Liouville-type results in two dimensions for stationary points of functionals with linear growth

In this note we mainly present results of Liouville-type for entire solutions $u$: $\mathbb{R}^{2}\to\mathbb{R}^{N}$ of the system

$$\operatorname{div}\big{[}\nabla F(\nabla u)\big{]}=0\quad\mbox{on $\mathbb{R}^{2}$}\,,$$

(1.1)

concentrating on the case of energy densities $F$: $\mathbb{R}^{2N}\to\mathbb{R}$ with linear growth.

To be precise we assume that $F$ is of class $C^{2}\big{(}\mathbb{R}^{2N}\big{)}$ satisfying with constants $M$, $\lambda$, $\Lambda>0$ and for some exponent $\mu>1$

	$\displaystyle|\nabla F(Z)|$	$\displaystyle\leq$	$\displaystyle M\,,$		(1.2)
	$\displaystyle\lambda(1+|Z|)^{-\mu}|Y|^{2}$	$\displaystyle\leq$	$\displaystyle D^{2}F(Z)(Y,Y)\leq\Lambda(1+|Z|)^{-1}|Y|^{2}$		(1.3)

for all $Y$, $Z\in\mathbb{R}^{2N}$, where the first inequality in (1.3) expresses the fact that $F$ is a $\mu$-elliptic integrand. Note that (1.2) and (1.3) exactly correspond to the requirements of Assumption 4.1 in [2] and as outlined in Remark 4.2 of this reference, conditions (1.2) and (1.3) imply that $F$ is of linear growth in the sense that

$$a|Z|-b\leq F(Z)\leq A|Z|+B\,,\quad Z\in\mathbb{R}^{2N}\,,$$

holds with constants $a$, $A>0$, $B$, $b$, $\geq 0$.

Note also that the “minimal surface case” is included by letting $F(Z):=(1+|Z|^{2})^{1/2}$. In this case we have the validity of (1.3) with the choice $\mu=3$, and two families of densities satisfying (1.2) and (1.3) with prescribed exponent $\mu>1$ are given by

$$F(Z):=\left\{\begin{array}[]{l}\displaystyle\int_{0}^{|Z|}\int_{0}^{s}(1+r)^{-\mu}\,{\rm d}r\,{\rm d}s\\[17.22217pt] \displaystyle\int_{0}^{|Z|}\int_{0}^{s}(1+r^{2})^{-\mu/2}\,{\rm d}r\,{\rm d}s\end{array}\right\}\,,\quad Z\in\mathbb{R}^{2N}\,.$$

Our results on the behaviour of global solutions of the Euler equations (1.1) with $\mu$-elliptic densities $F$ are as follows.

Before presenting the proof of Theorem 1.1 we wish to mention that there exists a variety of Liouville-type theorems for entire solutions $u$: $\mathbb{R}^{n}\to\mathbb{R}^{N}$, $n\geq 2$, $N\geq 1$, of systems of the form (1.1) (and even for nonhomogeneous systems not generated by a density $F$) assuming that $F$ is of superlinear growth. The interested reader should consult the references on this topic quoted for example in the textbooks [12], [13], [14], [16], [22] and [25]. A nice survey is also presented in [6].

Besides this more general discussion the validity of Liouville theorems for harmonic maps between Riemannian manifolds turned out to be a useful tool for the analysis of the geometric properties of the underlying manifolds. Without being complete we refer to [4], [5], [17], [18], [19], [20], [27] and [28].

Liouville theorems are also of interest in the setting of fluid mechanics, where in the stationary case (1.1) is replaced by a nonlinear variant of the Navier-Stokes equation with dissipative potential $F$ of superlinear growth and the incompressibility condition $\operatorname{div}u=0$ for the velocity field $u$: $\mathbb{R}^{n}\to\mathbb{R}^{n}$ has to be added. The validity of Liouville theorems has been established in the $2$-D-case, i.e. for $n=2$, for instance in the papers [3], [8], [9], [10], [11], [15], [21], [23], [29] and [30]. We like to mention that the case of potentials $F$ satisfying (1.2) and (1.3) is treated in [10] assuming $\mu<2$.

As it stands, the conclusions of Theorem 1.1 b) and c) are in the spirit of Bernstein’s theorem (see [1]) for nonparametric minimal surfaces, where in this particular setting conditions like (1.5) or (1.6) are seen to be superfluous. For completeness we specialize the Bernstein result obtained by Farina, Sciunzi and Valdinoci in Theorem 1.4 of their paper [7] to the case of linear growth integrands.

In the weak formulation of (1.1), i.e. in the equation

$$\int_{\mathbb{R}^{2}}\nabla F(\nabla u):\nabla\varphi\,{\rm d}x=0\,,\quad\varphi\in C^{1}_{0}\big{(}\mathbb{R}^{2},\mathbb{R}^{N}\big{)}\,,$$

(2.1)

the function $\varphi$ is replaced by $\partial_{\alpha}\varphi$ ($\alpha\in\{1,2\}$ fixed), where now $\varphi\in C^{2}_{0}\big{(}\mathbb{R}^{2},\mathbb{R}^{N})$ is assumed. With an integration by parts we obtain from (2.1)

$$\int_{rz^{2}}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla u,\nabla\varphi\big{)}\,{\rm d}x=0\,.$$

(2.2)

Now we choose $\varphi=\eta^{2}\partial_{\alpha}u\in C^{1}_{0}\big{(}\mathbb{R}^{2},\mathbb{R}^{N}\big{)}$ in (2.2), where $\eta\in C^{1}_{0}\big{(}\mathbb{R}^{2}\big{)}$, $\operatorname{spt}\eta\subset B_{2R}(0)$, $\eta\equiv 1$ on $B_{R}(0)$, $0\leq\eta\leq 1$, $|\nabla\eta|\leq cR^{-1}$. Then by Cauchy-Schwarz’s and Young’s inequality we have (summation w.r.t. $\alpha=1$, $2$)

	$\displaystyle\int_{B_{2R}(0)}\eta^{2}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x$				(2.3)
		$\displaystyle\leq$	$\displaystyle c\int_{B_{2R}(0)}D^{2}F(\nabla u)\big{(}\nabla\eta\otimes\partial_{\alpha}u,\nabla\eta\otimes\partial_{\alpha}u\big{)}\,{\rm d}x\,.$

The hypotheses (1.2) and (1.3) yield

	$\displaystyle\int_{B_{R}(0)}\big{(}1+|\nabla u|\big{)}^{-\mu}|\nabla^{2}u|^{2}\,{\rm d}x$	$\displaystyle\leq$	$\displaystyle cR^{-2}\int_{B_{2R}(0)-B_{R}(0)}\frac{|\nabla u|^{2}}{\sqrt{1+|\nabla u|^{2}}}\,{\rm d}x$		(2.4)
		$\displaystyle\leq$	$\displaystyle cR^{-2}\int_{B_{2R}(0)-B_{R}(0)}|\nabla u|\,{\rm d}x$

and using the auxiliary inequality (2.9) of Lemma 2.1 given below we obtain for any $\varepsilon>0$

	$\displaystyle\int_{B_{R}(0)}\big{(}1+|\nabla u|\big{)}^{-\mu}|\nabla^{2}u|^{2}\,{\rm d}x$				(2.5)
		$\displaystyle\leq$	$\displaystyle\frac{c}{R^{2}}\int_{B_{2R}(0)-B_{R}(0)}\Big{[}\varepsilon+c(\varepsilon)\big{(}\nabla F(\nabla u)-\nabla F(0)\big{)}:\nabla u\Big{]}\,{\rm d}x\,.$

With (2.1) we also have

$$\int_{\mathbb{R}^{2}}\big{(}\nabla F(\nabla u)-\nabla F(0)\big{)}:\nabla\varphi\,{\rm d}x=0\,,\quad\varphi\in C^{1}_{0}\big{(}\mathbb{R}^{2},\mathbb{R}^{N}\big{)}\,,$$

(2.6)

where we now choose $\varphi=\tilde{\eta}^{2}u$, $\tilde{\eta}\in C^{1}_{0}\big{(}\mathbb{R}^{2}\big{)}$, $\tilde{\eta}\equiv 1$ on $B_{2R}(0)-B_{R}(0)$, $\operatorname{spt}\eta\subset B_{5R/2}(0)-\overline{B}_{R/2}(0)$, $0\leq\tilde{\eta}\leq 1$, $|\nabla\tilde{\eta}|\leq c/R$.

With this choice (2.6) gives

	$\displaystyle\int_{\mathbb{R}^{2}}\big{(}\nabla F(\nabla u)-\nabla F(0)\big{)}:\nabla u\tilde{\eta}^{2}\,{\rm d}x$				(2.7)
		$\displaystyle=$	$\displaystyle-2\int_{\mathbb{R}^{2}}\tilde{\eta}\big{(}\nabla F(\nabla u)-\nabla F(0)\big{)}:(\nabla\tilde{\eta}\otimes u)\,{\rm d}x$
		$\displaystyle\leq$	$\displaystyle cR^{-1}\int_{B_{5R/2}(0)-B_{R/2}(0)}|u|\,{\rm d}x$
		$\displaystyle\leq$	$\displaystyle cR\sup_{B_{5R/2}(0)-\overline{B}_{R/2}(0)}|u|\,,$

where our assumption (1.2) is used.

By the definition of $\tilde{\eta}$ we obtain using (2.7)

	$\displaystyle\int_{B_{2R}(0)-B_{R}(0)}\big{(}\nabla F(\nabla u)-\nabla F(0)\big{)}:\nabla u\,{\rm d}x$				(2.8)
		$\displaystyle\leq$	$\displaystyle\int_{\mathbb{R}^{2}}\big{(}\nabla F(\nabla u)-\nabla F(0)\big{)}:\nabla u\tilde{\eta}^{2}\,{\rm d}x$
		$\displaystyle\leq$	$\displaystyle cR\sup_{B_{5R/2}(0)-\overline{B}_{R/2}(0)}|u|\,.$

If we insert (2.8) into inequality (2.5) and pass to the limit $R\to\infty$ recalling (1.4), we obtain for any $\varepsilon>0$

$$\int_{\mathbb{R}^{2}}\big{(}1+|\nabla u|\big{)}^{-\mu}|\nabla^{2}u|^{2}\,{\rm d}x\leq c\varepsilon\,,$$

hence $\nabla^{2}u\equiv 0$ and therefore we find $A\in\mathbb{R}^{2N}$, $a\in\mathbb{R}^{N}$ such that

$$u(x)=Ax+a\,.$$

Again we apply of the growth condition (1.4) and obtain $A=0$, hence the first part of Theorem 1.1 is established.

During the proof we made use of the elementary lemma

Proof of Lemma 2.1. We fix $\varepsilon>0$. If $|Z|\geq\varepsilon$ then

$$|Z|\theta\big{(}|Z|\big{)}\geq|Z|\theta(\varepsilon)\,,$$

which implies

$$|Z|\leq\theta^{-1}(\varepsilon)\,|Z|\,\theta\big{(}|Z|\big{)}\,,$$

and if $Z\in\mathbb{R}^{2N}$ is arbitrarily given, we have

$$|Z|\leq\varepsilon+\theta^{-1}(\varepsilon)\,|Z|\,\theta\big{(}|Z|\big{)}\,.$$

Moreover,

$$\theta\big{(}|Z|\big{)}\,|Z|\leq\big{[}\nabla F(Z)-\nabla F(0)\big{]}:Z\$$

(2.10)

easily follows from the first inequality in (1.3) as outlined in [2], formula (1), p. 98., and (2.10) gives (2.9). $\Box$

For Part b) we remark, that the idea of applying a Liouville argument to the derivatives of solutions, which are seen to solve an appropriate elliptic equation, has been successfully used by Moser [24], Theorem 6, with the result that entire solutions of the minimal surface equation with bounded gradients in fact must be affine functions in any dimension $n\geq 2$.

In our setting, i.e. for $n=2$ together with $N\geq 1$, one may just follow the arguments presented in [12], Chapter III, p. 82, for an elementary proof essentially based on the “hole-filling” technique.

In Theorem 1.1, Part b) turns out to be a corollary of Part c), which we now prove following some ideas given in [11].

As in the proof of the first part of Theorem 1.1 we obtain from (2.3) the following variant of inequality (2.4)

$$\int_{B_{R}(0)}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x\leq cR^{-2}\int_{B_{2R}(0)-B_{R}(0)}|\nabla u|\,{\rm d}x$$

(3.1)

and, as outlined after (2.4), (3.1) gives for all $R>0$ and with the choice $\varepsilon=1$

$$\int_{B_{R}(0)}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x\leq c\Big{[}1+R^{-1}\sup_{B_{5R/2}(0)-B_{R/2}(0)}|u|\Big{]}\,.$$

(3.2)

Inequality (3.2) shows, using (1.6),

$$\int_{\mathbb{R}^{2}}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x<\infty\,.$$

(3.3)

We finally claim that

$$\int_{\mathbb{R}^{2}}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x=0\,,$$

(3.4)

which gives $|\nabla^{2}u|=0$, hence the proof will be complete.

To prove (3.4) we again consider (2.2) and choose $\varphi$ as done after this inequality. We obtain with $T_{R}:=B_{2R}(0)-\overline{B}_{R/2}(0)$ using the Cauchy-Schwarz inequality

	$\displaystyle\int_{\mathbb{R}^{2}}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla u,\partial_{\alpha}\nabla u\big{)}\eta^{2}\,{\rm d}x$
		$\displaystyle=$	$\displaystyle-2\int_{T_{R}}D^{2}F(\nabla u)\big{(}\eta\partial_{\alpha}\nabla u,\nabla\eta\otimes\partial_{\alpha}u\big{)}\,{\rm d}x$
		$\displaystyle\leq$	$\displaystyle c\Bigg{[}\int_{T_{R}}\eta^{2}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x\Bigg{]}^{\frac{1}{2}}$
			$\displaystyle\cdot\Bigg{[}\int_{T_{R}}D^{2}F(\nabla u)\big{(}\nabla\eta\otimes\partial_{\alpha}u,\nabla\eta\otimes\partial_{\alpha}u\big{)}\,{\rm d}x\Bigg{]}^{\frac{1}{2}}$
		$\displaystyle=:$	$\displaystyle I_{1}(R)\cdot I_{2}(R)\,.$

We recall (3.3) which gives

$$I_{1}(R)\to 0\quad\mbox{as $R\to\infty$.}$$

Assumption (1.3) yields the estimate

$$I_{2}(R)\leq c\Bigg{[}R^{-2}\int_{T_{R}}|\nabla u|\,{\rm d}x\Bigg{]}^{\frac{1}{2}}\,.$$

thus we obtain (3.4), if we can prove

$$\int_{B_{R}(0)}|\nabla u|\,{\rm d}x\leq c\big{(}1+R^{2}\big{)}\,.$$

(3.5)

For (3.5) we use (2.9) (recall $\eta\equiv 1$ on $B_{R}(0)$) with the choice $\varepsilon=1$, hence (compare the derivation of (2.7))

	$\displaystyle\int_{B_{R}(0)}|\nabla u|\,{\rm d}x$	$\displaystyle\leq$	$\displaystyle|B_{R}(0)|+c\int_{B_{R}(0)}\big{[}\nabla F(\nabla u)-\nabla F(0)\big{]}:\nabla u\,{\rm d}x$
		$\displaystyle\leq$	$\displaystyle|B_{R}(0)|+c\int_{B_{2}R(0)}\eta^{2}\big{[}\nabla F(\nabla u)-\nabla F(0)\big{]}:\nabla u\,{\rm d}x$
		$\displaystyle\leq$	$\displaystyle c\Big{[}R^{2}+R\sup_{T_{R}}|u|\Big{]}$
		$\displaystyle=$	$\displaystyle cR^{2}\Bigg{[}1+\frac{1}{R}\sup_{T_{R}}|u|\Bigg{]}\,,$

and our hypothesis (1.6) gives (3.4), hence the proof of Theorem 1.1 is complete. $\Box$

We follow the lines of [7] by checking the hypotheses of Theorem 1.4 in this reference. We let

$$a(t):=\frac{g^{\prime}(t)}{t}\,,\quad t>0\,,$$

(4.1)

and observe that on account of (1.7) the function $a$ continuously extends to $t=0$ by letting $a(0)=g^{\prime\prime}(0)$. Obviously $a$ satisfies (1.2) from [7] ($g^{\prime}$ is strictly increasing and thereby positive on $(0,\infty)$ due to (1.7)), and since for any $t>0$ it holds (compare (4.1))

$$\lambda_{1}(t):=a(t)+ta^{\prime}(t)=g^{\prime\prime}(t)$$

(4.2)

we get (1.3) in [7]. At the same time assumption (A2) from [7] is obvious by formula (4.1) and our requirements concerning $g$. Moreover, the stability condition (see (1.11) in [7]) follows from

$$\int_{\mathbb{R}^{2}}\big{(}A(\nabla u)\nabla\phi\big{)}\cdot\nabla\phi\,{\rm d}x\geq 0$$

(4.3)

for any $\phi\in C^{1}_{0}(\mathbb{R}^{2})$ with matrix ($Z\in\mathbb{R}^{2}-\{0\}$)

$$A_{ij}(Z):=|Z|^{-1}a^{\prime}\big{(}|Z|\big{)}Z_{i}Z_{j}+a\big{(}|Z|\big{)}\delta_{ij}$$

by observing that (recall (4.1), (1.7))

	$\displaystyle\big{(}A(Y)Z\big{)}\cdot Z$	$\displaystyle=$	$\displaystyle|Y|^{-1}\Big{[}g^{\prime\prime}\big{(}|Y|\big{)}|Y|^{-1}-g^{\prime}\big{(}|Y|\big{)}|Y|^{-2}\Big{]}(Y\cdot Z)^{2}$
			$\displaystyle+|Y|^{-1}g^{\prime}\big{(}|Y|\big{)}|Z|^{2}$
		$\displaystyle\geq$	$\displaystyle g^{\prime}\big{(}|Y|\big{)}\Big{[}|Y|^{-1}|Z|^{2}-|Y|^{-3}(Y\cdot Z)^{2}\Big{]}\geq 0$

on account of $g^{\prime}\big{(}|Y|\big{)}>0$ for $Y\not=0$.

It remains to check (1.17) and (1.18) in [7]: since $g$ is convex (see (1.7)) and of linear growth (compare (1.8)) the boundedness of $g^{\prime}$ follows, hence we get (1.17) and by monotonicity $g^{\prime}_{\infty}:=\lim_{t\to\infty}g^{\prime}(t)$ exists in $(0,\infty)$. By (4.2) the function $\lambda_{1}$ defined in (4.2) (see (1.5) of [7]) is just $g^{\prime\prime}$ so that (1.18) is a consequence of (1.9) and the aforementioned limit behaviour of $g^{\prime}$ provided we assume $\mu\geq 3$. From Theorem 1.4 in [7] it follows that $u(x)=\tilde{u}(\omega\cdot x)$ for some $\omega\in\mathbb{R}^{2}$, $|\omega|=1$, and a function $\tilde{u}$: $\mathbb{R}\to\mathbb{R}$. Assuming $\omega=(1,0)$ (w.l.o.g.) we see that (1.1) implies

$$\frac{{\rm d}}{\,{\rm d}t}\Bigg{[}\frac{1}{|\tilde{u}^{\prime}|}g^{\prime}\big{(}|\tilde{u}^{\prime}|\big{)}\tilde{u}^{\prime}\Bigg{]}=0\,,$$

hence

$$|\tilde{u}^{\prime}|^{-1}g^{\prime}\big{(}|\tilde{u}^{\prime}|\big{)}\tilde{u}^{\prime}\equiv c$$

(4.4)

for some $c\in\mathbb{R}$.

Case 1, $c=0$. Recalling $g^{\prime}>0$ on $(0,\infty)$ equation (4.4) yields $\tilde{u}^{\prime}\equiv 0$ and we are done.

Case 2, $c\not=0$. Then (4.4) shows $\tilde{u}^{\prime}(t)\not=0$ for any $t\in\mathbb{R}$, thus $g^{\prime}\big{(}|\tilde{u}^{\prime}|\big{)}\equiv|c|$ and in conclusion

$$\tilde{u}^{\prime}(t)\in\Big{\{}-(g^{\prime})^{-1}\big{(}|c|\big{)},(g^{\prime})^{-1}\big{(}|c|\big{)}\Big{\}}$$

for any $t\in\mathbb{R}$. But this immediately implies the constancy of $\tilde{u}^{\prime}$ and our claim follows. ∎