Unique continuation principles in cones under nonzero Neumann boundary conditions

Abstract

We consider an elliptic equation in a cone, endowed with (possibly inhomogeneous) Neumann conditions. The operator and the forcing terms can also allow non-Lipschitz singularities at the vertex of the cone.

In this setting, we provide unique continuation results, both in terms of interior and boundary points.

The proof relies on a suitable Almgren-type frequency formula with remainders. As a byproduct, we obtain classification results for blow-up limits.

Introduction

In this article we consider an elliptic equation with Neumann boundary condition. The domain taken into consideration is a cone, and the equation and the boundary condition can be inhomogeneous and be singular at the origin.

The main results that we provide are of unique continuation type. Roughly speaking, we will show that if a solution vanishes at any order at the vertex of the cone, then the solution must necessarily vanish in a neighborhood of the vertex (and then everywhere, up to suitable assumptions).

The notion of vanishing can be framed both with respect to the convergence of points coming from the interior of the domain and, under the appropriate assumptions, with respect to the convergence of points coming from the boundary.

From these results, we also obtain classification results for the blow-up limits. The method of proof will rely on the special geometric structure of the cone, which is a set invariant under dilations and in which the normal on the side of the cone is perpendicular to the radial direction. The main analytic tool in use will be an appropriate type of frequency function. Differently from the classical case in [5], the choice of the frequency function in our case has to comprise additional quantities and reminders to deal with the forcing terms and possibly compensate for the singular behaviors near the vertex.

The mathematical setting in which we work is the following. We let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, with $n⩾2$, be a cone with vertex at the origin (namely, we assume that $x\in \mathrm{\Omega }$ if and only if $tx\in \mathrm{\Omega }$ for all $t>0$). We consider the spherical cap

$$\mathrm{\Sigma }=\{\frac{x}{|x|}:x\in \mathrm{\Omega }\}\subset {\mathbb{S}}^{n-1}$$

and we assume that Σ has $C^2$ boundary in ${\mathbb{S}}^{n-1}$.

We also take into account a positive function $A\in W^{1,1}(\mathrm{\Omega })$ such that

$$c⩽A(x)⩽\frac{1}{c}\text{ for some }c>0\text{ and a.e. }x\in \mathrm{\Omega }\text{.}$$

For every $r>0$ we denote $B_r=\{x\in {\mathbb{R}}^n:|x|<r\}$. We deal with weak solutions of the following partial differential equation in a neighbourhood of the vertex of the cone (to fix the notations we consider $\mathrm{\Omega }\cap B_1$) with possibly inhomogeneous Neumann datum:

$$\{\begin{array}{cc}\mathrm{div}(A(x)\mathrm{\nabla }u(x))=g(x,u(x)),\hfill & \text{ for every }x\in \mathrm{\Omega }\cap B_1,\hfill \\ A(x)\mathrm{\nabla }u(x)\cdot \nu (x)=f(x,u(x)),\hfill & \text{ for every }x\in B_1\cap \partial \mathrm{\Omega },\hfill \end{array}$$

where $\nu (x)$ denotes the exterior unit normal of Ω at $x\in \partial \mathrm{\Omega }$, $f\in C^1((\bar{\mathrm{\Omega }}\setminus \{0\})\times \mathbb{R})$, and $g:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function.

We say that a function $u\in H^1(B_1\cap \mathrm{\Omega })$ is a weak solution to (1.3) if, for all $\phi \in C_{\mathrm{c}}^{\infty }(B_1\cap \bar{\mathrm{\Omega }})$,

$$\underset{B_1\cap \mathrm{\Omega }}{\int }A(x)\mathrm{\nabla }u(x)\cdot \mathrm{\nabla }\phi (x)dx=-\underset{B_1\cap \mathrm{\Omega }}{\int }g(x,u(x))\phi (x)dx+\underset{B_1\cap \partial \mathrm{\Omega }}{\int }f(x,u(x))\phi (x)d{\mathcal{H}}_x^{n-1}.$$

As a technical observation, we point out that the integrals at the right hand side of the above identity are finite under the assumptions of Theorem 1.1 below in view of the Poincaré-type Inequality and the Trace Inequality proved in Corollary 2.3 and Lemma 2.5 respectively.

The use of Almgren-type frequency functions to study unique continuation properties of elliptic partial differential equations dates back to the pioneering contribution of Garofalo and Lin [9] and relies essentially on the possibility of deducing from the boundedness of the frequency quotient a doubling-type condition. Unique continuation from boundary points was investigated via Almgren-type monotonicity arguments in [1], [2], [8], [11], [16]. As far as elliptic equations with Neumann-type boundary conditions are concerned, we mention that in [15] boundary unique continuation theorems and doubling properties near the boundary were established under zero Neumann boundary conditions. The main novelty of the present paper is a strong unique continuation result for solutions whose restriction to the boundary vanishes at any order at the vertex under non-homogeneous Neumann boundary conditions, while in [15, Theorem 1.7] unique continuation from the boundary was proved for solutions vanishing on positive surface measure subsets of the boundary and satisfying a zero Neumann condition on such set. The achievement of such a result requires a combination of the monotonicity argument with a blow-up analysis for scaled solutions, in the spirit of [6], [7].

We now introduce the notation needed to define the frequency function for our setting. For $r>0$, we define

$$\begin{gathered} D(r):=r^{2-n}\underset{B_r\cap \mathrm{\Omega }}{\int }A(x)|\mathrm{\nabla }u(x){|}^{2}dx-r^{2-n}\underset{B_r\cap \partial \mathrm{\Omega }}{\int }f(x,u(x))u(x)d{\mathcal{H}}_x^{n-1} \\ +r^{2-n}\underset{B_r\cap \mathrm{\Omega }}{\int }g(x,u(x))u(x)dx \\ \text{and }H(r):=r^{1-n}\underset{\partial B_r\cap \mathrm{\Omega }}{\int }A(x)u^2(x)d{\mathcal{H}}_x^{n-1}=\underset{\mathrm{\Sigma }}{\int }A(ry)u^2(ry)d{\mathcal{H}}_y^{n-1}. \end{gathered}$$

We also introduce the “Almgren frequency function” in our framework, given by

$$\mathcal{N}(r):=\frac{D(r)}{H(r)}.$$

With this setting, the pivotal result that we obtain is an appropriate monotonicity formula with reminders, which we state as follows:

Theorem 1.1

Suppose that (1.2) holds and

$$|\mathrm{\nabla }A(x)\cdot x|⩽{\epsilon }_{r}A(x),\mathit{\text{for a.e. }}x\in B_r\cap \mathit{\Omega }\mathit{\text{, with}}\underset{r\searrow 0}{\lim }{\epsilon }_r=0,$$

$$|\mathrm{\nabla }A(x)|⩽\frac{CA(x)}{|x|},\mathit{\text{for a.e. }}x\in B_1\cap \mathit{\Omega }\mathit{\text{,}}$$

$$|f(x,t)|⩽CA(x)|x{|}^{\delta -1}|t|,\mathit{\text{for a.e. }}x\in \mathit{\Omega }\cap B_1\mathit{\text{ and any }}t\in \mathbb{R}\mathit{\text{,}}$$

$$|{\mathrm{\nabla }}_{x}f(x,t)|⩽CA(x)|x{|}^{\delta -2}|t|,\mathit{\text{for a.e. }}x\in \mathit{\Omega }\cap B_1\mathit{\text{ and any }}t\in \mathbb{R}\mathit{\text{,}}$$

$$\mathit{\text{and}}|g(x,t)|⩽CA(x)|x{|}^{\delta -2}|t|,\mathit{\text{for a.e. }}x\in B_1\cap \mathit{\Omega }\mathit{\text{ and any }}t\in \mathbb{R}\mathit{\text{,}}$$

for some $C>0$ and $\delta >0$.

Let also

$$F(x,t):=\underset{0}{\overset{t}{\int }}f(x,\tau )d\tau .$$

Let

$$u\in H^1(\mathit{\Omega }\cap B_1)\cap L^{\infty }(\mathit{\Omega }\cap B_1)$$

be a solution of (1.3) in the sense of (1.4), such that

$$u\not\equiv 0\mathit{\text{ in }}\mathit{\Omega }\cap B_r\mathit{\text{,}}$$

for all $r\in (0,1)$.

Then the following holds true.

• There exists $r_0>0$ such that

  $$H(r)>0\mathit{\text{and}}\mathcal{N}(r)+1>0\mathit{\text{for all }}r\in (0,r_0);$$

  in particular the function $\mathcal{N}$ defined in (1.6) is well defined on $(0,r_0)$.

• There exist $r_1\in (0,r_0)$ and $C_1>0$ such that

  $${\mathcal{N}}^{\prime }(r)⩾-C_1\max \{r^{\delta },{\epsilon }_r\}{r}^{-1}(2+\mathcal{N}(r))\mathit{\text{for all}}r\in (0,r_1).$$

• If also

  $$r\mapsto \frac{{\epsilon }_r}{r}\in L^1(0,r_1),$$

  then the limit

  $$\gamma :=\underset{r\searrow 0}{\lim }\mathcal{N}(r)$$

  exists, is finite and $\gamma ⩾0$.

We observe that the assumptions of Theorem 1.1 are very general and do not necessarily require the weight A to be Lipschitz continuous or the source terms f and g to be bounded. In particular, estimate (1.16) requires assumptions (1.7) and (1.8) which could be satisfied even by unbounded potentials, as for example $A(x)=\log |x|(\cos (x_n/|x|)-2)$. On the other hand, to prove that $\mathcal{N}$ is bounded and has finite limit as $r\to 0^{+}$ assumption (1.17) is also needed; we observe that (1.17) forces the boundedness of A but could be satisfied by non-Lipschitz continuous weights, like $A(x)=1+|x{|}^{\delta }$ with δ positive and small, for example.

The functions f and g can be singular as well, in accordance with (1.9) and (1.11). To allow all these possible singularities, it is crucial that the “frequency function” also takes into account the special behaviors of A, f and g, as in (1.5). Moreover, the special geometry of the cone Ω will turn out to be the cornerstone for our main estimates to hold, thus providing an interesting interplay between analytic and geometric properties of the problem.

We also observe that condition (1.14) is quite natural, since it requires that the solution is nontrivial in any neighborhood of the vertex of the cone. Furthermore, under the additional assumption that A is locally Lipschitz continuous, assumption (1.14) is satisfied by all nontrivial solutions, in light of the classical unique continuation principle in [10], see also [12] (similarly, if A satisfies a Muckenhoupt-type assumption, then (1.14) is a consequence of the unique continuation principle in [16], see also [9]).

From Theorem 1.1 and a “doubling property” method one obtains a number of results of unique continuation type. In this spirit, we first provide a unique continuation result from the vertex of the cone with respect to interior points:

Theorem 1.2

Let u be a solution of (1.3), under assumptions (1.2), (1.7), (1.8), (1.9), (1.10), (1.11), (1.13) and (1.17).

Assume also that u vanishes at the origin at any order with respect to interior points, namely that for every $k\in \mathbb{N}$

$$\underset{\mathit{\Omega }\ni x\to 0}{\lim }\frac{u(x)}{|x{|}^k}=0.$$

Then there exists $r>0$ such that

$$u\equiv 0\mathit{\text{ in }}\mathit{\Omega }\cap B_r\mathit{\text{.}}$$

If, in addition, A is locally Lipschitz continuous, then

$$u\equiv 0\mathit{\text{ in }}\mathit{\Omega }\cap B_1\mathit{\text{.}}$$

An interesting consequence of our Theorem 1.1 deals with blow-up limits. Namely, for each $\lambda >0$, we define

$$u_{\lambda }(x):=\frac{u(\lambda x)}{\sqrt{H(\lambda )}}.$$

We consider the Laplace-Beltrami operator ${\mathcal{L}}_{\mathrm{\Sigma }}:=-{\mathrm{\Delta }}_{{\mathbb{S}}^{n-1}}$ on the spherical cap Σ under null Neumann boundary conditions. By classical spectral theory, the spectrum of the operator ${\mathcal{L}}_{\mathrm{\Sigma }}$ is discrete and consists in a nondecreasing diverging sequence of eigenvalues $0={\lambda }_1(\mathrm{\Sigma })<{\lambda }_2(\mathrm{\Sigma })⩽\cdots ⩽{\lambda }_k(\mathrm{\Sigma })⩽\cdots$ with finite multiplicity.

In the following theorem we describe the limit profiles of the blowed-up family (1.22) in terms of the eigenvalues and the eigenfunctions of ${\mathcal{L}}_{\mathrm{\Sigma }}$.

Theorem 1.3

Let u be a solution of (1.3), under assumptions (1.2), (1.7), (1.8), (1.9), (1.10), (1.11), (1.13) and (1.17).

Assume that (1.14) holds true,

$$|f_t(x,t)|⩽C|x{|}^{\delta -1},\mathit{\text{for a.e. }}x\in \mathit{\Omega }\cap B_1\mathit{\text{ and any }}t\in \mathbb{R},$$

and that

$$\underset{x\to 0}{\lim }A(x)=1.$$

Then, up to a subsequence, as $\lambda \searrow 0$, we have that $u_{\lambda }$ converges strongly in $H^1(\mathit{\Omega }\cap B_1)$ to a function $\tilde{u}$ which is positively homogeneous and can be written in the form

$$\tilde{u}(x)=|x{|}^{\gamma }\psi \left(\frac{x}{|x|}\right),$$

where

$$\gamma =-\frac{n-2}{2}+\sqrt{{\left(\frac{n-2}{2}\right)}^2+{\lambda }_{k_0}(\mathit{\Sigma })}⩾0$$

for some $k_0\in \mathbb{N}\setminus \{0\}$ and ψ is an eigenfunction of the operator ${\mathcal{L}}_{\mathit{\Sigma }}$ associated to the eigenvalue ${\lambda }_{k_0}(\mathit{\Sigma })$ such that

$$\underset{\mathit{\Sigma }}{\int }{\psi }^2(x)d{\mathcal{H}}_x^{n-1}=1.$$

From Theorem 1.3, one can also obtain a unique continuation result from the vertex of the cone with respect to boundary points:

Theorem 1.4

Let u be a solution of (1.3), under assumptions (1.2), (1.7), (1.8), (1.9), (1.10), (1.11), (1.13), (1.17), (1.23) and (1.24).

Assume also that u vanishes at the origin at any order with respect to boundary points, namely that for every $k\in \mathbb{N}$

$$\underset{\partial \mathit{\Omega }\ni x\to 0}{\lim }\frac{u(x)}{|x{|}^k}=0.$$

Then there exists $r>0$ such that

$$u\equiv 0\mathit{\text{ in }}\mathit{\Omega }\cap B_r\mathit{\text{.}}$$

If, in addition, A is locally Lipschitz continuous, then

$$u\equiv 0\mathit{\text{ in }}\mathit{\Omega }\mathit{\text{.}}$$

We stress that while (1.19) is assumed for interior points, we have that hypothesis (1.27) focuses on boundary points.

The rest of the article is organized as follows. Section 2 presents a number of ancillary results, to be exploited in the proofs of the main theorems. In particular, we will collect there some observations on the geometry of the cone and suitable functional inequalities.

The proof of Theorem 1.1 is presented in Section 3 and will serve as a pivotal result for the main theorems of this paper. Namely, Theorem 1.2 will be proved in Section 4, Theorem 1.3 will be proved in Section 5, and Theorem 1.4 will be proved in Section 6.