Abstract
We consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold \(M^{2m}\). Such objects satisfy the elliptic system \([\varDelta ^m J, J]=0\) weakly. We prove a general regularity theorem for semilinear systems in critical dimensions (with critical growth nonlinearities), which includes the system of polyharmonic almost complex structures in dimension four and six.
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1 Introduction
Let (M, g) be a compact Riemannian manifold of dimension n with a compatible almost complex structure. Denote by \({\mathcal {J}}_g\) the space of smooth almost complex structures compatible with g, i.e., \(g(J\cdot , J\cdot )=g(\cdot , \cdot )\). Consider the following functional, for all \(m\in {\mathbb {N}}^+\), \(J\in {\mathcal {J}}_{g}\),
where \(\nabla \) and \(\varDelta \) are Levi-Civita connection and Laplace-Beltrami operator on (M, g), respectively, and dV denotes the volume element of (M, g). We call the critical points of functional \({\mathcal {E}}_{m}(J)\) m-harmonic almost complex structures. These objects are tensor-valued version of polyharmonic maps which have attracted quite some attention in recent years. When \(m=1\), the critical points of functional \({\mathcal {E}}_{m}(J)\) are also called harmonic almost complex structures introduced by Wood [17] in 1990s. We refer the reader to the recent survey [3] for the background and results in this subject. The first author have studied the existence and regularity of harmonic almost complex structures [7] from the point of view of geometric analysis. In this paper, we focus on the case of polyharmonic almost complex structures with \(m\ge 2\). Recall the definition of the Sobolev spaces of almost complex structures.
Definition 1
Suppose \((M^n,g)\) be an almost Hermitian manifold with compatible almost complex structures in \({\mathcal {J}}_g\). We define \(W^{k,p}({\mathcal {J}}_g)\) to be the closed subspace of \(W^{k,p}(T^*M \otimes TM)\) consisting of those sections \(J \in W^{k,p}(T^*M \otimes TM)\), which satisfy \(J^2=-id,\quad g(J\cdot , J\cdot )=g(\cdot , \cdot )\) almost everywhere.
Now, we state our main results.
-
Theorem 1 There always exists an energy-minimizer of \({\mathcal {E}}_{m}(J)\) in \(W^{m,2}({\mathcal {J}}_g)\).
-
Theorem 2 Suppose \(J\in W^{m,2}({\mathcal {J}}_{g})\) is a weakly m-harmonic almost complex structure on \((M^{2m}, g)\) with \(m\in \{2, 3\}\). Then J is Hölder-continuous.
For semilinear elliptic systems with critical growth nonlinearities, the most essential step towards the smoothness is to prove the Hölder continuity, such as the systems for (poly)harmonic maps, see for example [2, 5, 10, 15, 16] and references therein. It is well-known that a semilinear elliptic system with critical growth nonlinearities and at critical dimension might be singular [4, 9]. For weakly harmonic map, it can be even singular everywhere [13] when the dimension is three and above. The smooth regularity starts with Helein’s seminal result [10] for harmonic maps in dimension two where the special (algebraic) structure of the system plays a substantial role. New proofs and understanding of Helein’s seminal results can be found [1, 14]. The methods can be generalized to fourth order elliptic system in dimension four [2, 11]. General smooth regularity for biharmonic maps and polyharmonic maps have been obtained by [16] and [5], respectively.
We shall briefly compare our results with the results in the theory of (poly)harmonic maps. Theorem 1 is a standard practice in calculus of variations, while the main point is that the absolute energy-minimizer is not trivial due to its tensor-valued nature. The main result is to prove the Hölder regularity in Theorem 2 and our method is motivated by the work in [2] and [5]. In [2], the authors explore a special divergence structure of the biharmonic system into the spheres and our elliptic system shares some similarities. On the other hand, the tensor-valued nature makes our arguments much more complicated, mainly due to the fact that matrix multiplication is not commutative. We certainly believe that this divergence structure should hold for all weakly polyharmonic almost complex structures but we do not find a systematic way to argue that. Instead we only show that the elliptic system for polyharmonic almost complex structures has a desired divergence structure when \(m=2, 3\) by brutal computations. Given this divergence structure, our argument for Hölder regularity is quite different from the method used in [2], but more like a generalization of [5]. We use extension of maps (almost complex structures) instead of solving boundary value problem. Our methods are very general and work for all dimensions. A main difficulty is that the background metric is not necessarily Euclidean, while most results in the setting of polyharmonic maps (see [2, 5, 16] etc) only consider the Euclidean case. Even though the methods for semilinear system are expected to work similarly, the non-Euclidean background metric really leads to complicated computations and presentations. Once the Hölder regularity is assured, the proof of smoothness follows the strategy in [5].
The paper is organized as follows. In Sect. 2, we collect some facts for Lorentz spaces and Green’s functions. In Sect. 3, we establish the existence of the energy-minimizers and derive the Euler-Lagrange equations. Moreover we show that a weak limit of a sequence of weakly m-harmonic almost complex structures in \(W^{m,2}\) is still m-harmonic. In Sect. 4, we prove decay estimates for a class of semilinear elliptic equations in critical dimension and obtain the Hölder regularity of weakly m-harmonic almost complex structures on \((M^{2m},g)\) for \(m=2,3\). In Sect. 5, we generalize the higher regularity results of Gastel and Scheven [5] to prove smoothness of weakly m-harmonic almost complex structures. Appendix derive the divergence structures in detail for m-harmonic almost complex structures when \(m=2, 3\).
2 Preliminaries
In this section, we gather some facts that will be used later. First of all, let us denote by \(G(x)=c_m \ln |x|\) the fundamental solution for \(\varDelta ^m\) on \({\mathbb {R}}^{2m}\), where \(c_m\) is a suitable constant only dependent of m. We have the following lemma,
Lemma 1
Suppose \(k \in [1,2m]\) is a positive integer and \(p,q \in (1,\infty )\) satisfy
If \(f\in L^{q}({\mathbb {R}}^{2m})\), then we have
where C is a positive constant only dependent of m, k, q.
Proof
Since \(\nabla ^{2m}G\) is a Calderón-Zygmund kernel, (2) holds for \(k=2m\) and all \(p=q\in (1,\infty )\). For \(k=1,\cdots ,2m-1\), we have
where \(L^{\frac{2m}{k}, \infty }({\mathbb {R}}^{2m})\) is a Lorentz space. By the convolution inequality for Lorentz spaces (cf. [12] Theorem 2.6), we deduce that, for \(s\le p\)
The fact that \(k \in [1,2m-1]\) implies \(\frac{1}{p} < \frac{1}{q}\). Thus, we can choose \(s=q\). Moreover, there holds that \(L^p({\mathbb {R}}^{2m})=L^{p,p}({\mathbb {R}}^{2m})\) for all \(p\in (1,\infty )\) (cf. [18] Lemma 1.8.10), which implies (2). \(\square \)
For more details about Lorentz spaces, we refer the readers to [12, 18]. We also need the following standard fact about the elliptic operator \(\varDelta ^m\).
Lemma 2
Let \(B_1\) be the unit ball of \({\mathbb {R}}^n\). Suppose \(v(x)\in W^{m,2}(B_1) \cap L^\infty \) and \(f\in L^{\infty }(B_1)\). If v(x) satisfies \(\varDelta ^m v(x) =f(x)\) in distributional sense, then
where C is a positive constant only dependent of n
3 Existence of Energy-Minimizer and the Euler-Lagrange Equation
In this section, we establish the existence of the energy-minimizers of \({\mathcal {E}}_{m}(J)\), derive its Euler-Lagrange equation and define the weak solutions. Moreover we prove that a weak limit of a sequence of weakly m-harmonic almost complex structures with bounded \(W^{m,2}\) norm is still m-harmonic.
Theorem 1
There always exists an energy-minimizer of \({\mathcal {E}}_{m}(J)\) in \(W^{m,2}({\mathcal {J}}_g)\).
Proof
The proof is standard in calculus of variations. We include the details for completeness. Take a minimizing sequence \(J_k \in W^{m,2}({\mathcal {J}}_g)\) such that
Note that by interpolation inequality and integration by parts,
Hence, the sequence \(\{J_k\}\) is bounded in \(W^{m,2}\). This implies that there exists a subsequence, still denoted by \(J_k\), and \(J_0 \in W^{m,2}\), such that \(J_k\) converges weakly to \(J_0\) in \(W^{m,2}\) and \({\mathcal {E}}_{m}(J_0) \le \varliminf _{k \rightarrow \infty } {\mathcal {E}}_{m}(J_k).\) Moreover, \(J_k\) converges strongly to \(J_0\) in \(W^{m-1,2}\) and hence \(J_0 \in {\mathcal {J}}_{g}\). It follows that \(J_0\) is an energy-minimizer of the functional \({\mathcal {E}}_{m}(J)\). \(\square \)
Denote by \(T_{q}^{p}(M)\) the set of all (p, q) tensor fields on (M, g). There is a natural inner product on \(T_{q}^{p}(M)\) induced by g, denoted by \(\left\langle , \right\rangle \). In local coordinate \(\{x^{i}\}_{i=1}^{n}\), \(A\in T_{q}^{p}(M)\) can be expressed by
The inner product of \(A,B \in T^p_q(M)\) is given by
where \(g=g_{ij} dx^i \otimes dx^j\) and \((g^{ij})\) is the inverse of \((g_{ij})\). For \(A\in T_1^1(M)\), define the adjoint operator \(A^*\) of A by
where \({\mathfrak {X}}(M)\) is the set of all smooth vector fields on (M, g). In local coordinates, if \(A=A_{i}^{j}\partial _{x^{j}}\otimes dx^{i}\) we have \((A^*)_i^j=A_k^l g^{kj} g_{li}\).
Proposition 1
We have the following standard facts,
-
1.
For all \(A,B\in T_{q}^{p}(M)\), there holds
$$\begin{aligned} \int _{M}\left\langle \nabla A,\nabla B\right\rangle =-\int _{M}\left\langle A,\varDelta B\right\rangle . \end{aligned}$$ -
2.
For all \(A\in T_{1}^{1}(M)\) and \(X \in {\mathfrak {X}}(M)\), there holds \(\left( \nabla _X A\right) ^{*}=\nabla _X (A^{*}).\)
-
3.
For all \(A,B\in T_{1}^{1}(M)\), there holds \(\left\langle A,B\right\rangle =\left\langle A^{*},B^{*}\right\rangle .\)
-
4.
For all \(A,B,C\in T_{1}^{1}(M)\), there holds \(\left\langle A,BC\right\rangle =\left\langle B^{*}A,C\right\rangle =\left\langle AC^{*},B\right\rangle .\)
For \(A, B \in T^1_1(M)\), AB is regarded as the composition of linear maps, i.e., \(AB\in T^1_1(M)\). In local coordinate we have \((AB)_i^j=A_s^jB_i^s.\) With these notations, we have
Let \(\{ J(t)\}_{t \in (-\delta ,\delta )}\) be a \(C^1\) curve in \({\mathcal {J}}_g\) with \(J(0)=J\). Let \(S=\frac{dJ}{dt} |_{t=0}\). Such S is called an admissible variational direction of J in \({\mathcal {J}}_{g}\). Denote \({\mathcal {S}}_J\) to be the set of all admissible variational directions of J.
Proposition 2
We have
For any \(J\in {\mathcal {J}}_{g}\), define the operator \(\varPhi _J: T^1_1(M) \rightarrow S_J\) by
On each fiber of \(T^1_1 (M)\), \(\varPhi _J\) is precisely the orthogonal projection onto \((S_J)_x\), satisfying that for all \(T\in T^1_1(M)\) and \(S\in {\mathcal {S}}_J\),
Proposition 3
The Euler-Lagrange equation of functional \({\mathcal {E}}_{m}(J)\) is
Proof
Suppose \(J\in {\mathcal {J}}_{g}\) is a critical point of \({\mathcal {E}}_{m}(J)\). For any \(S\in {\mathcal {S}}_J\), we have
which implies \(\varPhi _J(\varDelta ^m J)=0.\) Equivalently we have
\(\square \)
An almost complex structure \(J\in W^{m,2}({\mathcal {J}}_{g})\) satisfying (5) in distributional sense is called weakly m-harmonic.
Proposition 4
A weakly m-harmonic almost complex structure J satisfies the following in distributional sense,
where \(g_s =\sum C_{k_1,k_2,k_3}\, \nabla ^{k_1} J \nabla ^{k_2} J \nabla ^{k_3} J\) for nonnegative integers \(k_1+k_2+k_3=2m-s, \;k_i \in [0,m]\) and \(C_{k_1,k_2,k_3}\in {\mathbb {Z}}.\) That is, for any \(T \in T^1_1(M)\), there holds
-
1.
when \(m=2k\), \(k\in {\mathbb {N}}^+\),
$$\begin{aligned} \int _M \left\langle \varDelta ^k J , \varDelta ^k T \right\rangle +\sum _{s=0}^{m-1} \int _M \left\langle g_s, \nabla ^s T \right\rangle =0 \end{aligned}$$(7) -
2.
when \(m=2k-1\), \(k\in {\mathbb {N}}^+\),
$$\begin{aligned} \int _M \left\langle \nabla \varDelta ^{k-1} J , \nabla \varDelta ^{k-1} T \right\rangle +\sum _{s=0}^{m-1} \int _M \left\langle g_s, \nabla ^s T \right\rangle =0 \end{aligned}$$(8)
For simplicity, we will give the exact meaning of \(\nabla ^s\) in the proof.
Proof
We focus on the case \(m=2k\) since the case \(m=2k-1\) is similar. Suppose J is weakly m-harmonic. Then for any \(S\in {\mathcal {S}}_J\), we have
Taking \(S=\varPhi _J(T)\) for any \(T\in T^1_1(M)\),
where
We describe the terms of \(R_1\) and \(R_2\) by taking a local orthonormal fields \(\{ e_i\}_{i=1}^n\) as follows,
where the symbol \(\alpha<\) means \(1\le \alpha _1\le \cdots \le \alpha _{k_1}\le k\) and we write
Then, we can rewrite \(R_1\) and \(R_2\) in the following,
Substituting the above into (10), we get (7) as follows,
\(\square \)
Proposition 5
A weak limit of a sequence of weakly m-harmonic almost complex structures with uniformly bounded \(W^{m,2}\) norm is still m-harmonic.
Proof
Let J be weakly m-harmonic. Then for any \(T\in T^1_1(M)\), there holds
-
1.
when \(m=2k\), \(k\in {\mathbb {N}}^+\)
$$\begin{aligned} \int _M \left\langle \varDelta ^k J, \left[ J, \varDelta ^k T\right] \right\rangle +\sum _{\begin{array}{c} k_1+k_2=m \\ 1\le k_1, k_2 \le m-1 \end{array}} \left\langle \varDelta ^k J, \left[ \nabla ^{k_1} J, \nabla ^{k_2}T\right] \right\rangle =0, \end{aligned}$$(11) -
2.
when \(m=2k-1\), \(k\in {\mathbb {N}}^+\)
$$\begin{aligned} \int _M \left\langle \nabla \varDelta ^{k-1} J, \left[ J, \nabla \varDelta ^{k-1} T\right] \right\rangle +\sum _{\begin{array}{c} k_1+k_2=m \\ 1\le k_1, k_2 \le m-1 \end{array}} \left\langle \nabla \varDelta ^{k-1} J, \left[ \nabla ^{k_1} J, \nabla ^{k_2}T\right] \right\rangle =0. \end{aligned}$$
We only prove the case \(m=2k\). Recall (9) holds for all \(T\in T^1_1(M)\),
By replacing T by JT, we derive
This implies (11) since
Now, suppose \(\{J_l\}\) is a sequence of weakly m-harmonic almost complex structures in \(W^{m,2}\) such that \(J_l \rightharpoonup J_0 \quad \text{ in }\,\, W^{m,2} \quad \text{ and } \quad \sup _l \Vert J_l\Vert _{W^{m,2}} <\infty .\) By Rellich-Kondrachov theorem, we know that \(J_l\) converges to \(J_0\) in \(W^{m-1,2}\). Hence \(J_0 \in W^{m,2}({\mathcal {J}}_{g})\). Since \(J_l \rightharpoonup J_0\) in \(W^{m,2}\), we have
Since \(J_l\) converges to \(J_0\) in \(W^{m-1,2}\), \(\sup _l \Vert J_l\Vert _{W^{m,2}} <\infty \) and
we have \(\lim _{l\rightarrow \infty } \int _M \left\langle \varDelta ^k J_l, [J_l-J_0, \varDelta ^k T]\right\rangle = 0.\) With (12) this implies
Similarly we conclude that, for all \(k_1+k_2=m\) and \(1\le k_1, k_2 \le m-1\),
Hence \(J_0\) is weakly m-harmonic and this completes the proof. \(\square \)
4 Decay Estimates and Hölder Regularity
In this section, we establish decay estimates for a class of semilinear elliptic equations in critical dimension and deduce the Hölder regularity of \(W^{m,2}\) m-harmonic almost complex structure on \((M^{2m},g)\) for \(m=2,3\). For simplicity, we use C to denote a uniform positive constant.
4.1 Decay Estimates for \(W^{2,2}\) Biharmonic Almost Complex Structure on \({\mathbb {R}}^4\)
First, we consider decay estimates for biharmonic almost complex structure defined on \(B_1\) in \({\mathbb {R}}^4\) as a special case. The presentation is much clearer and more streamlined for this case and the main ideas are essentially the same. Consider the biharmonic almost complex structure equation,
where \(J : B_1 \subset {\mathbb {R}}^4 \rightarrow M_4({\mathbb {R}})\) (the set of all \(4\times 4\) real matrices) satisfies
Proposition 6 asserts that the biharmonic almost complex structure equation admit a good divergence form. That is, for any given constant matrix \(\lambda _0\), biharmonic almost complex structure J satisfies
where \(T_{\lambda _0}\) is a linear combination of the following terms
where \(\alpha ,\beta ,\gamma ,\delta \) are multi-indices such that \(1\le |\alpha | \le 3\), \(0\le |\beta |,|\gamma |,|\delta |\le 2\), \(|\alpha |+|\beta |+|\gamma |=4\) and \(|\alpha |+|\delta |=4\). The notation \(A*B\) means the composition of terms A and B, such as AB and BA. Then we have the following,
Lemma 3
Suppose \(J \in W^{2,2}(B_1, M_4({\mathbb {R}}))\) is a weakly biharmonic almost complex structure on unit ball \(B_1 \subset {\mathbb {R}}^4\). Then, given any \(\tau \in (0,1)\), there exists \(\epsilon _0>0\) and \(\theta _0 \in (0, \frac{1}{2})\) such that if
then we have
where \(p_0=\frac{8}{3}\) and \(D_{p}(J,r):= \bigg (r^{p-4} \int _{B_r} |\nabla u|^p \bigg )^{\frac{1}{p}}.\)
Proof
We extend J to \({\widetilde{J}}\in W^{2,2}({\mathbb {R}}^4, M_4({\mathbb {R}})) \cap L^\infty \) such that
By the standard extension to \(J-\lambda _0\) in \(B_1\), there exists a function \({\widetilde{J}}-\lambda _0\) on \({\mathbb {R}}^{4}\) with compact support contained in \(B_2\) and satisfying
Since \({\widetilde{J}}-\lambda _0\) has a compact support, (17) implies \({\widetilde{J}}-\lambda _0 \in L^q({\mathbb {R}}^{4})\) for all \(q\in [1,\infty ]\). By Poincáre inequality (15) follows from (18). We obtain (16) by Poincáre inequality, Sobolev inequality and (19),
Note that \({\widetilde{J}}\) may not be almost complex structure outside \(B_1\). Now let \(G(x)=c \ln |x|\) be the fundamental solution for \(\varDelta ^2\) on \({\mathbb {R}}^{4}\), where c is a constant. Then \(\nabla ^{4} G\) is a Calderón-Zygmund kernel. Define
where
and \({\widetilde{T}}_{\lambda _0}\) is defined by replacing J by \({\widetilde{J}}\) in \(T_{\lambda _0}\) (see (13)) , and \(\alpha ,\beta ,\gamma ,\delta \) are multi-indices such that \(1\le |\alpha | \le 3\), \(0\le |\beta |,|\gamma |,|\delta |\le 2\), \(|\alpha |+|\beta |+|\gamma |=4\) and \(|\alpha |+|\delta |=4\). We claim that for \(E(J,1)\le 1\), there holds
We will prove the above inequality term by term. By Lemma 1, we have
where we let \(\frac{4}{s}:=\infty \) for \(s=0\), \(N_{\beta ,\gamma }\) stands for the number of non-zero elements in \(\{\beta ,\gamma \}\) and \(q_0, q_1 \in (1,\infty )\) satisfy
Since \(|\alpha |+|\beta |+|\gamma |=4\) and \(1\le |\alpha | \le 3\), we know that \(1\le N_{\beta ,\gamma }\le 2\) and hence such \(q_0\) and \(q_1\) exist. If \(E(J,1)\le 1\), there holds
By a similar argument, we also have
Combining (21) and (22), we deduce (20).
Finally, we turn to proving (14). Let \(v(x):=J(x)-\omega (x)\), then we know v(x) is biharmonic on unit ball \(B_1\), i.e., \(\varDelta ^2 v(x)=0\). Since \(\nabla v\) is also biharmonic, it follows from Lemma 2 (or see Lemma 6.2 in [5]) that there holds
Hence, for any \(\theta \in (0,\frac{1}{2})\) and \(E(u,1)\le 1\), there holds
Thus, for any given \(\tau \in (0,1)\), by choosing \(\theta =\theta _0\) and \(\epsilon _0\) sufficiently small, we obtain (14) for \(E(J,1)\le \epsilon _0\). \(\square \)
4.2 Decay Estimates for a Class of Semilinear Elliptic Equations
Consider the following semilinear elliptic equation for \(u: B_1 \subset {\mathbb {R}}^n\rightarrow {\mathbb {R}}^K\),
where \(\varPsi : {\mathbb {R}}^n \times {\mathbb {R}}^{nK}\times \cdots \times {\mathbb {R}}^{n^{2m-1}K} \rightarrow {\mathbb {R}}^K\) is smooth and \(B_1\) is the unit ball in \({\mathbb {R}}^n\) centered at origin. We can generalize the results in Sect,. 4.1 to (23) which admit a good divergence structure specified in the following,
Definition 2
We say that the equation (23) admits a good divergence form if for any fixed constant vector \(\lambda _0 \in {\mathbb {R}}^K\), \(\varPsi \) can be decomposed into \(\varPsi _H+\varPsi _L\), the highest order term \(\varPsi _H\) and the lower order term \(\varPsi _L\), which satisfy the following properties:
-
1.
\(\varPsi _H \) is a linear combination of the following terms
$$\begin{aligned} \nabla ^\alpha ((u-\lambda _0) * h_{\alpha , \beta }), \quad \text{ with } |h_{\alpha , \beta }| \le C \prod _{i=1}^{s} \big |\nabla ^{\beta _i} u \big |, \end{aligned}$$(24)where \(\alpha , \beta _i\) are multi-indices and \(\beta =(\beta _1, \cdots , \beta _s)\) such that
$$\begin{aligned}&|\alpha |+\sum _{i=1}^{s} |\beta _i|= 2m, \end{aligned}$$(25)$$\begin{aligned}&|\beta _i| \le m, \quad i=1, \cdots , s, \,\, s \in {\mathbb {N}}^+, \end{aligned}$$(26)$$\begin{aligned}&1 \le \sum _{i=1}^s|\beta _i| \le 2m-1, \end{aligned}$$(27) -
2.
\(\varPsi _L\) is a linear combination of the following three types of terms
$$\begin{aligned} \begin{aligned}&\nabla ^{\alpha }(a_{\alpha ,\gamma }(x)*\ell _{\alpha ,\gamma }), \quad \text{ with }\,\,\, |\ell _{\alpha ,\gamma }| \le C \prod _{i=1}^{s} \big |\nabla ^{\gamma _i} u \big |, \\&b_t(x)*\big (u(x)-\lambda _0 \big )*\ell _{0,t}, \quad \text{ with }\,\,\, |\ell _{0,t}| \le C |u|^t, \quad t \in {\mathbb {N}}, \\&c(x), \end{aligned} \end{aligned}$$(28)where \(\gamma =(\gamma _1, \cdots , \gamma _s)\), \(a_{\alpha ,\gamma }(x), b_t(x), c(x) \in C^{2m}(\overline{B_1}, {\mathbb {R}}^K)\) and
$$\begin{aligned}&|\alpha |+\sum _{i=1}^{s} |\gamma _i|\le 2m-1, \end{aligned}$$(29)$$\begin{aligned}&|\gamma _i| \le m, \quad i=1, \cdots , s, \,\, s \in {\mathbb {N}}^+, \end{aligned}$$(30)$$\begin{aligned}&\sum _{i=1}^{s} |\gamma _i| \ge 1. \end{aligned}$$(31)
Remark 1
-
1.
The condition (26) and (30) are natural for us to define the weak solution to (23) for \(u \in W^{m,2}\).
-
2.
The condition (27) plays an important role in proving the Hölder continuity of u in critical dimension \(n=2m\) under the structure (24) of \(\varPsi _H\).
-
3.
A trivial verification shows that the terms in the form
$$\begin{aligned} g(x)*\nabla ^{\alpha _1}u * \cdots * \nabla ^{\alpha _t}u \quad \text{ for } \,\, g(x) \in C^{4m}(\overline{B_1}, {\mathbb {R}}^K), \,\, \sum _i |\alpha _i| \le 2m-1 \end{aligned}$$can always be rewritten as a linear combination of terms (28).
For any ball \(B_r\) of radius r centered at origin in \({\mathbb {R}}^n\), any \(p>1\), and \(q_l\in (1,\infty )\) given by \(\frac{1}{q_l}=\frac{1}{2}-\frac{m-l}{n}\) for \(l=1,\cdots , m\) and \(n\ge 2m\), denote
Lemma 4
Suppose \(n=2m\) and \(u \in W^{m,2}(B_1, {\mathbb {R}}^K) \cap L^\infty \) satisfies (23) in distributional sense. If (23) admits a good divergence form and \(\Vert u\Vert _{L^\infty (B_1)} \le {\mathcal {B}}<\infty \), then, given any \(\tau \in (0,1)\), there exists \(\epsilon _0>0\) and \(\theta _0 \in (0, \frac{1}{2})\), which are only dependent of \(\tau , {\mathcal {B}}, m\), such that if \(E(u,1)\le \epsilon _0,\) then we have
where \(p_0=\frac{4m}{3} \in (1,2m)\) and
where \(a_{\alpha , \gamma }(x), b_t(x), c(x)\) are from (28) in lower order terms \(\varPsi _L\) of (23).
Proof
For simplicity, we denote by C a positive constant only dependent of \(\tau ,{\mathcal {B}},m\). Following the similar argument in the proof of Lemma 3, we can extend u to \({\widetilde{u}} \in W^{m,2}({\mathbb {R}}^{2m}, {\mathbb {R}}^K) \cap L^\infty \) such that
Of course, by a standard extension theorem to the functions \(a_{\alpha , \gamma }(x)\), \(b_t(x)\) \(\in C^{2m}(\overline{B_1}, {\mathbb {R}}^K)\) from the lower order term \(\varPsi _L\), there exist the corresponding functions \({\widetilde{a}}_{\alpha , \gamma }(x)\), \({\widetilde{b}}_t(x)\in C_0^{2m}({\mathbb {R}}^{2m}, {\mathbb {R}}^K)\) such that
Let \(G(x)=c_m \ln |x|\) be the fundamental solution for \(\varDelta ^m\) on \({\mathbb {R}}^{2m}\). Then \(\nabla ^{2m} G\) is a Calderón-Zygmund kernel. Let us define
where
We claim that, for \(p_0=\frac{4m}{3}\) and \(E(u,1)\le 1\), there holds
We will prove above inequality term by term. By Lemma 1, we have
where \(\frac{2m}{|\beta _i|}:=\infty \) for \(|\beta _i|=0\), \(n_\beta =\big |\{\beta _i: \beta _i \ne 0 \}\big |\ge 1\), and
Note that (27) implies \(q_{\alpha , \beta }\in (1,\infty )\). Hence, if \(E(u,1)\le 1\), there holds
Similarly, by Lemma 1, we obtain that
where \(\frac{2m}{|\gamma _i|}:=\infty \) for \(|\gamma _i|=0\), \(n_\gamma =\big |\{\gamma _i: \gamma _i \ne 0 \}\big |\ge 1\) due to (31), and
Note that (29) implies \(q_{\alpha ,\gamma }, q_1 \in (1,\infty )\). Hence, if \(E(u,1)\le 1\), there holds
Similar argument applies to terms \(\omega _{0,t}\) and yields
Combining (39), (40) and (41) gives (38). In the last we prove (34). Denote \(v(x):=u(x)-\omega (x)\), then v(x) satisfies \(\varDelta ^m v(x)=c(x)\) on \(B_1\) in distributional sense. By Lemma 2, we have
Hence, for any \(\theta \in (0,\frac{1}{2})\) and \(E(u,1) \le 1\), there holds
Thus, for any given \(\tau \in (0,1)\), by choosing \(\theta =\theta _0\) and \(\epsilon _0\) sufficiently small, we obtain (34) for \(E(u,1)\le \epsilon _0\). \(\square \)
4.3 Hölder Regularity for m-Harmonic Almost Complex Structure
In this subsection, we prove the Hölder regularity using the decay estimates above,
Theorem 2
Suppose \(J\in W^{m,2}({\mathcal {J}}_{g})\) is a weakly m-harmonic almost complex structure on \((M^{2m}, g)\) with \(m\in \{2, 3\}\). Then J is Hölder-continuous.
Since the Hölder regularity is a local property, we work on local coordinates on \((M^n, g)\). Let \(B_1\) be the unit ball of \({\mathbb {R}}^n\) and write g as a smooth metric on \(B_1\). First we consider the Euclidean case with \(g=g_0=\sum _i dx^i \otimes dx^i\) on \(B_1\). The general case is a small perturbation of the Euclidean case.
4.3.1 The Euclidean Case \((B_1, g_0)\)
In this case, an almost complex structure J on \(B_1\) can be regarded as a function in \(W^{m,2}(B_1, M_n({\mathbb {R}}))\) such that \(J^2=-id\) and \(J^T+J=0\), where \(M_n({\mathbb {R}})\) is the set of all real \(n\times n\) matrices and \(J^T\) is the transpose of matrix J. The inner product of \(A, B \in T^1_1(B_1)\) reads \(\langle A, B\rangle =\sum _{i,j=1}^n A^j_i B^j_i\) for \(A=A_i^j dx^i \otimes \frac{\partial }{\partial x^j}\) and \(B=B_i^j dx^i \otimes \frac{\partial }{\partial x^j}\). Thus, the inner product of (1, 1) tensor fields on \(B_1\) can be viewed as the inner product of two vectors in Euclidean space \({\mathbb {R}}^{n^2}\).
First we need to write the Euler-Lagrange equation in a good divergence form in the sense of Definition 2.
Lemma 5
Suppose J is a \(W^{m,2}\) weakly m-harmonic almost structure on \((B_1,g_0)\), \(m=2, 3\). Then J satisfies the following in distributional sense,
where \(\varPsi \) can be rewritten as a linear combination of the following terms, for any fixed constant matrix \(\lambda _0\in M_n({\mathbb {R}})\),
where \(\alpha ,\beta ,\gamma ,\delta \) are multi-indices such that \(1\le |\alpha | \le 2m-1\), \(0\le |\beta |,|\gamma |,|\delta |\le m\), \(|\alpha |+|\beta |+|\gamma |=2m\) and \(|\alpha |+|\delta |=2m\).
Lemma 5 will be proved in Appendix. Now we prove Theorem 2 for the Euclidean case \((B_1,g_0)\). First, we use the normalized energy E(J; x, r) defined by replacing u,\(B_r\) by J, \(B_r(x)\) respectively in (32). That is, due to \(n=2m\),
For any fixed \(R_0 \in (0,1)\), we have that for every \(\epsilon _0>0\), there exists \(r_0 \in (0, 1-R_0)\) such that
For \(x_0 \in {\overline{B}}_{R_0}\), \(J_{x_0, r_0}(x):=J(x_0+r_0x)\) is also a \(W^{m,2}\) m-harmonic almost structure on \((B_1,g_0)\) with
By Lemma 5, \(J_{x_0,r_0}\) admits a good divergence form (see Definition 2) with \(\varPsi _L=0\). Then it follows from Lemma 4 that by choosing suitable \(\epsilon _0>0\) in (43), there exists \(\theta _0 \in (0,\frac{1}{2})\) and \(p_0=\frac{4m}{3}\) such that
A standard iteration argument shows that there exists \(\alpha \in (0,1)\) such that
This, combined with the Morrey’s lemma, yields that \(J \in C^{0,\alpha }({\overline{B}}_{R_0})\), hence that \(J \in C^{0,\alpha }(B_1)\).
4.3.2 The General Case \((B_1, g)\)
In this subsection, we prove the Hölder regularity of the general case on \((B_1,g)\) by a perturbation method. We start by recalling the scaling invariance of the functional \({\mathcal {E}}_m(J)\) in critical dimension \(n=2m\). If \(g_{\lambda }:=\lambda ^2 g\) for some positive real number \(\lambda \), then \({\mathcal {E}}_m(J,g)= {\mathcal {E}}_{m}(J, g_{\lambda }),\) where \({\mathcal {E}}_{m}(J, g)=\int _M \big | \varDelta ^{\frac{m}{2}}_{g} J\big |^2 dV_{g}.\) It follows that if J is a weakly m-harmonic on (M, g), then J is also m-harmonic on \((M,g_{\lambda })\). If we take the geodesic normal coordinates on the unit geodesic ball centered at fixed point in \((M,g_{\lambda })\), then the metric \(g_{\lambda }\) in such local coordinates converges to the Euclidean metric in \(C^{\infty }(B_1)\) as \(\lambda \) goes to infinity. Hence, we can assume that, by a scaling if necessary, the metric g on \(B_1\) is sufficiently close to the Euclidean metric in the sense
where \(\delta _0\) is sufficiently small and will be determined later. Now we prove Theorem 2 in the general case \((B_1,g)\).
Firstly, we introduce an operator \({\mathfrak {m}}\) which maps a (1, 1) tensor field A on \((B_1,g)\) to a \(n\times n\) real matrix valued function,
where \(A=A_i^j dx^i \otimes \frac{\partial }{\partial x^j}\). In other words, A denotes tensor field and \(A_{{\mathfrak {m}}}\) denotes its coefficient matrix. Let us denote by \(\nabla \) the covariant derivative on \((B_1,g)\) and D the ordinary derivatives (i.e., \(D_k=\partial _k\)). Here it is necessary to emphasize the difference between the derivatives on tensor fields and matrix valued functions. For example, for \(A=A_i^j dx^i \otimes \partial _j\), we have
where \(\varGamma _{ij}^k\) denote the Christoffel symbols with respect to metric g. To simplify notation, we rewrite above equation as
where \(D_k A_{{\mathfrak {m}}}=D_k (A_i^j)=(D_k A_i^j)\). Similarly, there holds
Recall the m-harmonic almost complex structure equation (61), i.e.,
We will reduce above equation to a perturbation form of the Euclidean case step by step. As a example, we show how to handle the term \(\varDelta ^m J\). Repeated application of (45) yields
where \(L_1\) stands for the lower order terms in the following form
with \(i_\mu \ge 1, \mu =1, \cdots , s\), \(j \ge 0\) and \(j+\sum _{\mu =1}^s i_{\mu }=2m\). Let \(\varDelta _0= \sum _{i=1}^{2m} \partial ^2_i\) be the standard Laplace operator on Euclidean space \({\mathbb {R}}^{2m}\). Recall that for any smooth function f,
where we omit the terms \(g^{ij}\) in the expression \(Dg*Df\) due to boundedness of g. Then we have
where \(P_1\) stands for the perturbation term in the following form
and \(L_2\) also stands for the lower order terms and has the similar expression as \(L_1\). Hence, we obtain
where \({\widetilde{L}}_1=L_1+L_2\) has the following form
with \(i_\mu \ge 1, \mu =1, \cdots , s\), \(j \ge 0\) and \((\sum _{\mu =1}^s i_{\mu }) +j=2m\). Similar arguments apply to the nonlinear terms and yield
where \(T_s\) admits a good divergence form as \(\varPsi \) in Lemma 5, \(P_2\) stands for the perturbation terms
where \(b_{ijk}\) consists of \(|g^{st}-\delta ^{st}|\), \(0\le j,k \le m\) and \(i+j+k=2m\), and \({\widetilde{L}}_2\) stands for the lower order terms in the following form
with \(i_{\mu } \ge 1\) \(\mu =1, \cdots , s\), \(0\le j+k+l\le 2m-1\) and \(\big (\sum _{\mu =1}^s i_{\mu } \big )+ j+k+l=2m\). By the arguments above, we get the final reduced equation about \(J_{{\mathfrak {m}}}\)
where \({\mathcal {P}}=P_2 - P_1\) and \({\mathcal {L}}={\widetilde{L}}_2 - {\widetilde{L}}_1\). In other words, the nonlinear part of (48) consists of three types of terms: terms that admit a good divergence form, the perturbation terms and the lower order terms. Now recall the definition of E(u, r) and \(D_p(u,r)\) in (32) and (33) respectively. Then, we claim that for any given \(\tau \in (0,1)\), there exists \(\delta _0>0\), \(\epsilon _0>0\) and \(\theta _0 \in (0,\frac{1}{2})\) such that if the metric g satisfies (44) and \(E(J_{{\mathfrak {m}}},1)<\epsilon _0\), then we have
where \(p_0=\frac{4m}{3}\). The above claim is a direct consequence of Lemma 4 provided \({\mathcal {P}} \equiv 0\). Hence the key point is to prove that the inequality (38) in Lemma 4 still holds with additional nonlinear terms \({\mathcal {P}}\). We claim that, there holds
where \(\omega _{{\mathcal {P}}}(x):= \int _{{\mathbb {R}}^{2m}} G(x-y) {\mathcal {P}}({\widetilde{J}}_{{\mathfrak {m}}})(y) dy.\) We now turn to proving (50). Since
and
it follows from estimates for lower order terms in Lemma 4 that (50) holds for the terms \(a(x) D^{2m} J_{{\mathfrak {m}}}\) in (46). In the same manner, (50) also holds for the terms in (47). Hence the decay estimate (49) holds and it implies \(J_{{\mathfrak {m}}} \in C^{0,\alpha }(B_1)\) for some \(\alpha \in (0,1)\).
5 Higher Regularity for m-Harmonic Almost Complex Structures
We state the higher regularity results for a class of semilinear elliptic equations as a generalization in [5]. The proof follows essentially Proposition 7.1 in [5].
Theorem 3
Suppose \(n\ge 2m\) and \(u\in C^{0,\mu } \cap W^{m,2}(B_1,{\mathbb {R}}^K)\) satisfies
in distributional sense, where \(\varPsi \) can be divided into two parts: the highest order terms H and lower order terms L, i.e., \(\varPsi =H+L\), which admit the following structures:
Then, \(u\in C^\infty (B_1, {\mathbb {R}}^K)\).
Proof
Gastel and Scheven in [5] proved the theorem in the case \(\varPsi =H\). According to the proof of Proposition 7.1 in [5], it suffices to prove the following two claims in the case \(\varPsi =L\):
-
(1)
$$\begin{aligned} \sup _{B_\rho (x)\subset B_R} \rho ^{2m-n-2\mu } \int _{B_\rho (x)}|\nabla ^m u|^2<\infty , \quad \forall \, \,0<R<1, \end{aligned}$$(52)
-
(2)
For every non-integer \(\nu :=[\nu ]+\sigma \in (0,m)\), if \(u\in C^{[\nu ],\sigma }(B_1, {\mathbb {R}}^K)\) and
$$\begin{aligned} \sup _{B_\rho (x)\subset B_R} \rho ^{2m-n-2\nu } \int _{B_\rho (x)}|\nabla ^m u|^2<\infty , \quad \forall \,0<R<1, \end{aligned}$$(53)then we have that, for \(0\le k\le m-1\) and \(B_\rho (x) \subset B_R\), there holds
$$\begin{aligned} \bigg ( \rho ^{2m-n}\int _{B_\rho (x)} |{\widetilde{g}}_k|^{\frac{2m}{2m-k}} \bigg )^{\frac{2m-k}{2m}} \le C \rho ^{\frac{m+1}{m}\nu } \end{aligned}$$(54)
Before proceeding to prove claims, we make some conventions: fix \(R\in (0,1)\), always assume \(B_\rho (x) \subset B_R\), and C stand for the positive constants only dependent of \(m,n,\Vert u\Vert _{C^{0,\mu }(B_R)}\).
We first prove the Claim (1) by standard integral estimates. Since \(u\in C^{0,\mu }(B_1)\), we have
where \({\overline{u}}=\frac{1}{|B_\rho (x)|}\int _{B_\rho (x)} u(y)\). To simplify the proof in the following estimate, we assume \(\Vert u-{\overline{u}}\Vert _{L^\infty (B_\rho (x))} \le 1\).
By Gagliardo-Nirenberg interpolation inequality, we have that, for \(1\le l\le m-1\), there holds
It follows that
On the other hand, by Hölder’s inequality, it follows from (55) that, for \(1\le l \le m-1\) and \(q \in [1,\frac{2m}{l}]\)
We choose a cut-off function \(\eta \in C^\infty _0(B_\rho (x),[0,1])\) such that
Testing (51) with \(\eta ^{2m}(u-{\overline{u}})\), we compute
Let \(\epsilon _1, \epsilon _2\in (0,1)\) be constants to be chosen later. For \(I_0\), we obtain
For \(1\le k \le m-1\), we obtain
where we use (57) with \(q=2\) and Young’s inequality in the fourth inequality. Next, we estimate \(II_{00}\) as follows
where we use (56) in the fifth inequality, and \(1=\frac{1}{p_0}+\frac{1}{2m}\sum _i |\gamma _i|.\) Note that, due to \(\sum _i |\gamma _i| \le 2m-1\), it follows that \(p_0 \in [1, 2m]\). Similar arguments apply to \(II_{k,0}\) and we obtain, for \(1\le k \le m-1\),
By Hölder’s inequality and Young’s inequality, we have that, for \(1\le k \le m-1\),
where we apply (56) in second inequality. Now for \(1\le j <k\le m-1\), we have
where we use (57) with \(q=\frac{2m}{k}\) and (58) in the last inequality. Similarly, we obtain, for \(1\le k \le m-1\)
Combining above all estimates, we deduce that
Thus, by choosing \(\epsilon _1, \epsilon _2, \rho _0\) small enough, we have that, for all \(\rho \le \rho _0\), there holds
where \(\varepsilon < 2^{2m-n-2\mu }\) is a fixed positive number. A standard iteration argument implies (52).
The task is now to prove Claim (2). Since \(u \in C^{[\nu ],\sigma }(B_1)\) with \(\nu =[\nu ]+\sigma \), we know that, there exists a Taylor polynomials \(P_x\) at the points x such that
By Gagliardo-Nirenberg interpolation inequality and (53), we have that, for \(\nu < l \le m\), there holds
Let us compute
where
Combining above two identities yields
where \(n_{\nu }=\big |\{\gamma _i: |\gamma _i|>\nu \}\big |\).
We claim that
which implies (54). Obviously, (60) holds for \(n_\nu \ge 2\). For \(n_\nu =0\), (59) implies \(\frac{1}{q_0}=\frac{2m-k}{2m}\). Hence, for \(\nu \in (0,m)\)
For \(n_\nu =1\), (59) and the fact \(k+\sum _i |\gamma _i| \le 2m-1\) imply \(\frac{1}{q_0} \ge \frac{1}{2m}\). Hence, for \(\nu \in (0,m)\),
Thus, the claim (60) is proved. \(\square \)
As a direct consequence, Theorem 3 implies the smoothness of weakly m-harmonic almost complex structures.
Corollary 1
Suppose \(n\ge 2m\) and \(J \in C^{0,\alpha } \cap W^{m,2}\) is a weakly m-harmonic almost complex structure on \((M^n,g)\). Then J is smooth.
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Weiyong He is partially supported by National Science Foundation (No. 1611797). Ruiqi Jiang is partially supported by National Natural Science Foundation of China (No. 11901181).
Appendix
Appendix
In this section, we will rewrite m-harmonic almost complex structure equation in a good divergence form in the spirit of [2] to prove Lemma 5. First we have the following,
Lemma 6
The Euler-Lagrange equation \([\varDelta ^{m}J,J]=0\) is equivalent to
where \(T_m=JQ_m+Q_mJ\) and
Proof
This is a direct computation using the fact \(\varDelta (J^2)=0\). \(\square \)
Lemma 5 can be stated as follows,
Proposition 6
For \(m=2, 3\), \(T_m\) in Lemma 6 can be rewritten as
where \(T_{\lambda _0}\) is a linear combination of the following terms
where \(\alpha ,\beta ,\gamma ,\delta \) are multi-indices such that \(1\le |\alpha | \le 2m-1\), \(0\le |\beta |,|\gamma |,|\delta |\le m\), \(|\alpha |+|\beta |+|\gamma |=2m\) and \(|\alpha |+|\delta |=2m\).
In what follows, we always assume J is a square matrix valued function and satisfies \(J^2=-id\). In this situation, we know \(\nabla \lambda _0=0\) for every constant matrix \(\lambda _0\). The reason for emphasizing this point is that if we consider the constant matrix \(\lambda _0 \) as a (1,1) tensor field on \((B_1,g)\), then (1, 2) tensor field \(\nabla \lambda _0\) might not be zero.
1.1 The case m=2: biharmonic almost complex structure
By the definition of \(T_m\) in Theorem 6, we have
where \(Q_2=2 \nabla \varDelta J \nabla J + 2\nabla J \nabla \varDelta J+ 2\varDelta J \varDelta J + 2 \varDelta ( \nabla J )^2.\) Set
Thus, we obtain \(T_2=2{\mathbf {I}}+2\mathbf {II}+2\mathbf {III}.\) Firstly, we compute the term \({\mathbf {I}}\):
Since \(\nabla \bigg ( [\nabla \varDelta J, J] -[\varDelta J, \nabla J] \bigg )= [\varDelta ^2 J, J],\) we have
Now we compute the left-hand side of above equality:
Substituting above equality into (63) yields
We now turn to compute the term \(\mathbf {III}\). Since
and similarly \(\varDelta (\nabla J)^2 J = (\nabla J)^2 \varDelta J + T_{\lambda _0},\) we have
Now let us proceed to compute \(\mathbf {II}\):
where we used the fact \(\varDelta (J^2)=0\) which implies
On the other hand, we also have
Hence, we obtain
where in the last equality we used (65). Substituting (67) into (64), we get
which is the desired conclusion.
1.2 The Case m=3: 3-Harmonic Almost Complex Structure
By the definition of \(T_m\) in Theorem 6, we have
where
For simplicity, we collect some terms which are \(T_{\lambda _0}\) type and appear frequently in the following proof.
Lemma 7
The following terms are \(T_{\lambda _0}\) type terms for any given constant matrix \(\lambda _0\):
Proof
For simplicity, we only show how to rewrite the first term and the third term. Other terms can be handled in much the same way. The first term:
The third term:
\(\square \)
Note that we will emphasize the terms of \(T_{\lambda _0}\) type by underlining it in the following proof. Set
Then, we obtain \(T_3= 2{\mathbf {I}}+ \mathbf {II}+ 2\mathbf {III}+ 2\mathbf {IV}+ 2{\mathbf {V}}.\)
Step One: dealing with I. Now Let us compute the first term I:
Since \(\nabla \bigg ([\nabla \varDelta ^{2}J,J]-[\varDelta ^{2}J,\nabla J] +[\nabla \varDelta J,\varDelta J] \bigg )=[\varDelta ^3 J, J],\) we have
Now we compute the left-hand side of above equality.
where in the second equality from bottom we employ lemma 7. By substituting above equality into (68), we obtain
Since \(\nabla [\nabla J,[\varDelta ^{2}J,J]]=[\varDelta J,[\varDelta ^{2}J,J]]+[\nabla J,[\nabla \varDelta ^{2}J,J]]+[\nabla J,[\varDelta ^{2}J,\nabla J]],\) we deduce
By lemma 7, we can derive
and
where in the second equality from bottom we used (66). Substituting equalities (70) and (71) into equality (69) yields
Step Two: dealing with V and II. Firstly, we deal with fifth term V. It follows from Lemma 7 that
Since
and
we have
Next, we deal with the second term
where we have used (66) and (73).
Step Three: dealing with III Here we begin to deal with the third term:
Since
we have
Step Four: dealing with IV Since
we have
Step Five: divergence forms of nonlinearity Combining the equalities (72), (74), (75) and (76), we derive that
which completes the proof.
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He, W., Jiang, R. Polyharmonic Almost Complex Structures. J Geom Anal 31, 11648–11684 (2021). https://doi.org/10.1007/s12220-021-00695-0
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DOI: https://doi.org/10.1007/s12220-021-00695-0
Keywords
- Polyharmonic almost complex structures
- Regularity of Semilinear systems
- Critical growth of nonlinearities