On the nodal set of solutions to degenerate or singular elliptic equations with an application to s-harmonic functions

Abstract

This work is devoted to the geometric-theoretic analysis of the nodal set of solutions to degenerate or singular equations involving a class of operators including

$$L_a=\text{div}({\left|y\right|}^a\mathrm{\nabla }),$$

with $a\in (-1,1)$ and their perturbations.

As they belong to the Muckenhoupt class $A_2$, these operators appear in the seminal works of Fabes, Kenig, Jerison and Serapioni [1], [2], [3] and have recently attracted a lot of attention in the last decade due to their link to the localization of the fractional Laplacian via the extension in one more dimension [4]. Our goal in the present paper is to develop a complete theory of the stratification properties for the nodal set of solutions of such equations in the spirit of the seminal works of Hardt, Simon, Han and Lin [5], [6], [7].

Résumé

Ce travail est dévolu à l'analyse des propriétés géométriques des ensembles nodaux de solutions d'équations dégénérées/singulières ayant comme opérateur principal

$$L_a=\text{div}({\left|y\right|}^a\mathrm{\nabla }),$$

pour $a\in (-1,1)$. Comme le poids est dans la classe de Muckenhoupt $A_2$, ces opérateurs apparaissent à l'origine dans la série de travaux de Fabes, Kenig, Jerison and Serapioni [1], [2], [3] et ont récemment connu un renouveau de leur étude en raison du lien avec la laplacien fractionnaire, comme l'ont mis en évidence Caffarelli et Silvestre [4]. Notre but ici est de développer une théorie générale des propriétés de stratification de l'ensemble nodal des solutions de telles équations dans l'esprit des travaux de Hardt, Simon, Han et Lin [5], [6], [7].

Introduction

In the last decades the question of the structure of the nodal set of solutions of elliptic equations has in the centre of the attention of the scientific community (see e.g. [8], [6], [9], [7]), with a special focus on the measure theoretical features of its singular part, also in connection with the validity of a strong unique continuation principle, in order to ensure the existence of a finite vanishing order, as pointed out in [10], [11], [7]. Recently, major progress have been done on the study of nodal sets of eigenfunctions (or critical sets of harmonic functions) by Logunov and Malinnikova [12], [13], [14] in connection with conjectures by Yau and Nadirashvili.

In this paper we consider the nodal set in ${\mathbb{R}}^{n+1}$ of solutions of a class of degenerate-singular operators which have recently become very popular in connection with the study of the fractional powers of the Laplacian, and firstly studied in the pioneering works [1], [2], [3]. Given $a\in (-1,1)$ and $X=(x,y)\in {\mathbb{R}}_x^n\times {\mathbb{R}}_y$ we consider a class of operators including

$$L_a=\text{div}({\left|y\right|}^a\mathrm{\nabla }),$$

and their perturbations (here we denote by div and ∇ respectively the divergence and the gradient operator in ${\mathbb{R}}^{n+1}$). Our main purpose is to fully understand the local behaviour of $L_a$-harmonic functions near their nodal set and to develop a geometric analysis of its structure and regularity, in order to comprehend how the degenerate or singular character of the coefficients can affect the local picture of the nodal set itself. Thus, we introduce the notion of characteristic manifold Σ associated with the operator $L_a$, as the set of points where the coefficient either vanishes or blows up, and we study the properties of the nodal set $\mathrm{\Gamma }(u)$ of solutions to equation

$$-L_{a}u=0\text{in }{B}_1\subset {\mathbb{R}}^{n+1}.$$

In particular, since the operator $L_a$ is locally uniformly elliptic on ${\mathbb{R}}^{n+1}\setminus \mathrm{\Sigma }$, we restrict our attention on the structure of the nodal set neighbouring the characteristic manifold Σ, trying to understand the structural difference between the whole nodal set $\mathrm{\Gamma }(u)=\{x\in B_1,u(x)=0\}$ and its restriction on Σ.

As a further motivation, this analysis will be the starting point of the study of competition-diffusion systems of populations under an anomalous diffusion. More precisely, we can imagine that the characteristic manifold Σ is playing a major role in the diffusion phenomenon by penalizing or encouraging the diffusion across Σ, according to the value of $a\in (-1,1)$. Inspired by [15], [16], [17], [18], [19], in the case of strong competition, the limiting segregated configurations will satisfy a refection law which represents the only interaction between the different densities through the common free boundary. Thanks to this reflection property, the free boundary will be locally described as the nodal set of $L_a$-harmonic function.

As already mentioned, our operators belong to the class introduced in the 80s by Fabes, Jerison, Kenig and Serapioni in [1], [2], [3], where they established Hölder continuity of solutions within a general class of degenerate-singular elliptic operators $L=\text{div}(A(X)\mathrm{\nabla }\cdot )$ whose coefficient $A(X)=(a_{ij}(X))$ are defined starting from a symmetric matrix valued function such that

$$\lambda \omega (X){\left|\xi \right|}^2\le (A(X)\xi ,\xi )\le \mathrm{\Lambda }\omega (X){\left|\xi \right|}^2,\text{for some }\lambda ,\mathrm{\Lambda }>0,$$

where the weight ω may either vanish, or be infinite, or both. In particular, the prototypes of weights considered in their analysis where in the Muckenhoupt $A_2$-class, i.e. such that

$$\underset{B\subset {\mathbb{R}}^{n+1}}{\sup }(\frac{1}{\left|B\right|}\underset{B}{\int }\omega (X)\mathrm{d}X)(\frac{1}{\left|B\right|}\underset{B}{\int }{\omega }^{-1}(X)\mathrm{d}X)<\infty .$$

Our case correspond to the choice $\omega (X)=|y{|}^a$ and is Muckenhoupt whenever $a\in (-1,1)$. Note however that this class of $A_2$-weights is not the optimal one to have Hölder regularity as noticed in [1]. However, it provides a good model with applications for our purposes.

Our approach is based upon the validity of an Almgren and Weiss type monotonicity formulæ, the existence and uniqueness of non trivial tangent maps at every point of the nodal set, and on a complete classification of the possible homogeneous configurations appearing at the blow-up limit. Nevertheless, the starting point of our analysis relies on the decomposition of an $L_a$-harmonic function with respect to the orthogonal direction to the characteristic manifold Σ. Indeed, denoting by $H^{1,\beta }(B_1)$ the Sobolev space w.r.t. the measure $|y{|}^{\beta }dydx$, we have (see also [20], [21])

Proposition 1.1

Given $a\in (-1,1)$ and u an $L_a$-harmonic function in $B_1$, there exist two unique functions $u_e^a\in H^{1,a}(B_1),u_e^{2-a}\in H^{1,2-a}(B_1)$ symmetric with respect to Σ respectively $L_a$ and $L_{2-a}$ harmonic in $B_1$ and locally smooth, such that

$$u(X)=u_e^a(X)+u_e^{2-a}(X)y{\left|y\right|}^{-a}\mathit{\text{in}}{B}_1.$$

With this decomposition in mind, we can reduce the classification of the possible blow-up limits to the symmetric ones and eventually recover all the possible cases. In particular, it is worthwhile introducing a new notion of tangent field ${\mathrm{\Phi }}^{X_0}$ of u at a nodal point, which takes care of the different behaviour of both the symmetric and antisymmetric part of an $L_a$-harmonic function. Namely, by the decomposition and the Definition 5.7 of the notion of tangent map, i.e. the unique nonzero map ${\phi }^{X_0}\in {\mathfrak{B}}_k^a(u)$ such that

$$u_{X_0,r}(X)=\frac{u(X_0+rX)}{r^k}⟶{\phi }^{X_0}(X),$$

with k the vanishing order of u at $X_0$, we introduce the following concept, of crucial use in our results.

Definition 1.2

Let $a\in (-1,1),u$ be an $L_a$-harmonic function in $B_1$ and $X_0\in {\mathrm{\Gamma }}_k(u)\cap \mathrm{\Sigma }$, for some $k\ge \min \{1,1-a\}$. We define as tangent field of u at $X_0$ the unique nontrivial vector field ${\mathrm{\Phi }}^{X_0}\in {(H_{\text{loc}}^{1,a}({\mathbb{R}}^{n+1}))}^2$ such that

$${\mathrm{\Phi }}^{X_0}=({\phi }_e^{X_0},{\phi }_o^{X_0}),$$

where ${\phi }_e^{X_0}$ and ${\phi }_o^{X_0}$ are respectively the tangent map of the symmetric part $u_e$ of u and of the antisymmetric one $u_o$.

At first, the notion of tangent field allows us to describe the topology of the nodal set by proving in Proposition 5.19 a vectorial counterpart of the classic result of upper semi-continuity of the vanishing order. In order to define properly the relevant subsets, we define

$${\partial }_y^{a}u=\{\begin{array}{cc}{\left|y\right|}^a{\partial }_{y}u\hfill & \text{if }X\notin \mathrm{\Sigma }\hfill \\ {\lim }_{y\to 0}{\left|y\right|}^a{\partial }_{y}u(x,y)\hfill & \text{if }X\in \mathrm{\Sigma }\hfill \end{array}.$$

This quantity, as observed already in previous works, is the nontrivial one to be considered as far as the derivative in y is concerned.

In the light of this observation, it is natural to define the regular part $\mathcal{R}(u)$ and the singular part $\mathcal{S}(u)$ of the nodal set as follows:

$$\mathcal{R}(u)=\{X\in \mathrm{\Gamma }(u):{|{\mathrm{\nabla }}_{x}u(X)|}^2+{|{\partial }_y^{a}u(X)|}^2\ne 0\},\mathcal{S}(u)=\{X\in \mathrm{\Gamma }(u):{|{\mathrm{\nabla }}_{x}u(X)|}^2+{|{\partial }_y^{a}u(X)|}^2=0\}.$$

The next step is to develop a blow-up analysis in order to fully understand the structure of $\mathrm{\Gamma }(u)$ in ${\mathbb{R}}^{n+1}$ and its restriction on Σ. The following is a summary of our main result describing the stratified structure of both the regular and singular parts of the nodal set. A key point is the complete classification of the spectrum of the previously defined tangent fields.

Theorem 1.3

Let $a\in (-1,1),a\ne 0$ and u be an $L_a$-harmonic function in $B_1$. Then the regular set $\mathcal{R}(u)$ is locally a $C^{k,r}$ hypersurface on ${\mathbb{R}}^{n+1}$ in the variable $(x,y{\left|y\right|}^{-a})$ with

$$k=\lfloor \frac{2}{1-a}\rfloor \mathit{\text{and}}r=\frac{2}{1-a}-\lfloor \frac{2}{1-a}\rfloor .$$

On the other hand, there holds

$$\mathcal{S}(u)\cap \mathit{\Sigma }={\mathcal{S}}^{⁎}(u)\cup {\mathcal{S}}^a(u)$$

where ${\mathcal{S}}^{⁎}(u)$ is contained in a countable union of $(n-2)$-dimensional $C^1$ manifolds and ${\mathcal{S}}^a(u)$ is contained in a countable union of $(n-1)$-dimensional $C^1$ manifolds. Moreover

$${\mathcal{S}}^{⁎}(u)=\bigcup _{j=0}^{n-2}{\mathcal{S}}_j^{⁎}(u)\mathit{\text{and}}{\mathcal{S}}^a(u)=\bigcup _{j=0}^{n-1}{\mathcal{S}}_j^a(u),$$

where both ${\mathcal{S}}_j^{⁎}(u)$ and ${\mathcal{S}}_j^a(u)$ are contained in a countable union of j-dimensional $C^1$ manifolds.

In the last Section of this paper, we provide applications of our results in the context of nonlocal elliptic equations. In particular, inspired by [4], [21], [22], we exploit the local realisation of the fractional Laplacian, and more generally of fractional power of divergence form operator L with Lipschitz leading coefficient, in order to study the structure and the regularity of the nodal set of ${(-L)}^s$-harmonic functions, for $s\in (0,1)$. More precisely, we combine the extension technique with a geometric reduction introduced in [23] and exploited in the seminal papers [10], [11]. This will allow us to extend our analysis to fractional powers ${(-{\mathrm{\Delta }}_g)}^s$ of the Laplace-Beltrami operator on a Riemannian manifold M, also for the case of Lipschitz metrics g.

Our results show some genuinely nonlocal features in the Taylor expansion of ${(-L)}^s$-harmonic functions near their zero set and their deep impact on the structure of the nodal set itself. We prove that the first term of the Taylor expansion of an ${(-L)}^s$-harmonic function is either an homogeneous harmonic polynomial or any possible homogeneous polynomial. In particular, this implies

Theorem 1.4

Given L, a divergence form operator with Lipschitz leading coefficients, and $s\in (0,1)$, let u be ${(-L)}^s$-harmonic in $B_1$. Then there holds

$$\mathcal{S}(u)={\mathcal{S}}^{⁎}(u)\cup {\mathcal{S}}^s(u)$$

where ${\mathcal{S}}^{⁎}(u)$ is contained in a countable union of $(n-2)$-dimensional $C^1$ manifolds and ${\mathcal{S}}^s(u)$ is contained in a countable union of $(n-1)$-dimensional $C^1$ manifolds. Moreover

$${\mathcal{S}}^{⁎}(u)=\bigcup _{j=0}^{n-2}{\mathcal{S}}_j^{⁎}(u)\mathit{\text{and}}{\mathcal{S}}^s(u)=\bigcup _{j=0}^{n-1}{\mathcal{S}}_j^s(u),$$

where both ${\mathcal{S}}_j^{⁎}(u)$ and ${\mathcal{S}}_j^s(u)$ are contained in a countable union of j-dimensional $C^1$ manifolds.

Finally, we prove what can be seen as the nonlocal counterpart of a conjecture that Lin proposed in [7]. Following his strategy, we give an explicit estimate on the $(n-1)$-Hausdorff measure of the nodal set $\mathrm{\Gamma }(u)$ of s-harmonic functions in terms of the Almgren frequency previously introduced. We have

Theorem 1.5

Given $s\in (0,1)$, let u be an s-harmonic function in $B_1$ and $0\in \mathit{\Gamma }(u)$. Then

$${\mathcal{H}}^{n-1}(\mathit{\Gamma }(u)\cap B_{\frac{1}{2}})\le C(n,s)N,$$

where v is the $L_a$-harmonic extension of u in $B_1^{+}$ and $N=N(0,v,1)$ is the frequency defined by

$$N=\frac{\underset{B_1^{+}}{\int }{\left|y\right|}^a{\left|\mathrm{\nabla }v\right|}^2\mathrm{d}X}{\underset{\partial B_1^{+}}{\int }{\left|y\right|}^a{v}^2\mathrm{d}\sigma }.$$

The paper is organized as follows. In Section 2 we prove some preliminary result about $L_a$-harmonic functions. Next, in Section 3, we prove the validity of an Almgren's type monotonicity formula which allows in Section 4 to prove the existence of blow-up limit in every point of the nodal set $\mathrm{\Gamma }(u)$.

Finally, in Section 5 we prove a Weiss type monotonicity formula, which allows to introduce the notion of tangent map and tangent field at every point of the nodal set. In Section 6 we present some useful result on the stratification of the nodal set and finally in Section 7 we prove a general result on the regularity of the whole nodal set $\mathrm{\Gamma }(u)$ and on its restriction on the characteristic manifold Σ. The last two Sections are devoted to the applications of the previous results to solutions of fractional powers of divergence form operator, with Lipschitz leading coefficient. In particular, in Section 8 we apply our technique in order to study the nodal set of s-harmonic function and, more generally, of solutions of ${(-L)}^s$ operators and more general nonlocal equations, and in Section 10 we give a new estimate of the Hausdorff measure of the nodal set of s-harmonic functions.