A family of nonlocal degenerate operators: maximum principles and related properties

Abstract

We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles, and their relation with suitably defined principal eigenvalues. We also show a Hopf type Lemma, the existence of solutions for the corresponding Dirichlet problem, and representation formulas in some particular cases.

Appendices

Appendix A: Representation formulas

We recall that in [10] some representation formulas for $\mathcal {I}_k^\pm$ and $\mathcal {J}_k^\pm$ are given, and Liouville type theorems are proved. Here, we extend their results to the operators $\mathcal {K}_{k_1, k_2}^-$ with $k <N$. However, it does not seem trivial to adapt the arguments to the full general case, precisely when a competition between different terms in the sum arises. Nonetheless, one has

Lemma A.1

Let $k < N$. Assume $u(x)=\tilde{g}(|x|^2) \in C^2(\mathbb {R}^N ) \cap L^\infty (\mathbb {R}^N)$ such that $\tilde{g}$ is convex. Then for any $x \ne 0$

$$\begin{aligned} \mathcal {K}_{k_1,k_2}^- u(x) = \left( \mathcal {J}_{\{ \eta _j^1 \}_{j=1}^{k_1} } u(x) + \mathcal {J}_{\{ \eta _j^2 \}_{j=1}^{k_2} } u(x) \right) = \frac{C_{k_1, s}+C_{k_2, s}}{C_{1, s}} \; \mathcal {I}_{x^\perp } u(x) \end{aligned}$$

(A.1)

where $\cup _{i=1}^2 \{ \eta _j^i \}_{j=1}^{k_i}$ is any orthonormal basis of $\mathbb {R}^k$ such that each $\eta _j^i$ is orthogonal to x, whereas $x^\perp$ is any unit vector orthogonal to x.

Remark A.2

In particular, if $k_i=1$ for any $i=1,2$, namely when $\mathcal {K}^-_{1, 2}=\mathcal {I}_2^-$, we recover [10, Theorem 3.4].

Proof

The proof immediately follows recalling that, by [10, Proposition 5.1, Lemma 3.3] for any $V_i, W_i \in \mathcal {V}_{k_i}$ such that $x \perp \langle W_i \rangle$ one has

$$\begin{aligned} \mathcal {J}_{W_i} u(x) \le \mathcal {J}_{V_i} u(x). \end{aligned}$$

Then

$$\begin{aligned} \mathcal {J}_{\{ \eta _j^1 \}_{j=1}^{k_1} } u(x) + \mathcal {J}_{\{ \eta _j^2 \}_{j=1}^{k_2} } u(x) \le \mathcal {J}_{V_1} u(x)+ \mathcal {J}_{V_2} u(x), \end{aligned}$$

where $\cup _{i=1, 2} \{ \eta _j^i \}_{j=1}^{k_i}$ is any orthonormal basis of $\mathbb {R}^k$ such that each $\eta _j^i$ is orthogonal to x. Hence

$$\begin{aligned} \mathcal {K}_{k_1, k_2}^- u(x) = \mathcal {J}_{\{ \eta _j^1 \}_{j=1}^{k_1} } u(x) + \mathcal {J}_{\{ \eta _j^2 \}_{j=1}^{k_2} } u(x). \end{aligned}$$

Moreover, we know by [10] that for any $V_i = \{ \xi _1^i, \dots , \xi _{k_i}^i \}$ ,

$$\begin{aligned} \mathcal {J}_{V_i} u(x)= \frac{C_{k_i, s}}{2} f_i(\Phi ) \end{aligned}$$

(A.2)

with

$$\begin{aligned}f_i(\Phi )=\int _{\mathbb {R}^{k_i}} \frac{\tilde{g} (|x|^2+|\tau |^2+2|x|\tau _1 \sin (\Phi )) + \tilde{g}(|x|^2+|\tau |^2 - 2|x| \tau _1 \sin (\Phi )) - 2 \tilde{g}(|x|^2) }{\left( \sum _{j=1}^{k_i} \tau _j^2 \right) ^{\frac{k_i+2s}{2}}} \, d\tau \end{aligned}$$

and $\Phi$ is the angle between x and $\hat{\xi }$, the unit vector orthogonal to $\langle V_i \rangle$ in the $k_i+1$ dimensional space generated by $\xi _1^i, \dots , \xi _{k_i}^i$ and x, see [10, Proposition 5.1]. Also,

$$\begin{aligned} f_i(0)=2 \int _0^{+\infty } (\tilde{g}(|x|^2+r^2)-\tilde{g}(|x|^2)) r^{-1-2s}, \end{aligned}$$

(A.3)

which in particular means that it does not depend on i.

We now recall (A.2) and (A.3) to conclude that for any $i=1,2$ and any $\cup _{i=1,2} \{ \eta _j^i \}_{j=1}^{k_i}$ orthonormal basis of $\mathbb {R}^k$ such that each $\eta _j^i$ is orthogonal to x,

$$\begin{aligned} \mathcal {J}_{\{ \eta _j^i \}_{j=1}^{k_i} } u(x) = \frac{C_{k_i, s}}{2} f_i(0)= \frac{C_{k_i, s}}{C_{1, s}} \mathcal {I}_{x^\perp } u(x). \end{aligned}$$

This yields (A.1).

As a corollary, we get, following the same arguments as in [10, Theorem 4.7], the next Liouville type result.

Corollary A.3

Assume $k < N$. Then, for any $p \ge 1$ there exist positive classical solutions of the equation

$$\begin{aligned} \mathcal {K}_{k_1, k_2}^- u(x) + u^p(x)=0 \text { in } \mathbb {R}^N. \end{aligned}$$

If $p \in (0, 1)$, then there exist nonnegative viscosity solutions $u \not \equiv 0$.

Precisely,

$$\begin{aligned} u(x)=\frac{\alpha }{(a^2 + |x|^2)^{\frac{s}{p-1}}} \end{aligned}$$

is a solution for the case $p>1$, for any $a \ne 0$ and for a suitable $\alpha$. On the other hand, a solution if $p=1$ is given by

$$\begin{aligned} u(x)=be^{-\beta |x|^2} \end{aligned}$$

with $b >0$ and $\beta$ suitably chosen. Finally, the case $p\in (0,1)$ can be treated exploiting the function

$$\begin{aligned} u(x)=\alpha (R^2-|x|^2)^{\frac{s}{1-p}}_+ \end{aligned}$$

for a suitable $\alpha >0$ and for any $R>0$.

Appendix B: Generalizations

In this section, we consider a more general operator inspired by $\mathcal {K}_{k_1, k_2}^\pm$, and we briefly comment on how to extend results of the previous sections to this new class of operators. As a particular case, we will recover some results for the operators $\mathcal {J}_k^\pm$, for which we also refer to [10].

Choose $1\le \ell \le N$. Let $1 \le k_1 \le \dots \le k_\ell \le N$ such that

$$\begin{aligned} k:=\sum _{j=1}^\ell k_j \end{aligned}$$

satisfies $1 \le k \le N$. We define for any $u \in L^\infty (\mathbb {R}^N) \cap C^2(\Omega )$

$$\begin{aligned} \mathcal {K}_{k_1, \dots , k_\ell }^+ u(x):=\sup _{ V_1 \in \mathcal {V}_{k_1} } \sup _{V_2 \in \mathcal {V}_{k_2}(V_1)} \cdots \sup _{V_\ell \in \mathcal {V}_{k_\ell }(V_1, \dots , V_{\ell -1})} \sum _{i=1}^\ell \mathcal {J}_{V_i} u(x), \end{aligned}$$

where for any $t=2, \dots , \ell$

$$\begin{aligned} \mathcal {V}_{k_t}(V_1, \dots , V_{t-1})= \left\{ \{\xi _i\}_{i=1}^{k_t} \in \mathcal {V}_{k_t}: \xi _i \cdot \xi =0 \text { for all } i=1, \dots , k_t, \, \forall \xi \in \cup _{j=1}^{t-1} V_j \right\} . \end{aligned}$$

Analogously, we define $\mathcal {K}_{k_1, \dots , k_\ell }^-$ taking the infimums in place of the supremums. This clearly extends $\mathcal {K}_{k_1, k_2}^\pm$, just take $\ell =2$ in the definition above.

We can give the definition of weak solution as in Definition 2.4, and also the analog of Remarks 2.2 and 2.5 hold true. In particular, $\mathcal {K}_ {k_1, \dots , k_\ell }^\pm \rightarrow \mathcal {P}_k^\pm$ as $s \rightarrow 1^-$. Also, we point out the following.

Remark B.1

Taking in the definition above $\ell =1$, one has

$$\begin{aligned} \mathcal {K}_{k}^+ u(x)=\sup _{V\in \mathcal {V}_k} \mathcal {J}_V u(x)= \mathcal {J}_k^+u(x). \end{aligned}$$

In particular, if $\ell =1$, and $k=1$, then $\mathcal {K}_1^{\pm } =\mathcal {J}_1^\pm =\mathcal {I}_1^\pm$, whereas if $\ell =1$, and $k=N$, then $\mathcal {K}_{N}^{\pm } =\mathcal {J}_N^\pm =-(-\Delta )^s$.

On the other hand, take $k_1= \dots = k_\ell =1$. Then $k=\ell$ and

$$\begin{aligned} \mathcal {K}_{1, \dots , 1}^+ u(x)&=\sup _{\xi ^1: |\xi ^1|=1} \sup _{\begin{array}{c} \xi ^2: |\xi ^2|=1\\ \xi ^1\cdot \xi ^2=0 \end{array}} \cdots \sup _{\begin{array}{c} \xi ^\ell : |\xi ^\ell |=1\\ \xi ^\ell \cdot \xi ^j=0, j=1, \dots , \ell -1 \end{array}} \sum _{j=1}^\ell \mathcal {J}_{\xi ^j} u(x)\\&= \sup _{\xi ^1, \dots , \xi ^\ell \in \mathcal {V}_\ell } \sum _{j=1}^\ell \mathcal {I}_{\xi ^j} u(x)= \mathcal {I}_k^+u(x). \end{aligned}$$

In particular, if $\ell =N$, then $\mathcal {K}_{1, \dots , 1}^\pm =\mathcal {I}_N^\pm$.

We finally notice that if $\ell \ne 1$, then $\mathcal {K}_{k_1, \dots , k_\ell }^\pm$ does not coincide with the fractional Laplacian, even if $k=N$.

Results in the previous sections can be extended to these operators, up to some technicalities. In what follows, we state for convenience of the reader the main results, and we sketch the proofs in case they significantly differ from the case $\ell =2$.

As a first observation, the map $x \mapsto \mathcal {K}_{k_1, \dots , k_\ell }^\pm u(x)$ is in general not continuous under the assumption $u \in C^2(\Omega ) \cap L^\infty (\mathbb {R}^N)$, except from the case $\ell =1$, $k=N$ (here, $\mathcal {K}_{k_1, \dots , k_\ell }^\pm$ coincides with the fractional Laplacian, which is continuous). One can verify as in Sect. 3 that a counterexample is given by the function

$$\begin{aligned} u(x)=\left\{ \begin{array}{rl} 0 &{}{} \text{ if }\quad |x|\le 1,\quad \text{ or }\quad \exists i=1, \dots , \ell \quad \text{ s.t. } \quad x \in \langle e_j \rangle _{j \in \mathcal {A}_i}\\ -1 &{}{} \text{ otherwise, } \end{array}\right. \end{aligned}$$

defined on $\Omega =B_1(0)$, where

$$\begin{aligned} \mathcal {A}_i=\left\{ \sum _{j=1}^{i-1} k_j +1, \dots , \sum _{j=1}^{i} k_j \right\} , \, i=1, \dots , \ell . \end{aligned}$$

As in Proposition 3.1, also in this more general case we can recover continuity once we impose a global regularity on u.

Moreover, the supremum in the definition of $\mathcal {K}_{k_1, \dots , k_\ell }^+$ is in general not attained. The proof is slightly more delicate than the one given in Sect. 3, hence we give here all the details. Let

$$\begin{aligned} u(x)= {\left\{ \begin{array}{ll} e^{-\langle x, e_N \rangle } &{}\text { if } \sum _{i=1}^{N-k_1-1} \langle x, e_i \rangle ^2=0, \langle x, e_N \rangle>0 \text { and }\left| x\right| >1\\ 0 &{}\text { otherwise}. \end{array}\right. }\end{aligned}$$

Let us first assume $k_1=k_2=1$. Fix any orthonormal set of $\mathbb {R}^k$. Thus, there are at most two vectors in this set which belong to the space $\sum _{i=1}^{N-k_1-1} \langle x, e_i \rangle ^2=0$. We distinguish three possible situations:

• (i) If they both belong to $\langle V_j \rangle$ of dimension $k_1+1=2$ (if one such space exists), then this is the only space which makes a significant contribution, as $\mathcal {J}_{V_i}u(0)=0$ for $i \ne j$. We then use (3.4) to conclude that

  $$\begin{aligned} \sum _{i=1}^\ell \mathcal {J}_{V_i} u(0) = \mathcal {J}_{V_j} u(0) \le \frac{1}{4s}. \end{aligned}$$

  (B.1)

• (ii) If one of the two vectors belongs to a space of dimension $k_1=1$, and the other one to a space of dimension $>1$, then

  $$\begin{aligned} \sum _{i=1}^\ell \mathcal {J}_{V_i} u(0) \le \frac{1}{2s}, \end{aligned}$$

  (B.2)

  again by (3.4).

• (iii) Finally, if they belong to two different spaces of dimension $k_1=1$, then we have a similar situation as in [9], and

  $$\begin{aligned} \sum _{i=1}^\ell \mathcal {J}_{V_i} u(0) \le \int _{1}^\infty \frac{1+ e^{-\tau }}{\tau ^{2s+1}} \, d\tau . \end{aligned}$$

Then we conclude that for any orthonormal set of $\mathbb {R}^k$

$$\begin{aligned} \sum _{i=1}^\ell \mathcal {J}_{V_i} u(0) \le \max \left\{ \frac{1}{4s}, \frac{1}{2s}, \int _1^\infty \frac{1+ e^{-\tau }}{\tau ^{2s+1}} \, d\tau \right\} = \int _1^\infty \frac{1+ e^{-\tau }}{\tau ^{2s+1}} \, d\tau . \end{aligned}$$

From this we deduce

$$\begin{aligned} \mathcal {K}_{k_1, \dots , k_\ell }^+ u(0)= \int _1^\infty \frac{1+ e^{-\tau }}{|\tau |^{2s+1}} \, d\tau , \end{aligned}$$

as we can find a sequence which converges to this value, arguing as in [9]. However, this is not attained.

We now take into account the situation in which $1=k_1<k_2$, or $k_1>1$. Again, for any orthonormal set of $\mathbb {R}^k$ there are at most $k_1+1$ vectors such that $\sum _{i=1}^{N-k_1-1} \langle x, e_i \rangle ^2=0$. If they all belong to one of the spaces of dimension $k_1+1$ (if it exists), then (B.1) holds. If $k_1$ of them appear in a space of dimension $k_1$, then (B.1) holds if $k_1>1$, whereas one has (B.2) if $k_1=1$. In any other case, the sum is identically 0. Thus, by similar arguments as above,

$$\begin{aligned} \mathcal {K}_{k_1, \dots , k_\ell }^+ u(0)= {\left\{ \begin{array}{ll} \frac{1}{4s} &{}\text { if } k_1>1 \\ \frac{1}{2s} &{}\text { if } k_1=1, \end{array}\right. } \end{aligned}$$

and the supremum is not attained, see also Sect. 3.

As an extension of results in Sect. 4, we obtain a comparison principle, and the following

Theorem B.2

The following conclusions hold.

• (i) The operators $\mathcal {K}_{k_1, \dots , k_\ell }^-$, with $k < N$, do not satisfy the strong minimum principle in $\Omega$.

• (ii) The operators $\mathcal {K}_{k_1, \dots , k_\ell }^-$ with $k=N$ satisfy the strong minimum principle in $\Omega$.

• (iii) The operators $\mathcal {K}_{k_1, \dots , k_\ell } ^+$ satisfy the following implication

  $$\begin{aligned} \mathcal {K} u(x) \le 0 \text { in } \Omega , \quad u \ge 0 \text { in }\mathbb {R}^N \; \Rightarrow \; u > 0 \text { in } \Omega \text { or } u \equiv 0 \text { in } \mathbb {R}^N. \end{aligned}$$

  In particular, they satisfy the strong minimum principle.

• (iv) Let $k<N$, or $k=N$ and $\ell >1$. There exist functions u such that $\mathcal {K}_{k_1, \dots , k_\ell } ^- u \le 0$ in $\Omega$, $u \equiv 0$ in $\overline{\Omega }$, and $u \not \equiv 0$ in $\mathbb {R}^N \setminus \overline{\Omega }$, namely these operators do not satisfy the implication in (iii).

Remark B.3

The case $k=N$, $\ell =1$, not treated in item (iv), corresponds to the fractional Laplacian, for which the implication in (iii) is already known [20, Corollary 4.2].

Remark B.4

If $k_1= \dots =k_\ell =1$, we recover the results for $\mathcal {I}_k^\pm$ for which we refer to [9]. We wish to cite here also the recent paper [6], in which the geometry of the sets of minima for supersolutions of equations involving the operators $\mathcal {I}_k^\pm$ is characterized.

We stress that the result above includes the case of the operator $\mathcal {J}_k^\pm$, for which we get

Corollary B.5

One has

• (i) The operator $\mathcal {J}_k^-$, with $k < N$, does not satisfy the strong minimum principle in $\Omega$.

• (ii) The operators $\mathcal {J}_k^+$ satisfy the following implication

  $$\begin{aligned} \mathcal {J} u(x) \le 0 \text { in } \Omega , \quad u \ge 0 \text { in }\mathbb {R}^N \; \Rightarrow \; u > 0 \text { in } \Omega \text { or } u \equiv 0 \text { in } \mathbb {R}^N. \end{aligned}$$

  In particular, they satisfy the strong minimum principle.

Following the arguments in Sects. 5 and 6, we get an Hopf-type lemma and also existence of solutions to the corresponding Dirichlet problem, as detailed in the next

Proposition B.6

Let f be a bounded continuous function, and let $\Omega$ be a uniformly convex domain. Then there exists a unique function $u \in C(\mathbb {R}^N)$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {K}_{k_1, \dots , k_\ell }^\pm u = f(x) &{}\text { in } \Omega \\ u=0 &{}\text { in } \mathbb {R}^N \setminus \Omega . \end{array}\right. } \end{aligned}$$

Let us now define

$$\begin{aligned} \bar{\mu }^\pm _{k_1, \dots , k_\ell }=\sup \Big \{\mu :\,\exists v\in LSC(\Omega )\cap L^\infty (\mathbb {R}^N),\,\inf _\Omega v>0,\,v\ge 0\;\text {in}\quad \mathbb {R}^N,\;\\ \mathcal {K}_{k_1, \dots , k_\ell }^\pm v+\mu v\le 0 \text { in } \Omega \Big \}. \end{aligned}$$

Then one has

Theorem B.7

The operators $\mathcal {K}_{k_1, \dots , k_\ell }^\pm (\cdot )+\mu \cdot$ satisfy the maximum principle for $\mu <\bar{\mu }^\pm _{k_1, \dots , k_\ell }$.

Moreover, if we define

$$\begin{aligned} \mu _{k_1, \dots , k_\ell }^\pm = \sup \Big \{ \mu :\, \exists v \in LSC(\Omega )\cap L^\infty (\mathbb {R}^N), v>0 \text{ in } \Omega , v \ge 0 \text{ in } \mathbb {R}^N, \\ \quad \mathcal {K}_{k_1, \dots , k_\ell }^\pm v + \mu v \le 0 \text{ in } \Omega \Big \}, \end{aligned}$$

suitable analogs of Propositions 6.5 and 6.6 hold, from which we get

Theorem B.8

Let $\Omega$ be a uniformly convex domain. The operator

$$\begin{aligned} \mathcal {K}^+_{k_1, \dots , k_\ell } + \mu \end{aligned}$$

satisfies the maximum principle if and only if $\mu< \mu _{k_1, \dots , k_\ell }^+ < +\infty$, and correspondingly

$$\begin{aligned} \mathcal {K}_{k_1, \dots , k_\ell }^- + \mu \end{aligned}$$

satisfies the maximum principle if and only if $\mu< \mu _{k_1, \dots , k_\ell }^- < +\infty$ if $k=N$, and for any $\mu \in \mathbb R$ if $k < N$.

We finally notice that also Appendix A can be adapted to this class of operators.

Another possible generalization includes operators of the form

$$\begin{aligned} \inf _{ V_1 \in \mathcal {V}_{k_1} } \sup _{V_2 \in \mathcal {V}_{k_2}(V_1)} \left( \mathcal {J}_{V_1} u(x)+\mathcal {J}_{V_2} u(x) \right) , \end{aligned}$$

(B.3)

or

$$\begin{aligned} \sup _{ V_1 \in \mathcal {V}_{k_1} } \inf _{V_2 \in \mathcal {V}_{k_2}(V_1)} \left( \mathcal {J}_{V_1} u(x)+\mathcal {J}_{V_2} u(x) \right) . \end{aligned}$$

(B.4)

Recall, see also Remark 2.1 above, that this class of $\inf$-$\sup$ operators have a natural game theoretical interpretation. Some of the results in the present work can be extended to this class of operators, following the same arguments above and also recalling that

$$\begin{aligned} \inf \inf \le \inf \sup \le \sup \sup , \qquad \inf \inf \le \sup \inf \le \sup \sup . \end{aligned}$$

For instance, it is an immediate consequence of Proposition 5.2 that a Hopf type lemma holds for (B.3) and (B.4). Also, we point out the following

Proposition B.9

The operators (B.3) and (B.4) satisfy the following implication

$$\begin{aligned} \mathcal {K} u(x) \le 0 \text { in } \Omega , \quad u \ge 0 \text { in }\mathbb {R}^N \; \Rightarrow \; u > 0 \text { in } \Omega \text { or } u \equiv 0 \text { in } \mathbb {R}^N. \end{aligned}$$

In particular, they satisfy the strong minimum principle.

Proof

Let us consider the operator (B.4), similar arguments hold for the other one. Take u which satisfies the assumptions of the minimum principle, and assume there exists $x_0 \in \Omega$ such that $u(x_0)=0$. Choose any orthonormal basis of $\mathbb {R}^N$ $\{ \xi _1, \dots , \xi _{N} \}$. Thus, using $u \ge 0$,

$$\begin{aligned} 0 \ge \sup _{V_1 \in \mathcal {V}_{k_1}} \mathcal {J}_{V_1} u(x_0) \ge C_{k_1, s} \int _{\mathbb {R}^{k_1}} \frac{u(x_0 + \sum _{j=1}^{k_1} \tau _j \xi _j ) }{|\tau |^{k_1+2s}} \, d\tau . \end{aligned}$$

Hence, we conclude that $u \equiv 0$ on every $k_1$ dimensional space $\langle V_1\rangle + x_0$. Since the directions $\xi _i$ are arbitrary, we get $u \equiv 0$ on $\mathbb {R}^N$.

However, it is not completely clear whether, for instance, the Perron method or the continuity result in Proposition 3.1 can be extended to this situation. Also, one could try to treat even more general operators, in which a combination of infimums and supremums appears. We leave the complete study of the rich structure of these $\inf$-$\sup$ operators as an interesting open problem.