Approximation of Elliptic Equations with Interior Single-Point Degeneracy and Its Application to Weak Unique Continuation Property

The unique continuation properties for uniformly elliptic equations have been extensively studied in the literature ([1, 2, 3, 7, 9, 10, 12, 17, 18, 19, 21, 22, 23, 24, 26, 27]). There are two types of unique continuation properties: the strong unique continuation property (SUCP) and the weak unique continuation property (WUCP). Below, we briefly recall these two properties.

Let $P(x,\partial)$ be a uniformly elliptic operator. The strong unique continuation property (SUCP) states that if $P(x,\partial)u=0$ in a domain $\Omega\subset\mathbb{R}^{N}$, and there exists a point $x_{0}\in\Omega$ such that $u$ vanishes to infinite order, meaning

where $B_{r}(x_{0})$ denotes a ball in $\Omega$ centered at $x_{0}$ with radius $r$, then $u\equiv 0$ in $\Omega$.

The weak unique continuation property (WUCP) states that if $P(x,\partial)u=0$ in $\Omega$, and $u=0$ on an open subset $\omega\subset\Omega$, then $u\equiv 0$ in $\Omega$. It is easy to see that the WUCP requires less stringent conditions compared to the SUCP.

Furthermore, by introducing quantitative descriptions, we can obtain the quantitative weak unique continuation property (QWUCP), which provides quantitative estimates of a solution’s local behavior, refining the conditions of the WUCP. Specifically, the QWUCP can be described as the form

where $D\subset\Omega$ is an open domain satisfying certain conditions (e.g., boundary conditions), and $C>0$ is a constant independent of $u$.

Notably, unique continuation does not hold universally for all uniformly elliptic equations ([27]). Furthermore, the analysis of unique continuation properties becomes significantly more challenging for degenerate elliptic equations compared to uniformly elliptic ones. Currentlty, there are two methods — the three-ball theorem ([1, 17, 23, 24, 25, 26, 30]) and Carleman estimates ([2, 3, 7, 21, 22, 27]), which are effective in dealing with certain special cases ([3, 15, 16, 19, 20]).

The three-ball theorem states that for a harmonic (or subharmonic) function $u(x)$ defined in a region containing three concentric balls $B_{r_{1}}$, $B_{r_{2}}$ and $B_{r_{3}}$ with $r_{1}<r_{2}<r_{3}$, the maximum value $H(r)$ of $u(x)$ on the intermediate sphere $B_{r_{2}}$ can be bounded by a weighted geometric mean of the maximum values on the inner and outer spheres:

where $\mu\in(0,1)$ is determined by the radii. The three-ball theorem is developed on the basis of the double-ball theorem, which was originally introduced by Garofalo and Lin in [17]. The authors in [26] provides a detailed introduction to the double-ball theorem, the three-ball theorem, and their applications in unique continuation. In [16], the authors primarily investigates the unique continuation properties of a specific class of second-order elliptic operators that degenerate on manifolds of arbitrary codimension, using the double-ball theorem. The focus is on the model operator

in $\mathbb{R}^{n}\times\mathbb{R}^{m}$, which is elliptic outside a degeneracy manifold $(\left\{0\right\}\times\mathbb{R}^{m})$ but degenerates on it. The authors establishes SUCP using Carleman estimates and introduces a quantitative version of SUCP that bypasses Carleman estimates, instead relying on the double-ball theorem. Similarly, in [29], the double-ball theorem is also applied to study the unique continuation properties of solutions to degenerate Schrödinger equations influenced by singular potentials and weighted settings. In [4], SUCP is established for a class of degenerate elliptic operators with Hardy-type potentials using Carleman estimates. This work extends the results of [16] but does not yield a quantitative conclusion. Notably, the three-ball theorem appears to be more effective for studying quantitative weak unique continuation properties.

In this paper, we shall consider the quantitative weak unique continuation properties for the elliptic equation with degenerate interior point by approximation. It is well known that the solution spaces of degenerate elliptic equations belong to weighted Sobolev spaces ([9, 10, 12, 13, 28]). A natural approach is to approximate a solution of a degenerate elliptic equation by a sequence of solutions to uniformly elliptic equations ([9, 10]). This method is feasible in weighted spaces and applies to high-dimensional cases, but it heavily relies on the Calderón-Zygmund decomposition, which can compromise certain desirable properties of the weight function. For instance, the approximating weight functions may lack differentiability, which is crucial when using the three-ball theorem to prove QWUCP, requiring the approximating weight functions to be at least Lipschitz continuous. Another approximation approach, similar to that in [5, 6, 31], involves constructing a non-degenerate coefficient $|x+\epsilon|^{\alpha}$ over the entire domain $\Omega$ to approximate the degenerate coefficient $|x|^{\alpha}$. However, this method is suitable for one-dimensional degenerate equations but not for the high-dimensional problems we aim to study. For the problem we consider in this paper, local estimates are required to approximate the solution (see Lemma 3.3).

While the idea of approximation has been utilized in many works, our method is fundamentally different from those in the existing literature. First, one of our main contributions is the introduction of an alternative approximation method for a specific class of weight functions with a single degenerate interior point. Our approximation is achieved by constructing a carefully designed non-degenerate weight function to approximate the degenerate weight function within a small local region $B_{\epsilon}$ rather than the entire domain $\Omega$. This ensures that the weight function remains differentiable in high-dimensional settings. For a detailed discussion, refer to Section 3. Second, in the proof of QWUCP, we consider two cases: $0\in\omega$ and $0\notin\omega$. For case $0\in\omega$, the result is obtained using the three-ball theorem at both degenerate and non-degenerate points. For the more challenging case $0\notin\omega$, we apply Schauder estimates to address the difficulties arising from the degenerate point being excluded from $\omega$. Finally, we derive a quantitative WUCP result.

It is worth noting that in most works (see [5, 6, 31]), the SUCP is typically achieved using the double-ball theorem. However, this paper employs the more robust three-ball theorem. Although we do not present results on SUCP here, we have demonstrated it in another working paper using an annular estimate method.

We organize the paper as follows: In Section 2, we present several well-posedness results. In Section 3, we provide a detailed explanation of the construction of the approximation and introduce the preliminary lemmas required for proving the three-ball theorem at the degenerate point. In Section 4, we establish the three-ball theorem at the degenerate point and prove QWUCP for two cases:$0\in\omega$ and $0\notin\omega$.

Let us consider the following equation

where $\Omega\subset\mathbb{R}^{N}$ ($N\geq 2$) is a domain containing the origin ($0\in\Omega$), and its boundary $\partial\Omega$ is of class $C^{2}$. The weight function is given by $w=|x|^{\alpha}$, with a fixed $\alpha\in(0,2)$, and $f$ is a given function such that $f\in L^{2}(\Omega;w^{-1})$. The weighted Sobolev space $L^{2}(\Omega;w)$ defined for every $w>0$ almost everywhere as:

The inner product on $L^{2}(\Omega;w)$ is

and the norm on $L^{2}(\Omega;w)$ is

It is well known that $(L^{2}(\Omega;w),(\cdot,\cdot)_{L^{2}(\Omega;w)})$ (see [14]) is a Hilbert space and $(L^{2}(\Omega;w),\|\cdot\|_{L^{2}(\Omega;w)})$ is a Banach space.

Set

where $\frac{\partial u}{\partial x_{i}},i=1,\cdots,N$ are the distribution partial derivatives, the inner product on $H_{w}^{1}(\Omega)$ is

and the norm is

Define

where $\mathcal{D}(\Omega)=C_{0}^{\infty}(\Omega)$ is the space of test functions. Denote by $H_{w}^{-1}(\Omega)$ the dual space of $H_{w,0}^{1}(\Omega)$. This space is a subspace of $\mathcal{D}^{\prime}(\Omega)$, the space of distributions on $\Omega$. It is well known that $(H_{w}^{1}(\Omega),(\cdot,\cdot)_{H_{w}^{1}(\Omega)})$ forms a Hilbert space, while $(H_{w}^{1}(\Omega),\|\cdot\|_{H_{w}^{1}(\Omega)})$ is a Banach space.

Next, we aim to establish some well-posedness results for equation (2.1). First, we introduce some notations that will be used:

Similar to the proof of Lemma 3.1 in [32] or Proposition 2.1 (1) in [28], we can easily derive the following weighted Hardy inequality.

From (2.2), it is evident that if $u\in H_{w,0}^{1}(\Omega)$, then $u\in L^{2}(\Omega)$. $\square$

From Lemma 2.1, it is also evident that space $H_{w,0}^{1}(\Omega)$ is embedded into space $L^{2}(\Omega)$. Next, we will prove that this embedding is compact.

Since $H_{w,0}^{1}(\Omega)$ is continuously embedded in $L^{2}(\Omega)$ by Lemma 2.1, $L^{2}(\Omega)^{*}\subset$ $H_{w,0}^{1}(\Omega)^{*}$ and hence $\left\{u_{n}\right\}$ converges weakly to zero in $L^{2}(\Omega)$.

Consider $\epsilon>0$. If $\left\{u_{n}\right\}$ does not converge weakly to zero in $W^{1,2}\left(\Omega^{\epsilon}\right)$, there exist $f\in W^{1,2}\left(\Omega^{\epsilon}\right)^{*}$, a subsequence $\left\{u_{n_{k}}\right\}$ and $\delta>0$ such that $\left|f\left(u_{n_{k}}\right)\right|\geq\delta$ for all $n_{k}$. Passing to a further subsequence if necessary, we can suppose that $\left\{u_{n_{k}}\right\}$ converges weakly to an element $v$ in $W^{1,2}\left(\Omega^{\epsilon}\right)$. Thus $\left\{u_{n_{k}}\right\}$ converges weakly to $v$ in $L^{2}\left(\Omega^{\epsilon}\right)$ and so $v=0$ a.e. on $\Omega^{\epsilon}$ since $\left\{u_{n}\right\}$ converges weakly to zero in $L^{2}(\Omega)$ and hence also on $L^{2}\left(\Omega^{\epsilon}\right)$. But then $f\left(u_{n_{k}}\right)\rightarrow f(v)=f(0)=0$ as $n_{k}\rightarrow\infty$, contradicting with $\left|f\left(u_{n_{k}}\right)\right|\geq\delta$ for all $n_{k}$. Hence $\left\{u_{n}\right\}$ converges weakly to zero in $W^{1,2}\left(\Omega^{\epsilon}\right)$ and therefore $\left\|u_{n}\right\|_{L^{2}\left(\Omega^{\epsilon}\right)}\rightarrow 0$ as $n\rightarrow\infty$ . From the above it follows that

But from [8] and (2.3) in Corollary 2.2, we have

then (taking $q>2$)

The weak convergence of $\left\{u_{n}\right\}$ in $H_{w,0}^{1}(\Omega)$ and (2.6) imply that this sequence is bounded in $L^{q}(\Omega)$, then (note that $q>2$)

Letting $\epsilon\rightarrow 0^{+}$ in (2.5) shows that $\left\|u_{n}\right\|_{L^{2}(\Omega)}\rightarrow 0$ as $n\rightarrow\infty$, completing the proof. $\square$

Next, we use the Lax-Milgram theorem to show that equation (2.1) has a unique weak solution $u\in H_{w,0}^{1}(\Omega)$ in the sense of

for all $v\in H_{w,0}^{1}(\Omega)$.

On one hand, we have

and $B_{w}[u,u]=\|u\|_{H_{w,0}^{1}(\Omega)}$. On the other hand, we have

by Cauchy inequality and (2.3) in Corollary 2.2, i.e., $f:H_{w,0}^{1}(\Omega)\rightarrow\mathbb{R}$ is a bounded linear functional on $H_{w,0}^{1}(\Omega)$.

Finally, by the Lax-Milgram theorem, we obtain that there exists a unique $u\in H_{w,0}^{1}(\Omega)$ satisfying (2.1). $\square$

Our approach is to approximate the solution of a degenerate equation by a sequence of solutions to non-degenerate equations that satisfy the uniform ellipticity condition.

Let

Then it is clear that $w_{\epsilon}\in C^{0,1}(\overline{\Omega})$ since $\alpha\in(0,2)$, $w_{\epsilon}$ is a radial convex function on $\mathbb{R}^{N}$ and nondecreasing on $[0,\infty)$, and $(\frac{\epsilon}{2})^{\alpha}\leq w_{\epsilon}\leq\epsilon^{\alpha}$ in $B_{\epsilon}$, and

It is worth noting that, our approximation methods is different from the one used in other literature (see [5, 6, 31]), such as setting a non-degenerate coefficient $|x+\epsilon|^{\alpha}$ to approximate the degenerate coefficient $|x|^{\alpha}$, which takes the form $|x+\epsilon|^{\alpha}$ over the entire domain $\Omega$. However, in this paper, our setup of $w_{\epsilon}$ ensures that the approximate coefficients do not depend on $\epsilon$ outside $B_{\epsilon}$ while approximating the original degenerate coefficient within $B_{\epsilon}$. This allows us to achieve better estimates of the solution and obtain improved regularity results, even in the high-dimensional case.

For each $k\in\mathbb{N}$, we denote $w_{\frac{1}{k}}$ by $w_{k}$ and consider the following approximate equation

with $f_{k}\in L^{2}(\Omega;w_{k}^{-1})$. We say that $u_{k}\in H_{w_{k},0}^{1}(\Omega)$ is a weak solution of (3.8), if

for all $v\in H_{w_{k},0}^{1}(\Omega)$.

We note that $H_{w_{k},0}^{1}(\Omega)=H_{0}^{1}(\Omega)$ since $(\frac{1}{k})^{\alpha}\leq w_{k}\leq m^{\alpha}$ (see (2.3)) for each $k\in\mathbb{N}$, where $H_{0}^{1}(\Omega)$ is the classical Sobolev spaces.

As Lemma 2.1, we provide a proof of the Hardy inequality for the non-degenerate equation.

Since $w_{\epsilon}\in C^{0,1}(\overline{\Omega})$ (i.e., $w_{\epsilon}\in W^{1,\infty}(\Omega)$), we have

Then

by $\alpha\in(0,2)$ and $|x|^{2}\leq\frac{3}{4}|x|^{2}+\frac{1}{4}\epsilon^{2}$ on $B_{\epsilon}$. This shows that

Finally, by (3.9) and $\frac{1}{4}\epsilon^{2}\leq\frac{3}{4}|x|^{2}+\frac{1}{4}\epsilon^{2}\leq m^{2}$ in $\Omega$ (see (2.3) for $m$), we get (3.10). $\square$

To prove that the solution of the non-degenerate equation converges weakly in the solution space to the solution of the degenerate equation, we first show that the approximate solutions are bounded.

Now, we are ready to show the existence of the solution for degenerate equation (2.1) by using the approximation via the solutions of the non-degenerate equations.

Now, we prove $\widetilde{u}_{0}$ satisfies the equation (2.1) with $f\in L^{2}(\Omega;w^{-1})$.

Let $\psi\in\mathcal{D}(\Omega)$. Let $\epsilon>0$. By (3.11), there exists $k_{0}\in\mathbb{N}$, such that

Since $u_{k}$ is a solution of (3.8) for each $k\in\mathbb{N}$, we have

Note that $w_{k}=w$ on $\Omega\setminus B_{\frac{1}{k}}$ for $k\geq k_{0}$, we have

By Lemma 3.2 we have

where $C_{\psi,N,\alpha}>0$ is a constant that depends only on $\psi,\Omega,N$ and $\alpha$, and $w_{k}(B_{\frac{1}{k}})^{\frac{1}{2}}=\left(\int_{B_{\frac{1}{k}}}w_{k}\mathrm{d}x\right)^{\frac{1}{2}}$. Hence we can assume

On the other hand, by $w\leq w_{k}$, and by the same argument as for (3.16), we have

hence we also can assume

From (3.13), (3.14), (3.15), (3.17) and (3.18), we obtain