Monotonicity-based inversion of fractional semilinear elliptic equations with power type nonlinearities

Abstract

We investigate the monotonicity method for fractional semilinear elliptic equations with power type nonlinearities. We prove that if-and-only-if monotonicity relations between coefficients and derivatives of the Dirichlet-to-Neumann map hold. Based on the strong monotonicity relations, we study a constructive global uniqueness for coefficients and inclusion detection for the fractional Calderón type inverse problem. Meanwhile, we can also derive the Lipschitz stability with finitely many measurements. The results hold for any $n\ge 1$.

Appendices

Appendix A. The $L^p$-estimate for the fractional laplacian

Let us review the other estimates for solutions to the fractional Laplacian: The $L^p$ estimate. Before doing so, let us recall some fundamental properties for the Riesz potential.

Proposition A.1

(Riesz potential) For $0<s<1$ with $n>2s$. Let V and F satisfy

$$\begin{aligned} V=(-\Delta )^{-s}F \text { in }\mathbb {R}^n, \end{aligned}$$

in the sense that V is the Riesz potential of order 2s of the function F.

• (a) If $F\in L^1(\mathbb {R}^n)$, then there exists a constant $C>0$ depending only on n and s such that

  $$\begin{aligned} \Vert V \Vert _{L^p_{\mathrm {w}}(\mathbb {R}^n)} \le C\Vert F \Vert _{L^1(\mathbb {R}^n)}, \end{aligned}$$

  where $L^p_{\mathrm {w}}$ denotes the weak-$L^p$ norm and $p=\frac{n}{n-2s}$.

• (b) For $r\in (1,\frac{n}{2s})$, $F\in L^r(\mathbb {R}^n)$, then there exists a constant $C>0$ depending only on n, s, and r such that

  $$\begin{aligned} \Vert V \Vert _{L^p(\mathbb {R}^n)} \le C\Vert F \Vert _{L^r(\mathbb {R}^n)}, \end{aligned}$$

  where $p=\frac{nr}{n-2rs}$.

• (c) For $r\in (\frac{n}{2s}, \infty )$, then there exists a constant $C>0$ depending only on n, s, and r such that

  $$\begin{aligned} {[}u]_{C^\alpha (\mathbb {R}^n)}\le C\Vert F \Vert _{L^r(\mathbb {R}^n)}, \end{aligned}$$

  where $\alpha =2s-\frac{n}{p}$ and $[u]_{C^\alpha (\mathbb {R}^n)}$ is the seminorm given in Sect. 2.

Proof

Parts (a) and (b) are classical results for the Riesz potential, and the proof can be found in Stein’s book [61, Chapter V]. For (c), we refer readers to [61, p.164] and [16]. $\square$

Furthermore, we have the following Hölder estimate for the fractional Laplacian, which was shown in [56, Proposition 1.7]. We state the result in the following proposition and the proof can be found in [56].

Proposition A.2

($C^\beta$-estimate) For $n\ge 1$, $0<s<1$, let $\Omega \subset \mathbb {R}^{n}$ be a bounded domain with $C^{1,1}$ boundary $\partial \Omega$. Let $h\in C^\alpha (\Omega _e)$ for some $\alpha \in (0,1)$. Let w be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s w=0 &{} \text { in }\Omega , \\ w=h &{} \text { in }\Omega _e. \end{array}\right. } \end{aligned}$$

Then the solution $w\in C^\beta (\mathbb {R}^n)$, where $\beta =\min \{s,\alpha \}$, and

$$\begin{aligned} \Vert w \Vert _{C^\beta (\mathbb {R}^n)}\le C\Vert h \Vert _{C^\alpha (\Omega _e)}, \end{aligned}$$

for some constant $C>0$ depending only on $\Omega$, $\alpha$, and s.

The following proposition was also proved in [56], which is an important result in the proof of our Runge approximation (Theorem 3.2). We state the result and prove it for the sake of completeness. The proof is based on the preceding properties of the Riesz potential, the maximum principle for the fractional Laplacian and the $C^\beta$-estimate (Proposition A.2).

Proposition A.3

For $n\ge 1$, $0<s<1$, let $\Omega \subseteq \mathbb {R}^{n}$ be a bounded domain with $C^{1,1}$ boundary $\partial \Omega$. For $F\in L^r (\Omega )$, let v be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s v =F &{} \text { in }\Omega , \\ v=0 &{} \text { in }\Omega _e, \end{array}\right. } \end{aligned}$$

(A.1)

then we have:

• (a) Let $n>2s$, $r=1$, and $p\in [1, \frac{n}{n-2s})$ be an arbitrary number, then there exists a constant $C>0$ independent of v and F such that

  $$\begin{aligned} \Vert v \Vert _{L^p(\Omega )}\le C\Vert F \Vert _{L^1(\Omega )}. \end{aligned}$$

• (b) Let $n>2s$, $r\in (1,\frac{n}{2s})$ and $p=\frac{nr}{n-2rs}$, then there exists a constant $C>0$ independent of v and F such that

  $$\begin{aligned} \Vert v \Vert _{L^p(\Omega )}\le C\Vert F \Vert _{L^p (\Omega )}. \end{aligned}$$

• (c) Let $n>2s$, $r\in (\frac{n}{2s},\infty )$, and $\beta =\min \left\{ s, 2s-\frac{n}{r} \right\}$, then there exists a constant $C>0$ independent of v and F such that

  $$\begin{aligned} \Vert v \Vert _{C^\beta (\Omega )}\le C\Vert F \Vert _{L^r (\Omega )}. \end{aligned}$$

• (d) Let $n=1$, $s\in [\frac{1}{2},1)$, $r\ge 1$, and any $p<\infty$, then there exists a constant $C>0$ independent of v and F such that

  $$\begin{aligned} \Vert v \Vert _{L^p(\Omega )}\le C\Vert F \Vert _{L^r (\Omega )}. \end{aligned}$$

Proof

(a) Let us extend the function F by 0 outside $\Omega$, and we still denote the function as F. Let V be the solution of

$$\begin{aligned} (-\Delta )^s V=|F| \text { in }\mathbb {R}^n, \end{aligned}$$

so that $V=(-\Delta )^{-s}|F|$ in $\mathbb {R}^n$, where $(-\Delta )^{-s}|F|$ is the Riesz potential of |F|. By the definition of the Riesz potential, we have $V\ge 0$ in $\Omega _e$. Via the maximum principle, we obtain that $|v|\le V$ in $\Omega$. By applying Proposition A.1, one can see that

$$\begin{aligned} \Vert v \Vert _{L^q_{\mathrm {w}}(\Omega )}\le \Vert V \Vert _{L^q_{\mathrm {w}}(\Omega )}\le C\Vert F \Vert _{L^1(\Omega )}, \end{aligned}$$

if $F\in L^1(\Omega )$ and for some constant $C>0$ independent of v and F. Thus, one has

$$\begin{aligned} \Vert v \Vert _{L^r(\Omega )} \le C\Vert F \Vert _{L^1(\Omega )}, \end{aligned}$$

for some constant $C>0$ independent of v and F. This proves (a).

(b) Similarly, the proof of (b) can be completed by using the result (b) in Proposition A.1 and the maximum principle for the fractional Laplacian as before. Furthermore, when $r=\frac{n}{2s}$, it is easy to see that $F\in L^{r}(\Omega )\subset L^{\widetilde{r}}(\Omega )$, for any $\widetilde{r}\in [1,r]$ (since $\Omega$ is bounded). We still have the $L^p$ estimate for the solution in the borderline case $r=\frac{n}{2s}$.

(c) Let us write $v=\widetilde{v}+ w$, where $\widetilde{v}$ and w are given by

$$\begin{aligned} \widetilde{v} =(-\Delta )^{-s}F \text { in }\mathbb {R}^n, \end{aligned}$$

(A.2)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s w =0 &{} \text { in }\Omega , \\ w=\widetilde{v} &{} \text { in }\Omega _e. \end{array}\right. } \end{aligned}$$

(A.3)

By using (A.2) and Proposition A.1 (c), there exists a constant $C>0$ depending only on n, s, and r such that

$$\begin{aligned} {[}\widetilde{v}]_{C^\alpha (\mathbb {R}^n)}\le C\Vert F \Vert _{L^r(\mathbb {R}^n)}, \quad \text { where }\alpha =2s-\frac{n}{r}. \end{aligned}$$

Since $\Omega$ is bounded and F is compactly supported, one has $\widetilde{v}$ decays at infinity. This implies that

$$\begin{aligned} \Vert \widetilde{v} \Vert _{C^\alpha (\mathbb {R}^n)}\le C\Vert F \Vert _{L^r(\mathbb {R}^n)}, \quad \text { where }\alpha =2s-\frac{n}{r}, \end{aligned}$$

(A.4)

for some constnat $C>0$ depending only on n, s, r and $\Omega$.

On the other hand, we can apply Proposition A.2 to derive the Hölder estimate for the solution w of (A.3) that

$$\begin{aligned} \Vert w \Vert _{C^\beta (\mathbb {R}^n)}\le C\Vert \widetilde{v} \Vert _{C^\alpha (\Omega _e)}, \end{aligned}$$

(A.5)

for some constant $C>0$ depending only on $\Omega$, $\alpha$, and s, where

$$\begin{aligned} \beta =\min \{\alpha , s\}=\min \left\{ s, 2s-\frac{n}{r}\right\} . \end{aligned}$$

Combining with (A.4) and (A.5), we can obtain the Hölder estimate for the solution $v=\widetilde{v}+w$ such that (c) holds. Moreover, since $v\in C^\beta (\overline{\Omega })$ with $v=0$ in $\Omega _e$, we must have $v\in L^p(\mathbb {R}^n)$ for any $p\ge 1$.

(d) Notice that for $s<\frac{1}{2}$, we have $n=1>2s$ automatically, such that the case (d) holds by applying the results either (a) or (b). On the other hand, for the case $1=n\le 2s$, this implies that $\frac{1}{2}\le s <1$. Under this situation, any bounded domain is of the form $\Omega =(a,b)\subset \mathbb {R}$. By [2], the Green function G(x, y) for the exterior value problem (A.1) is explicit. Furthermore, $G(\cdot ,y)\in L^\infty (\Omega )$ when $s>\frac{1}{2}$ and $G(x,y)\in L^r(\Omega )$ for any $r<\infty$ when $s=\frac{1}{2}$. Therefore, one has

$$\begin{aligned} \Vert v \Vert _{L^\infty (\Omega )}\le C\Vert F \Vert _{L^1(\Omega )}, \end{aligned}$$

for some constant $C>0$ independent of v and F, where $n<2s$. For the case $n=2s$, we have either

$$\begin{aligned} \Vert v \Vert _{L^p(\Omega )}\le C\Vert F \Vert _{L^1(\Omega )}, \quad \text { for all }p<\infty , \end{aligned}$$

or

$$\begin{aligned} \Vert v \Vert _{L^\infty (\Omega )}\le C\Vert F \Vert _{L^r(\Omega )}, \quad \text { for }r>1, \end{aligned}$$

for some constant $C>0$ independent of v and F. $\square$

Remark A.4

From the $L^p$ estimate of s-harmonic functions, we have:

• (a) No matter what exponent $r\ge 1$ and what space dimension n are, for any $F\in L^r(\Omega )$ with $\Omega \subset \mathbb {R}^n$ in the statement of Proposition A.3, then we can always conclude that the solution v of (A.1) must belong to $L^p(\mathbb {R}^n)$, for some $p> 1$.

• (b) Moreover, since the domain $\Omega$ is bounded in $\mathbb {R}^n$, then we can confine the exponent p in the region $p \in (1,2)$. The condition $p \in (1,2)$ plays an essential role in order to prove Proposition 3.3 (see [9, Section 4] for more detailed discussions about the strong uniqueness of the s-harmonic function). Meanwhile, we also need to use the $L^p$-estimate to prove the Runge approximation via the strong uniqueness for the fractional Laplacian in Sect. 3.

Appendix B. The maximum principle

We review the known maximum principle for the fractional Laplacian in the end of this work. These results were shown in [5, 54] for the fractional Laplacian equation and [45, 46] for the fractional Schrödinger equation. For the sake of convenience, we state the results as follows.

Proposition B.1

(The maximum principle) Let $\Omega \subset \mathbb {R}^n$, $n\ge 1$ be a bounded domain with Lipschitz boundary $\partial \Omega$, and $0<s<1$. Let $v\in H^s(\mathbb {R}^n)$ be the unique solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s v=F &{} \text { in }\Omega , \\ v=g &{}\text { in }\Omega _e. \end{array}\right. } \end{aligned}$$

Suppose that $0\le F\in L^\infty (\Omega )$ in $\Omega$ and $0\le g \in L^\infty (\Omega _e)$ in $\Omega _e$. Then $v\ge 0$ in $\Omega$. Moreover, if $g\not \equiv 0$ in $\Omega _e$, then $v>0$ in $\Omega$.