Polyharmonic k− Hessian equations in R N
aa r X i v : . [ m a t h . A P ] J u l POLYHARMONIC k − HESSIAN EQUATIONS IN R N PEDRO BALODIS, CARLOS ESCUDEROA
BSTRACT . This work is focused on the study of the nonlinear elliptic higher orderequation ( − ∆) m u = S k [ − u ] + λf, x ∈ R N , where the k − Hessian S k [ u ] is the k th elementary symmetric polynomial of eigen-values of the Hessian matrix of the solution and the datum f belongs to a suitablefunctional space. This problem is posed in R N and we prove the existence of atleast one solution by means of topological fixed point methods for suitable valuesof m ∈ N . Questions related to the regularity of the solutions and extensions ofthese results to the nonlocal setting are also addressed. On the way to construct theseproofs, some technical results such as a fixed point theorem and a refinement of thecritical Sobolev embedding, which could be of independent interest, are introduced.
1. I
NTRODUCTION
The goal of this work is to develop an analytical framework for the study of thefamily of higher order equations(1) ( − ∆) m u = S k [ − u ] + λf, x ∈ R N , where m, N, k ∈ N , λ ∈ R and the datum f : R N −→ R belongs to a suitablefunctional space, to be made precise in the following. The nonlinearity in this equationis the k − Hessian S k [ u ] = σ k (Λ) , where σ k (Λ) = X i < ···
Date : July 25, 2018.
Key words and phrases.
Higher order elliptic equations, k − Hessian type equations, Existence of so-lutions, Fixed point methods, Functional inequalities, Harmonic analysis of partial differential equations.2010
MSC: 35G20, 35G30, 35J60, 35J61, 42B35, 42B37, 46E30, 46E35, 46N20. for k = N . In fact, such an equation S k [ u ] = f, is denominated the k − Hessian equation, and it, together with related problems, hasbeen intensively studied during the last years [12, 13, 34, 38, 50, 54, 55, 56, 57, 58,59, 60, 61, 62, 63, 64, 65]. It is interesting to note that the analytical approach to thisproblem has required the assumption of a series of geometric constraints in order topreserve the ellipticity of the nonlinear k − Hessian operator [65]. Such constraints arenot needed in the case of full equation (1) [20], what makes this sort of problem analternative viewpoint to the interesting nonlinear k − Hessian operator.A second source of motivation is the rise of studies focused on polyharmonic prob-lems in recent times [1, 3, 16, 18, 19, 27, 28, 29, 39]. While boundary value prob-lems for polyharmonic operators have already been considered with different typesof interesting nonlinearities in these and different works, the history of polyharmonic k − Hessian equations is still short [20, 21, 22, 23, 24, 25, 26]. At this point, it is im-portant to stress the natural character of this sort of nonlinearity in the polyharmonicframework. Indeed, the k − Hessians, ≤ k ≤ N , form a basis of the vector spaceof polynomial invariants of the Hessian matrix under the orthogonal group O ( N ) ofdegree lower or equal to N , at least for regular enough u [47]. So on one hand thesenonlinearities give rise to genuinely polyharmonic semilinear equations with no pos-sible harmonic analogue, what makes them an excellent candidate to push forwardthe theory of polyharmonic boundary value problems. While on the other hand, thesehigher order equations are some of the simplest ones compatible with the ideas of in-variance with respect to rotations and reflections widespread in the realm of physicalmodeling.Yet another interesting property that motivates us to study equation (1) is its in-triguing dependence on the boundary conditions, as already noted in [25]. We studiedin [20] this family of equations on bounded domains subject to Dirichlet boundaryconditions. In this work we are interested on the “boundary value problem” ( − ∆) m u = S k [ − u ] + λf, x ∈ R N , (2a) u ( x ) → , when | x | → ∞ . (2b)First of all we have to state what do we mean by this “boundary condition”; in fact,this constitutes a very important remark : we say that a solution “vanishes at infinity”if it belongs to some L p ( R N ) , ≤ p < ∞ , although we cannot give any reasonablepointwise meaning to such an affirmation. Note that this is the only way in which anexistence theory `a la Calder´on-Zygmund can be pushed forward. Of course, if a func-tion pointwise vanishes at infinity, we will also say that it “vanishes at infinity”. Notealso that the nonlinearity is S k [ − u ] rather than S k [ u ] ; that is, the nonlinearity is exactlythe coefficient of the monomial of degree N − k within the characteristic polynomialof the Hessian matrix. We have considered such a form to be in complete agreementwith the structure of the equation in [20]. However, this assumption was needed in thisreference in order to construct the variational approach to the existence of solutionsemployed there. Our present approach relies on a topological fixed point argumentand would work exactly in the same way if we substituted the current nonlinearity by S k [ u ] . This, among other things, highlights the fact that the present existence proofsare genuinely different from previously used arguments. OLYHARMONIC k − HESSIAN EQUATIONS 3
We now present our main result:
Theorem 1.1.
Problem (2a) - (2b) has at least one weak solution in the following cases: (a) f ∈ L p ( R N ) , < p < N k , m = 1 + N ( k − / (2 pk ) ∈ N , N > k , (b) f ∈ L ( R N ) , m = 1 + N ( k − / (2 k ) ∈ N , N > k , (c) f ∈ H ( R N ) , m = 1 + N ( k − / (2 k ) ∈ N , N > k , (d) f ∈ H ( R N ) , m = 1 + N ( k − / (2 k ) ∈ N , N = 2 k ,provided | λ | is small enough. Then, respectively (a) u ∈ ˙ W m − ǫ,Np/ ( N − ǫp ) ( R N ) ∀ ≤ ǫ ≤ m , (b) u ∈ ˙ W m − ǫ,N/ ( N − ǫ ) ( R N ) ∀ < ǫ ≤ m , (c) u ∈ ˙ W m − ǫ,N/ ( N − ǫ ) ( R N ) ∀ ≤ ǫ ≤ m , (d) u ∈ ˙ W m − ǫ,N/ ( N − ǫ ) ( R N ) ∀ ≤ ǫ ≤ m .Moreover, in case (b), D m u ∈ L , ∞ ( R N ) , in case (c), D m u ∈ H ( R N ) and, in case(d), D m u ∈ H ( R N ) and u ∈ C ( R N ) . Also, for a smaller enough | λ | , the solutionis locally unique in cases (a), (b) and (c).Proof. The statement follows as a consequence of Theorems 6.6, 6.7, 6.9, 7.2, 7.3 andCorollary 9.4. (cid:3)
Remark 1.2.
Note that, in case (d), m = k always, so problem (2a)-(2b) reduces to ( − ∆) k u = S k [ − u ] + λf, x ∈ R k ,u ( x ) → , when | x | → ∞ , for any k ≥ . Remark 1.3.
It is important to note that our methods are applicable to more generalfamilies of nonlinearities. Denote by R jk ( · ) the j − th principal minor of order k . Thepresent results hold as well if we substituted S k ( − u ) by R jk ( − u ) in equation (2a)for any j . In fact, the nonlinearities S k ( − u ) are just a particular linear combinationof these R jk ( − u ) ; and our theory could be constructed actually for any linear combi-nation of them. This comes from the fact that we need two main ingredients in ourproofs: weak continuity of the maps S k and the fact that they also preserve the L p and Hardy spaces the datum f belongs to. The same holds, for example, for the maps R jk , see [14, 31], and for any linear combination of them by linearity. Our main atten-tion lies, however, in the operators S k described before due to their simple geometricmeaning which is at least not as evident for the operators R jk or their arbitrary linearcombinations.Now we describe the remainder of the article. In section 2 we introduce the func-tional framework we need in our proofs and some notation. In section 3 we devel-oped the theory that corresponds to the linear counterpart of problem (2a)-(2b). Insection 4 we state and prove a topological fixed point theorem that will be the mainabstract tool for proving existence of solutions to our differential problem. In sec-tion 5 we prove a refinement of the classical critical Sobolev embedding that will besubsequently needed in the following section. These last two sections could be of in-dependent interest and, as such, they have been written in a self-contained fashion.Our main existence results come in section 6, and the local uniqueness results in sec-tion 7. A nonlocal extension of Theorem 1.1 is proven in section 8 and, finally, some P. BALODIS, C. ESCUDERO further results regarding the weak continuity of the branch of solutions and some extraregularity for the critical case (d) are described in section 9.2. F
UNCTIONAL F RAMEWORK AND N OTATION
In order to build the existence theory for our partial differential equation we needto introduce the Hardy space H in R N [52] and its dual, the space of functions ofbounded mean oscillation. Definition 2.1.
Let Φ ∈ S ( R N ) , where S ( R N ) denotes the Schwartz space, be afunction such that R R N Φ dx = 1 . Define Φ s := s − N Φ( x/s ) for s > . A locallyintegrable function f is said to be in H ( R N ) if the maximal function M f ( x ) := sup s> | Φ s ∗ f ( x ) | belongs to L ( R N ) . We define the norm k f k H ( R N ) = kM f k . Remark 2.2.
There are several equivalent definitions of this space, see [51].Now we introduce the space of functions of bounded mean oscillation [51].
Definition 2.3.
A locally integrable function f is said to be in BMO( R N ) if the semi-norm (or norm in the quotient space of locally integrable functions modulo additiveconstants) k f k BMO( R N ) := sup Q | Q | Z Q | f ( x ) − f Q | dx, where | Q | is the Lebesgue measure of Q , f Q = | Q | R Q f ( x ) dx and the supremum istaken over the set of all cubes Q ⊂ R N , is finite.We also need the pre-dual of the Hardy space H ( R N ) . Definition 2.4.
We define
VMO( R N ) as the closure of C ( R N ) in BMO( R N ) , with k f k VMO( R N ) = k f k BMO( R N ) ∀ f ∈ VMO( R N ) .The following functional spaces will also be useful in the construction of the exis-tence theory. Definition 2.5.
We define the homogeneous Sobolev space ˙ W j,p ( R N ) as the space ofall measurable functions u that are j times weakly derivable and whose weak deriva-tives of j − th order obey k D j u k p < ∞ , where k · k p denotes the norm of L p ( R N ) , ≤ p ≤ ∞ , j ∈ N .In our derivations we will need the following operators. Definition 2.6.
We define the Riesz transforms in R N : R x j ( f )( x ) = Γ (cid:0) n +12 (cid:1) π ( n +1) / P. V. Z R N x j − y j | x − y | n +1 f ( y ) dy. Remark 2.7.
The normalization of the Riesz transforms is chosen in such a way that F [ R x j ( f )]( ξ ) = πi ξ j | ξ | F ( f )( ξ ) . OLYHARMONIC k − HESSIAN EQUATIONS 5
Finally, we introduce two definitions relating to real numbers and their relationships.
Definition 2.8.
Let x α , y α ∈ R ( α ∈ A , A some set). We write x ≪ y ( x = { x α } α ∈ A , y = { y α } α ∈ A ) whenever there exists a positive constant c such that x α ≤ cy α forevery α ∈ A . Definition 2.9.
We denote R + := { x ∈ R | x ≥ } .3. L INEAR T HEORY
This section is devoted to the study of the linear problem(4) ( − ∆) m u = λf, x ∈ R N , where m ∈ N and we consider the “boundary condition” u → when | x | → ∞ . Proposition 3.1.
Equation (4) has a unique solution in the following cases: (a) f ∈ L p ( R N ) , < p < N m , m < N/ , (b) f ∈ L ( R N ) , m < N/ , (c) f ∈ H ( R N ) , m < N/ , (d) f ∈ H ( R N ) , m = N/ .Then, respectively (a) u ∈ L q ( R N ) ∩ ˙ W m,p ( R N ) , (b) u ∈ ˙ W m − ǫ,N/ ( N − ǫ ) ( R N ) ∀ < ǫ ≤ m , (c) u ∈ L q ′ ( R N ) ∩ ˙ W m, ( R N ) , (d) u ∈ L ∞ ( R N ) ∩ ˙ W m, ( R N ) ,where q = N p/ ( N − mp ) and q ′ = N/ ( N − m ) . Moreover, in case (b), D m u ∈ L , ∞ ( R N ) , and, in all cases, the map f u is continuous.Proof. The proof focuses on the range m ≥ since the case m = 1 is classical.S TEP ( − ∆) m G = δ , x ∈ R N , where δ is the unit Dirac mass centered at the origin. The explicit solution to thisequation is well known [29]:(6) G ( x ) = − log | x | NV N m − Γ( N/ m − if N = 2 m, N/ − m ) NV N m Γ( N/ m − | x | N − m in other case , where V N = π N/ / Γ(1 + N/ is the volume of the N − dimensional unit ball, andalways under the assumption N ≥ m .The unique solution to equation (4) is given by the convolution(7) u = λ G ∗ f. Now we justify that this is a well defined function in a suitable functional space.S
TEP
N > m we have G ∝ | x | m − N , therefore G defines a Newtonian potential I m ( f ) = Z R N G ( x − y ) f ( y ) dy, P. BALODIS, C. ESCUDERO and, as such, k I m ( f ) k q ≪ k f k p , see [32], and therefore k u k q ≪ | λ | k f k p , where q = N p/ ( N − mp ) , in case (a). Cases (b) and (c) follow analogously.For N = 2 m we have G ∝ log | x | and since in this case f ∈ H ( R N ) , and log | x | ∈ BMO ( R N ) , it follows that k u k ∞ ≪ | λ | k f k H ( R N ) . S TEP u it suffices to show that D m G defines a singular integraloperator [32]. Note that ∆ | x | − α = ( α + 2 − N ) α | x | α +2 ∀ α > . If we denote C α,N := α ( α + 2 − N ) and K N,m := G ( x ) | x | N − m whenever N > m ,we have ( − ∆) m − G ( x ) = ( − m K N,m C N − m,N C N − m − ,N · · · C N − ,N | x | − N . On the other hand, it is easy to check that ∂ x j x k | x | − N = (2 − N ) | x | δ jk − N x j x k | x | N +2 . Note also the average of the numerator over the unit sphere I jk = Z S N − ( | x | δ jk − N x j x k ) dw = δ jk | S N − | − N Z S N − w j w k dw = 0 . We denote ∂ jk := ∂ x j x k and define the operator T j,k ( f ) := ∂ jk ( − ∆) m − u, which is clearly a singular integral operator in R N . Consider now a multi-index α , | α | = 2 m , and so ∂ α u = R j R k · · · R j m − R k m − T j,k ( f ) , where R j n is the Riesz transform with respect to the j n − th coordinate, ≤ j n , n ≤ N . This operator is a product of singular integral operators and therefore a singularintegral operator itself. This completes the proof in the case N > m .In the case N = 2 m it is enough to consider G ( x ) = C N log | x | and ∆ G ( x ) = C N N − | x | , and to apply the same reasoning as before. (cid:3) Corollary 3.2.
The unique solution found in Proposition 3.1 fulfils: • u ∈ ˙ W m − ǫ,Np/ ( N − ǫp ) ( R N ) ∀ ≤ ǫ ≤ m in case (a). • u ∈ ˙ W m − ǫ,N/ ( N − ǫ ) ( R N ) ∀ ≤ ǫ ≤ m in case (c). • u ∈ ˙ W N − ǫ,N/ ( N − ǫ ) ( R N ) ∀ ≤ ǫ ≤ N in case (d). • D m u ∈ H ( R N ) in cases (c) and (d). Remark 3.3.
The strict inequality p < N/ (2 m ) in case (a) of Proposition 3.1 is sharp,see [32]. OLYHARMONIC k − HESSIAN EQUATIONS 7
Remark 3.4.
Note that for an odd
N < m the formula for G is still given by thesecond line of (6). For an even N < m we have G ( x ) = ( − m − N/ − N V N m − Γ( N/ m − N/ m − | x || x | N − m . In particular, note that G never decays to zero when | x | → ∞ whenever N ≤ m . Remark 3.5.
Following the previous remark, note that G is not unique since its prop-erty (5) is invariant with respect to the addition of a m − polyharmonic function. How-ever, if we consider the condition G → when | x | → ∞ , then the above formulas be-come the unique solution whenever N > m , and the set of solutions becomes emptyif N ≤ m . Moreover, it is not clear how to fix uniqueness in this latter case [29]. Inconsequence, it is clear that formula (7) gives the unique solution to problem (4) for N > m . For N = 2 m we take this formula as the definition of unique solution, butsee Remark 3.8 below. Lemma 3.6.
Let v be a m − harmonic function in R N . If v ∈ BMO ( R N ) , then v isconstant.Proof. By definition, v being m − harmonic means ( − ∆) m v = 0 . TransformingFourier this equation yields | k | m ˆ v ( k ) = 0 , and since v ∈ BMO ( R N ) then ˆ v ( k ) ∈ S ∗ ( R N ) , where S ∗ ( R N ) denotes the space ofSchwartz distributions. This equation implies the support of ˆ v supp (ˆ v ) ⊂ { } , and therefore ˆ v = X | α |≤ ℓ C α ∂ α δ , for some ℓ ∈ N , C α ∈ R , and where α denotes a N − dimensional multi-index.Consequently v is polynomial of degree ℓ or lower. We conclude invoking the John-Nirenberg theorem, that implies that functions showing a super-logarithmic growth donot belong to BMO ( R N ) , see [32]. (cid:3) Remark 3.7.
The proof of Lemma 3.6 actually implies that any m − harmonic functionin R N showing a sub-linear growth when | x | → ∞ is constant. Remark 3.8.
Following Remark 3.5, we note that a way to fix the uniqueness of thefundamental solution in the critical case N = 2 m is to impose an at most logarithmicgrowth when | x | → ∞ . According to Lemma 3.6 this fixes the fundamental solutionexcept for the presence of an additive constant. Of course, as we are looking forsolutions in BMO ( R N ) , and the seminorm of this space is invariant with respect tothe addition of a constant, this fixes uniqueness in the corresponding quotient space inwhich this seminorm becomes a norm. In other words, the solution to (4), u = λ G ∗ f ,is unique even if we considered G as a one-parameter family of fundamental solutionsindexed by an additive constant, given that functions in the Hardy space H ( R N ) havezero mean. Note also that our definition of solution does not guarantee a priori thatthe solution will obey the “boundary condition” in any reasonable sense. However, itobeys it in the pointwise sense, which is the strongest possible sense. This is justifiedby Theorem 9.3 and Corollary 9.4 below. P. BALODIS, C. ESCUDERO
4. A
TOPOLOGICAL FIXED POINT THEOREM
We now state the fixed point theorem that will allow us to construct the existencetheory for our partial differential equation. This result can be regarded as a corollaryof the more general Schauder-Tychonoff theorem [4]. For the reader convenience weinclude a proof of the result, which is independent of the proof present in [4].
Theorem 4.1.
Let Y be a real dual Banach space with separable predual and let Υ ⊂ Y be non-empty, convex and weakly −∗ sequentially compact. If there exist aweakly −∗ sequentially continuous map Z : Υ −→ Υ then Z has at least one fixedpoint.Proof. By our hypothesis, every convex, bounded and weakly −∗ sequentially closedset in Y is compact (by the Theorem of Banach-Alaoglu) , and moreover, the trace overthat set of the weak −∗ topology is metrizable. As a result, such a set can be considereda compact metrizable space with respect to that topology; notice in particular thatcompactness is equivalent to sequential compactness for such Υ .Let us recall how this metric is defined: if we denote by X ∗ ≡ Y our dual Banachspace, and { y n } n ≥ is a denumerable dense subset of the closed unitary ball B of thepredual X , we define another seminorm k · k ∗ in X ∗ as k x k ∗ = X n ≥ − n |h x, y n i| , x ∈ X ∗ . It is readily checked that the standard norm k · k X ∗ dominates this seminorm, andbecause of the density of the set { y n } n ≥ over the unit ball of X and the fact that theweak −∗ topology is Hausdorff, it is indeed a norm, and it is not hard to prove thatit induces the weak −∗ topology over strongly closed balls of X ∗ , or, more generally,over strongly closed convex sets of X ∗ (which are known to be weak −∗ sequentiallycompact). Now, since Υ is weakly −∗ compact then it is totally bounded in the metricwhich induces the weak −∗ topology and also bounded with respect to the strong ornorm topology. Therefore for any δ > we may choose a finite set { v , · · · , v n δ | v i ∈ Υ , ≤ i ≤ n δ } such that Υ ⊂ [ ≤ i ≤ n δ B v i ( δ ) , where B v i ( δ ) is the open ball in Y (open with respect to the metric induced by k · k ∗ )whose center is v i and whose radius is δ . Consider Υ δ := ( n δ X i =1 c i v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c i ∈ R + ∧ n δ X i =1 c i = 1 ) . Note, however, that strongly closed, convex and bounded is not enough. To see this, consider Y = M ( R n ) the space of finite Radon measures, which is the dual of ( C ( R n ) , k · k ∞ ) . Now, consider themap T : Y 7→ Y given by T ( µ ) = µ ∗ µ . It is not difficult to show that this non-linear map is weak −∗ sequentially continuous, and maps the simplex S = { µ | µ ≥ , k µ k Y = 1 } into itself. This is a convexweak −∗ closed and bounded set, and T maps S into itself; the delta function is the unique fixed pointof it, but T also maps into itself S ′ = { µ | µ ≥ , k µ k Y = 1 , µ absolutely continuous w . r . t . dx } = L ( R , dx ) ∩ S , which is strongly closed, convex and bounded, but without fixed points. OLYHARMONIC k − HESSIAN EQUATIONS 9
The convexity of Υ guarantees Υ δ ⊂ Υ . We introduce the projector P δ : Υ −→ Υ δ , P δ [ v ] := P n δ i =1 λ i ( v ) v i P n δ i =1 λ i ( v ) , λ i ( v ) := d ( v, Υ \ B v i ( δ )) , where d ( · , · ) is the distance induced by the norm k · k ∗ . Any of the functions λ i ( v ) isLipschitz continuous and non-negative, and at least one of these functions is positive:indeed, if v ∈ B v i ( δ ) , then, it is immediate that λ i ( v ) ≥ δ .Therefore the sum of all of them is positive, and we obtain as a result that thisprojection is well defined and continuous for v ∈ Υ . Moreover, as a consequence ofthe triangle inequality, we have, for v ∈ Υ ,(8) kP δ [ v ] − v k ∗ ≤ P n δ i =1 λ i ( v ) k v i − v k ∗ P n δ i =1 λ i ( v ) ≤ δ, since, for a given ≤ i ≤ n δ , either v ∈ B v i ( δ ) , in whose case k v − v i k ∗ < δ orelse v / ∈ B v i ( δ ) , in whose case λ i ( v ) = 0 (meaning that P δ [ v ] can be thought of as ansmall perturbation of the identity map over the set Υ in the metric induced by k · k ∗ );it is clear also that P δ [ v ] maps the set Υ to the finite-dimensional set Υ δ .Now we define the map Z δ : Υ δ −→ Υ δ , Z δ ( v ) := P δ [ Z ( v )] , which is well defined whenever v ∈ Υ δ and continuous. Since Υ δ is the closed convexhull of the set { v , · · · , v n δ } then it is homeomorphic to the closed unit ball in R j δ forsome j δ ≤ n δ . Now invoke the Brouwer fixed point theorem [46] to see there exists atleast one fixed point, v δ ∈ Υ δ , of Z δ .Taking a sequence < δ k → and select for each k ≥ a fixed point v k ∈ Υ δ k ⊂ Υ of Z δ j . By weak −∗ compactness of Υ , there exists a subsequence v k j , j ≥ of thesequence v k , k ≥ which is weak −∗ convergent to some v ∈ Υ , or in other terms, k v − v k j k ∗ → , j → ∞ . Let us check that v is a fixed point of Z : k v − Z ( v ) k ∗ = k ( v − v k j ) + ( P δ kj ( Z ( v k j )) − Z ( v k j ))) + ( Z ( v k j ) − Z ( v )) k ∗ [since v k j = Z δ kj ( v k j ) = P δ kj ( Z ( v k j ))] ≤ k v − v k j k ∗ + k P δ kj ( Z ( v k j )) − Z ( v k j ) k ∗ + k Z ( v k j ) − Z ( v ) k ∗ ≤ k v − v k j k ∗ + δ k j + k Z ( v k j ) − Z ( v ) k ∗ [by equation (8) ] → , j → ∞ , where, in the last step, we use the weak −∗ sequential continuity of the map Z . So, k v − Z ( v ) k ∗ = 0 , which is equivalent to v = Z ( v ) , as claimed. (cid:3)
5. R
EFINEMENT OF THE CRITICAL S OBOLEV EMBEDDING
In this section we introduce a series of preparatory results which are needed inour existence proofs. These constitute in fact a refinement of the classical Sobolevembedding at the critical dimensional index. Consequently, this section has an intereston its own, and therefore we have written it in a self-contained fashion.
Theorem 5.1.
Consider the homogeneous Sobolev space X = ˙ W ,N ( R N ) = { f ∈ S ′ ( R N ) : |∇ f | ∈ L N ( R N ) } , normed by k f k X = k|∇ f |k L N ( R N ) . Then we have forall spatial dimensions N ≥ : (1) There exists a finite constant C such that for all f ∈ X , k f k BMO ( R N ) ≤ C k f k X . (2) If, in addition, |∇ f | ∈ H N ( R N ) , we have f ∈ VMO ( R N ) . In any event thereexists some absolute and finite C , such that given a ball B = B r ( x ) , r > , x ∈ R N , | f − f B | B ≤ C k|∇ f |k L N ( B ) ; f B := 1 | B | Z B f dx. Remark 5.2.
While Part (1) of this theorem is classical, we shall give a proof of it forthe sake of completeness.
Remark 5.3. As |∇ f | N dx can be regarded as a finite and absolutely continuous mea-sure with respect to Lebesgue measure dx , for any ε > , ∃ δ > such that if < r ≤ δ , | f − f B | B ≤ ε , where r is the radius of B . Remark 5.4.
For any dimension N ≥ , H N ( R N ) = L N ( R N ) . So, an imme-diate corollary of this theorem can be stated as follows: ∀ N ≥ , ˙ W ,N ( R N ) ⊆ VMO ( R N ) , with continuous inclusion. Remark 5.5.
Note on the other hand that H ( R N ) ( L ( R N ) . It is also easy tofind functions f : R −→ R such that f ∈ ˙ W , ( R ) and f VMO ( R ) (such as f ( · ) = arctan( · ) ). But however it holds that ˙ W , ( R ) ⊂ AC ( R ) ∩ L ∞ ( R ) . Remark 5.6.
The space VMO ( R N ) can be defined either intrinsically as the spaceof those BMO ( R N ) functions such that for any given ε > , there exists δ > and R > such that if a ball B = B r ( x ) has radius smaller that δ or bigger than R ,then | f − f B | B ≤ ε or extrinsically as the closure of the space C ( R N ) under theBMO ( R N ) norm; as Claim (2) of our theorem shows, any function in our space X isvery close to be a VMO ( R N ) function and the averages of the mean oscillation oversmall balls are always small. This is an intrinsical estimate, but to close the proof ofthe claim we shall hinge on the extrinsical description of VMO ( R N ) instead. Proof.
The key ingredient in Part (1) of the above Theorem is Poincar´e inequality:given a ball B and an exponent ≤ p ≤ ∞ , we have, for some finite C = C ( p, B ) ,(9) k f − f B k L p ( B ) ≤ C ( p, B ) k|∇ f |k L p ( B ) , f ∈ C ( B ) . The above inequality can be closed to all the (inhomogeneous) Sobolev spaces W ,p ( B ) in the range ≤ p < ∞ by an standard density argument; in the case p = N ,it is easily checked that equation (9) is scale invariant, meaning that the constant C N ( B ) := C ( N, B ) indeed only depends on N , and not on the ball B r ( x ) we are in.In other words, we have(10) k f − f B k L N ( B ) ≤ C N k|∇ f |k L N ( B ) , f ∈ X. OLYHARMONIC k − HESSIAN EQUATIONS 11
From this, the continuous embedding in Claim (1) follows: fix f ∈ X and B a ball in R N . Then we have | f − f B | B ≤ k f − f B k L N ( B ) ≤ C N k|∇ f |k L N ( B ) ≤ C N k|∇ f |k L N ( R N ) , where the first inequality follows by H ¨older inequality and the second by (10); so tak-ing the supremum over all balls in R N we find Claim (1) of our theorem follows andmoreover the same argument yields the sharper estimate | f − f B | B ≤ C N k|∇ f |k L N ( B ) .Now we remind the definition of the (real) Hardy space H p ( R N ) , < p < ∞ ; firstfix a bump function ϕ ∈ C ∞ c ( R N ) with total mass one, and consider the mollifiers ϕ t := t − n ϕ ( t − · ) , t > . Then we have the following: Definition 5.7.
The Hardy space H p ( R N ) is the space of those tempered distributions f ∈ S ∗ ( R N ) such that the maximal operator M ∗ f = sup t> | ( ϕ t ∗ f ) | ∈ L p ( R N ) . Remark 5.8.
Notice that this definition in fact does not depend on the choice of ϕ .Now we use the following Lemmata: Lemma 5.9.
For < p < ∞ , the space D of Schwartz functions such that ˆ f issupported away from the origin is dense in H p ( R N ) .Proof. We begin with the case < p < ∞ . Then H p ( R N ) = L p ( R N ) , as a Corollaryof the L p boundedness of the Hardy-Littlewood Maximal operator (which dominatespointwise the auxiliary M ∗ f maximal operator). If we define S t ( f ) := f ∗ ϕ t , as itis the convolution of a Schwartz distribution and a Schwartz function, it is C ∞ (see,e. g., Grafakos [32]); and S t ( f ) ∈ L p ( R N ) ∩ C ∞ ( R N ) because | S t ( f )( x ) | ≤ M ∗ f ( x ) .Since S t ( f ) → f, t ց , both in L p ( R N ) and pointwise almost everywhere (which isa corollary of the Lebesgue Differentiation Theorem and the Dominated ConvergenceTheorem), it follows that L p ( R N ) ∩ C ∞ ( R N ) is dense in L p ( R N ) . Fix now Θ ∈ C ∞ c ( R N ) such that Θ = 1 if | x | ≤ / and Θ = 0 if | x | ≥ and consider the operator R s ( f )( x ) = f ( x )Θ( sx ) , s > . It is immediate that R s ( f ) → f, s ց , again both in L p and pointwise. More-over, R s ( f ) → , s → ∞ , in L p and pointwise for x = 0 . For a given ε > , ∃ t > such that k f − S t ( f ) k L p ( R N ) ≤ ε/ . For such t > , ∃ s > suchthat k S t ( f ) − R s [ S t ( f )] k L p ( R N ) ≤ ε/ so that k f − R s [ S t ( f )] k L p ( R N ) ≤ ε . As R s [ S t ( f )] ∈ C ∞ c ( R N ) ⊂ S ( R N ) , it follows that S ( R N ) is dense in L p ( R N ) ,
. Then, ∃ g ∈ S ( R N ) with k f − g k p ≤ ε/ . Consider the operators M s ( f ) given by [ M s ( f )] ∧ := ˆ f − R s ( ˆ f ) = [1 − Θ( s · )] ˆ f , so supp [ M s ( f )] ∧ ⊂ { ξ ∈ R N : | ξ | ≥ / (2 s ) } . By Fourier Inversion M s ( f ) = f − (cid:0) ˇΘ s ∗ f (cid:1) ; ˇΘ s ( · ) := s − N ˇΘ( s − · ) . Since the Fourier transform preserves S ( R N ) , it follows that M s ( f ) ∈ D , s > , if f ∈ S ( R N ) . And since ˇΘ t , t > , define, like the family ϕ t , a standard approximation of identity, it follows that M s ( f ) → f in L p as s → ∞ . Picking s > so that k h − M s ( h ) k p ≤ ε/ , we obtain k f − M s ( h ) k p ≤ ε , which concludes the proof ofthe Lemma in the range < p < ∞ .In the case < p ≤ , the result follows as a corollary of the Littlewood-Paleysquare function characterization of the spaces H p ( R N ) ; we refer to Grafakos [32],Chapter 6, for the details. (cid:3) Lemma 5.10.
Let
Λ = ( − ∆) / in the spectral sense (see also section 8). Then, forany N ≥ , Λ − : H N ( R N ) −→ BMO ( R N ) boundedly.Proof. Given f ∈ H N ( R N ) and a ball B in R N , for N ≥ , using the H ¨older inequal-ity and the Poincar´e inequality for the exponent N , | Λ − f − (Λ − f ) B | B ≤ C N (cid:13)(cid:13)(cid:12)(cid:12) ∇ [Λ − ( f )] (cid:12)(cid:12)(cid:13)(cid:13) L N ( B ) ≤ C N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 | R j ( f ) | / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L N ( R N ) ≤ C ′ N k f k H N ( R N ) , where R j is the j − th Riesz Transform. The last estimate follows since it is a classicalresult in Fourier Analysis that (cid:13)(cid:13)(cid:13)(cid:13)(cid:16)P Nj =1 | R j ( f ) | (cid:17) / (cid:13)(cid:13)(cid:13)(cid:13) L p ( R N ) is equivalent to the normof the Hardy space H p ( R N ) in any dimension N and for any exponent ≤ p < ∞ (we refer again to Grafakos [32]). (cid:3) Now we can finish the proof of the main theorem of this section: given f ∈ X ,there exists a sequence g j ∈ D such that g j → Λ f, j → ∞ in H N ( R N ) (byLemma 5.9). Now because g j ∈ D , [Λ − ( g j )] ∧ ( ξ ) = c N | ξ | − ˆ g j ( ξ ); ξ = 0 . Sincefor g ∈ D ⊂ L p ( R N ) , ≤ p ≤ ∞ , Λ − f ∈ L q ( R N ) , q > N by the classicalSobolev Embedding Theorem, and this rules out the possibility of a singular support at ξ = 0 of (Λ − g ) ∧ . As a result, for g ∈ D , (Λ − g ) ∧ is also in D ; it follows that Λ − g isa Schwartz function, and since the Fourier transform preserves this class, so it belongstoo to S ( R N ) ⊂ C ( R N ) . Since by Lemma 5.10, Λ − : H N ( R N ) → BMO ( R N ) iscontinuous, given f ∈ X , f belongs to the closure of C ( R N ) in BMO ( R N ) , which isVMO ( R N ) . (cid:3)
6. E
XISTENCE RESULTS
Now we introduce the general theoretical framework in which our existence resultsfollow. For the sake of clarity, we divide this section into three subsections correspond-ing each to the different type of data we are interested in. Our key theoretical tool willbe the combination of the results we have proven in the previous sections with suitableweak continuity properties of the k − Hessian. We note that related properties werestudied in the past by several authors, see for instance [2, 5, 6, 7, 8, 9, 15, 17, 30, 33,35, 36, 37, 40, 41, 42, 43, 44, 45, 49].
OLYHARMONIC k − HESSIAN EQUATIONS 13 H data. We start this first subsection introducing a series of technical resultswhich will be of use in the remainder of the section.
Lemma 6.1. If ψ ∈ ˙ W m − δ,N/ ( N − δ ) ( R N ) ∀ ≤ δ ≤ m − for m = 1 + N ( k − / (2 k ) ∈ N then S k [ ψ ] ∈ H ( R N ) .Proof. It is clear that S k [ ψ ] ∈ L ( R N ) for ψ ∈ ˙ W m − δ,N/ ( N − δ ) ( R N ) ∀ ≤ δ ≤ m − as a direct consequence of a suitable Sobolev embedding when necessary.The improved regularity in the statement follows from the divergence form of the k − Hessian (see equation (13) below) and Theorem I in [31], see also [14, 15]. (cid:3)
Remark 6.2.
We find admissible values of m whenever • N is a multiple of k . • N is odd, N is a multiple of k and k is odd.For example, when N = 2 k we always find the admissible value m = N/ . Notealso that, as we are assuming N, k ≥ , then m ≥ , so we are always treating withpolyharmonic, rather than harmonic, problems. Proposition 6.3. S k [ · ] is weakly −∗ sequentially continuous from ˙ W m, ( R N ) to theHardy space H ( R N ) , provided m = 1 + N ( k − / (2 k ) ∈ N . That is, if ψ n ⇀ ψ ; weakly in ˙ W m, ( R N ) , then S k [ ψ n ] ∗ ⇀ S k [ ψ ]; weakly −∗ in H ( R N ) . Proof.
Since (cid:2)
VMO ( R N ) (cid:3) ∗ = H ( R N ) the statement means that whenever ψ n ⇀ ψ weakly in ˙ W m, ( R N ) then Z R N ϕ S k [ ψ n ] dx → Z R N ϕ S k [ ψ ] dx ∀ ϕ ∈ VMO ( R N ) . Note that S k [ ψ n ] , S k [ ψ ] ∈ H ( R N ) by Lemma 6.1. We start proving weak sequentialcontinuity in the sense of distributions(11) ψ n ⇀ ψ weakly in ˙ W m, ( R N ) ⇒ S k [ ψ n ] → S k [ ψ ] in D ∗ ( R N ) . Fix φ ∈ C ∞ c ( R N ) and compute(12) Z R N φ S k [ ψ n ] dx = − k X i,j Z R N φ i ( ψ n ) j S ijk [ ψ n ] dx, where we have used integration by parts and the divergence form of the k − Hessian(13) S k [ ψ ] = 1 k X i,j ∂ x i ( ψ x j S ijk [ ψ ]) , see [65], where S ijk ( D ψ ) = ∂∂a ij σ k [Λ( A )] (cid:12)(cid:12)(cid:12)(cid:12) A = D ψ , where Λ( A ) are the eigenvalues of the N × N matrix A which entries are a ij , and weremind the definition of the k − Hessian S k [ ψ ] = σ k (Λ) where σ k (Λ) = X i < ··· .So it follows that Z R N ϕ S k [ ψ n ] dx − Z R N ϕ S k [ ψ ] dx = Z R N ϕ ǫ S k [ ψ n ] dx − Z R N ϕ ǫ S k [ ψ ] dx + Z R N ( ϕ − ϕ ǫ ) S k [ ψ n ] dx − Z R N ( ϕ − ϕ ǫ ) S k [ ψ ] dx. Since S k [ ψ n ] and S k [ ψ ] are uniformly bounded in H ( R N ) , we can estimate (cid:12)(cid:12)(cid:12)(cid:12)Z R N ϕ S k [ ψ n ] dx − Z R N ϕ S k [ ψ ] dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ n k S k [ ψ n ] k H ( R N ) + k S k [ ψ ] k H ( R N ) o ×k ϕ − ϕ ǫ k VMO ( R N ) + (cid:12)(cid:12)(cid:12)(cid:12)Z R N ϕ ǫ S k [ ψ n ] dx − Z R N ϕ ǫ S k [ ψ ] dx (cid:12)(cid:12)(cid:12)(cid:12) , and lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z R N ϕ S k [ ψ n ] dx − Z R N ϕ S k [ ψ ] dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cǫ + o (1) . OLYHARMONIC k − HESSIAN EQUATIONS 15
The statement follows by the arbitrariness of ǫ . (cid:3) Corollary 6.4. S k [ · ] is weakly −∗ continuous from the homogeneous Sobolev space ˙ W m − δ,N/ ( N − δ ) ( R N ) ∀ ≤ δ < m − to H ( R N ) , provided m = 1 + N ( k − / (2 k ) ∈ N . Corollary 6.5.
The k − Hessian S k [ · ] is weakly −∗ continuous from the homogeneousSobolev space ˙ W m − δ,N/ ( N − δ ) ( R N ) ∀ ≤ δ < m − to M ( R N ) , where M ( R N ) isthe set of (signed) Radon measures, provided m = 1 + N ( k − / (2 k ) ∈ N . That is, if ψ n ⇀ ψ ; weakly in ˙ W m − δ,N/ ( N − δ ) ( R N ) , then S k [ ψ n ] ∗ ⇀ S k [ ψ ]; weakly −∗ in M ( R N ) . Now we state the main result of this subsection:
Theorem 6.6.
Let m = 1 + N ( k − / (2 k ) ∈ N . Then problem (2a) - (2b) has atleast one weak solution in ˙ W m, ( R N ) for any N ≥ and any N/ ≥ k ≥ ( N, k ∈ N ) provided | λ | is small enough and f ∈ H ( R N ) . Moreover any suchsolution u ∈ ˙ W m − ǫ,N/ ( N − ǫ ) ( R N ) ∀ ≤ ǫ ≤ m and D m u ∈ H ( R N ) .Proof. Consider w ∈ ˙ W m, ( R N ) . Then S k [ − w ] ∈ H ( R N ) by Lemma 6.1 and theequation ( − ∆) m u = S k [ − w ] + λf, x ∈ R N ,u → , when | x | → ∞ , has a unique solution u ∈ ˙ W m − ǫ,N/ ( N − ǫ ) ( R N ) ∀ ≤ ǫ ≤ m such that D m u ∈H ( R N ) by Corollary 3.2. So the map T : H ( R N ) −→ H ( R N ) v v ′ = S k (cid:2) ( − ∆) − m ( − v ) (cid:3) + λf, is well defined and moreover k v ′ k H ( R N ) ≪ k S k (cid:2) ( − ∆) − m ( − v ) (cid:3) k H ( R N ) + | λ |k f k H ( R N ) ≪ k ( − ∆) − m ( − v ) k k ˙ W m, ( R N ) + | λ |k f k H ( R N ) ≪ k v k k H ( R N ) + | λ |k f k H ( R N ) , by the triangle inequality in the first step, Lemma 6.1 in the second, and Proposition 3.1in the third. Now consider the particular case v = 0 , i. e. v = λf , then obviously k v k H ( R N ) = | λ |k f k H ( R N ) and v ′ − v = S k (cid:2) ( − ∆) − m ( − v ) (cid:3) , x ∈ R N . Therefore k v ′ − v k H ( R N ) = k S k (cid:2) ( − ∆) − m ( − v ) (cid:3) k H ( R N ) ≪ k ( − ∆) − m ( − v ) k k ˙ W m, ( R N ) ≪ k v k k H ( R N ) ≪ h k v − v k H ( R N ) + k v k H ( R N ) i k = h k v − v k H ( R N ) + | λ |k f k H ( R N ) i k . Consequently it is clear that T will map the ball B = n v ∈ H ( R N ) : k v − v k H ( R N ) ≤ R o into itself provided we choose R and | λ | small enough.Now assume ψ j ∗ ⇀ ψ in H ( R N ) , therefore (cid:10) ( − ∆) m φ, ( − ∆) − m ψ j (cid:11) −→ (cid:10) ( − ∆) m φ, ( − ∆) − m ψ (cid:11) for any fixed φ ∈ VMO ( R N ) , or equivalently D ˆ φ, ( − ∆) − m ψ j E −→ D ˆ φ, ( − ∆) − m ψ E , for any fixed ˆ φ ∈ I − m ( VMO )( R N ) , with the obvious definition of I − m ( VMO )( R N ) (see for instance [53]). By Corollary 3.2 ( − ∆) − m ψ j ∈ ˙ W m − ,N/ ( N − ( R N ) , but weneed ( − ∆) − m ψ j ⇀ ( − ∆) − m ψ in ˙ W m − ,N/ ( N − ( R N ) ; note that the first mode ofconvergence does not, in principle, trivially imply the second. On the other hand thetwo facts { ˙ W m − ,N/ ( N − ( R N ) } ∗ = ˙ W − m,N ( R N ) and Remark 5.4 imply that, for N ≥ , the first mode of convergence indeed implies the second. As a consequence ofthis and Corollary 6.4 the map T is weakly −∗ continuous, and consequently by The-orem 4.1 it has a fixed point. The existence of solution follows from u = ( − ∆) − m v and Proposition 3.1. The regularity follows by Sobolev embeddings. (cid:3) Summable Data.
An analogous existence theorem can still be proven for data f ∈ L ( R N ) . Theorem 6.7.
Let m = 1 + N ( k − / (2 k ) ∈ N . Then problem (2a) - (2b) has at leastone weak solution in ˙ W m − ǫ,N/ ( N − ǫ ) ( R N ) ∀ < ǫ ≤ m for any N ≥ and any N/ > k ≥ ( N, k ∈ N ) provided | λ | is small enough and f ∈ L ( R N ) . Moreoverany such solution fulfills D m u ∈ L , ∞ ( R N ) .Proof. Consider w ∈ ˙ W m − ,N/ ( N − ( R N ) . Then S k [ − w ] ∈ H ( R N ) by Lemma 6.1and Remark 6.2, and the equation ( − ∆) m u = S k [ − w ] + λf, x ∈ R N ,u → , when | x | → ∞ , has a unique solution u ∈ ˙ W m − ǫ,N/ ( N − ǫ ) ( R N ) ∀ < ǫ ≤ m such that D m u ∈ L , ∞ ( R N ) by Proposition 3.1. So the map T : ˙ W m − ,N/ ( N − ( R N ) −→ ˙ W m − ,N/ ( N − ( R N ) w u = ( − ∆) − m S k [ − w ] + λ ( − ∆) − m f, OLYHARMONIC k − HESSIAN EQUATIONS 17 is well defined and furthermore for g := ( − ∆) − m f k u k ˙ W m − ,N/ ( N − ( R N ) ≪ (cid:13)(cid:13) ( − ∆) − m S k [ − w ] (cid:13)(cid:13) ˙ W m − ,N/ ( N − ( R N ) + | λ | k g k ˙ W m − ,N/ ( N − ( R N ) ≪ k w k k ˙ W m − ,N/ ( N − ( R N ) + | λ | k g k ˙ W m − ,N/ ( N − ( R N ) , by the triangle inequality and Proposition 3.1 in the first step, and Lemma 6.1 andCorollary 3.2 in the second. Now consider the particular case w = 0 , i. e. u = λ ( − ∆) − m f , then clearly k u k ˙ W m − ,N/ ( N − ( R N ) = | λ | k g k ˙ W m − ,N/ ( N − ( R N ) and u − u = ( − ∆) − m S k [ − w ] , x ∈ R N . Therefore k u − u k ˙ W m − ,N/ ( N − ( R N ) = (cid:13)(cid:13) ( − ∆) − m S k [ − w ] (cid:13)(cid:13) ˙ W m − ,N/ ( N − ( R N ) ≪ k w k k ˙ W m − ,N/ ( N − ( R N ) ≪ h k w − u k ˙ W m − ,N/ ( N − ( R N ) + k u k ˙ W m − ,N/ ( N − ( R N ) i k = h k w − u k ˙ W m − ,N/ ( N − ( R N ) + | λ | k g k ˙ W m − ,N/ ( N − ( R N ) i k . Consequently it is clear that T maps the ball B = n w ∈ ˙ W m − ,N/ ( N − ( R N ) : k w − u k ˙ W m − ,N/ ( N − ( R N ) ≤ R o into itself given that we choose R and | λ | small enough.Corollary 6.4 implies the convergence property(14) h φ, S k [ ψ n ] i −→ h φ, S k [ ψ ] i , for any fixed φ ∈ VMO ( R N ) given that ψ n ⇀ ψ in ˙ W m − ,N/ ( N − ( R N ) . By equa-tion (14) we get (cid:10) ( − ∆) m φ, ( − ∆) − m S k [ ψ n ] (cid:11) −→ (cid:10) ( − ∆) m φ, ( − ∆) − m S k [ ψ ] (cid:11) , for any fixed φ ∈ VMO ( R N ) , or in other terms D ˆ φ, ( − ∆) − m S k [ ψ n ] E −→ D ˆ φ, ( − ∆) − m S k [ ψ ] E , for any fixed ˆ φ ∈ I − m ( VMO )( R N ) , as in the previous subsection. This mode ofconvergence is not, in principle, equivalent to the one we need: ( − ∆) − m S k [ ψ n ] ⇀ ( − ∆) − m S k [ ψ ] in ˙ W m − ,N/ ( N − ( R N ) . However using { ˙ W m − ,N/ ( N − ( R N ) } ∗ =˙ W − m,N ( R N ) and Remark 5.4 we find for N ≥ that the second mode of conver-gence follows as a consequence of the first.Given our assumption N ≥ we get that the map T is weakly continuous in ˙ W m − ,N/ ( N − ( R N ) (and thus it is weakly −∗ continuous), so by Theorem 4.1 it has a fixed point. The regularity follows from Proposition 3.1 and a classical bootstrapargument. (cid:3) Remark 6.8.
Note that the space ˙ W m − ,N/ ( N − ( R N ) is not a Banach space since k · k ˙ W m − ,N/ ( N − ( R N ) is a seminorm rather than a norm. Note however that the nullsubspace of k · k ˙ W m − ,N/ ( N − ( R N ) is composed by the polynomials of degree smalleror equal to m − . So we can consider ˙ W m − ,N/ ( N − ( R N ) as the quotient spacewhich equivalence classes are closed with respect to the addition of one such polyno-mial. Since in Theorem 6.7 we are proving the existence of solutions that vanish atinfinity, and the set of polynomials that vanish at infinity has a unique element that isidentically zero, the use of norm k · k ˙ W m − ,N/ ( N − ( R N ) in the proof of Theorem 6.7is meaningful.6.3. L p data. We now state the complementary result that assumes our datum f ∈ L p ( R N ) . Theorem 6.9.
Let m = 1 + N ( k − / (2 pk ) ∈ N . Then problem (2a) - (2b) has atleast one weak solution in ˙ W m − ǫ,Np/ ( N − ǫp ) ( R N ) ∀ ≤ ǫ ≤ m for any N ≥ and any N/ > k ≥ ( N, k ∈ N ) provided | λ | is small enough and f ∈ L p ( R N ) , < p < N/ (2 k ) .Proof. The proofs mimics that of Theorem 6.6 with the space ˙ W m,p ( R N ) playingthe role of both ˙ W m − ,N/ ( N − ( R N ) and ˙ W m, ( R N ) , except for the proof of weakconvergence. Therefore we will only include this part here.In this case ψ ∈ ˙ W m,p ( R N ) ֒ → ˙ W ,kp ( R N ) , so we need to prove Z R N ϕ S k [ ψ n ] dx → Z R N ϕ S k [ ψ ] dx ∀ ϕ ∈ L q ( R N ) , where p − + q − = 1 (and so q > ). We again start proving weak continuity in thesense of distributions(15) ψ n ⇀ ψ weakly in ˙ W m,p ( R N ) ⇒ S k [ ψ n ] → S k [ ψ ] in D ∗ ( R N ) . We fix φ ∈ C ∞ c ( R N ) and calculate(16) Z R N φ S k [ ψ n ] dx = − k X i,j Z R N φ i ( ψ n ) j S ijk [ ψ n ] dx, where we have used integration by parts and the divergence form of the k − Hessian.Now we take the limit lim n →∞ Z R N φ S k [ ψ n ] dx = − lim n →∞ k X i,j Z R N φ i ( ψ n ) j S ijk [ ψ n ] dx, = − k X i,j Z R N φ i ( ψ ) j S ijk [ ψ ] dx, = Z R N φ S k [ ψ ] dx, where the first and third equalities follow from (16) and the second from the Rellich-Kondrachov theorem which for ψ n ⇀ ψ weakly in ˙ W m,p ( R N ) OLYHARMONIC k − HESSIAN EQUATIONS 19 implies ψ n → ψ strongly in ˙ W , (2 N − pk ) pk ( k − / [(2 N − pk ) pk − N − pk )]) loc ( R N ) and ψ n → ψ strongly in ˙ W , (2 N − pk ) pk/ (2 N − pk ) loc ( R N ) . Thus (15) is proven.Since C ∞ c ( R N ) is dense in L q ( R N ) , p − + q − = 1 , we select an approximatingfamily ϕ ǫ ∈ C ∞ c ( R N ) of ϕ ∈ L q ( R N ) such that k ϕ − ϕ ǫ k L q ( R N ) ≤ ǫ for any ǫ > .So it holds that Z R N ϕ S k [ ψ n ] dx − Z R N ϕ S k [ ψ ] dx = Z R N ϕ ǫ S k [ ψ n ] dx − Z R N ϕ ǫ S k [ ψ ] dx + Z R N ( ϕ − ϕ ǫ ) S k [ ψ n ] dx − Z R N ( ϕ − ϕ ǫ ) S k [ ψ ] dx. Given that S k [ ψ n ] and S k [ ψ ] are bounded in L p ( R N ) , we can establish the estimate (cid:12)(cid:12)(cid:12)(cid:12)Z R N ϕ S k [ ψ n ] dx − Z R N ϕ S k [ ψ ] dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ n k S k [ ψ n ] k L p ( R N ) + k S k [ ψ ] k L p ( R N ) o ×k ϕ − ϕ ǫ k L q ( R N ) + (cid:12)(cid:12)(cid:12)(cid:12)Z R N ϕ ǫ S k [ ψ n ] dx − Z R N ϕ ǫ S k [ ψ ] dx (cid:12)(cid:12)(cid:12)(cid:12) , and lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z R N ϕ S k [ ψ n ] dx − Z R N ϕ S k [ ψ ] dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cǫ + o (1) . Therefore the arbitrariness of ǫ guarantees that, if ψ n ⇀ ψ ; weakly in ˙ W m,p ( R N ) , then S k [ ψ n ] ⇀ S k [ ψ ]; weakly in L p ( R N ) , and so the statement follows. (cid:3) Remark 6.10.
The lower bounds for the values of N in Theorems 6.6, 6.7 and 6.9 areeasily proven using the inequalities in the statement of Proposition 3.1. Also, it is easyto find examples of m , N , k and p for which the statements of these theorems apply.7. L OCAL U NIQUENESS
In this section we prove existence and local uniqueness of a solution under morerestrictive conditions. We start concentrating on the case that corresponds to Theo-rem 6.7.
Definition 7.1.
Let u be a weak solution to problem (2a)-(2b) and W a Banach space.If there exists a ρ > such that this solution is unique in the ball ˜ B ρ ( u ) = { v ∈ W : k u − v k W ≤ ρ } , then we say that u is locally unique in W . Theorem 7.2.
Let m = 1 + N ( k − / (2 k ) ∈ N . Then problem (2a) - (2b) has at leastone weak solution in ˙ W m − ǫ,N/ ( N − ǫ ) ( R N ) ∀ < ǫ ≤ m for any N ≥ and any N/ > k ≥ ( N, k ∈ N ) provided | λ | is small enough and f ∈ L ( R N ) . Moreoverany such solution fulfills D m u ∈ L , ∞ ( R N ) and at least one is locally unique in ˙ W m − ,N/ ( N − ( R N ) .Proof. Consider w , w ∈ ˙ W m − ,N/ ( N − ( R N ) . Then S k [ − w , ] ∈ H ( R N ) byLemma 6.1 and the equations ( − ∆) m u , = S k [ − w , ] + λf, x ∈ R N ,u , → , when | x | → ∞ , have a unique solution u , ∈ ˙ W m − ,N/ ( N − ( R N ) by Proposition 3.1. Now we cansubtract them ( − ∆) m ( u − u ) = S k [ − w ] − S k [ − w ] , x ∈ R N ,u − u → , when | x | → ∞ , and find a unique solution u − u ∈ ˙ W m − ,N/ ( N − ( R N ) such that k u − u k ˙ W m − ,N/ ( N − ( R N ) ≪ k S k [ − w ] − S k [ − w ] k , by the same proposition. Now using S k [ ψ ] = 1 k X i,j ∂ x i ( ψ x j S ijk [ ψ ]) = 1 k X i,j ψ x i x j S ijk [ ψ ] , since X i ∂ x i S ijk [ ψ ] = 0 ∀ ≤ j ≤ N, for any smooth function ψ [40], yields k u − u k ˙ W m − ,N/ ( N − ( R N ) ≪ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i,j ( w ) x i x j S ijk [ w ] − k X i,j ( w ) x i x j S ijk [ w ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≪ h k w k ˙ W m − ,N/ ( N − ( R N ) + k w k ˙ W m − ,N/ ( N − ( R N ) i k − ×k w − w k ˙ W m − ,N/ ( N − ( R N ) , after arguing by approximation in the first step and using Sobolev and triangle inequal-ities, and a reasoning akin to that in the proof of Theorem 1 in [7], in the second. Weknow the map T : ˙ W m − ,N/ ( N − ( R N ) −→ ˙ W m − ,N/ ( N − ( R N ) w , u , , OLYHARMONIC k − HESSIAN EQUATIONS 21 is well defined and also maps the ball B = n w ∈ ˙ W m − ,N/ ( N − ( R N ) : k w − u k ˙ W m − ,N/ ( N − ( R N ) ≤ R o into itself provided we choose R and | λ | small enough by the proof of Theorem 6.6.Therefore k u − u k ˙ W m − ,N/ ( N − ( R N ) ≪ h | λ |k f k L ( R N ) + R i k − ×k w − w k ˙ W m − ,N/ ( N − ( R N ) < k w − w k ˙ W m − ,N/ ( N − ( R N ) , where we have used the triangle inequality and Proposition 3.1 in the first step and havechosen sufficiently smaller R and | λ | in the second. Thus the existence and uniquenessof the solution follows by the application of the Banach fixed point theorem and theregularity by Proposition 3.1 and a classical bootstrap argument. (cid:3) We can now state the corresponding result for f ∈ L p ( R N ) . Theorem 7.3.
Let m = 1 + N ( k − / (2 pk ) ∈ N . Then problem (2a) - (2b) has atleast one weak solution in ˙ W m − ǫ,Np/ ( N − ǫp ) ( R N ) ∀ ≤ ǫ ≤ m for any N ≥ and any N/ > k ≥ ( N, k ∈ N ) provided | λ | is small enough and f ∈ L p ( R N ) , < p < N/ (2 k ) . Moreover, at least one of these solutions is locally unique in ˙ W m,p ( R N ) .Proof. Follows analogously to that of Theorem 7.2. (cid:3)
Remark 7.4.
The proof of Theorem 7.2 is not applicable to the case f ∈ H ( R N ) and k = N/ ; for the existence theory under these hypotheses the reader is referredto Theorem 6.6. On the other hand assuming f ∈ H ( R N ) and k < N/ allows oneto reproduce this proof identically with the slight improvement in regularity D m u ∈H ( R N ) . 8. N ONLOCAL PROBLEMS
In this section we extend our results for problem (1) to(17) Λ n u = S k [ − u ] + λf, x ∈ R N , where Λ is a pseudo-differential operator defined in the following way. Definition 8.1.
The pseudo-differential operator
Λ := √− ∆ , where the square root isinterpreted in the sense of the Spectral Theorem. Remark 8.2.
The operator Λ is well defined since − ∆ is essentially self-adjoint in C ∞ c ( R N ) ⊂ L ( R N ) [48]. Remark 8.3.
The operator Λ n is a differential, and thus local, operator when n iseven; in this case we actually have Λ n = ( − ∆) n/ . If n is odd then Λ n is a nonlocalpseudo-differential operator. Proposition 8.4. Λ f = F − [2 π | η |F ( f )] . Proof.
This is an immediate consequence of the spectral representation of the Lapla-cian in terms of the Fourier transform: − ∆ f = F − [4 π | η | F ( f )] . (cid:3) Definition 8.5.
We define G n,N ∈ S ∗ ( R N ) in the following way: • If < n < N , it is the unique solution to Λ n G n,N = δ that obeys G n,N ( x ) → when | x | → ∞ . • If n = N , it is the unique solution to Λ n G n,N = δ in BMO ( R N ) . Proposition 8.6.
The distribution G n,N is given by the exact formulas: • If < n < N , G n,N ( x ) = C n,N | x | N − n , C n,N = 2 − n π − N/ Γ (cid:0) N − n (cid:1) Γ (cid:0) n (cid:1) . • If n = N , G N,N ( x ) ≡ G N ( x ) = C N log | x | ,C N = (2 − N )(2 π ) N − π − N/ Γ (cid:0) N − (cid:1) , N ≥ − (2 π ) − , N = 2 π − , N = 1 . Proof.
We use F (Λ n G n,N ) = F ( δ ) = 1 , to find F ( G n,N )( ξ ) = (2 π | ξ | ) − n ∀ ξ ∈ R N \ { } . When < n < N , F ( G n,N )( ξ ) is well defined in L ( R N ) + L ∞ ( R N ) , and thereforeit is well defined as a Schwartz distribution. Now, an argument akin to that in the proofof Lemma 3.6 yields that indeed F ( G n,N )( ξ ) = (2 π | ξ | ) − n in S ∗ ( R N ) . The statementfollows by Fourier inversion.If n = N , then F ( G N )( ξ ) = (2 π | ξ | ) − N ∀ ξ ∈ R N \ { } . Therefore in this case F ( G N ) L loc ( R N ) and it does not even define a singularintegral operator. Consequently our approach in this case will be different; lets startwith R , in this case Λ G ( x ) = δ ⇐⇒ − ∆ G ( x ) = δ , and so G ( x ) = − (2 π ) − log | x | . Now focus in N ≥ and compute Λ n log | x | = Λ n − (Λ log | x | )= Λ n − [( − ∆) log | x | ]= Λ n − (cid:18) − n | x | (cid:19) . OLYHARMONIC k − HESSIAN EQUATIONS 23
By means of Fourier transform we find F (Λ n log | x | )( ξ ) = (2 π | ξ | ) n − F (cid:18) − n | x | (cid:19) ( ξ )= (2 π | ξ | ) n − (2 − n ) π − n/ Γ( n/ − | ξ | − ( n − =: C − N ; note that C N is always well defined for N ≥ . Therefore F [Λ n ( C N log | x | )]( ξ ) = 1 ⇐⇒ Λ n ( C N log | x | ) = δ . It only rests to show that Λ G ( x ) = δ in R . We remind the reader that G ( x ) ∝ log | x | , d log | x | /dx = x − and that x − defines a Schwartz distribution when inter-preted as a principal value; in this case F (cid:20) P. V. (cid:18) x (cid:19)(cid:21) ( ξ ) = iπ sgn ( ξ ) . Now compute iπ sgn ( ξ ) = F (cid:18) d log | x | dx (cid:19) ( ξ ) , = 2 πi ξ F (log | x | ) ( ξ ) ⇒ F (log | x | ) ( ξ ) = 12 | ξ | if ξ = 0 . Clearly, | ξ | − S ∗ ( R ) , and therefore F (log | x | ) ( ξ ) has to be interpreted as a renor-malization of (2 | ξ | ) − . Now consider F (cid:16) Λ / log | x | (cid:17) ( ξ ) = (2 π | ξ | ) / F (log | x | ) ( ξ )= (2 π | ξ | ) / (2 | ξ | ) − = r π | ξ | − / , if ξ = 0 . Regularizing the singularity of F (log | x | )( ξ ) at the origin and letting theregularization parameter go to zero we find F (cid:16) Λ / log | x | (cid:17) ( ξ ) = r π | ξ | − / in S ∗ ( R ) . Finally F (Λ log | x | ) ( ξ ) = F h Λ / (cid:16) Λ / log | x | (cid:17)i ( ξ )= (2 π | ξ | ) / r π | ξ | − / = π = F ( πδ ) , in S ∗ ( R ) . (cid:3) Proposition 8.7.
The distribution G n,N ( x ) is well defined and, in particular: • If G ( x ) solves Λ n G = δ , < n < N , and G ( x ) → when | x | → ∞ , then G = G n,N . • If G ( x ) solves Λ N G = δ and G ( x ) ∈ BMO ( R N ) , then G − G N is constant,i. e. G ≡ G N in BMO ( R N ) .Proof. The existence of this distribution was proven in the previous Proposition andits uniqueness follows analogously as in the proof of Lemma 3.6. (cid:3)
Theorem 8.8.
Let n ∈ Z , < n ≤ N , and Λ n u = f in R N . Then ∂ αx u = A n,N R α ( f ) for some constant A n,N , where | α | = n , the monomial ∂ αx = ∂ x j · · · ∂ x jn , R α = R x j · · · R x jn and R x , · · · , R x n are the correspondingRiesz transforms in R N .Proof. We start with the subcritical case < n < N : u ( x ) = ( G n,N ∗ f )( x ) ≡ Λ − n f. We can write Λ n − u = Λ − f = C N Z R N f ( y ) | x − y | N − dy, and thus ∂ x j Λ n − u = C N (1 − N ) P. V. Z R N x j − y j | x − y | N +1 f ( y ) dy = D N R x j ( f ) , where D N = 0 since N ≥ . Therefore, ∂ αx u = ∂ x j ( ∂ x j · · · ∂ x j )Λ − n (Λ n − u )= ( ∂ x j · · · ∂ x jn )Λ − n ( ∂ x j Λ n − u )= D N R x j · · · R x jn ( R x j f )= D N R α ( f ) . Now we move to the case n = N ≥ . We know u = G N ∗ f where G N = C N log | x | . Then Λ N − u = Λ N − ( − ∆ u )= Λ N − [( − ∆ G N ) ∗ f ] , where − ∆ G N = C N (2 − N ) | x | − . Therefore Λ N − u = C N Z R N f ( y ) | x − y | N − dy, where C N = 0 and the rest of the proof follows as in the previous case.When n = N = 2 we write u = − (2 π ) − log | x | ∗ f ( x ) and therefore ∂ x j u = − π Z R x j − y j | x − y | f ( y ) dy = − π Z R y j | y | f ( x − y ) dy. OLYHARMONIC k − HESSIAN EQUATIONS 25
Finally we have Λ ∂ x j u ( x ) ∝ P. V. Z R y j | y | f ( x − y ) dy ∝ R x j ( f )( x ) , and thus ∂ x j ∂ x k u = ( R x k Λ) ∂ x j u ∝ R x j R x k u. The case n = N = 1 comes from the fact that u ( x ) = 1 π Z R log | x − y | f ( y ) dy, and the fact that u ′ ( x ) = 1 π P. V. Z R x j − y j | x − y | f ( y ) dy, which is the Hilbert transform of f . (cid:3) Corollary 8.9.
The linear equation Λ n u = λf, x ∈ R N , has a unique solution in the following cases: (a) f ∈ L p ( R N ) , < p < Nn , n < N , (b) f ∈ L ( R N ) , n < N , (c) f ∈ H ( R N ) , n < N , (d) f ∈ H ( R N ) , n = N .Then, respectively (a) u ∈ ˙ W n − ǫ,Np/ ( N − ǫp ) ( R N ) ∀ ≤ ǫ ≤ n , (b) u ∈ ˙ W n − ǫ,N/ ( N − ǫ ) ( R N ) ∀ < ǫ ≤ n , (c) u ∈ ˙ W n − ǫ,N/ ( N − ǫ ) ( R N ) ∀ ≤ ǫ ≤ n , (d) u ∈ ˙ W N − ǫ,N/ ( N − ǫ ) ( R N ) ∀ ≤ ǫ ≤ N .Moreover, in case (b), D n u ∈ L , ∞ ( R N ) and, in cases (c) and (d), D n u ∈ H ( R N ) . Now we state the main result of this section. Of course,
N, k ∈ N always and wealso assume N > k . Theorem 8.10.
Equation (17) has at least one weak solution in the following cases: (a) f ∈ L p ( R N ) , < p < N k , n = 2 + N ( k − / ( pk ) ∈ N , (b) f ∈ L ( R N ) , n = 2 + N ( k − /k ∈ N , (c) f ∈ H ( R N ) , n = 2 + N ( k − /k ∈ N ,provided | λ | is small enough. Then, respectively (a) u ∈ ˙ W n − ǫ,Np/ ( N − ǫp ) ( R N ) ∀ ≤ ǫ ≤ n , (b) u ∈ ˙ W n − ǫ,N/ ( N − ǫ ) ( R N ) ∀ < ǫ ≤ n , (c) u ∈ ˙ W n − ǫ,N/ ( N − ǫ ) ( R N ) ∀ ≤ ǫ ≤ n .Moreover, in case (b), D n u ∈ L , ∞ ( R N ) and, in case (c), D n u ∈ H ( R N ) . Also, fora smaller enough | λ | , the solution is locally unique in every case.Proof. The proof follows as a consequence of corollary 8.9 and going through thesame arguments as in Section 6 and 7. (cid:3)
Remark 8.11.
The case N = 2 k was already examined in Theorem 6.6.
9. F
URTHER RESULTS
Our previous results imply the weak continuity of the branch of solutions that de-parts from u = 0 and λ = 0 under certain conditions. Theorem 9.1.
Let
Φ : D (Φ) ⊂ B −→ B v u ( v ) , where u is the unique solution to Λ n u = S k [ − u ] + λf, x ∈ R N ,v = Λ − n f , B = ˙ W n,p ( R N ) , f ∈ L p ( R N ) and the rest of hypotheses as in Theo-rem 8.10. Then Φ is weakly continuous, i. e. ∀ { v j } j ⊂ D (Φ) such that v j ⇀ v weakly in B , it holds that Φ( v j ) ⇀ Φ( v ) weakly in B . Proof.
Take D (Φ) to be the ball in B used in Theorem 7.3. Then we know Φ is welldefined and moreover Φ : D (Φ) −→ D (Φ) . We rewrite our equation u j = Λ − n ( S k [ − u j ]) + λv j ; we know that for every v j ∈ D (Φ) there exist a unique solution u j ∈ D (Φ) . Nowtake the limit j → ∞ and we conclude by weak continuity of S k [ · ] in L p ( R N ) , see theproof of Theorem 6.9. (cid:3) We also have a comparatively weaker result for summable data.
Theorem 9.2.
Let
Φ : D (Φ) ⊂ B −→ B v u ( v ) , where u is the unique solution to Λ n u = S k [ − u ] + λf, x ∈ R N ,v = Λ − n f , B = ˙ W n − ,N/ ( N − ( R N ) , f ∈ L ( R N ) and the rest of assumptions as inTheorem 8.10. Then Φ is weakly continuous, i. e. ∀ { v j } j ⊂ D (Φ) such that v j ⇀ v weakly in B , it holds that Φ( v j ) ⇀ Φ( v ) weakly in B . Proof.
The proof follows as the proof of Theorem 9.1 combined with the argumentsregarding weak continuity in the proof of Theorem 6.7. (cid:3)
In the following we will improve our regularity results from sections 3 and 8 andguarantee that the solution of the critical case obeys the boundary conditions.
Theorem 9.3.
Let f ∈ H ( R N ) , then Λ − N f ∈ C ( R N ) and Λ − N : H ( R N ) −→ C ( R N ) is bounded. OLYHARMONIC k − HESSIAN EQUATIONS 27
Proof.
We already know from Proposition 3.1 that k Λ − N f k L ∞ ( R N ) ≪ k f k H ( R N ) .Now let a be a L ∞ -atom for H ( R N ) , i. e. a ∈ H ( R N ) and • There exists a R N − cube Q ⊂ R N , that is Q = c ( Q ) + ℓ ( Q ) Q with c ( Q ) ∈ R N , Q = [ − / , / N and ℓ ( Q ) > , such that a is supported on Q , • k a k L ∞ ( Q ) ≤ | Q | − , • R Q a dx = 0 .We start proving that Λ − N a ∈ C ( R N ) . Let x ∈ R N be such that | x − c ( Q ) | ≥ √ N ℓ .Then Λ − N a ( x ) = C N Z Q log | x − y | a ( y ) dy = C N Z Q [log | x − y | − log | x − c ( Q ) | ] a ( y ) dy, after the use of the first and third defining properties of a in the first and second equal-ities respectively. If y ∈ Q , then | x − y | = | [ x − c ( Q )] − [ y − c ( Q )] |≥ | x − c ( Q ) | − | y − c ( Q ) |≥ | x − c ( Q ) | > , where we have used | y − c ( Q ) | ≤ √ N ℓ ≤ | x − c ( Q ) | . The same reasoning leads to conclude ≤ | x − y || x − c ( Q ) | ≤ , and then | x − y || x − c ( Q ) | = 1 + t, | t | ≤ . The triangle inequality again gives | | x − y | − | x − c ( Q ) | | ≤ | y − c ( Q ) | , which implies | t | ≤ | y − c ( Q ) || x − c ( Q ) | , and then (cid:12)(cid:12)(cid:12)(cid:12) log (cid:20) | x − y || x − c ( Q ) | (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | y − c ( Q ) || x − c ( Q ) | . Therefore (cid:12)(cid:12) Λ − N a ( x ) (cid:12)(cid:12) ≪ Z Q | y − c ( Q ) || x − c ( Q ) | | a ( y ) | dy ≪ | x − c ( Q ) | Z Q | y − c ( Q ) | | Q | − dy ≪ ℓ ( Q ) | x − c ( Q ) | . Since this last estimate holds for | x − c ( Q ) | ≥ √ N ℓ and (cid:12)(cid:12) Λ − N a ( x ) (cid:12)(cid:12) ≪ k a k H ( R N ) ≪ , it follows that (cid:12)(cid:12) Λ − N a ( x ) (cid:12)(cid:12) ≪ ℓ ( Q ) ℓ ( Q ) + | x − c ( Q ) | ∀ x ∈ R N , which proves the decay in the limit | x | → ∞ .To prove continuity of Λ − N a ( x ) choose x, h ∈ R N to find (cid:12)(cid:12) Λ − N a ( x + h ) − Λ − N a ( x ) (cid:12)(cid:12) = C N (cid:12)(cid:12)(cid:12)(cid:12)Z Q (log | x + h − y | − log | x − y | ) a ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≪ k a k L ∞ Z Q | log | x + h − y | − log | x − y | | dy = Z Q (cid:12)(cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12)(cid:12) x − c + hℓ − z (cid:12)(cid:12)(cid:12)(cid:12) − log (cid:12)(cid:12)(cid:12)(cid:12) x − cℓ − z (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) dz =: F (cid:18) x − cℓ , hℓ (cid:19) , where we have used the change of variables y = ℓz + c in the previous to last step.It is enough to prove continuity of F and we may assume < | h | ≤ . Since Q ⊂ B √ N/ (0) =: B , we have F ( x, h ) ≤ Z B | log | x + h − y | − log | x − y | | dy = | h | N Z | h | − B (cid:12)(cid:12) log | x ′ + h ′ − u | − log | x ′ − u | (cid:12)(cid:12) du, after the change of variables y = | h | u , and where x ′ = x/ | h | and h ′ = h/ | h | ∈ S N − .If | x | ≥ √ N then | x ′ − u | ≥ | x ′ | − | u | ≥ √ N / (2 | h | ) for u ∈ | h | − B . Therefore log | x ′ + h ′ − u | − log | x ′ − u | = log (cid:12)(cid:12)(cid:12)(cid:12) x ′ − u | x ′ − u | + h ′ | x ′ − u | (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) | h ′ || x ′ − u | (cid:19) = O ( | h | ) . Then F ( x, h ) ≪ | h | N Z | h | − B | h | du ≪ | h | , which proves continuity in this case. OLYHARMONIC k − HESSIAN EQUATIONS 29 If | x | ≤ √ N then B − x ⊂ B √ N/ (0) = 3 B and F ( x, h ) ≤ | h | N Z B | h | − (cid:12)(cid:12) log | z + h ′ | − log | z | (cid:12)(cid:12) dz = | h | N Z { B | h | − }∩{| z |≤ } (cid:12)(cid:12) log | z + h ′ | − log | z | (cid:12)(cid:12) dz + | h | N Z { B | h | − }∩{| z |≥ } (cid:12)(cid:12) log | z + h ′ | − log | z | (cid:12)(cid:12) dz =: I + I . after the change of variables y = x + | h | z in the first step. The first term can beestimated as follows I ≤ | h | N Z | z |≤ (cid:12)(cid:12) log | z + h ′ | − log | z | (cid:12)(cid:12) dz ≪ | h | N , since the integral can be bounded by a constant independent of h ′ . For the second termwe find I = | h | N Z ≤| z |≤ √ N/ (2 | h | ) (cid:12)(cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12)(cid:12) z | z | + h ′ | z | (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) dz ≪ ( | h | log (cid:16) | h | (cid:17) , n = 1 | h | n > , because the integrand is O (cid:0) | z | − (cid:1) . Summing up: (cid:12)(cid:12) Λ − N a ( x + h ) − Λ − N a ( x ) (cid:12)(cid:12) ≪ ( min n , | h | h (cid:16) | h | (cid:17)io , n = 1min { , | h |} , n > ∀ x, h ∈ R N , such that < | h | ≤ / .Therefore Λ − N : H at ( R N ) −→ C ( R N ) , where H at ( R N ) is the set of all finite linear combinations of L ∞ ( R N ) -atoms for H ( R N ) . Since H at ( R N ) is dense in H ( R N ) for f ∈ H ( R N ) there exists f j ∈H at ( R N ) such that f j → f in H ( R N ) , and therefore Λ − N f j → Λ − N f in L ∞ ( R N ) .Uniform convergence guarantees that Λ − N f is not only bounded but also continuous.Now we prove that Λ − N f ( x ) → when | x | → ∞ . Uniform convergence of Λ − N f j ( x ) to Λ − N f ( x ) implies that there exists a J ∈ N such that for j ≥ J itholds that | Λ − N f ( x ) − Λ − N f j ( x ) | ≤ ǫ/ ∀ x ∈ R N . Now fix such a j ≥ J . Since Λ − N f j ( x ) → when | x | → ∞ , then there exist < R < ∞ such that for | x | ≥ R itholds that | Λ − N f j ( x ) | ≤ ǫ/ . In consequence for | x | ≥ R , | Λ − N f ( x ) | = | Λ − N f ( x ) − Λ − N f j ( x ) + Λ − N f j ( x ) |≤ | Λ − N f ( x ) − Λ − N f j ( x ) | + | Λ − N f j ( x ) |≤ ǫ. (cid:3) Corollary 9.4.
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The k − Hessian equation , Lectures Notes in Mathematics (2009) 177–252.Pedro BalodisDepartamento de Matem´aticasUniversidad Aut´onoma de Madrid [email protected] &Carlos EscuderoDepartamento de Matem´aticasUniversidad Aut´onoma de Madrid&Carlos EscuderoDepartamento de Matem´aticasUniversidad Aut´onoma de Madrid