv=0,so A cannot be strictly positive definite.This implies detA=0.

Lemma 3.1 implies that if problem(2.38)with ν=0 has a convex solution,then the solution also satisfies the classical homogeneous Monge-Ampère equation

From this observation,we conclude the following.

Proposition 3.2There exist a bounded domain ? and a continuous function g defined on ?? such that problem(2.38)with ν =0 has no solution in the class of convex functions.

ProofLet U ? R2be a bounded domain andThen,? is a bounded domain in R3.Let g be a restriction of the functionto the set ??.Then,φ is a solution of(3.2).This solution is unique[4,9].Now,suppose that φ1is a convex solution of(2.38)with the same boundary data g.By Lemma 3.1,φ1is also a solution of(3.2),that is,.But

Thus,problem(2.38)with boundary data g does not have a convex solution.

Remark 3.3The existence of generalized solution for the homogeneous Dirichlet problem(3.2)for strictly convex domains is well known,for example,[4,Theorem 1.5.2].Clearly,the difference between convexity and strict convexity of the domain is not an issue here.A similar counter-example can be produced,for example,on the unit ball in R3:Let g be the restriction of the functionto the unit sphere ??.Then,proceeding as in the proof of Proposition 3.2,problem(2.38)with boundary data g does not have a convex solution.

Remark 3.4Actually,g in the proof of Proposition 3.2 can be written asThen,is the unique concave solution to problem(2.38)and also a concave solution to problem(3.2),unique by the concave analog to Theorem 2.10.

Remark 3.5For boundary dataon the unit sphere ?,there is neither a concave nor a convex solution to problem(2.38).Indeed,asg(x)can be written as

3.2 Non-existence of bi-planar convex generalized solutions

We now re fine the construction to show that relaxing the constraint from convexity to bi-planar convexity does not help:there exists a smooth boundary condition for the Dirichlet problem such that the bi-planar Monge-Ampère equation does not have a generalized solution.

Theorem 3.6Let ? be the unit ball centered at the origin and λ ∈ [0,8)be fixed.Then,the problem

has no generalized solution in the class of continuous bi-planar convex functions.

ProofSuppose that problem(3.6)has a generalized solutionBi-planar convexity implies that each of the measures μ1and μ2is positive.Asμ1+μ2= λ,we have

in the sense of measure for i=1,2.

Now,consider two classical Monge-Ampère equations.First, fix|x2|<1 and consider

The smooth function

solves(3.8).Furthermore,φ1is convex,therefore it is the unique generalized solution;see,for example,[4,Corollary 1.4.7].Similarly,we fix|x1|<1 and consider the problem

Here,the unique generalized solution is the smooth convex function

We further set

In particular,for x=(0,0,0),we get

Remark 3.7Note that problem(3.6)has a classical solution,

for any λ ≥ 0.When λ ≥ 8,this function is convex,hence bi-planar convex.When 0≤ λ <8,thenis not positive semi-definite so that φ is not bi-planar convex.However,even in this case,problem(3.6)satisfies the ellipticity condition(1.5)for the bi-planar Monge-Ampère equation.

4 Discussion

Our counter-examples show that ellipticity for the bi-planar Monge-Ampère equation is a substantially weaker condition than bi-planar convexity,while bi-planar convexity implies ellipticity.In contrast,for the classical Monge-Ampère equation,ellipticity and convexity coincide,which makes Aleksandrov generalized solutions a useful concept.For the bi-planar equation,there is a“gap”between ellipticity and bi-planar convexity,so that the requirements of convex analysis,necessary to obtain generalized solutions in the sense of Aleksandrov,over-constrain the system.Therefore,a useful solution concept for the bi-planar Monge-Ampère equation requires a different setting,possibly in more traditional function space setting as,for example,in[6,7].This question is left open for future work.