Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon

Abstract

In an open, bounded Lipschitz polygon $\Omega \subset \mathbb {R}^2$, we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in $\Omega$. The boundary conditions on each edge of $\partial \Omega$ are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in $H^1(\Omega )$, quantized tensor formats and certain deep neural networks.

1 Introduction

The efficient numerical approximation of solutions of elliptic PDEs in corner domains has received much attention in the past decades. It was motivated by many applications in engineering and the sciences, by the development of the finite element methods (FEM) and their analysis, and by the advance of elliptic regularity theory in corner domains. As is well known by now (see, e.g., [8, 19] and the references there), mathematical statements of high regularity of solutions in Sobolev spaces require either the use of the corner weights (as in [8, 19]) or the use of Besov–Triebel–Lizorkin spaces with summability indices $0<p<1$ as developed e.g. in [10] and the references there. The former regularity results facilitate the development of optimal order FEM approximations on so-called graded meshes whereas the latter are at the core of approximation classes for adaptive FEM (AFEM). See, e.g., [21] and the references there.

These developments pertained to the so-called h-FEM, which achieves convergence by (possibly adaptive) mesh refinement, at fixed polynomial order of the elements. An alternative concept is furnished by the more general, so-called hp-FEM. There, mesh refinements and polynomial degree increase are combined. It has been proved in 80ies in a number of landmark papers by Babuška and Guo that hp-FEM can achieve exponential rates of convergence for linear elliptic PDEs in polygonal domains, with analytic data (source term and inhomogeneous boundary data). A key ingredient in the theory is the weighted analytic regularity of solutions. Regularity results of this type in corner domains for linear elliptic PDEs appeared also in the 80ies. We mention only [2, 5, 9].

While analyticity of solutions of nonlinear, elliptic PDEs with analytic data (coefficients, nonlinearity, domain) are classical (e.g. [20, Chap. 5.8] and the references there), results on analytic regularity for nonlinear elliptic PDE in corner domains and with singular nonlinearity appeared recently in [11, 15, 16, 18].

In particular, in [16] weighted, analytic regularity of solutions to nonlinear Schrödinger eigenvalue problems with cubic nonlinearity and with singular potential exhibiting a point singularity at the origin was established. This problem arises in models of electron structure in atoms. In [16], analytic regularity is quantified in terms of weight functions being given as powers of the distance to the origin, which is assumed to coincide with the position of the nucleus.

To establish weighted analytic regularity and exponential convergence of hp-FE discretizations for a class of semilinear, scalar elliptic PDEs in a polygon $\Omega$ with analytic nonlinearity, and subject to analytic data, is the topic of the present paper.

This is achieved by localization of the PDE near corners of $\Omega$. As in [16], a scale of corner-weighted Sobolev spaces with radial weight functions, being powers of the distance to the corners of $\Omega$, is employed. The bootstrapping argument which quantifies the derivative bounds on the weak solution of the semilinear boundary value problem is an adaptation of the arguments in [16], with particular modifications to handle arbitrary, polynomial growth at infinity of the nonlinear term. A novel combinatorial inequality is used in the inductive bootstrapping argument to maintain tight, analytic, bounds on the corner-weighted norms of weak solutions with respect to the differentiation order.

The layout of the present paper is as follows. In Sects. 1.1 and 1.2 we provide a variational formulation of the semilinear elliptic boundary value problem, recapitulate notation and, in Sect. 1.3, definitions and basic properties of corner-weighted function spaces of Sobolev type. In Sect. 2, we recapitulate corner-weighted regularity shift results for the linear Poisson equation. Section 3 then contains the proof of the main result of the present paper: weighted analytic regularity for the weak solutions of the semilinear elliptic PDE with analytic in the polygon $\overline{\Omega }$ forcing. The final Sect. 4 then addresses some direct consequences from the analytic regularity results: exponential approximability of the weak solution by hp-finite element methods, and exponential bounds on Kolomogorov n-widths of the (nonlinear) solution manifold. Appendix A contains some auxiliary results supporting the proof of the main result.

1.1 Problem formulation

Let $\Omega \subset \mathbb {R}^2$ be a polygon with $n\ge 3$ vertices $c_i$ and n straight open edges $\Gamma _i$. We assume vertices and edges to be enumerated in clockwise order, with indexing modulo n, i.e. $c_i = c_{i+n}$ for all $i\in \mathbb {Z}$.

For $1\le i\le n$, $\Gamma _i$ connects $c_i$ and $c_{i+1}$ so that $\partial \Gamma _i = \{ c_i,c_{i+1}\}$. We denote by $\omega _i\in (0,2\pi )$ the internal angle at $c_i$. In particular, then, the polygon $\Omega$ has a Lipschitz boundary $\Gamma = \partial \Omega$ [8].

We study the analytic regularity of solutions of the following semilinear elliptic PDE in $\Omega$

$$\begin{aligned} -\Delta u+ \lambda u^{2k+1} = f \quad \text{ in } \quad \Omega . \end{aligned}$$

(1.1)

Here, $\lambda \ge 0$ and $k\in \mathbb {N}_0$, with the case $\lambda =0$ corresponding to the linear Poisson equation, which was studied in [2] and the case $k=0, \lambda >0$ corresponding to a linear, reaction-diffusion boundary value problem.

The PDE (1.1) is completed by boundary conditions: on edge $\Gamma _i$, we impose either homogeneous Dirichlet or homogeneous Neumann BCs.

$$\begin{aligned} \gamma _0(u) = 0 \quad \text{ or }\quad \gamma _1(u) = 0 \quad \text{ on } \;\; \Gamma _i . \end{aligned}$$

(1.2)

Here, $\gamma _0$ and $\gamma _1$ are the weak trace and normal derivative operators, respectively. We denote the BCs in (1.2) abstractly as $B(u)=0$, with the boundary operator $B|_{\Gamma _i} \in \{ \gamma _0,\gamma _1 \}$ depending on whether $\Gamma _i$ is a Dirichlet or Neumann edge. We collect the indices $i\in \{1,\ldots ,n\}$ corresponding to Dirichlet edges in the index set ${\mathcal {D}}$, and the remaining indices in ${\mathcal {N}}$ (with membership in ${\mathcal {D}}$ or ${\mathcal {N}}$ again understood modulo n), so that ${\mathcal {D}}$ and ${\mathcal {N}}$ are a partition of $\{1,\ldots ,n\}$, and $B|_{\Gamma _i} = \gamma _0$ for $i\in {\mathcal {D}}$. We assume throughout ${\mathcal {D}}\ne \emptyset$, i.e.

$$\begin{aligned} \text{ there } \text{ is } \text{ at } \text{ least } \text{ one } \text{ edge }\, \Gamma _i\, \text{ where }\, \gamma _0(u) = 0\;. \end{aligned}$$

(1.3)

With these conventions, we set

$$\begin{aligned} H^1_D(\Omega ) = \{ v\in H^1(\Omega ): \gamma _0(v) = 0 \;\text{ on }\; \Gamma _i, i\in {\mathcal {D}}\}\;. \end{aligned}$$

(1.4)

Due to (1.3), $\sum _{i\in {\mathcal {D}}} |\Gamma _i| > 0$, and there holds the Poincaré inequality: there exists a constant $C>0$ such that

$$\begin{aligned} \forall v\in H^1_D(\Omega ): \quad \Vert v \Vert _{L^2(\Omega )} \le C \Vert \nabla v \Vert _{L^2(\Omega )} \;. \end{aligned}$$

(1.5)

In particular, therefore, on $H^1_D(\Omega )$, the expression $\Vert \nabla v \Vert _{L^2(\Omega )}$ is a norm.

We will show that given data f in a corner-weighted, analytic space $B^0_{\underline{\beta }}(\Omega )\cap L^2(\Omega )$, any generalized solution $u\in H^1_D(\Omega )$ to (1.1) will be contained in a corresponding analytic space $B^2_{\underline{\beta }}(\Omega )$. Here, the corner-weighted analytic function classes $B^2_{\underline{\beta }}$ and $B^0_{\underline{\beta }}$ will be introduced later in Sect. 1.3.1.

The notion “generalized solution” refers to variational solutions which are defined as follows.

Given $f\in L^2(\Omega )$, we seek $u\in H^1_D(\Omega )$ such that

$$\begin{aligned} \forall v\in H^1_D(\Omega ): \quad \int _{\Omega }\nabla u\cdot \nabla v + \lambda u^{2k+1}v \ d{\varvec{x}} = \int _{\Omega }fv \ d{\varvec{x}} . \end{aligned}$$

(1.6)

Following the proof of [23, Proposition 27.21] and using the property that the nonlinear term $u^{2k+1}$, is strictly monotone (see, e.g., [23, Example 25.5]), it can be shown that for every $f\in L^2(\Omega )$ there exists a unique generalized solution $u\in H^1_D(\Omega )$ of (1.6).

The proof that $u\in B^2_{\underline{\beta }}(\Omega )$ for $f \in B^0_{\underline{\beta }}(\Omega )\cap L^2(\Omega )$ will be based on a local regularity-shift result in a sector obtained for the linear Poisson problem in [2] and a corner-weighted $L^2$-estimate of (the derivatives of) the nonlinearity $\lambda u^{2k+1}$.

1.2 Notation

We denote $\mathbb {N}= \{1,2,3,\ldots \}$ the natural numbers, and $\mathbb {N}_0 = \mathbb {N}\cup \{0\} = \{0,1,2,\ldots \}$. For any multi-index $\alpha =(\alpha _1,\alpha _2)\in \mathbb {N}^2_0$, we write $\partial ^{\alpha }=\partial _x^{\alpha _1}\partial _y^{\alpha _2}$, $\mathcal {D}^{\alpha }=\partial ^{\alpha _1}_r\partial ^{\alpha _2}_{\theta }$ and $\vert \alpha \vert =\alpha _1+\alpha _2$. Factorials $\alpha !$ are defined as $\alpha ! = \alpha _1!\alpha _2!$ with the convention $0!:=1$. We denote with an underline n-dimensional tuples $\underline{\beta }= (\beta _1, \dots , \beta _n)\in \mathbb {R}^n$. We suppose that for multi-indices and n-dimensional tuples, arithmetic operations and inequalities such as $\underline{\gamma }< \underline{\beta }$ are understood component-wise: e.g., $\underline{\beta }+ k = (\beta _1+k, \ldots , \beta _n+k)$ for all $k\in \mathbb {N}$; furthermore, we indicate, e.g., $\underline{\beta }> 0$ if $\beta _i>0$ for all $i\in \{1,\dots ,n\}$. For $a\in \mathbb {R}$, we denote its nonnegative real part as $[a]_+ = \max (0,a)$. For a nonnegative integer k, we denote by $\mathbb {N}_{>k} =\{ n\in \mathbb {N}: n > k \}$ and by $\mathbb {N}_{\ge k} = \{ n\in \mathbb {N}: n\ge k\}$.

For any $\alpha ,\gamma \in \mathbb {N}^2_0$ or $i,j\in \mathbb {N}$, we denote by $\delta _{\alpha ,\gamma }$ or $\delta _{i,j}$ the Kronecker function which equals 1 if the two parameters are identical and which vanishes otherwise.

Given an angle $\omega \in (0,2\pi )$ and a radius $\delta \in (0,+\infty ]$, we define the sector $Q_{\delta ,\omega }$ with vertex at the origin

$$\begin{aligned} Q_{\delta ,\omega }:=\{(r,\theta )\in \mathbb {R}^2\vert r\in (0,\delta ),\theta \in (0,\omega )\}. \end{aligned}$$

(1.7)

For any corner $c_i$ and a radius $\delta \in (0,\min (\frac{1}{4}\min _{i,j\in \{1,2,\ldots ,n\},i\ne j}d(c_i,c_j),1))$, we set

$$\begin{aligned} Q_{\delta ,\omega _i}(c_i):=c_i+\{(r,\theta )\in \mathbb {R}^2\vert r\in (0,\delta ),\theta \in (0,\omega _i)\}. \end{aligned}$$

(1.8)

Here the polar coordinate system is assumed to be such that the half line $c_i+\{\theta =0\}$ contains $\Gamma _{i-1}$ (so that $c_i+\{\theta =\omega _i\}$ contains $\Gamma _{i}$).

1.3 Function spaces

For $x\in \Omega$ and for $i\in \{1,\ldots ,n\}$, let $r_i(x):= {\text {dist}}(x, c_i)$. We recall from [2] the n-tuple of corner-weight exponents $\underline{\beta }= (\beta _1,\ldots ,\beta _n) \in (0,1)^n$ and the corresponding corner weight function

$$\begin{aligned} \Phi _{\underline{\beta }} (x):= \prod _{i=1}^nr_i^{\beta _i}(x), \quad x\in \Omega \;. \end{aligned}$$

1.3.1 Weighted spaces in the whole domain $\Omega$

For any $k,l\in \mathbb {N}_0$ with $k\ge l$ and for any $\underline{\beta }\in (0,1)^n$, we introduce corner-weighted norms $\Vert v\Vert _{H^{k,l}_{\underline{\beta }}(\Omega )}$ by

$$\begin{aligned} \Vert v\Vert ^2_{H^{k,l}_{\underline{\beta }}(\Omega )} := \Vert v\Vert ^2_{H^{l-1}(\Omega )} + \sum _{\vert \alpha \vert = l}^k \Vert \Phi _{\underline{\beta }+\vert \alpha \vert -l}\partial ^{\alpha }v\Vert ^2_{L^2(\Omega )}, \end{aligned}$$

(1.9)

where the term $\Vert v\Vert ^2_{H^{l-1}(\Omega )}$ is dropped if $l=0$. See [2, Sec.1.2].

We also define the following weighted analytic function classes

$$\begin{aligned}&B^l_{\underline{\beta }}(\Omega ):=\Bigg \{ v\in \bigcap \limits _{k\ge l}H^{k,l}_{\underline{\beta }}(\Omega ): \exists C, A>0 {} {} \text{ s.t. } \Vert \Phi _{\underline{\beta }+ \vert \alpha \vert -l} \partial ^{\alpha }v\Vert _{L^2(\Omega )} \nonumber \\ {}&\le CA^{\vert \alpha \vert -l}(\vert \alpha \vert -l)!, \, \forall \vert \alpha \vert \ge l\Bigg \}. \end{aligned}$$

(1.10)

1.3.2 Weighted spaces in a sector

In a sector $Q_{\delta ,\omega }$, we define, for all $k\in \mathbb {N}_0$ and $\beta \in \mathbb {R}$, the corner-weighted space $W^k_\beta (Q_{\delta ,\omega })$ of functions v with finite norm $\Vert v \Vert _{W^k_\beta (Q_{\delta ,\omega })}$ given by

$$\begin{aligned} \Vert v \Vert ^2_{W^k_\beta (Q_{\delta ,\omega })} = \sum _{\vert \alpha \vert \le k} \Vert r^{\beta -k+\alpha _1} \mathcal {D}^{\alpha }v \Vert ^2_{L^2(Q_{\delta ,\omega })}. \end{aligned}$$

(1.11)

For $k, l\in \mathbb {N}_0$ with $k\ge l$ and for $\beta \in \mathbb {R}$, $\mathcal {H}^{k,l}_\beta (Q_{\delta , \omega })$ denote the space of functions with finite norm

$$\begin{aligned} \Vert v \Vert ^2_{\mathcal {H}^{k,l}_\beta (Q_{\delta ,\omega })} := \Vert v \Vert ^2_{H^{l-1}(Q_{\delta ,\omega })} + \sum _{l\le \vert \alpha \vert \le k}\Vert r^{\alpha _1+\beta - l} \mathcal {D}^{\alpha }v\Vert ^2_{L^2(Q_{\delta ,\omega })}, \end{aligned}$$

where the first term is omitted if $l=0$.

For $l \in \mathbb {N}_0$ and $\beta \in \mathbb {R}$, the weighted analytic class in polar coordinates is defined by

$$\begin{aligned} \mathcal {B}^l_\beta (Q_{\delta ,\omega })&= \Bigg \{ v\in \bigcap \limits _{k\ge l}\mathcal {H}^{k,l}_{\beta }(Q_{\delta ,\omega }) : \exists C, A>0~\text {s.t.}~\Vert r^{\alpha _1 + \beta -l} \mathcal {D}^{\alpha }v\Vert _{L^2(Q_{\delta , \omega })}\nonumber \\{}&{} \le CA^{\vert \alpha \vert -l}(\vert \alpha \vert -l)!, \, \forall \vert \alpha \vert \ge l \Bigg \}. \end{aligned}$$

(1.12)

The definition of the spaces $H^{k, l}_\beta (Q_{\delta ,\omega })$ and $B^l_{\beta }(Q_{\delta ,\omega })$ follows from replacing $\Phi _{\underline{\beta }+\vert \alpha \vert -l}$ in (1.9) and (1.10) with $r^{\beta +\vert \alpha \vert -l}$.

We require the following two lemmas regarding the relation between those spaces in $Q_{\delta ,\omega }$.

Lemma 1.1

Let $0<\delta \le 1$, $\omega \in (0, 2\pi )$, $\beta \in (0,1)$. Then the following equivalence relations hold for any $l\in \{0,1,2\}$ and $\mathbb {N}_0\ni k\ge l$:

$$\begin{aligned}{} & {} v\in H^{k,l}_{\beta }(Q_{\delta ,\omega })\Leftrightarrow v\in \mathcal {H}^{k,l}_{\beta }(Q_{\delta ,\omega }), \;\; v\in B^{l}_{\beta }(Q_{\delta ,\omega })\Leftrightarrow v\in \mathcal {B}^{l}_{\beta }(Q_{\delta ,\omega }),\\{} & {} v\in H^{1,1}_{\beta }(Q_{\delta ,\omega })\Leftrightarrow v\in W^1_{\beta }(Q_{\delta ,\omega }). \end{aligned}$$

For a proof we refer to [2, Theorem 1.1, Theorem 2.1, Lemma A.2].

Lemma 1.2

Let $0<\delta \le 1$, $\omega \in (0, 2\pi )$, $\beta \in (0,1)$. Then the following imbedding relations hold:

• 1. $W^2_{\beta }(Q_{\delta ,\omega })\subset H^{2, 2}_\beta (Q_{\delta ,\omega })\subset C^0(\overline{Q_{\delta ,\omega }})$.

• 2. If $v\in H^{2, 2}_\beta (Q_{\delta ,\omega })$ and $v((0,0))=0$, then $v\in W^2_{\beta }(Q_{\delta ,\omega })$.

For the proof of this lemma, see [2, Lemma 1.1, Lemma A.1, Lemma A.2] and [3, Section 2].

2 Poisson problem in a sector

The inductive proof of weighted, analytic regularity will require a $W^{2}_{\beta }$-regularity shift for the linear principal part of the differential operator in the problem (1.1). By localization, this regularity shift estimate is only required locally, in the vicinity of each corner. Consider thus functions u which satisfy the following Poisson problem in any finite sector $Q_{\delta ,\omega }$ with $0< \delta < \infty$,

$$\begin{aligned} -\Delta u = f\qquad \text {in }Q_{\delta ,\omega },\quad B(u) = 0\quad \text {on }\partial Q_{\delta ,\omega }\setminus \{r=\delta \}. \end{aligned}$$

(2.1)

Following the proofs of [2, Lemma 2.2-2.8, Theorem 2.1] item by item, we have the following result.

Proposition 2.1

Let $\beta \in (0,1)$ such that $\beta > 1-\frac{\pi }{\omega }$ for either Dirichlet or Neumann BCs, i.e. if $B|_{\partial Q_{\delta ,\omega }{\setminus } \{r=\delta \}} \in \{\gamma _0, \gamma _1\}$ and assume that $\beta > 1 - \frac{\pi }{2\omega }$ for mixed boundary conditions, i.e. if $B|_{\partial Q_{\delta ,\omega }\cap \{\theta =0\}}=\gamma _0$ and $B|_{\partial Q_{\delta ,\omega }\cap \{\theta =\gamma \}}=\gamma _1$. Furthermore, in (2.1) assume $f\in L_{\beta }(Q_{\delta ,\omega })$.

Then there exists a constant $C_{sec}>1$ such that any u which satisfies (2.1) weakly satisfies $u\in H^{2,2}_{\beta }(Q_{\delta ,\omega })$ and there holds the a-priori estimate

$$\begin{aligned} \Vert u-u(0,0)\Vert _{W^{2}_{\beta }(Q_{\delta /2,\omega })} \le C_{sec}(\Vert f\Vert _{L_{\beta }(Q_{\delta ,\omega })}+\Vert u\Vert _{H^1(Q_{\delta ,\omega }\setminus Q_{\delta /2,\omega })}). \end{aligned}$$

(2.2)

3 Weighted analytic regularity of the solution

Applying [20, Lemma 5.8.6, Lemma 5.8.6’] we derive the analyticity of u and of $\lambda u^{2k+1}$ in the interior of $\Omega$ and up to analytic parts of the boundary $\partial \Omega$.

Proposition 3.1

For any $0<\delta \le \frac{1}{4}\min _{i,j\in \{1,2,\ldots ,n\},i\ne j}d(c_i,c_j)$, any solution u to (1.1) and, for this u, $\lambda u^{2k+1}$ for $k\in \mathbb {N}$ are analytic in $\overline{\Omega \setminus (\cup ^n_{i=1}Q_{\delta /2,\omega _i}(c_i))}$.

By the Sobolev embedding $H^{1}(\Omega )\hookrightarrow L^q(\Omega )$ valid for any $q\in (1,+\infty )$ and by the Hölder inequality, one obtains that for any $u\in H^1(\Omega )$ and any $\underline{\beta }\in (0,1)^n$, $\lambda u^{2k+1}\in L^2(\Omega )\subset L_{\underline{\beta }}(\Omega )$. Therefore, we can move $\lambda u^{2k+1}$ to the right-hand side in (1.1) and consider

$$\begin{aligned} -\Delta u = f-\lambda u^{2k+1} \;\text{ in }\;\Omega ,\;\;\; B(u) = 0 \;\text{ on }\;\partial \Omega . \end{aligned}$$

(3.1)

Now Lemma 1.2 and Proposition 2.1 imply

Lemma 3.2

Let $\delta \in (0,\frac{1}{4}\min _{i,j\in \{1,2,\ldots ,n\},i\ne j}d(c_i,c_j))$ and let $f\in L_{\underline{\beta }}(\Omega )$ where $\underline{\beta }\in (0,1)^n$ satisfies that for any $i\in \{1,2,\ldots ,n\}$, $\beta _i > 1 - \frac{\pi }{\omega _i}$ if $\{i-1,i\}\subset {\mathcal {D}}$ or $\{i-1,i\}\subset {\mathcal {N}}$ and $\beta _i > 1 - \frac{\pi }{2\omega _i}$ otherwise.

Then any solution $u\in H^1_D(\Omega )$ to (1.1) satisfies $u\vert _{Q_{\delta ,\omega _i}(c_i)}\in \mathcal {H}^{2,2}_{\beta _i}(Q_{\delta ,\omega _i}(c_i))\subset C^0(\overline{Q_{\delta ,\omega _i}(c_i)})$ for $i\in \{1,2,\ldots ,n\}$.

3.1 Analytic estimates on the nonlinearity

In this subsection we examine the estimate on derivatives of $\lambda u^{2k+1}$. For this purpose we study $\mathcal {D}^{\alpha }(\lambda u^{2k+1})$.

The case $k=0$ is straightforward. If $k>0$, then by generalized Faà di Bruno formula [6, 14], the derivatives of u will take a complicated form. To describe it, we introduce the concept of decomposition of a multi-index $\alpha \in \mathbb {N}_0^2$. We say that $\alpha \in \mathbb {N}^2_0$ is decomposed into a finite number s of nonzero parts $p^1,\ldots ,p^s\in \mathbb {N}^2_0$ with multiplicities $m_1,\ldots ,m_s\in \mathbb {N}$ if

$$\begin{aligned} \alpha =\sum _{i=1}^{s} m_ip^i \end{aligned}$$

holds and all $p^i$ are distinct. Set $\underline{P}=(p^1,\ldots ,p^s)$ and $\varvec{M}=\{m_1,\ldots ,m_s\}$, we call the triple $(s,\underline{P},\varvec{M})$ a decomposition of $\alpha$. The total multiplicity of $\varvec{M}$ is $m:=\sum ^s_{i=1}m_i$.

The generalized Faà di Bruno formula states that for any $\alpha \in \mathbb {N}^2_0$ and, for any function $g(\cdot ):\mathbb {R}\rightarrow \mathbb {R}$ and for any $u=u(r,\theta )$ with sufficient smoothness, $\mathcal {D}^{\alpha }g(u)$ takes the form

$$\begin{aligned} \mathcal {D}^{\alpha }g(u)=\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha }}C_{(s,\underline{P},\varvec{M})}\frac{d^mg(u)}{d u^m}\prod _{i=1}^s(\mathcal {D}^{p^i}u)^{m_i}. \end{aligned}$$

(3.2)

Here $\mathscr {D}_{\alpha }$ is the set of all possible decompositions of $\alpha$ and

$$\begin{aligned} C_{(s,\underline{P},\varvec{M})}=\alpha !\prod ^{s}_{i=1}\left( \frac{1}{m_i!}\left( \frac{1}{p^i!}\right) ^{m_i}\right) >0, \end{aligned}$$

which depends only on the specific triple $(s,\underline{P},\varvec{M})$.

In the presently considered case $g(u)={\lambda }u^{2k+1}$, so it suffices to consider decompositions satisfying $m\le 2k+1$.

Lemma 3.2 implies $L^{\infty }$-boundedness of $\frac{d^m u^{2k+1}}{d u^m}$ for any $m\in \mathbb {N}$ in $Q_{\delta ,\omega _i}(c_i)$ for $i\in \{1,\ldots ,n\}$. To estimate the weighted-$L^2$ norm of $\mathcal {D}^{\alpha }({\lambda }u^{2k+1})$ based on (3.2) near each corner, we bound all individual terms $\prod _{i=1}^s(\mathcal {D}^{p^i}u)^{m_i}$ and the combinatorial constants $C_{(s,\underline{P},\varvec{M})}$. For the first step, we need the following two lemmas which provide weighted interpolation estimates in a sector. The proofs of these lemmas are along the lines proposed in [11, Lemma 4.2], and are based on dyadic decomposition of the sector and scaling of an interpolation inequality in domains satisfying a cone condition(see [1]). These techniques are useful in treating singularities in a corner.

Lemma 3.3

Assume given $\delta \in (0,+\infty )$, $\omega \in (0,2\pi ]$, $k\in \mathbb {N}$ and $\beta \in (0,1)$. Then, there exists a constant $C_{int}=C_{int}(\delta ,\omega ,k,\beta )>0$ such that for any function $\phi :Q_{\delta ,\omega }\rightarrow \mathbb {R}$ for which there exists $\alpha \in \mathbb {N}^2_0$ satisfying, for any $l\in \mathbb {N}$ with $2\le l\le 2k+1$,

$$\begin{aligned} \max _{\vert \eta \vert \le 1}\Vert r^{\beta -2+\alpha _1+\eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(Q_{\delta ,\omega })} < \infty \;, \end{aligned}$$

there holds the following bound

$$\begin{aligned}&\Vert r^{\frac{\beta }{l}+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^{2l}(Q_{\delta ,\omega })} \le C_{int}\Vert r^{\beta -2+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{\delta ,\omega })}^{\frac{1}{l}}\\&\quad \cdot \left( \sum _{\vert \eta \vert \le 1}\Vert r^{\beta -2+\alpha _1+\eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(Q_{\delta ,\omega })}^{\frac{l-1}{l}} +\alpha _1^{\frac{l-1}{l}}\Vert r^{\beta -2+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{\delta ,\omega })}^{\frac{l-1}{l}}\right) . \end{aligned}$$

The proof of this lemma is given in Appendix A.

Lemma 3.4

Let $\delta \in (0,+\infty )$, $\omega \in (0,2\pi ]$, $k\in \mathbb {N}$ and $\beta \in (0,1)$.

Then there exists a constant $C_t=C_t(\delta ,\omega ,k,\beta )>1$ such that for all $\phi \in \mathcal {H}^{2,2}_{\beta }(Q_{\delta ,\omega })$ with $\Vert \phi -\phi (0,0)\Vert _{W^2_{\beta }(Q_{\delta ,\omega })}<1$ and such that there exist $A,E>1$ and $i\in \mathbb {N}$ satisfying

$$\begin{aligned} \Vert r^{\beta -2+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{\delta ,\omega })} \le A^{\vert \alpha \vert -2}E^{\alpha _2}(\vert \alpha \vert -2)!,\quad \forall \alpha \in \mathbb {N}^2_0: 2\le \vert \alpha \vert \le i+1, \end{aligned}$$

it holds for any $1\le \vert \alpha \vert \le i$ and any $2\le l\le 2k+1$ that,

$$\begin{aligned} \Vert r^{\beta /l+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^{2l}(Q_{\delta ,\omega })} \le C_t A^{\vert \alpha \vert -1}E^{\alpha _2+1}(\vert \alpha \vert -1)!. \end{aligned}$$

Proof

We fix $\delta ,\omega ,k,\beta$ and any $\phi$ satisfying the conditions in this lemma with some $A,E>1$ and $i\in \mathbb {N}$.

By lemma 3.3, there exists $C_{int}>0$ depending on $\delta ,\omega ,k,\beta$ such that for any $1\le \vert \alpha \vert \le i$ and any $2\le l\le 2k+1$,

$$\begin{aligned}&\Vert r^{\frac{\beta }{l}+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^{2l}(Q_{\delta ,\omega })} \le C_{int}\Vert r^{\beta -2+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{\delta ,\omega })}^{\frac{1}{l}}\\&\qquad \cdot \left( \sum _{\vert \eta \vert \le 1}\Vert r^{\beta -2+\alpha _1+\eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(Q_{\delta ,\omega })}^{\frac{l-1}{l}} +\alpha _1^{\frac{l-1}{l}}\Vert r^{\beta -2+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{\delta ,\omega })}^{\frac{l-1}{l}}\right) \\&\quad \le C_{int}\max \big (A^{\vert \alpha \vert -2}E^{\alpha _2}(\vert \alpha \vert -2)!,1\big )^{\frac{1}{l}}\cdot \big (3(A^{\vert \alpha \vert -1}E^{\alpha _2+1}(\vert \alpha \vert -1)!)^{\frac{l-1}{l}}\\&\qquad +\max \big (A^{\vert \alpha \vert -2}E^{\alpha _2}(\vert \alpha \vert -1)!,1\big )^{\frac{l-1}{l}}\big )\\&\quad \le 4C_{int}A^{\vert \alpha \vert -1}E^{\alpha _2+1}(\vert \alpha \vert -1)!. \end{aligned}$$

This implies that $C_t:=4C_{int}$ satisfies all conditions of this lemma. $\square$

We investigate the constant $C_{(s,\underline{P},\varvec{M})}$ in (3.2). In the estimation of higher-order derivatives of the quadratic nonlinearity $({\varvec{u}}\cdot \nabla ){\varvec{u}}$ for the Navier–Stokes equation in [11, Lemma 4.5], another kind of combinatorial constant $\left( {\begin{array}{c}\alpha \\ \eta \end{array}}\right)$ for $\alpha ,\eta \in \mathbb {N}^2_0$ appears in the expansion of higher-order derivatives. Their control with respect to the differentiation order is achieved in [11] using a combinatorial identity.

Here, however, we do not derive a particular combinatorial identity that is best suited to bound the nonlinearity in problem (1.1). Instead, the following lemma provides sufficient control of $C_{(s,\underline{P},\varvec{M})}$. Its statement needs the introduction of the following auxiliary variable: given a multi-index $\alpha \in \mathbb {N}^2_0$ and $A,E>0$, we define

$$\begin{aligned} I_{[\alpha ,A,E]} := \sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\prod _{i=1}^s(A^{|p^i|-1}E^{p^i_2+1}(|p^i|-1)!)^{m_i}. \end{aligned}$$

(3.3)

Then the following estimate holds.

Lemma 3.5

Let $A>E>0$. Then for any $\vert \alpha \vert \ge 1$ and any $\vert \eta \vert =1$,

$$\begin{aligned} I_{[\alpha +\eta ,A,E]}\le (\vert \alpha \vert +1) AE^{\eta _2} I_{[\alpha ,A,E]}. \end{aligned}$$

See Appendix B for the proof.

We are ready to present the following corner-weighted regularity estimate on the nonlinearity.

Lemma 3.6

(weighted regularity estimate on the nonlinearity) Fix $\delta \in (0,1)$, $\omega \in (0,2\pi ]$, ${\lambda \in \mathbb {R}}$, $k\in \mathbb {N}_0$ and $\beta \in (0,1)$.

There exists a constant $C_{non}=C_{non}(\delta ,\omega ,\beta ,{\lambda },k)>0$ such that for any $\phi \in \mathcal {H}^{2,2}_{\beta }(Q_{\delta ,\omega })$ with $\Vert \phi \Vert _{\mathcal {H}^{2,2}_{\beta }(Q_{\delta ,\omega })}<1$ and $\Vert \phi -\phi (0,0)\Vert _{W^2_{\beta }(Q_{\delta ,\omega })}<1$ for which there exist $i\in \mathbb {N}$ and constants $A>E>1$ such that, for $2\le \vert \alpha \vert \le i+1$, hold the bounds

$$\begin{aligned} \Vert r^{\beta -2+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{\delta ,\omega })} \le A^{\vert \alpha \vert -2}E^{\alpha _2}(\vert \alpha \vert -2)! \;, \end{aligned}$$

there holds, for $1\le \vert \alpha \vert \le i$,

$$\begin{aligned} \Vert r^{\beta +\alpha _1}\mathcal {D}^{\alpha }(\lambda \phi ^{2k+1})\Vert _{L^2(Q_{\delta ,\omega })} \le C_{non} A^{\vert \alpha \vert -1}E^{\alpha _2+1}\vert \alpha \vert ! \;. \end{aligned}$$

(3.4)

We remark that due to item 1 in Lemma 1.2, the value of $\phi$ at the point $\{r=0\}$ is well-defined.

Proof

Without loss of generality we assume that $\lambda >0$. With the condition $\Vert \phi \Vert _{\mathcal {H}^{2,2}_{\beta }(Q_{\delta ,\omega })}<1$ and Lemma 1.2 we may assume $\max _{0\le j\le 2k+1}\Vert \frac{\partial ^j \phi ^{2k+1}}{\partial \phi ^j}\Vert _{L^{\infty }(Q_{\delta ,\omega })}\le K$ for some $K=K(\delta ,\omega ,\beta ,k)>0$. The assumptions $\delta <1$ and $\Vert \phi -\phi (0,0)\Vert _{W^2_{\beta }(Q_{\delta ,\omega })}<1$ imply, for any $1\le \vert \alpha \vert \le i+1$,

$$\begin{aligned}{} & {} \Vert r^{\beta +\alpha _1}\mathcal {D}^{\alpha }({\lambda }\phi )\Vert _{L^2(Q_{\delta ,\omega })} \le {\lambda }\Vert r^{\beta -2+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{\delta ,\omega })} \\{} & {} \quad \le {\lambda }\max (A^{\vert \alpha \vert -2}E^{\alpha _2}(\vert \alpha \vert -2)!,1) \le {\lambda }A^{\vert \alpha \vert -1}E^{\alpha _2+1}(\vert \alpha \vert -1)!, \end{aligned}$$

so the case $k=0$ is verified for any $C_{non}\ge \lambda$.

Consider the case $k>0$. By (3.2), the generalized Hölder inequality and Lemma 3.4, for any $1\le \vert \alpha \vert \le i$ we have,

$$\begin{aligned}&\Vert r^{\beta +\alpha _1}\mathcal {D}^{\alpha }({\lambda }\phi ^{2k+1})\Vert _{L^2(Q_{\delta ,\omega })}\\&\quad \le \lambda \Vert \sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m \le 2k+1} C_{(s,\underline{P},\varvec{M})}\frac{\partial ^m\phi ^{2k+1}}{\partial \phi ^m} \prod _{i=1}^s\left( r^{\frac{\beta }{m}+p^i_1}\mathcal {D}^{p^i}\phi \right) ^{m_i}\Vert _{L^2(Q_{\delta ,\omega })}\\&\quad \le \lambda K\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1} C_{(s,\underline{P},\varvec{M})}\Vert \prod _{i=1}^s\left( r^{\frac{\beta }{m}+p^i_1}\mathcal {D}^{p^i}\phi \right) ^{m_i}\Vert _{L^2(Q_{\delta ,\omega })}\\&\quad \le \lambda K\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1} C_{(s,\underline{P},\varvec{M})}\prod _{i=1}^s\Vert r^{\frac{\beta }{m}+p^i_1}\mathcal {D}^{p^i}\phi \Vert _{L^{2m}(Q_{\delta ,\omega })}^{m_i}\\&\quad \le \lambda KC_t^{2k+1}\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1} C_{(s,\underline{P},\varvec{M})}\prod _{i=1}^s\left( A^{|p^i|-1}E^{p^i_2+1}(|p^i|-1)!\right) ^{m_i} \\&\quad = \lambda KC_t^{2k+1}I_{[\alpha ,A,E]}. \end{aligned}$$

It suffices to show that there exists a constant $C_{non}>0$ such that for any $1\le \vert \alpha \vert \le i$

$$\begin{aligned} {\lambda }KC_t^{2k+1}I_{[\alpha ,A,E]} \le C_{non} A^{\vert \alpha \vert -1}E^{\alpha _2+1}\vert \alpha \vert !. \end{aligned}$$

(3.5)

It is easy to verify that the only possible decomposition for $\alpha$ with $\vert \alpha \vert =1$ is $\alpha =1\cdot \alpha$ and $C_{(1,\{\alpha \},\{1\})}=1$: it holds $\mathcal {D}^{\alpha }\phi ^{2k+1}=\frac{\partial \phi ^{2k+1}}{\partial \phi }\cdot \mathcal {D}^{\alpha }\phi$. Therefore, for $\vert \alpha \vert =1$ and for any $C_{non}\ge {\lambda }KC_t^{2k+1}$,

$$\begin{aligned} \begin{aligned} \lambda KC_t^{2k+1}I_{[\alpha ,A,E]}&= \lambda KC_t^{2k+1}C_{(1,\{\alpha \},\{1\})}A^0E^{\alpha _2+1}(0!)\\&=\lambda KC_t^{2k+1} E^{\alpha _2+1}\le C_{non}E^{\alpha _2+1}. \end{aligned} \end{aligned}$$

We show now that for any $C_{non}\ge \lambda KC_t^{2k+1}$, (3.5) holds for $1\le \vert \alpha \vert \le i$. The case $\vert \alpha \vert =1$ is already checked from the above equality and we consider all $\alpha$ such that $1\le \vert \alpha \vert \le i$ by mathematical induction. Assume now that (3.5) is true for $1\le \vert \alpha \vert \le j<i$. For any $\vert \alpha \vert =j+1$, we select $\vert \eta \vert =1$ such that $\alpha -\eta \in \mathbb {N}^2_0$. Then Lemma 3.5 implies that

$$\begin{aligned} \begin{aligned} \lambda KC_t^{2k+1}I_{[\alpha ,A,E]}&\le \lambda KC_t^{2k+1}(j+1)AE^{\eta _2}I_{[\alpha -\eta ,A,E]}\\&\le (j+1)AE^{\eta _2}(\lambda KC_t^{2k+1}I_{[\alpha -\eta ,A,E]})\\&\le (j+1)AE^{\eta _2}\times C_{non}A^{j}E^{(\alpha _2-\eta _2)+1}j!\\&\le C_{non}A^{j+1}E^{\alpha _2+1}(j+1)!. \end{aligned} \end{aligned}$$

Therefore (3.5) holds for $\vert \alpha \vert =j+1$. Repeating this step validates (3.5) for $1\le \vert \alpha \vert \le i$. Unifying the case $k>0$ and $k=0$ shows that we could set $C_{non}:={\lambda }\max (KC_t^{2k+1},1)$ to satisfy the requirement of this lemma. $\square$

3.2 Weighted analytic regularity near corners

By Lemmas 1.2 and 3.2, we may fix $\delta \in (0,\min (\frac{1}{4}\min _{i,j\in \{1,2,\ldots ,n\},i\ne j}d(c_i,c_j),1))$ such that $\Vert u-u(c_i)\Vert _{W^2_{\beta }(Q_{\delta ,\omega _i}(c_i))}<1$ at each corner.

We are now in position to prove local weighted analytic regularity estimates near all corners. The inductive claim used here is similar to the one shown in the proof of [11, Lemma 4.7].

Lemma 3.7

Let $\underline{\beta }\in (0,1)^n$ such that that for any $i\in \{1,2,\ldots ,n\}$, $\beta _i > 1 - \frac{\pi }{\omega _i}$ if $\{i-1,i\}\subset {\mathcal {D}}$ or $\{i-1,i\}\subset {\mathcal {N}}$ and $\beta _i > 1 - \frac{\pi }{2\omega _i}$ otherwise. Furthermore, let $u\in H^1_D(\Omega )$ be the weak solution to (1.1) with right hand side $f\in B^0_{\underline{\beta }}(\Omega )\cap L^2(\Omega )$.

Then there exists $A_u, E_u>1$ such that for all $i\in \{1,2,\ldots ,n\}$,

$$\begin{aligned} \Vert r^{\beta _i-2+\alpha _1}\mathcal {D}^{\alpha }u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\le A_u^{\vert \alpha \vert -2}E_u^{[\alpha _2-2,0]_+}(\vert \alpha \vert -2)!\quad \forall \alpha \in \mathbb {N}^2_0: \vert \alpha \vert \ge 2. \end{aligned}$$

Proof

In each sector $Q_{\delta ,\omega _i}(c_i)$, we rewrite (1.1), (1.2) as

$$\begin{aligned} -(\partial ^2_r+\frac{1}{r}\partial _r+\frac{1}{r^2}\partial ^2_{\theta }) u = f-{\lambda }u^{2k+1}\quad \text {in }Q_{\delta ,\omega _i}(c_i), \quad B(u) = 0\quad \text {on }\widehat{\Gamma }_i, \end{aligned}$$

(3.6)

where $\widehat{\Gamma }_i=\partial Q_{\delta ,\omega _i}(c_i)\cap \partial \Omega$, $f\in B^0_{\underline{\beta }}(\Omega )$. Lemma 1.1 and Proposition 3.1 then imply that there exists $A_1>1$ (depending on $\lambda$) such that for any $\alpha \in \mathbb {N}^2_0$,

$$\begin{aligned} \Vert r^{\beta _i+\alpha _1} \mathcal {D}^{\alpha }({\lambda }u^{2k+1})\Vert _{L^2(Q_{\delta ,\omega _i}(c_i)\setminus Q_{\delta /2,\omega _i}(c_i))}&\le A_1^{\vert \alpha \vert } \vert \alpha \vert !, \end{aligned}$$

(3.7a)

$$\begin{aligned} \Vert r^{\beta _i-2+\alpha _1} \mathcal {D}^{\alpha }(r^2f)\Vert _{L^2(Q_{\delta ,\omega _i}(c_i))}&\le A_1^{\vert \alpha \vert } \vert \alpha \vert !, \end{aligned}$$

(3.7b)

and, for all $j\in \mathbb {N}_0$,

$$\begin{aligned} \Vert r^{j} \partial _r^j u\Vert _{H^1(Q_{\delta ,\omega _i}(c_i)\setminus Q_{\delta /2,\omega _i}(c_i))} \le A_1^{j}j!. \end{aligned}$$

(3.7c)

Define the constants

$$\begin{aligned} A_u=\max (4C_{sec}A_1,108(C_{sec}C_{non}+1),162C_{non}). \end{aligned}$$

(3.8a)

and

$$\begin{aligned} E_u=18, \end{aligned}$$

(3.8b)

Our proof will be based on the following induction assumption.

Induction assumption For $j_1 \in \mathbb {N}_{\ge 2}$ and $j_2\in \mathbb {N}$ with $j_2\le j_1$, we say $H_{j_1,j_2}$ holds if

$$\begin{aligned} \Vert r^{\beta _i-2+\alpha _1}\mathcal {D}^{\alpha }u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}{} & {} \le A_u^{\vert \alpha \vert -2}E_u^{[\alpha _2-2,0]_+}(\vert \alpha \vert -2)!\nonumber \\{} & {} \forall \alpha \in \mathbb {N}^2_0: \left\{ \begin{aligned}&2\le \vert \alpha \vert \le j_1-1,\\&\qquad \text {or}\\&\vert \alpha \vert =j_1\text { and }\alpha _2 \le j_2. \end{aligned} \right. \end{aligned}$$

(3.9)

Here $A_u$ and $E_u$ are the constants in (3.8a) and (3.8b).

Then $H_{2,2}$ holds since $\Vert u-u(c_i)\Vert _{W^2_{\beta _i}(Q_{\delta ,\omega _i}(c_i))}<1$.

Strategy of the proof: The proof consists of two steps:

• 1. We will show that for any $j\in \mathbb {N}_{\ge 2}$,

  $$\begin{aligned} H_{j, j} \implies H_{j+1, 2}. \end{aligned}$$

  (3.10)

• 2. We show

  $$\begin{aligned} \forall j\in \mathbb {N}_{\ge 3} \;\; \forall l\in \mathbb {N}, 2\le l < j:\quad H_{j, l} \implies H_{j, l+1}. \end{aligned}$$

  (3.11)

Combining (3.10) and (3.11), we obtain that

$$\begin{aligned} H_{j, j} \implies H_{j+1, j+1}, \end{aligned}$$

(3.12)

We infer from (3.12) that $H_{j, j}$ is verified for all $j\in \mathbb {N}_{\ge 2}$. This will conclude the proof.

Step 1: verification of (3.10)

We will show equivalently that for any $j\in \mathbb {N}$,

$$\begin{aligned} H_{j+1, j+1} \implies H_{j+2, 2}. \end{aligned}$$

If $k\ge 1$, then by Lemma 3.6 there exists $C_{non}>1$ such that for any $\alpha \in \mathbb {N}_0^2$ with $\vert \alpha \vert \le j$

$$\begin{aligned} \Vert r^{\beta _i+\alpha _1}\mathcal {D}^{\alpha }({\lambda }u^{2k+1})\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\le C_{non}A_u^{j-1}E_u^{\alpha _2+1}j!, \end{aligned}$$

and if $k=1$, then

$$\begin{aligned}{} & {} \Vert r^{\beta _i+\alpha _1}\mathcal {D}^{\alpha }u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\le \Vert r^{\beta _i-2+\alpha _1}\mathcal {D}^{\alpha }u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\{} & {} \quad \le A_u^{j-2}E_u^{[\alpha _2-2]_+}j!\le C_{non}A_u^{j-1}E_u^{\alpha _2+1}j!. \end{aligned}$$

Define $v=r^j\partial _r^j u$. Then v solves the boundary value problem

$$\begin{aligned} -\left( \partial ^2_r+\frac{1}{r}\partial _r+\frac{1}{r^2}\partial ^2_{\theta }\right) v= & {} r^{j-2}\partial _r^{j}(r^2(f-{\lambda }u^{2k+1}))\qquad \text {in }Q_{\delta ,\omega _i}(c_i),\nonumber \\ B(v)= & {} 0\quad \text {on }\widehat{\Gamma }_i. \end{aligned}$$

(3.13)

Proposition 2.1 and (3.7a)–(3.7c) now imply

$$\begin{aligned}&\sum _{\vert \eta \vert =2}\Vert r^{\beta _i-2+\eta _1}\mathcal {D}^{\eta }v\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))} \le \Vert v-v(0,0)\Vert _{W^{2}_{\beta _i}(Q_{\delta /2,\omega _i}(c_i))}\\&\quad \le C_{sec}(\Vert r^{\beta _i+j-2}\partial _r^{j}(r^2(f-{\lambda }u^{2k+1}))\Vert _{L^{2}(Q_{\delta /2,\omega _i}(c_i))}+\Vert v\Vert _{H^1(Q_{\delta ,\omega _i}(c_i)\setminus Q_{\delta /2,\omega _i}(c_i))})\\&\quad \le C_{sec}(\Vert r^{\beta _i+j-2}\partial _r^{j}(r^2f)\Vert _{L^{2}(Q_{\delta /2,\omega _i}(c_i))} +\Vert r^{\beta _i+j}\partial _r^{j}({\lambda }u^{2k+1})\Vert _{L^{2}(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +j\Vert r^{\beta _i+j-1}\partial _r^{j-1}({\lambda }u^{2k+1})\Vert _{L^{2}(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +j(j-1)\Vert r^{\beta _i+j-2}\partial _r^{j-2}({\lambda }u^{2k+1})\Vert _{L^{2}(Q_{\delta /2,\omega _i}(c_i))}+\Vert v\Vert _{H^1(Q_{\delta ,\omega _i}(c_i)\setminus Q_{\delta /2,\omega _i}(c_i))})\\&\quad \le C_{sec}(A_1^j j!+3C_{non}A_u^{j-1}E_u j!+A_1^j j!)\\&\quad \le C_{sec}(2A_1^j j!+3C_{non}A_u^{j-1}E_u j!). \end{aligned}$$

For all $\eta \in \mathbb {N}_0^2$ with $| \eta | = 2$, it holds

$$\begin{aligned} \mathcal {D}^{\eta }v = r^j \partial _r^j\mathcal {D}^{\eta }u + \eta _1 j r^{j-1} \partial _r^{j+\eta _1-1} \partial _{\theta }^{\eta _2} u + [\eta _1-1]_+ j(j-1)r^{j-2} \partial _r^ju. \end{aligned}$$

Therefore, for all $\eta \in \mathbb {N}_0^2$ with $| \eta | = 2$,

$$\begin{aligned}&\Vert r^{\beta _i-2+j+\eta _1}\mathcal {D}^{\eta }\partial _r^ju\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\quad \le \sum _{\vert \eta \vert =2}\Vert r^{\beta _i-2+\eta _1}\mathcal {D}^{\eta }v\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}+2j\Vert r^{\beta _i-2+j+\eta _1}\partial _r^{j+\eta _1-1}\partial _{\theta }^{\eta _2} u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +j(j-1)\Vert r^{\beta _i-2+j}\partial _r^ju\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\quad \le C_{sec}(2A_1^j j!+3C_{non}A_u^{j-1}E_u j!)+3A_u^{j-1}j!\\&\quad \le 2C_{sec}A^j_1j!+(3C_{sec}C_{non}+3)A_u^{j-1}E_uj!\\&\quad \le A_u^j j!, \end{aligned}$$

which validates (3.10).

Step 2: proof of (3.11)

We now fix $l\in \{2, \ldots , j-1\}$ and assume that $H_{j, l}$ holds true. This implies, as before, that for any $1\le \vert \alpha \vert \le j-2$

$$\begin{aligned} \Vert r^{\beta _i+\alpha _1}\mathcal {D}^{\alpha }({\lambda }u^{2k+1})\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\le C_{non}A_u^{\vert \alpha \vert -1}E_u^{\alpha _2+1}\vert \alpha \vert !. \end{aligned}$$

So we have

$$\begin{aligned}&\Vert r^{\beta _i-2+(j-l-1)}\partial _r^{j-l-1}\partial _{\theta }^{l-1}(r^2{\lambda }u^{2k+1})\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\quad \le \Vert r^{\beta _i+(j-l-1)}\partial _r^{j-l-1}\partial _{\theta }^{l-1}({\lambda }u^{2k+1})\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +2(j-l-1)\Vert r^{\beta _i+(j-l-2)}\partial _r^{j-l-2}\partial _{\theta }^{l-1}({\lambda }u^{2k+1})\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +(j-l-1)(j-l-2)\Vert r^{\beta _i+(j-l-3)}\partial _r^{j-l-3}\partial _{\theta }^{l-1}(\lambda u^{2k+1})\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\quad \le 3C_{non}A_u^{j-3}E_u^{l}(j-2)!. \end{aligned}$$

Multiply the first equation of (3.6) by $r^2$ and differentiate the product by $\partial _r^{j-l-1}\partial _{\theta }^{l-1}$ to obtain

$$\begin{aligned}&-(r^2\partial _r^{j-l+1}\partial _{\theta }^{l-1}+2(j-l-1)\partial _r^{j-l}\partial _{\theta }^{l-1}+(j-l-1)(j-l-2)\partial _r^{j-l-1}\partial _{\theta }^{l-1}\\&\qquad +r\partial _r^{j-l}\partial _{\theta }^{l-1}+(j-l-1)\partial _r^{j-l-2}\partial _{\theta }^{l-1}+\partial _r^{j-l-1}\partial _{\theta }^{l+1})u\\&\quad =\partial _r^{j-l-1}\partial _{\theta }^{l-1}(r^2(f-(\lambda u^{2k+1}))). \end{aligned}$$

This is equivalent to

$$\begin{aligned}&\partial _r^{j-l-1}\partial _{\theta }^{l+1}u\\&\quad =-(r^2\partial _r^{j-l+1}\partial _{\theta }^{l-1}+2(j-l-1)\partial _r^{j-l}\partial _{\theta }^{l-1}+(j-l-1)(j-l-2)\partial _r^{j-l-1}\partial _{\theta }^{l-1}\\&\qquad +r\partial _r^{j-l}\partial _{\theta }^{l-1}+(j-l-1)\partial _r^{j-l-2}\partial _{\theta }^{l-1})u-\partial _r^{j-l-1}\partial _{\theta }^{l-1}(r^2(f-(\lambda u^{2k+1}))). \end{aligned}$$

Therefore

$$\begin{aligned}&\Vert r^{\beta _i-2+(j-l-1)}\partial _r^{j-l-1}\partial _{\theta }^{l+1}u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\quad \le \Vert r^{\beta _i-2+(j-l+1)}\partial _r^{j-l+1}\partial _{\theta }^{l-1}u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +2(j-l-1)\Vert r^{\beta _i-2+(j-l)}\partial _r^{j-l}\partial _{\theta }^{l-1}u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +(j-l-1)(j-l-2)\Vert r^{\beta _i-2+(j-l-1)}\partial _r^{j-l-1}\partial _{\theta }^{l-1}u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +\Vert r^{\beta _i-2+(j-l-1)}\partial _r^{j-l}\partial _{\theta }^{l-1}u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +(j-l-1)\Vert r^{\beta _i-2+(j-l-1)}\partial _r^{j-l-1}\partial _{\theta }^{l-1}u\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +\Vert r^{\beta _i-2+(j-l-1)}\partial _r^{j-l-1}\partial _{\theta }^{l-1}(r^2f)\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\qquad +\Vert r^{\beta _i-2+(j-l-1)}\partial _r^{j-l-1}\partial _{\theta }^{l-1}({\lambda }r^2u^{2k+1})\Vert _{L^2(Q_{\delta /2,\omega _i}(c_i))}\\&\quad \le A_u^{j-2} E_u^{[l-3]_+}(j-2)!+2A_u^{j-3}E_u^{[l-3]_+}(j-2)!\\&\qquad +A_u^{j-4}E_u^{[l-3]_+}(j-2)!+A_u^{j-3}E_u^{[l-3]_+}(j-3)!\\&\qquad +A_u^{j-4}E_u^{[l-3]_+}(j-3)!+A_1^{j-2}(j-2)!+3C_{non}A_u^{j-3}E_u^{l}(j-2)!\\&\quad \le 6A_u^{j-2} E_u^{[l-3]_+}(j-2)!+A_1^{j-2}(j-2)!+3C_{non}A_u^{j-3}E_u^l(j-2)!\\&\quad \le A_u^{j-2}E_u^{l-1}(j-2)!=A_u^{j-2}E_u^{[l-1]_+}(j-2)!\;. \end{aligned}$$

Therefore (3.11) holds true. The proof is completed by applying the strategy to show (3.12). $\square$

3.3 Weighted analytic regularity in the polygon

The main result of this paper is now a straightforward consequence of the corner-weighted, analytic estimates of solutions and classical results on interior and boundary regularity.

Theorem 3.8

Let $\underline{\beta }\in (0,1)^n$ such that that for any $i\in \{1,2,\ldots ,n\}$, $\beta _i > 1 - \frac{\pi }{\omega _i}$ if $\{i-1,i\}\subset {\mathcal {D}}$ or $\{i-1,i\}\subset {\mathcal {N}}$ and $\beta _i > 1-\frac{\pi }{2\omega _i}$ otherwise. Furthermore, let $u\in H^1_D(\Omega )$ be the weak solution to (1.1) with right hand side $f\in B^0_{\underline{\beta }}(\Omega )\cap L^2(\Omega )$. Then $u\in B^2_{\underline{\beta }}(\Omega )$.

Proof

We have analyticity of u in the interior and up to analytic parts of the boundary. In addition, Lemmas 3.2 and 3.7 show that $u\in \mathcal {B}^2_{\beta }(Q_{\delta /2,\omega _i}(c_i))$ at each corner $c_i$. Using Lemma 1.1 and combining these two claims we conclude the proof. $\square$

4 Exponential approximability

The weighted, analytic regularity of solutions in Theorem 3.8 implies, via well-known results on approximation properties of hp-FEM in [7, 9, 22] exponential approximability by finite-dimensional spaces of continuous, piecewise polynomial functions of the solution u of (1.1) with data $f\in B^0_{\underline{\beta }}(\Omega )\cap L^2(\Omega )$. Exponential approximability also holds for several other approximation methods: for Reduced Basis and for Model Order Reduction methods, as the Kolmogorov n-width in $H^1(\Omega )$ of the solution set of (1.1), (1.2) for data $f\in B^0_{\underline{\beta }}(\Omega )\cap L^2(\Omega )$ decreases exponentially as $n\rightarrow \infty$. It also implies corresponding exponential expressivity of solution sets by certain deep neural networks [17] and exponential tensor-rank bounds for tensor-structured approximation schemes [13].

Theorem 4.1

Assume that $\Omega$ is a polygon with $n\ge 3$ straight sides. Consider the nonlinear, elliptic PDE (1.1), (1.2) for analytic data

$$\begin{aligned} f\in A:= \{ f \in B^0_{\underline{\beta }}(\Omega )\cap L^2(\Omega ): \Vert f \Vert _{L^2(\Omega )} \le 1\}, \end{aligned}$$

with the corner-weight parameters $\beta _i$ as in Theorem 3.8. Denote by S the solution map of (1.1), (1.2).

Then, there exists a sequence $\{ V_p \}_{p\ge 1}$ of so-called hp-finite element subspaces of continuous, piecewise polynomial functions $v_p$ of total degree at most p on a sequence $\{ {\mathcal {T}}_p \}_{p\ge 1}$ of nested, regular, partitions ${\mathcal {T}}_p$ of $\Omega$ into triangles T which are obtained from O(p) steps of geometric mesh refinement towards the corners of $\Omega$, such that there holds, for certain constants $b_A, C_A>0$ depending on A,

$$\begin{aligned} \forall u\in S(A): \quad \inf _{v_p\in V_p} \Vert u - v_p \Vert _{H^1(\Omega )} \le C_A\exp (-b_A (\textrm{dim}V_p)^{1/3}). \end{aligned}$$

Furthermore, for every $n\in \mathbb {N}$, the Kolmogorov n-width $d_n$ of the solution set S(A) in $H^1(\Omega )$ is exponentially small: there holds

$$\begin{aligned} d_n(S(A); H^1(\Omega )) \le C_A\exp (-b_A n^{1/3}). \end{aligned}$$

In addition, for each $u\in S(A)$, there exists a collection of feedforward neural networks $\{ \Phi _{\varepsilon ,u} \}_\varepsilon$ with ReLU activation that can represent solutions $u\in S(A)$ of (1.1) with data $f\in B^0_{\underline{\beta }}(\Omega )\cap L^2(\Omega )$ with exponential expressivity in terms of the neural network size $M(\Phi _{\varepsilon ,u})$ and depth $L(\Phi _{\varepsilon ,u})$ to accuracy $\varepsilon >0$ in $H^1(\Omega )$, i.e. their function-realizations $\textrm{R}(\Phi _{\varepsilon ,u})$ satisfy

$$\begin{aligned} \Vert u\!-\!\textrm{R}(\Phi _{\varepsilon ,u}) \Vert _{H^1(\Omega )} \le \varepsilon ,\, M(\Phi _{\varepsilon ,u}) \le O( |\log (\varepsilon )|^5), \;\; L(\Phi _{\varepsilon ,u}) \le O( |\log (\varepsilon )\log (|\log (\varepsilon )|)|). \end{aligned}$$

Proof

By Theorem 3.8, $S(A)\subset B^2_{\underline{\beta }}(\Omega )$. Then, there exists a sequence $\{ V_p \}_{p\ge 1}$ of hp-Finite Element spaces of continuous, piecewise polynomial functions $v_p$ of total degree at most p on a sequence $\{ {\mathcal {T}}_p \}_{p\ge 1}$ of nested, regular, simplicial partitions ${\mathcal {T}}_p$ of $\Omega$ which are geometrically refined towards the corners of $\Omega$ such that there exists a constant $c>0$ so that for all $p\in \mathbb {N}$ holds

• (i) $\#({\mathcal {T}}_p) \le cp$,

• (ii) $n_p = \textrm{dim}(V_p) \le cp^3$,

• (iii) $\sup _{f\in S(A)} \inf _{v_p \in V_p} \Vert S(f) - v_p \Vert _{H^1(\Omega )} \le c\exp (-bp)$.

We refer, e.g., to [7] for a self-contained proof. This proves the first assertion.

With this (hp-FEM convergence) result in hand, we may bound the Kolmogorov n-width of the set $S(A)\subset \subset H^1(\Omega )$ as

$$\begin{aligned} \begin{array}{lll} d_n(S(A), H^1(\Omega )) &{}=&{} \displaystyle \inf _{V_n\subset H^1(\Omega ): \textrm{dim}(V_n)=n} \sup _{u\in S(A)} \inf _{v_n\in V_n} \Vert u-v_n \Vert _{H^1(\Omega )}\\ &{}\le &{} \displaystyle \sup _{u\in S(A)} \inf _{v_p\in V_p} \Vert u-v_p \Vert _{H^1(\Omega )} \le C\exp (-b p). \end{array} \end{aligned}$$

Here, the infimum in the definition of $d_n$ is taken over all subspaces of $H^1_{D}(\Omega )$ of finite dimension not larger than n, and we used that $S(A)\subset B^2_{\underline{\beta }}(\Omega )$, and property (iii) of the hp-FEM.

The second assertion then follows with property (ii) of the hp-FEM.

The final statement on the expression rates of deep ReLU neural networks follows once more from the inclusion $S(A)\subset B^2_{\underline{\beta }}(\Omega )$ with [17, Theorem 5.6]. $\square$

The exponential bound on the Kolmogorov n-width in $H^1(\Omega )$ of the solution manifold S(A) implies corresponding convergence rates of so-called reduced basis approximations which are generated by greedy searches. We refer to [12] and to the references there.

5 Conclusion

We summarize the main results of the present work, and indication directions for further research. Given analytic data f and g in (1.1), we established the analytic regularity of the solution u for the semilinear elliptic equation (1.1) in a polygon with homogeneous Dirichlet and Neumann boundary conditions. The analytic regularity shifts are shown in scales of corner-weighted spaces of Kondrat’ev type.

The analysis developed here is also capable of dealing with other similar semilinear elliptic problems. As an example, it is possible to study the analytic regularity of the solution to (1.1) with $\lambda u^{2k+1}$ replaced by any polynomial g(u). For this we only need to modify Lemma 3.6 and the corresponding proof so that they are suitable for any polynomial $g(\phi )$ rather than $\lambda \phi ^{2k+1}$. Another possibility would be studying the solution of (1.1) in a curvilinear domain or with $-\Delta u$ replaced by a general linear, divergence form second order elliptic operator $L(\cdot )$ defined by $L(u)=-\nabla \cdot (A(x)\nabla u)+b(x)\cdot \nabla u$ with analytic in $\overline{\Omega }$ coefficient matrix A(x) and advection field b(x). The analytic regularity of the solution u also reveals the potential to develop exponentially convergent numerical approximation methods such as hp-FEM, or reduced basis approximations based on subspace sequences obtained via greedy algorithms [4]. It also implies the exponential convergence of quantized, tensor-formatted approximations [13].

Appendices

Appendix A: Proof of Lemma 3.3

We firstly set $\delta =1$. Consider the dyadic partition given by the sets

$$\begin{aligned} S^j := \left\{ x\in Q_{1, \omega } : 2^{-j-1}< r(x) < 2^{-j}\right\} , \quad j\in \mathbb {N}_0, \end{aligned}$$

and denote the linear maps $\Psi _j: S^j \rightarrow S^0$ representing homothetic scaling. Denote $\widehat{\phi }_j:= \phi \circ \Psi _j^{-1}: S^0 \rightarrow \mathbb {R}$ and write $\widehat{\mathcal {D}}^{\alpha }$ for differentiation with respect to polar coordinates $(r, \theta )$ in $S^0$. Then, by scaling, for any $q\in [1, \infty )$ and any $\gamma \in \mathbb {R}$,

$$\begin{aligned} \Vert r^{\gamma + \alpha _1} \mathcal {D}^{\alpha }\phi \Vert _{L^q(S^j)} = 2^{-j(\gamma +2/q)} \Vert r^{\gamma +\alpha _1} \widehat{\mathcal {D}}^{\alpha }\widehat{\phi }\Vert _{L^q(S^0)}. \end{aligned}$$

(A.1)

Furthermore, the following interpolation inequality holds in $S^0$ [1, Theorem 3]: there exists $C_0>0$ depending on $k,\omega$ such that for any $2\le l\le 2k+1$, it holds that

$$\begin{aligned} \forall v \in H^1(S^0): \quad \Vert v \Vert _{L^{2l}(S^0)} \le C_0 \Vert v \Vert _{H^1(S^0)}^{1-1/l} \Vert v\Vert _{L^2(S^0)}^{1/l} . \end{aligned}$$

(A.2)

Moreover, there also holds for all $v \in H^1(S^0)$,

$$\begin{aligned} \Vert v\Vert _{H^1(S^0)} \le 2\sqrt{3}\left( \Vert v\Vert _{L^2(S^0)} +\Vert \partial _rv\Vert _{L^2(S^0)} +\Vert \partial _{\theta }v\Vert _{L^2(S^0)} \right) . \end{aligned}$$

(A.3)

To check this inequality, we have, by elementary calculus,

$$\begin{aligned} \partial _{x_1}=\cos \theta \partial _r-\frac{\sin \theta }{r}\partial _{\theta }, \quad \partial _{x_2}=\sin \theta \partial _r+\frac{\cos \theta }{r}\partial _{\theta }. \end{aligned}$$

These relations yield the following bounds:

$$\begin{aligned} \Vert \partial _{x_1} v\Vert _{L^2(S^0)}\le \Vert \cos \theta \partial _rv\Vert _{L^2(S^0)}+\left\| \frac{\sin \theta }{r}\partial _{\theta }v\right\| _{L^2(S^0)}\le \Vert \partial _rv\Vert _{L^2(S^0)}+2\Vert \partial _{\theta }v\Vert _{L^2(S^0)}, \end{aligned}$$

and

$$\begin{aligned} \Vert \partial _{x_2} v\Vert _{L^2(S^0)}\le \Vert \sin \theta \partial _rv\Vert _{L^2(S^0)}+\left\| \frac{\cos \theta }{r}\partial _{\theta }v\right\| _{L^2(S^0)}\le \Vert \partial _rv\Vert _{L^2(S^0)}+2\Vert \partial _{\theta }v\Vert _{L^2(S^0)}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert v\Vert ^2_{H^1(S^0)}&\le \Vert v\Vert ^2_{L^2(S^0)}+\Vert \partial _{x_1}v\Vert ^2_{L^2(S^0)}+\Vert \partial _{x_2}v\Vert ^2_{L^2(S^0)} \\&\le \Vert v\Vert ^2_{L^2(S^0)}+\big (\Vert \partial _rv\Vert _{L^2(S^0)}+2\Vert \partial _{\theta }v\Vert _{L^2(S^0)}\big )^2+\big (\Vert \partial _rv\Vert _{L^2(S^0)}+2\Vert \partial _{\theta }v\Vert _{L^2(S^0)}\big )^2 \\&\le \Vert v\Vert ^2_{L^2(S^0)}+2\Vert \partial _rv\Vert ^2_{L^2(S^0)}+8\Vert \partial _{\theta }v\Vert ^2_{L^2(S^0)}+8\Vert \partial _rv\Vert _{L^2(S^0)}\cdot \Vert \partial _{\theta }v\Vert _{L^2(S^0)} \\&\le \Vert v\Vert ^2_{L^2(S^0)}+6\Vert \partial _rv\Vert ^2_{L^2(S^0)}+12\Vert \partial _{\theta }v\Vert ^2_{L^2(S^0)}\\&\le 12\big ( \Vert v\Vert _{L^2(S^0)} +\Vert \partial _rv\Vert _{L^2(S^0)} +\Vert \partial _{\theta }v\Vert _{L^2(S^0)}\big )^2. \end{aligned}$$

Taking the square root on both sides leads to (A.3). Combining (A.2) and (A.3) and choosing $v=r^{\alpha _1}\mathcal {D}^{\alpha }\phi$ give

$$\begin{aligned}&\Vert r^{\alpha _1} \mathcal {D}^{\alpha }\phi \Vert _{L^{2l}(S^0)} \\&\quad \le 12^{\frac{1}{4}} C_0 \Vert r^{\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(S^0)}^{1/l}\left( \sum _{\vert \eta \vert \le 1} \Vert \mathcal {D}^{\eta }(r^{\alpha _1}\mathcal {D}^{\alpha }\phi ) \Vert _{L^2(S^0)}\right) ^{1-1/l}\\&\quad \le 12^{\frac{1}{4}} C_0 \Vert r^{\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(S^0)}^{1/l}\left( \sum _{\vert \eta \vert \le 1} \Vert r^{\alpha _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(S^0)} + \alpha _1 \Vert r^{\alpha _1-1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(S^0)}\right) ^{1-1/l}. \end{aligned}$$

Therefore, using the bound $2^{-|a|}\le r(x)^{a} \le 2^{|a|}$ valid for all $x\in S^0$ and all $a\in \mathbb {R}$,

$$\begin{aligned}&\Vert r^{\beta /l + \alpha _1} \mathcal {D}^{\alpha }\phi \Vert _{L^{2l}(S^0)}\\&\quad \le 2^{\beta /l + 2-\beta } 12^{\frac{1}{4}}C_0 \Vert r^{\beta -2 +\alpha _1 }\mathcal {D}^{\alpha }\phi \Vert _{L^2(S^0)}^{1/l}\\&\qquad \times \left( \sum _{\vert \eta \vert \le 1} \Vert r^{\beta -2+\alpha _1 + \eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(S^0)}+ \alpha _1 \Vert r^{\beta -2+ \alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(S^0)}\right) ^{1-1/l}. \end{aligned}$$

Set $C_1:=2^{(\beta -2)/(2k+1) + 2-\beta }\cdot 12^{\frac{1}{4}}\cdot C_0$. Scaling back to $S^j$ and using (A.1), we arrive at

$$\begin{aligned}&\Vert r^{\beta /l + \alpha _1} \mathcal {D}^{\alpha }\phi \Vert _{L^{2l}(S^j)}= 2^{-j(\frac{\beta }{l}+1/l)}\Vert r^{\beta /l + \alpha _1} \widehat{\mathcal {D}}^{\alpha }\widehat{\phi }\Vert _{L^{2l}(S^0)}\\&\quad \le 2^{-j(\beta /l+1/l)}C_1 \Vert r^{\beta -2 +\alpha _1 }\widehat{\mathcal {D}}^{\alpha }\widehat{\phi }\Vert _{L^2(S^0)}^{1/l}\\&\qquad \times \left( \sum _{\vert \eta \vert \le 1} \Vert r^{\beta -2+\alpha _1 + \eta _1}\widehat{\mathcal {D}}^{\alpha +\eta }\widehat{\phi }\Vert _{L^2(S^0)}+ \alpha _1 \Vert r^{\beta -2+ \alpha _1}\widehat{\mathcal {D}}^{\alpha }\widehat{\phi }\Vert _{L^2(S^0)}\right) ^{1-1/l}\\&\quad \le 2^{-j(\beta /l+1/l)}4C_1 \Vert r^{\beta -2 +\alpha _1 }\widehat{\mathcal {D}}^{\alpha }\widehat{\phi }\Vert _{L^2(S^0)}^{1/l}\\&\qquad \times \left( \sum _{\vert \eta \vert \le 1} \Vert r^{\beta -2+\alpha _1 + \eta _1}\widehat{\mathcal {D}}^{\alpha +\eta }\widehat{\phi }\Vert _{L^2(S^0)}^2+ \alpha _1^2 \Vert r^{\beta -2+ \alpha _1}\widehat{\mathcal {D}}^{\alpha }\widehat{\phi }\Vert _{L^2(S^0)}^2\right) ^{1/2-1/{2l}}\\&\quad \le 2^{-j(\beta /l+1/l-(\beta -2+1))}4C_1 \Vert r^{\beta -2 +\alpha _1 }\mathcal {D}^{\alpha }\phi \Vert _{L^2(S^j)}^{1/l}\\&\qquad \times \left( \sum _{\vert \eta \vert \le 1} \Vert r^{\beta -2+\alpha _1 + \eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(S^j)}^2 + \alpha _1^2 \Vert r^{\beta -2+ \alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(S^j)}^2\right) ^{1/2-1/{2l}}. \end{aligned}$$

Setting $C_2:=(\beta +1)/l-(\beta -2+1)>1-\beta >0$, we have

$$\begin{aligned}&\sum _{j\in \mathbb {N}_0}\Vert r^{\beta /l + \alpha _1} \mathcal {D}^{\alpha }\phi \Vert _{L^{2l}(S^j)}^{2l}\\&\quad \le \frac{1}{1-2^{-2lC_2}}(4C_1)^{2l}\sum _{j\in \mathbb {N}_0}\Vert r^{\beta -2 +\alpha _1 }\mathcal {D}^{\alpha }\phi \Vert _{L^2(S^j)}^{2}\\&\qquad \times \sum _{j\in \mathbb {N}_0}\left( \sum _{\vert \eta \vert \le 1} \Vert r^{\beta -2+\alpha _1 + \eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(S^j)}^2+ \alpha _1^2 \Vert r^{\beta -2+ \alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(S^j)}^2\right) ^{l-1}\\&\quad \le \frac{1}{1-2^{-2lC_2}}(4C_1)^{2l}\Vert r^{\beta -2 +\alpha _1 }\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{1,\omega }(c))}^{2}\\ {}&\qquad \times \left( \sum _{j\in \mathbb {N}_0}( \sum _{\vert \eta \vert \le 1} \Vert r^{\beta -2+\alpha _1 + \eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(S^j)}^2+ \alpha _1^2 \Vert r^{\beta -2+ \alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(S^j)}^2)\right) ^{l-1}\\&\quad \le \frac{1}{1-2^{-2lC_2}}(4C_1)^{2l}\Vert r^{\beta -2 +\alpha _1 }\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{1,\omega }(c))}^{2}\\ {}&\qquad \times \left( \sum _{\vert \eta \vert \le 1} \Vert r^{\beta -2+\alpha _1 + \eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(Q_{1,\omega }(c))}^2+ \alpha _1^2 \Vert r^{\beta -2+ \alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{1,\omega }(c))}^2\right) ^{l-1}\\&\quad \le \frac{1}{1-2^{-4(1-\beta )}}(4C_1)^{2l}\Vert r^{\beta -2 +\alpha _1 }\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{1,\omega }(c))}^{2}\\ {}&\qquad \times \left( \sum _{\vert \eta \vert \le 1} \Vert r^{\beta -2+\alpha _1 + \eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(Q_{1,\omega }(c))}^2+ \alpha _1^2 \Vert r^{\beta -2+ \alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{1,\omega }(c))}^2\right) ^{l-1}\\&\quad \le \frac{1}{1-2^{-4(1-\beta )}}(4C_1)^{2l}\Vert r^{\beta -2 +\alpha _1 }\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{1,\omega }(c))}^{2}\\&\qquad \times \left( \sum _{\vert \eta \vert \le 1}\Vert r^{\beta -2+\alpha _1 + \eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(Q_{1,\omega }(c))}^{\frac{l-1}{l}} + \alpha _1^{\frac{l-1}{l}} \Vert r^{\beta -2+ \alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{1,\omega }(c))}^{\frac{l-1}{l}}\right) ^{2l}. \end{aligned}$$

Here we used the fact that $\Vert \cdot \Vert _{l^p}\le \Vert \cdot \Vert _{l^1}$ for any $p\ge 1$ where $\Vert \cdot \Vert _{l^p}$ denotes the $l^p$-norm of a sequence. Taking the 2lth root on both sides concludes the proof of the bound in $Q_{1,\omega }(c)$ with $C_{int}=(\frac{1}{1-2^{-4(1-\beta )}})^{\frac{1}{2l}}(4C_1)$.

To deal with the case $\delta \ne 1$, we define $\Phi _{\delta }:Q_{\delta ,\omega }\rightarrow Q_{1,\omega }(c)$ as a homothetic mapping, denote $\widetilde{\phi }:=\phi \circ \Phi _{\delta }^{-1}: Q_{1,\omega }(c)\rightarrow \mathbb {R}$ and write $\widetilde{\mathcal {D}}^{\alpha }$ as the differentiation with respect to polar coordinates $(r,\theta )$ in $Q_{\delta ,\omega }$. We observe that, for any $\phi :Q_{\delta ,\omega }\rightarrow \mathbb {R}$, $2\le l\le 2k+1$ and any $\alpha \in \mathbb {N}^2_0$,

$$\begin{aligned} \Vert r^{\frac{\beta }{l}+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^{2l}(Q_{\delta ,\omega })}=\delta ^{\frac{\beta }{l}+\frac{1}{l}} \Vert r^{\frac{\beta }{l}+\alpha _1}\widetilde{\mathcal {D}}^{\alpha }\widetilde{\phi }\Vert _{L^{2l}(Q_{1,\omega }(c))} \end{aligned}$$

and

$$\begin{aligned} \Vert r^{\beta -2+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{\delta ,\omega })}=\delta ^{\beta -2+\frac{1}{l}}\Vert r^{\beta -2+\alpha _1}\widetilde{\mathcal {D}}^{\alpha }\widetilde{\phi }\Vert _{L^2(Q_{1,\omega }(c))}. \end{aligned}$$

By applying the scaling to $\phi$ and using our result for $\delta =1$, we have, for general $\delta >0$,

$$\begin{aligned}&\Vert r^{\frac{\beta }{l}+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^{2l}(Q_{\delta ,\omega })} \le C_{int}\cdot \delta ^{\frac{\beta }{l}-\beta +2}\Vert r^{\beta -2+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{\delta ,\omega })}^{\frac{1}{l}}\\&\quad \cdot \left( \sum _{\vert \eta \vert \le 1}\Vert r^{\beta -2+\alpha _1+\eta _1}\mathcal {D}^{\alpha +\eta }\phi \Vert _{L^2(Q_{\delta ,\omega })}^{\frac{l-1}{l}} +\alpha _1^{\frac{l-1}{l}}\Vert r^{\beta -2+\alpha _1}\mathcal {D}^{\alpha }\phi \Vert _{L^2(Q_{\delta ,\omega })}^{\frac{l-1}{l}}\right) . \end{aligned}$$

Therefore we could choose $C_{int}=\delta ^{\frac{\beta }{l}-\beta +2}(\frac{1}{1-2^{-4(1-\beta )}})^{\frac{1}{2l}}(4C_1)$ to complete the proof. $\square$

Appendix B: Proof of Lemma 3.5

We fix $\alpha$ and $\eta$ as in the statement of the lemma. There exists a polynomial $\psi (r,\theta )$ of at most order $\vert \alpha \vert +\vert \eta \vert$ such that for any $|\gamma |\ge 1$ with $\gamma \le \alpha +\eta$

$$\begin{aligned} \mathcal {D}^{\gamma }\psi (0,0)=A^{|\gamma |-1}E^{\gamma _2+1}(|\gamma |-1)!, \end{aligned}$$

(B.1)

and there exists a polynomial $g(\psi )$ of at most order $2k+1$ such that

$$\begin{aligned} \frac{\partial ^i g(\psi )}{\partial \psi ^i}|_{\psi =\psi (0,0)}=1 \end{aligned}$$

for any $0<i\le 2k+1$. The construction of above polynomials is based on Hermite interpolation: we take $\psi (r,\theta )$ as an example to explain it.

For any $\gamma \le \alpha +\eta$, the following polynomial of order $|\gamma |$

$$\begin{aligned} \psi _{\gamma }(r,\theta )=\frac{1}{\gamma !}r^{\gamma _1}\theta ^{\gamma _2} \end{aligned}$$

satisfies for any $\alpha \in \mathbb {N}^2_0$,

$$\begin{aligned} \mathcal {D}^{\alpha }\psi (r,\theta )|_{(r,\theta )=(0,0)} = \delta _{\alpha ,\gamma }. \end{aligned}$$

Therefore

$$\begin{aligned} \psi =\sum _{\gamma \le \alpha +\eta ,|\gamma |\ge 1}A^{|\gamma |-1}E^{\gamma _2+1}(|\gamma |-1)!\psi _{\gamma } \end{aligned}$$

will be a satisfactory choice for the required polynomial.

The identity (3.2) yields

$$\begin{aligned} \mathcal {D}^{\alpha }g(\psi )|_{(r,\theta )=(0,0)}= & {} \sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\prod _{i=1}^s(A^{|p^i|-1}E^{p^i_2+1}(|p^i|-1)!)^{m_i}\\= & {} I_{[\alpha ,A,E]} \end{aligned}$$

and

$$\begin{aligned} \mathcal {D}^{\alpha +\eta }g(\psi )|_{(r,\theta )=(0,0)}&=\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha +\eta },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\prod _{i=1}^s(A^{|p^i|-1}E^{p^i_2+1}(|p^i|-1)!)^{m_i}\\&=I_{[\alpha +\eta ,A,E]}. \end{aligned}$$

Now we examine $\mathcal {D}^{\alpha +\eta }g(\psi )|_{(r,\theta )=(0,0)}$ in a different way: observe that

$$\begin{aligned}&\mathcal {D}^{\alpha +\eta }g(\psi )|_{(r,\theta )=(0,0)}=\mathcal {D}^{\eta }(\mathcal {D}^{\alpha }g(\psi ))|_{(r,\theta )=(0,0)}\\&\quad =\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\left( \mathcal {D}^{\eta }\prod _{i=1}^s(\mathcal {D}^{p^i}\psi )^{m_i}\right) |_{(r,\theta )=(0,0)}\\&\qquad +\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\mathcal {D}^{\eta }\psi |_{(r,\theta )=(0,0)}\cdot \prod _{i=1}^s (\mathcal {D}^{p^i}\psi |_{(r,\theta )=(0,0)})^{m_i}\\&\quad =\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\sum _{i=1}^s\left( m_i\prod _{j=1}^s\left( \mathcal {D}^{p^i}\psi |_{(r,\theta ) =(0,0)}\right) ^{m_j-\delta _{i,j}}\cdot \mathcal {D}^{p^i+\eta }\psi |_{(r,\theta )=(0,0)}\right) \\&\qquad +\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\mathcal {D}^{\eta }\psi |_{(r,\theta )=(0,0)}\cdot \prod _{i=1}^s (\mathcal {D}^{p^i}\psi |_{(r,\theta )=(0,0)})^{m_i}\\&\quad =\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\sum _{i=1}^s\left( m_i\prod _{j=1}^s(\mathcal {D}^{p^i}\psi |_{(r,\theta )=(0,0)})^{m_j} \cdot \left( \frac{\mathcal {D}^{p^i+\eta }\psi }{\mathcal {D}^{p^i}\psi }\right) |_{(r,\theta )=(0,0)}\right) \\&\qquad +\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\mathcal {D}^{\eta }\psi |_{(r,\theta )=(0,0)}\cdot \prod _{i=1}^s(\mathcal {D}^{p^i}\psi |_{(r,\theta ) =(0,0)})^{m_i}\\&\quad =\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\prod _{j=1}^s(\mathcal {D}^{p^i}\psi |_{(r,\theta )=(0,0)})^{m_j} \sum _{i=1}^s\left( m_i\cdot \left( \frac{\mathcal {D}^{p^i+\eta }\psi }{\mathcal {D}^{p^i}\psi } \right) |_{(r,\theta )=(0,0)}\right) \\&\qquad +\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\mathcal {D}^{\eta }\psi |_{(r,\theta )=(0,0)}\cdot \prod _{i=1}^s (\mathcal {D}^{p^i}\psi |_{(r,\theta )=(0,0)})^{m_i}. \end{aligned}$$

Therefore, by using (B.1) and noting that $A>E>0$ we have (observe that all derivatives at zero are positive):

$$\begin{aligned}&\mathcal {D}^{\alpha +\eta }g(\psi )|_{(r,\theta )=(0,0)}\\&\quad \le \sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}AE^{\eta _2}\sum ^s_{i=1}(m_i|p^i|)C_{(s,\underline{P},\varvec{M})}\prod _{j=1}^s(\mathcal {D}^{p^i}\psi |_{(r,\theta )=(0,0)})^{m_i}\\&\qquad +\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}E^{\eta _2+1}C_{(s,\underline{P},\varvec{M})}\prod _{j=1}^s(\mathcal {D}^{p^i}\psi |_{(r,\theta )=(0,0)})^{m_i}\\&\quad =(\vert \alpha \vert AE^{\eta _2}+E^{\eta _2+1})\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\prod _{j=1}^s(\mathcal {D}^{p^i}\psi |_{(r,\theta )=(0,0)})^{m_i}\\&\quad \le (\vert \alpha \vert +1) AE^{\eta _2}\sum _{(s,\underline{P},\varvec{M})\in \mathscr {D}_{\alpha },m\le 2k+1}C_{(s,\underline{P},\varvec{M})}\prod _{j=1}^s(\mathcal {D}^{p^i}\psi |_{(r,\theta )=(0,0)})^{m_i}\\&\quad =(\vert \alpha \vert +1) AE^{\eta _2} I_{[\alpha ,A,E]}. \end{aligned}$$

which was to be proved. $\square$