Global existence and partial regularity for the p-harmonic flow

Abstract

We show a global existence for the Cauchy problem with large initial data for the p-harmonic flow between two smooth, compact Riemannian manifolds. We devise new monotonicity type formulas of a local scaled energy and establish a partial regularity for the solution. The partial regularity obtained is almost optimal, comparing with that of the corresponding stationary case. The p-harmonic flow obtained also converges to a p-harmonic map along a certain time sequence tending to infinity.

6. Appendix

1.1 Appendix A   A global existence and regularity of a weak solution of (2.1).

Proof od Lemma 6

We use the Galerkin method and the monotonicity trick for p-Laplace operator to solve the Cauchy problem (2.1). The proof is standard and we can refer to [6, Theorem 1.5 and its proof, pp. 29–31].

Regularity of a weak solution. Let $u = u_{K, \, \epsilon }$ be a weak solution of (2.1). The lower-order term is bounded by the definition of $\chi$ as

$$\begin{aligned} \Big | K \, \chi ^\prime \left( \text{ dist }^2 (p, \, {\mathcal{N}})\right) \text{ dist } (p, \, {\mathcal{N}}) D_p \text{ dist } (p, \, {\mathcal{N}}) \Big | \le C \, K \, \delta _{\mathcal{N}} \sup _ {s > 0} \big |\chi ^\prime (s)\big | \end{aligned}$$

and thus, we can apply the Hölder regularity result for the evolutionary p-Laplace operator in [12, Theorem $1.1^\prime$, p. 256] (also see [26]) to find that the solution u and its gradient are locally Hölder continuous on ${\mathcal{M}}_\infty$. We also have that the second derivative is integrable : $\big (\epsilon + |D u|^2\big )^{\frac{p - 2}{2}} |D^2 u|^2$ is locally integrable in ${\mathcal{M}}_\infty$ and that the gradient Du is locally bounded in ${\mathcal{M}}_\infty$ (see [12, Proposition 3.1, p. 223; Theorem 5.1 , p. 238]. Then, expanding the principal part of the p-Laplace operator, the solution u is also satisfies the linear parabolic systems with Hölder continuous coefficients and lower-order terms almost everywhere. Thus, it follows from the Schauder regularity theory that u, Du, $D^2 u$ and $\partial _t u$ are locally Hölder continuous in ${\mathcal{M}}_\infty$. $\square$

1.2 Appendix B   Energy inequality and maximum principle

We present the proof of Lemmata 7 and 8.

Proof of Lemma 7

The energy inequality (3.1) is shown to be valid in the proof of Lemma 6. However, as a priori estimates for regular solutions of (2.1), we naturally multiply (2.1) by $\partial _t u \, \sqrt{|g|}$ and integrate by parts on space variable in ${\mathcal{M}}_T$ for any $T > 0$. $\square$

Proof of Lemma 8

We multiply (2.1) by $\sqrt{|g|} u \big (|u|^2 - H^2\big )_+$ and integrate in ${\mathcal{M}}_\infty$, where $\big (f\big )_+$ is the positive part of a function f. Since the support of $\chi ^\prime$ is in $\mathcal{O}_{2 \delta _{\mathcal{N}}} ({\mathcal{N}}) \subset B (H)$, $\chi ^\prime (\mathrm{dist}^2 (u, \, {\mathcal{N}}))$ is zero in $\mathrm{I}\!\mathrm{R}^l {\setminus } B (H, \, 0)$. Also $u_0 \in {\mathcal{N}} \subset B (H)$. Hence, we have

$$\begin{aligned}&\displaystyle { \frac{1}{4} \int _{\mathcal{M}} \big (|u (t)|^2 - H^2\big )_+ \, d {\mathcal{M}} }\\&\quad \displaystyle { + \int _{{\mathcal{M}}_t} \big (\epsilon + |D u|_g^2\big )^{\frac{p - 2}{2}} \left( \frac{1}{2} \big |D \big (|u|^2 - H^2\big )_+\big |_g^2 + |D u|_g^2 \big (|u|^2 - H^2\big )_+ \right) \, d {\mathcal{M}} d t } = 0 \quad ; \\&\displaystyle { \frac{1}{4} \int _{\mathcal{M}} \big (|u (t)|^2 - H^2\big )_+^2 \, d {\mathcal{M}} \le 0 } \end{aligned}$$

and thus, $|u (t)| \le H$ in ${\mathcal{M}}$ and any $t \ge 0$. $\square$

1.3 Appendix C   Proof of the Bochner estimate.

Proof of Lemma 9

In the proof, for brevity, let the regularized p-energy density be

$$\begin{aligned} \displaystyle { f = f (u) : = \frac{1}{p} \big (\epsilon + |D u|^2\big )^{\frac{p}{2}}. } \end{aligned}$$

In the general case in ${\mathcal{M}}$, the terms containing the spatial derivative of $g_{\alpha \beta }$ only appear and are bounded by $C \, (\epsilon + |D u|^2)^{p - 1}$. In fact, in (6.5) below, by a direct computation, we have the terms with derivatives of the metric

$$\begin{aligned}&- \frac{ D_\gamma \sqrt{|g|} }{|g|} \, g^{\gamma \mu } D_\mu u \cdot D_\alpha \big ( (p \, f)^{1 - \frac{2}{p}} \sqrt{|g|} g^{\alpha \beta } D_\beta u \big ) \nonumber \\&\quad + D_\alpha \big ( (p f)^{1 - \frac{2}{p}} D_\gamma (\sqrt{|g|} g^{\alpha \beta }) g^{\gamma \mu } D_\mu u \cdot D_\beta u \big ) \nonumber \\&\displaystyle { \quad - \frac{1}{2} \, D_\alpha \big ( (p f)^{1 - \frac{2}{p}} \sqrt{|g|} g^{\alpha \beta } D_\beta g^{\gamma \mu } D_\gamma u \cdot D_\mu u \big ) } \nonumber \\&\displaystyle { \quad - \frac{1}{\sqrt{|g|}} (p f)^{1 - \frac{2}{p}} \left\{ D_\alpha g^{\gamma \mu } D_\gamma (\sqrt{|g|} g^{\alpha \beta }) D_\mu u \cdot D_\beta u \right. } \nonumber \\&\displaystyle { \quad + \frac{p - 2}{2} \sqrt{|g|} g^{\alpha \beta } \frac{ D_\alpha g^{\gamma \mu } D_\mu u \cdot D_\beta u }{ (p f)^{\frac{2}{p}} } \big ( D_\gamma g^{\mu \nu } D_\mu u \cdot D_\nu u + 2 g^{\mu \nu } D_\mu D\gamma u \cdot D_\nu u \big ) } \nonumber \\&\displaystyle { \quad + \sqrt{|g|} g^{\alpha \beta } D_\alpha g^{\gamma \mu } D_\mu u \cdot D_\beta D_\gamma u + D_\gamma (\sqrt{|g|} g^{\alpha \beta } g^{\gamma \mu } D_\alpha D_\mu u \cdot D_\beta u } \nonumber \\&\displaystyle { \left. \quad \, + \frac{p - 2}{2} \sqrt{|g|} g^{\gamma \mu } g^{\alpha \beta } D_\alpha D_\mu u \cdot D_\beta u \frac{ D_\gamma g^{\mu \nu } D_\mu u \cdot D_\nu u }{ (p f)^{\frac{2}{p}} } \right\} , } \end{aligned}$$

(6.1)

which are bounded by such terms as

$$\begin{aligned} \displaystyle { C \, |D g^{\alpha \beta }|^2 \, f; \quad C \, |D g^{\alpha \beta }| |g^{\alpha \beta }| \, f^{1 - \frac{1}{p}} \, |D^2 u| } \end{aligned}$$

(6.2)

with a positive constant C depending only on $\mathcal{M}$, p and m. Here the 1st term is controllable lower-order one. By Cauchy’s inequality with $c > 0$, the 2nd term are estimated above as

$$\begin{aligned} \displaystyle { C \, |D g^{\alpha \beta }| |g^{\alpha \beta }| \, f^{1 - \frac{1}{p}} \, |D^2 u| \le c f^{1 - \frac{2}{p}} \, |D^2 u|^2 + C (c^{- 1}) \, |D g^{\alpha \beta }|^2 |g^{\alpha \beta }|^2 \, f, } \end{aligned}$$

of which the 1st term with a small $c >0$ in the right hand side is abosrbed into the squared 2nd derivative term of the solution in (6.5) below, and the 2nd term is a controllable one. The controllable terms $C \, f$ above are multiplied by $(p f)^{1 - \frac{2}{p}}$ in (6.6) below, and thus, becomes $C \, f^{2 (1 - \frac{1}{p})}$.

Hereafter, we assume that the metric $g = \left( g_{\alpha \beta }\right)$ is the identity matrix.

Since u, Du and $D^2 u$ are continuous in $\mathrm{I}\!\mathrm{R}^m_\infty$, it holds in the distribution sense that

$$\begin{aligned} D u \cdot D \Big (f^{1 - \frac{2}{p}} D u\Big )= & {} \, D_\alpha \Big ( \mathcal{A}^{\alpha \beta } D_\beta f \Big ) - (p \, f)^{1 - \frac{2}{p}} \big |D^2 u\big |^2 \nonumber \\& -\, (p - 2) (p \, f)^{1 - \frac{4}{p}} \big |D \frac{1}{2} |D u|^2\big |^2.\qquad \end{aligned}$$

(6.3)

Hereafter the summation convention over repeated indices is used. Since $\chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}})\big ) = 2 (\delta _{\mathcal{N}})^2$ for $u \in \mathrm{I}\!\mathrm{R}_l {\setminus } \mathcal{O}_{2 \delta _{\mathcal{N}}}$, $D \Big (D_u \chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}})\big )\Big ) = 0$ if $\text{ dist } (u, \, {\mathcal{N}}) > 2 \delta _{\mathcal{N}}$ and then, we have (3.4) by (2.1). We treat the case that $\text{ dist } (u, \, {\mathcal{N}}) \le 2 \delta _{\mathcal{N}}$. Noting that $\chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}})\big )$ is smooth, by a direct calculation we have

$$\begin{aligned}&\displaystyle { D u \cdot D \Big (\frac{K}{2} \, D_u \chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}})\big )\Big ) = \frac{K}{2} \, \left( D u^i \cdot D u^j \right) D_{u^i} D _{u^j} \chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}})\big ) \quad ; } \nonumber \\&\displaystyle { D_{u^i} D _{u^j} \chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}})\big ) = 2 \, D_{u^i} \text{ dist } (u, \, {\mathcal{N}}) D_{u^j} \text{ dist } (u, \, {\mathcal{N}}) \left( \chi ^\prime + 2 { \text{ dist }}^2 (u, \, {\mathcal{N}}) \chi ^{\prime \prime } \right) } \nonumber \\&\quad \displaystyle { + \, 2 \chi ^\prime \, \text{ dist } (u, \, {\mathcal{N}}) D_{u^i} D _{u^j} \text{ dist } (u, \, {\mathcal{N}}), } \end{aligned}$$

(6.4)

where the arguments in $\chi ^\prime$ are omitted. By (6.3) and (6.4) with (2.1), we have

$$\begin{aligned}&\displaystyle { \partial _t \frac{1}{2} (p \, f)^{\frac{2}{p}} - D_\alpha \Big ( \mathcal{A}^{\alpha \beta } D_\beta f \Big ) + (p \, f)^{1 - \frac{2}{p}} \big |D^2 u\big |^2 + (p - 2) (p \, f)^{1 - \frac{4}{p}} \big |D \frac{1}{2} |D u|^2\big |^2 } \nonumber \\&\quad \displaystyle { + \, C_0 \, K |D \text{ dist } (u, \, {\mathcal{N}})|^2 \left( \chi ^\prime + 2 { \text{ dist }}^2 (u, \, {\mathcal{N}}) \chi ^{\prime \prime } \right) } \nonumber \\&\quad \displaystyle { + \, C_0 \, K \chi ^\prime \, \text{ dist } (u, \, {\mathcal{N}}) D u^i \cdot D u^j D_{u^i} D_{u^j} \text{ dist } (u, \, {\mathcal{N}}) = 0. } \end{aligned}$$

(6.5)

Furthermore, multiplying (6.5) by $(p \, f)^{1 - \frac{2}{p}}$, we obtain

$$\begin{aligned}&\displaystyle { \partial _t f - D_\alpha \Big ( (p \, f)^{1 - \frac{2}{p}} \, \mathcal{A}^{\alpha \beta } D_\beta f \Big ) + (p \, f)^{2 - \frac{4}{p}} \big |D^2 u\big |^2 + (p - 2) (p \, f)^{2 - \frac{6}{p}} \big |D \frac{1}{2} |D u|^2\big |^2 } \nonumber \\&\quad \displaystyle { + \, C_0 \, (p \, f)^{1 - \frac{2}{p}} \, K |D \text{ dist } (u, \, {\mathcal{N}})|^2 \left( \chi ^\prime + 2 { \text{ dist }}^2 (u, \, {\mathcal{N}}) \chi ^{\prime \prime } \right) }\nonumber \\&\quad \displaystyle { + \, C_0 \, (p \, f)^{1 - \frac{2}{p}} \, K \chi ^\prime \, \text{ dist } (u, \, {\mathcal{N}}) D u^i \cdot D u^j D_{u^i} D_{u^j} \text{ dist } (u, \, {\mathcal{N}}) \le 0, } \end{aligned}$$

(6.6)

where we use the fact that

$$\begin{aligned} \displaystyle { \mathcal{A}^{\alpha \beta } D_\beta f D_\alpha (p \, f)^{1 - \frac{2}{p}} = p^{- \frac{2}{p}} (p - 2) \, \mathcal{A}^{\alpha \beta } D_\beta f D_\alpha f \ge 0. } \end{aligned}$$

By differentiation of the penalty term

$$\begin{aligned} \displaystyle { g = g (u) : = \frac{K}{2} \chi \big ({ \text{ dist }}^2 (u, \, {\mathcal{N}})\big ) } \end{aligned}$$

and (6.4), we have

(6.7)

where in particular, multiplying (2.1) by the derivative of penalty term we compute

(6.8)

By the support of $\chi ^{\prime \prime }$, we have

$$\begin{aligned} \text{ dist }^2 (u, \, {\mathcal{N}}) \left| \chi ^{\prime \prime }\right| \le 100 \, \sup \left| \chi ^\prime \right| \chi \end{aligned}$$

and thus, the 2nd terms in the 2nd line of (6.6) and the 3rd line of (6.7), and the 3rd term in the 3rd line of (6.8) are estimated above by

$$\begin{aligned}&2 \, K \, (p \, f)^{ 1 - \frac{2}{p} } \, { \text{ dist }}^2 (u, \, {\mathcal{N}}) \, \left| \chi ^{\prime \prime } \right| \, \left( (C_0 + 1) \big | D \text{ dist } (u, \, {\mathcal{N}}) \big |^2\right. \nonumber \\&\qquad \left. + \, \big | \mathcal{A}^{\alpha \beta } D_\alpha \text{ dist } (u, \, {\mathcal{N}}) D_\beta \text{ dist } (u, \, {\mathcal{N}}) \big | \right) \nonumber \\&\quad \displaystyle { \le C \, \left( 1 + C_0\right) f \, g } \end{aligned}$$

(6.9)

with a positive constant C depending only on p and $\chi$, because

$$\begin{aligned} \left| D_u \text{ dist } (u, \, {\mathcal{N}})\right| = 1; \quad \left| D \text{ dist } (u, \, {\mathcal{N}}) \right|= & {} \left| D u \cdot D_u \text{ dist } (u, \, {\mathcal{N}}) \right| \\\le & {} |D u| \le (p \, f)^{\frac{2}{p}}. \end{aligned}$$

By Schwarz’s and Cauchy’s inequalities, the terms in the 3rd lines of (6.6) and in the 4th line of (6.7), (6.8) are bounded by

$$\begin{aligned}&\displaystyle { C ({\mathcal{N}}) \, \left( C_0 + p - 1\right) \, (p \, f)^{1 - \frac{2}{p}} |D u|^2 \, K \chi ^\prime \, \text{ dist } (u, \, {\mathcal{N}}) } \nonumber \\&\quad \displaystyle { = \, C ({\mathcal{N}}) \, \left( C_0 + p - 1\right) \, (p \, f)^{1 - \frac{2}{p}} \, |D u|^2 \left| D_u g\right| } \nonumber \\&\quad \displaystyle { \le \, \left( \frac{C_0}{2} + 1\right) \left| D_u g\right| ^2 + C^2 ({\mathcal{N}}) \left( \frac{C_0}{2} + \frac{(p - 1)^2}{4} \right) (p \, f)^{2 - \frac{4}{p}} |D u|^4, } \end{aligned}$$

(6.10)

where by a positive constant $C ({\mathcal{N}})$ depending on a bound for the curvature of ${\mathcal{N}}$, we have the boundedness for any $u \in {\mathcal{N}}$

$$\begin{aligned} \displaystyle { \left| \big ( 2 \delta ^{\alpha \beta } + \mathcal{A}^{\alpha \beta } \big ) D_\alpha u^i D_\beta u^j \, D_{u^i} D_{u^j} \text{ dist } (u, \, {\mathcal{N}}) \right| \le C ({\mathcal{N}}) \, |D u|^2, } \end{aligned}$$

(6.11)

of which the validity will be shown later.

The terms in the 5th line of (6.7) are bounded by

$$\begin{aligned} \displaystyle { 2 (p - 2) \, (p \, f)^{1 - \frac{2}{p}} \left| D^2 u\right| \left| D_u g\right| \le \frac{1}{2} (p \, f)^{2 - \frac{4}{p}} \left| D^2 u\right| ^2 + 2 (p - 2)^2 \, \left| D_u g\right| ^2. } \end{aligned}$$

(6.12)

Gathering (6.9), (6.10) and (6.12) in (6.6) and (6.7), respectively, we obtain

$$\begin{aligned}&\displaystyle { \partial _t e (u) - D_\alpha \left( (p \, f)^{1 - \frac{2}{p}} \mathcal{A}^{\alpha \beta } D_\beta e (u) \right) } \nonumber \\&\qquad \displaystyle { + \, (p \, f)^{ 2 \left( 1 - \frac{2}{p}\right) } \big |D^2 u\big |^2 + (p - 2) (p \, f)^{ 2 \left( 1 - \frac{3}{p}\right) } \big |D \frac{1}{2} |D u|^2\big |^2 + C_2 \, \big |D_u g\big |^2 } \nonumber \\&\quad \displaystyle { \le \, C^2 ({\mathcal{N}}) \left( \frac{C_0}{2} + \frac{(p - 1)^2}{4} \right) (p \, f)^{2 \left( 1 - \frac{2}{p}\right) } |D u|^4 + C \, \left( 1 + C_0\right) f \, g } \end{aligned}$$

(6.13)

and thus, from (6.13), the desired inequality (3.4) is obtained, if the constant $C_0$ is so large that

$$\begin{aligned} \displaystyle { C_2 : = \frac{C_0}{2} - 1 - \frac{25 (p - 2)^2}{2} > 0. } \end{aligned}$$

(6.14)

We present the proof of (6.11). We follow the argument as in [2, Theorems 3.1 and 3.2, their proofs, pp. 704–707] (also refer to [1, Theorem 2.2]). $\square$

Lemma 21

There exists a positive constant C depending only on a bound of curvatures of $\mathcal{N}$ such that, for any $u \in \mathcal{O}_{2 \delta _{\mathcal{N}}}$ and $q \in \mathrm{I}\!\mathrm{R}^l \cong \mathcal{T}_u \mathrm{I}\!\mathrm{R}^l$,

$$\begin{aligned} \big | q^i \, q^j \, D_{u^i} D_{u^j} \text{ dist } (u, \, {\mathcal{N}}) \big | \le C \, |q|^2. \end{aligned}$$

(6.15)

Proof

For any $u \in \mathcal{O}_{2 \delta _{\mathcal{N}}}$ such that $u \notin {\mathcal{N}}$, we make parallel transformation with the direction $\big (u - \pi _{\mathcal{N}} (u)\big )/\big |u - \pi _{\mathcal{N}} (u)\big |$ and follow the following argument. Therefore, we treat the case that $u \in {\mathcal{N}}$ and thus, $\pi _{\mathcal{N}} (u) = u$. For any $v \in \mathcal{O}_{2 \delta _{\mathcal{N}}}$ let us put

$$\begin{aligned} \displaystyle { d (v) : = \text{ dist } (v, \, {\mathcal{N}}); \quad \eta (v) = \frac{1}{2} d (v)^2. } \end{aligned}$$

We know that the squared distance function $\eta (v)$ is smooth on $v \in \mathcal{O}_{2 \delta _{\mathcal{N}}}$. Let $q \in \mathrm{I}\!\mathrm{R}^l$ be any vector in $\mathrm{I}\!\mathrm{R}^l$ and be fixed. Under the orthogonal decomposition of $\mathrm{I}\!\mathrm{R}^l$ with respect to the tangent space $\mathcal{T}_u {\mathcal{N}}$ at $u \in {\mathcal{N}}$, $\mathrm{I}\!\mathrm{R}^l = \mathcal{T}_u {\mathcal{N}} \oplus (\mathcal{T}_u {\mathcal{N}})^\bot$, we set as

$$\begin{aligned} \displaystyle { q = q_\tau + q_\nu ; \quad p : = \frac{q_\nu }{\big |q_\nu \big |}; \quad f (t) : = d (u + t p) \quad \text{ for } \text{ any } t \in (0, \, 2 \delta _{\mathcal{N}}], } \end{aligned}$$

(6.16)

where p is the unit normal vector along the normal component of q and f(t) is the distance to $\mathcal{N}$ measured along p. Then, we compute as

$$\begin{aligned} \displaystyle { D_v d (u + t p) : = \left. D_v d (v) \right| _{v = u + t p} = p; \quad \frac{d f}{d t} (t) = p \cdot D_u d (u + t p) = |p|^2 = 1 \qquad } \end{aligned}$$

(6.17)

and, also, for any $v \in \mathcal{O}_{2 \delta _{\mathcal{N}}}$, $v \notin {\mathcal{N}}$,

$$\begin{aligned}&\displaystyle { D_v \eta (v) = d (v) D_u d (v); \quad } \displaystyle { D_{v^i} D_{v^j} \eta (v) = d (v) D_{v^i} D_{v^j} d (v) + D_{v^i} d (v) D_{v^j} d (v) } \nonumber \\&\quad \displaystyle { \Longleftrightarrow D_{v^i} D_{v^j} d (v) = \frac{ D_{v^i} D_{v^j} \eta (v) - D_{v^i} d (v) D_{v^j} d (v) }{d (v)}. } \end{aligned}$$

(6.18)

Thus, letting, for any $t \in (0, \, 2 \delta _{\mathcal{N}}]$,

$$\begin{aligned} \displaystyle { D_{v^i} D_{v^j} d (u + t p) : = \left. D_{v^i} D_{v^j} d (v) \right| _{v = u + t p}; \quad D_{v^i} D_{v^j} \eta (u + t p) : = \left. D_{v^i} D_{v^j} \eta (v) \right| _{v = u + t p}, } \end{aligned}$$

we have

$$\begin{aligned} \displaystyle { q^i D_{v^i} D_{v^j} d (u + t p) q^j = \frac{ q^i D_{v^i} D_{v^j} \eta (u + t p) q^j - \big |q \cdot D_v d (u + t p)\big |^2 }{d (u + t p)}, } \end{aligned}$$

where by (6.17) we have, as $t \searrow 0$,

$$\begin{aligned}&\displaystyle { q^i D_{v^i} D_{v^j} \eta (u + t p) q^j \rightarrow q^i D_{v^i} D_{v^j} \eta (u) q^j = \big |q_\nu \big |^2; }\\&\displaystyle { \big |q \cdot D_v d (u + t p)\big |^2 = (q \cdot p)^2 = \big |q_\nu \big |^2 \rightarrow \big |q_\nu \big |^2; \quad d (u + t p) \rightarrow 0 } \end{aligned}$$

and the 1st convergence is valid because $D_{v^i} D_{v^j} \eta (u) = \left. D_{v^i} D_{v^j} \eta (v)\right| _{v = u}$ is the orthogonal projection on $(\mathcal{T}_u {\mathcal{N}})^\bot$ (see [2, Theorem 3.1, p.704]). Therefore, from l’Hospital’s theorem, we obtain

$$\begin{aligned}&\displaystyle { q^i D_{v^i} D_{v^j} d (u) q^j : = \exists \, \lim _{t \searrow 0} \left( q^i D_{v^i} D_{v^j} d (u + t p) q^j \right) }\\&\quad \displaystyle { = \lim _{t \searrow 0} \frac{ q^i q^j D_{v^i} D_{v^j} D_{v^k} \eta (u + t p) p^k - 2 q \cdot D_v (p \cdot D_v d (u + t p)) \, q \cdot D_v d (u + t p) }{p \cdot D_v d (u + t p)} }\\&\quad \displaystyle { = q^i q^j D_{v^i} D_{v^j} D_{v^k} \eta (u) p^k, } \end{aligned}$$

where we use that $\displaystyle { p \cdot D_v d (u + t p) = \big |p\big |^2 = 1 }$ and $\displaystyle { D_v (p \cdot D_u d (u + t p)) = 0. }$ Thus, we have

$$\begin{aligned} \displaystyle { \left| q^i D_{v^i} D_{v^j} d (u) q^j \right| = \left| q^i q^j D_{v^i} D_{v^j} D_{v^k} \eta (u) p^k \right| \le C \, \big |q\big |^2, } \end{aligned}$$

where in the last inequality the positive constant C depends only on a bound of curvatures of $\mathcal{N}$ (see [2, Remark 3.3; Theorem 3.5, its proof, pp. 707–709]). $\square$

1.4 Appendix D   Proof of the gradient boundedness.

Here we demonstrate the proof of Lemma 10, relying on Moser’s iteration method as usual. Such estimate has been originally done for the evolutionary p-Laplacian system with controllable growth lower-order terms, by DiBenedetto, developing the intrinsic scaling transformation to the evolutionary p-Laplace operator (refer to [10, 12]).

However, the emphasis here is to make localization by use of the cut-off function $\mathcal C$.

Proof of Lemma 10

In the following we use the same notation as in Lemma 10.

By use of a scaling transformation intrinsic to the evolutionary p-Laplace operator

$$\begin{aligned} \displaystyle { t = t_0 + L^{2 - p} (r_0)^2 s; \quad x = x_0 + r_0 y, } \end{aligned}$$

(6.19)

we now rewrite (3.4) in Lemma 9 by the scaled solution v on $Q (1, \, 1) : = Q (1, \, 1) (0)$,

$$\begin{aligned} \displaystyle { v (s, \, y) = \frac{u \left( t_0 + L^{2 - p} (r_0)^2 s, \, x_0 + r_0 y\right) }{L \, r_0} } \end{aligned}$$

satisfying in $Q (1, \, 1)$

$$\begin{aligned}&\partial _s v - \frac{1}{\sqrt{|g|}} D_\alpha \left( \big (L^{- 2} \epsilon + |D v|^2\big )^{\frac{p - 2}{2}} \sqrt{|g|} \, g^{\alpha \beta } D_\beta v \right) \nonumber \\&\quad = - C_0 \, \frac{K/L^p}{2} D_v \chi \left( \text{ dist }^2 \big (L r_0 v, \, {\mathcal{N}}\big )\right) . \end{aligned}$$

(6.20)

We put the notation

$$\begin{aligned}&\displaystyle { {\bar{\epsilon }} = L^{- 2} \epsilon ; } \displaystyle { \quad {\bar{K}} = L^{- p} K; } \\&\displaystyle { f = f (v) : = \frac{1}{p} \big ({\bar{\epsilon }} + |D v|^2\big )^{\frac{p}{2}}; } \displaystyle { \quad g = g (v) : = \frac{{\bar{K}}}{2} \chi \big ({ \text{ dist }}^2 (L r v, \, {\mathcal{N}})\big ); }\\&\displaystyle { e (v) = f (v) + g (v); } \displaystyle { \quad \mathcal{B}^{\alpha \beta } = g^{\alpha \beta } + \frac{(p - 2) \, g^{\alpha \gamma } \, g^{\beta \mu } D_\gamma v \cdot D_\mu v }{{\bar{\epsilon }} + |D v|^2}. } \end{aligned}$$

As in “Appendix C”, we assume that the metric $(g_{\alpha \beta })$ is the identity matrix and, in the general case in ${\mathcal{M}}$, the terms with the derivative of $g_{\alpha \beta } (x_0 + r y)$ appear and are bounded by $C \, f (v)^{2 (1 - \frac{1}{p})}$, as in (6.1) and (6.2), where we note that $D_y g^{\alpha \beta } (x_0 + r y) = r D g^{\alpha \beta } (x)$ and $r \le 1$. We proceed to the same computation as (6.6) and (6.7), where the quantities appeared are transformed to the corresponding ones, respectively, defined by the scaled solution v as above. Now we will look at the transformed estimation for the scaled solution v.

By the support of $\chi ^{\prime \prime }$, we have

$$\begin{aligned} \displaystyle { \text{ dist }^2 (L r_0 v, \, {\mathcal{N}}) \left| \chi ^{\prime \prime }\right| \le 100 \, \sup \left| \chi ^\prime \right| \, \chi } \end{aligned}$$

and thus, the corresponding terms for the scaled solution v to 2nd terms in the 2nd line of (6.6) and the 3rd line of (6.7), and the 3rd term in the 3rd line of (6.8) are estimated above by

$$\begin{aligned}&\displaystyle { 2 \, \bar{K} \, (p \, f)^{1 - \frac{2}{p}} \, { \text{ dist }}^2 (L r_0 v, \, {\mathcal{N}}) \, \left| \chi ^{\prime \prime }\right| \times } \nonumber \\&\qquad \displaystyle { \times \left( (C_0 + 1) \big | D \text{ dist } (L r_0 v, \, {\mathcal{N}}) \big |^2 + \, \mathcal{B}^{\alpha \beta } D_\alpha \text{ dist } (L r_0 v, \, {\mathcal{N}}) D_\beta \text{ dist } (L r_0 v, \, {\mathcal{N}}) \right) } \nonumber \\&\quad \displaystyle { \, \, \le C \, \left( 1 + C_0\right) \, (p \, f)^{1 - \frac{2}{p}} \, g } \end{aligned}$$

(6.21)

with a positive constant C depending only on p and $\chi$, because, by the definition of $r_0$,

$$\begin{aligned}&\displaystyle { D_v \text{ dist } (L r_0 v, \, {\mathcal{N}}) = L r_0 \left. D_u \text{ dist } (u, \, {\mathcal{N}}) \right| _{u = L r_0 v}; \quad \left| D_v \text{ dist } (L r_0 v, \, {\mathcal{N}}) \right| = L r_0 ; } \\&\displaystyle { \left| D \text{ dist } (L r_0 v, \, {\mathcal{N}}) \right| \le L r_0 \, |D v| \le L r_0 \, \frac{1}{L} \Vert D u\Vert _{\mathrm{L}^\infty (Q_0)} \le C. } \end{aligned}$$

By Schwarz’s and Cauchy’s inequality, the terms in the 3rd line of (6.6) and in the 4th line of (6.7), (6.8) are estimated as

$$\begin{aligned}&\displaystyle { C ({\mathcal{N}}) \, \left( C_0 + p - 1\right) \, L^2 (r_0)^2 \, (p \, f)^{1 - \frac{2}{p}} |D v|^2 \, {\bar{K}} \chi ^\prime \, \text{ dist } (L r_0 v, \, {\mathcal{N}}) } \nonumber \\&\quad \displaystyle { \le C ({\mathcal{N}}) \, C \, \left( C_0 + p - 1\right) \, (p \, f)^{1 - \frac{2}{p}} |D v| \, \left| D_v g\right| } \nonumber \\&\quad \displaystyle { \le \, \left( \frac{C_0}{2} + 1\right) \left| D_v g\right| ^2 + (C ({\mathcal{N}}) C)^2 \left( \frac{C_0}{2} + \frac{(p - 1)^2}{4} \right) (p \, f)^{2 - \frac{4}{p}} |D v|^2, } \end{aligned}$$

(6.22)

where by the definition of $r_0$ as before, and a positive constant $C ({\mathcal{N}})$ depending on a bound for the curvature of ${\mathcal{N}}$, we compute as

$$\begin{aligned}&\displaystyle { L r_0 \, |D v| \le L r_0 \, \frac{1}{L} \Vert D u\Vert _{\mathrm{L}^\infty (Q_0)} \le C; }\\&\displaystyle { D_v g = L r_0 \, {\bar{K}} \, \chi ^\prime \, \text{ dist } (L r_0 v, \, {\mathcal{N}}) \left. D_u \text{ dist } (u, \, {\mathcal{N}}) \right| _{u = L r_0 v}; \quad \left| \left. D_u \text{ dist } (u, \, {\mathcal{N}}) \right| _{u = L r_0 v} \right| = 1; } \\&\displaystyle { D_{v^i} D_{v^j} \text{ dist } (L r_0 v, \, {\mathcal{N}}) = L^2 (r_0)^2 \, \left. D_{u^i} D_{u^j} \text{ dist } (u, \, {\mathcal{N}}) \right| _{u = L r_0 v}; }\\&\displaystyle { \left| \big ( 2 \delta ^{\alpha \beta } + \mathcal{B}^{\alpha \beta } \big ) D_\alpha v^i D_\beta v^j \left. D_{u^i} D_{u^j} \text{ dist } (u, \, {\mathcal{N}}) \right| _{u = L r_0 v} \right| \le C ({\mathcal{N}}) |D v|^2. } \end{aligned}$$

The terms in 5th line of (6.7) are bounded by

$$\begin{aligned} \displaystyle { 2 (p - 2) \, (p \, f)^{1 - \frac{2}{p}} \left| D^2 v\right| \left| D_v g\right| \le \frac{1}{2} (p \, f)^{2 - \frac{4}{p}} \left| D^2 v\right| ^2 + 2 (p - 2)^2 \, \left| D_v g\right| ^2. } \end{aligned}$$

(6.23)

Gathering all of the estimations above yields

$$\begin{aligned}&\displaystyle { \partial _t e (v) - \sum _{\alpha , \, \beta = 1}^m D_\alpha \left( (p \, f)^{1 - \frac{2}{p}} \mathcal{B}^{\alpha \beta } D_\beta e (v) \right) + \, C_1 \, (p \, f)^{2 - \frac{4}{p}} \big |D^2 v\big |^2 } \nonumber \\&\quad \displaystyle { \le \, C^\prime ({\mathcal{N}}) \left( \frac{C_0}{2} + \frac{(p - 1)^2}{4} \right) (p \, f)^{2 - \frac{4}{p}} |D v|^2 + C \, \left( 1 + C_0\right) (p \, f)^{1 - \frac{2}{p}} \, g. } \qquad \end{aligned}$$

(6.24)

Finally we make Moser’s iteration estimate by (6.24) and scaling back to have (3.6). Now, taking care of localization by the cut off function $\mathcal C$, we proceed to the estimations.

Let $B (\rho ) = B (\rho , \, 0)$ be a ball in $\mathrm{I}\!\mathrm{R}^m$ with radius $\rho \le \min \{1, \, R_{\mathcal{M}}/2, \, T^{1/\lambda _0}\}$ and center of origin. Let $0< r < \rho$. We use local parabolic cyllinders $Q (r^2, \, r) = (- r^2, 0) \times B (r)$ and $Q (\rho ^2, \, \rho ) = (- \rho ^2, 0) \times B (\rho )$, Let $\eta$ be a smooth real-valued function on $\mathrm{I}\!\mathrm{R}^m$ such that $0 \le \eta \le 1$, the support of $\eta$ is contained in $B (\rho )$ and $\eta = 1$ on B(r). Let $\sigma = \sigma (t)$ be a smooth real-valued function on $\mathrm{I}\!\mathrm{R}$ such that $0 \le \sigma \le 1$, $\sigma = 1$ on $[- r^2, \, \infty )$ and $\sigma = 0$ on $(- \infty , \, - \rho ^2]$. We denote by the original notation the scaled function under (6.19). Put

$$\begin{aligned} \displaystyle { \mathcal{C} (s, \, y) = \left( (t_0 + L^{2 - p} \, (r_0)^2 \, s + R^{\lambda _0})^{1/\lambda _0} - |x_0 + r_0 \, y| \right) _+, \quad (s, \, y) \in Q (1, 1), } \end{aligned}$$

and also write as $z = (s, \, y) \in Q (1, 1)$ and $d z = d {\mathcal{M}} d s$.

Put $w = e (v)$ in the Bochner type estimate (6.24). Let $\alpha$ be nonnegative number and use the test function $w^\alpha \eta ^2 \sigma \, \mathcal{C}^q \, \sqrt{|g|}$ in the weak form of (6.24). After a routine computation we have the so-called reverse Poincaré inequality

$$\begin{aligned}&\displaystyle { \sup _{ - r^2< \tau < 0 } \int \limits _{ \{\tau \} \times B (\rho ) } w^{\alpha + 1} \eta ^2 \sigma \, \mathcal{C}^q \, d {\mathcal{M}} \, + \, \int \limits _{ Q (\rho ^2, \, \rho ) } \, \left| D w^{ \frac{\alpha }{2} + 1 - \frac{1}{p} } \right| ^2 \eta ^2 \sigma \, \mathcal{C}^q \, d z } \nonumber \\&\quad \le \displaystyle { C \, (\alpha + p)^3 \int \limits _{ Q (\rho ^2, \, \rho ) } \, \left\{ w^{\alpha + 1} \eta ^2 \, \partial _t \sigma + w^{ \alpha + 2 - \frac{2}{p} } \left( \eta ^2 + |D \eta |^2 \right) \sigma \right\} \mathcal{C}^q \, d z, } \end{aligned}$$

(6.25)

where we compute as

$$\begin{aligned}&\displaystyle { \left| D w\right| \le |D v| \, \left( (p \, f)^{1 - \frac{2}{p}} \, \big |D^2 v\big | + \big |D_v g\big | \right) ; }\\&\displaystyle { \left| D w^{\frac{\alpha }{2} + 1 - \frac{1}{p}} \right| ^2 \le C \, (\alpha + p)^2 \, w^\alpha \, \left( (p \, f)^{2 - \frac{4}{p}} \big |D^2 v\big |^2 + \, \big |D_v g\big |^2 \right) . } \end{aligned}$$

Applying the Sobolev embedding $W^{1, 2}_0 (B (\rho )) \rightarrow L^{2 m/(m - 2)} (B (\rho ))$ we have

$$\begin{aligned} \displaystyle { \left( \int _{B (\rho )} \left( w^{ \frac{\alpha }{2} + 1 - \frac{1}{p} } \, \eta \, \mathcal{C}^{ \frac{q}{2} } \right) ^{ \frac{2 m}{m - 2} } \, d {\mathcal{M}} \right) ^{ \frac{m - 2}{2 m} } \le C \, \left( \int _{B (\rho )} \left| D \big ( w^{ \frac{\alpha }{2} + 1 - \frac{1}{p} } \, \eta \, \mathcal{C}^{ \frac{q}{2} } \big ) \right| ^2 \, d {\mathcal{M}} \right) ^{ \frac{1}{2} }, } \end{aligned}$$

which is combined with (6.25) and yields

$$\begin{aligned}&\displaystyle { \sup _{ - r^2< \tau < 0 } \int \limits _{\{\tau \} \times B (r)} (w (\tau ))^{\alpha + 1} \, \mathcal{C}^q \, d {\mathcal{M}} + \int \limits _{- r^2}^0 \left( \int \limits _{B (r)} w^{\frac{2 m}{m + 2} \frac{\alpha }{2} + 1 - \frac{1}{p}} \, \mathcal{C}^{\frac{m q}{m - 2}} \, d {\mathcal{M}} \right) ^{\frac{m - 2}{m}} \, d t } \nonumber \\&\quad \displaystyle { \le \, \frac{C \, (\alpha + p)^3}{(\rho - r)^2} \, \int \limits _{ Q (\rho ^2, \, \rho ) } \left( w^{\alpha + 1} + w^{\alpha + 2 - \frac{2}{p}} \right) \, \mathcal{C}^q \, d z. } \end{aligned}$$

(6.26)

By Hölder’s inequality and (6.26) we compute as

$$\begin{aligned}&\displaystyle { \int \limits _{Q (r^2, \, r)} w^{ \alpha + 2 - \frac{2}{p} + \frac{2 (\alpha + 1)}{m} } \, \mathcal{C}^{\frac{q (m + 2)}{m}} \, d z }\\&\quad \displaystyle { \le \int \limits _{- r^2}^0 \left( \int \limits _{B (r)} w^{\alpha + 1} \, \mathcal{C}^q \, d {\mathcal{M}} \right) ^{\frac{2}{m}} \left( \int \limits _{B (r)} w^{\frac{m (\alpha + 2 - 2 \, p^{- 1})}{m - 2}} \, \mathcal{C}^{\frac{q m}{m - 2}} \, d {\mathcal{M}} \right) ^{\frac{m - 2}{m}} \, d t } \\&\quad \displaystyle { \le \left( \sup _{- r^2< \tau < 0} \int \limits _{B (r)} w (\tau )^{\alpha + 1} \, \mathcal{C}^q \, d {\mathcal{M}} \right) ^{\frac{2}{m}} \int \limits _{- r^2}^0 \left( \int \limits _{B (r)} w^{\frac{m (\alpha + 2 - 2 \, p^{- 1})}{m - 2}} \, \mathcal{C}^{\frac{q m}{m - 2}} \, d {\mathcal{M}} \right) ^{\frac{m - 2}{m}} \, d t }\\&\quad \displaystyle { \le \left( \frac{C \, (\alpha + p)^3}{(\rho - r)^2} \, \int _{Q (\rho ^2, \, \rho )} w^{\alpha + 2 - \frac{2}{p}} \, \mathcal{C}^q \, d z \, + \frac{ C \, (\alpha + p)^3 \, |Q (\rho ^2, \, \rho )|}{(\rho - r)^2} \right) ^{\frac{m + 2}{m}}, } \end{aligned}$$

where we use a simple inequality valid for $\alpha \ge 0$

$$\begin{aligned} \displaystyle { w^{\alpha + 1} }= & {} \displaystyle { \left( \chi _{\{w \ge 1\}} + \chi _{\{w< 1\}} \right) w^{\alpha + 1} }\\\le & {} \displaystyle { \chi _{\{w \ge 1\}} \, w^{\alpha + 2 - \frac{2}{p}} + \chi _{\{w < 1\}} }\\\le & {} \displaystyle { w^{\alpha + 2 - \frac{2}{p}} + 1 } \end{aligned}$$

and also estimate the derivative of $\mathcal{C}$ as

$$\begin{aligned} \displaystyle { \left| D \mathcal{C} (s, \, y) \right| }= & {} \displaystyle { \left| D \left( (t_0 + L^{2 - p} (r_0)^2 \, s + R^{\lambda _0})^{1/\lambda _0} - |x_0 + r_0 y| \right) _+ \right| }\\= & {} \displaystyle { \left| - \frac{x_0 + r_0 y}{|x_0 + r_1^\prime y|} \, r_0 \right| \chi _{\{ |x_0 + r_0 y|< t_0; L^{2 - p} (r_0)^2 s + R^{\lambda _0}\} } (s, y) }\\\le & {} \left( (t_0 + L^{2 - p} (r_0)^2 s + R^{\lambda _0})^{1/\lambda _0} - |x_0 + r_0 y| \right) _+ \times \\&\quad \times \frac{r_0}{ (t_0 + L^{2 - p} (r_0)^2 s + R^{\lambda _0})^{1/\lambda _0} - |x_0 + r_0 y| } \\\le & {} \displaystyle { \left( (t_0 + L^{2 - p} (r_0)^2 s + R^{\lambda _0})^{1/\lambda _0} - |x_0 + r_0 y| \right) _+ } \\= & {} \displaystyle { \mathcal{C} (s, \, y); } \\ \displaystyle { \left| \partial _s \mathcal{C} (s, \, y) \right| }= & {} \displaystyle { \left| \partial _s \left( (t_0 + L^{2 - p} (r_0)^2 s + R^{\lambda _0})^{1/\lambda _0} - |x_0 + r_0 y| \right) _+ \right| } \\= & {} \displaystyle { \left| \frac{1}{\lambda _0} \, (t_0 + L^{2 - p} (r_0)^2 s + R^{\lambda _0})^{1/\lambda _0 - 1} \, L^{2 - p} (r_0)^2 \right| \chi _{ \{ |x_0 + r_0 y|< t_0 + L^{2 - p} (r_0)^2 s + R^{\lambda _0} \} } (s, y) }\\\le & {} \displaystyle { \frac{1}{\lambda _0} \, \chi _{ \{ |x_0 + r_0 y| < t_0 + L^{2 - p} (r_0)^2 s + R^{\lambda _0} \} } (s, y) \, \left( (\rho _0)^{\lambda _0} \right) ^{1/\lambda _0 - 1} \, (\rho _0)^{\lambda _0} }\\\le & {} \displaystyle { \frac{\rho _0}{\rho _0 /2} \, \frac{(\rho _0)^{\lambda _0}}{(\rho _0)^{\lambda _0}} \, \left( (t_0 + L^{2 - p} (r_0)^2 s + R^{\lambda _0})^{1/\lambda _0} - |x_0 + r_0 y| \right) _+ }\\= & {} \displaystyle { 2 \, \mathcal{C} (s, \, y), } \end{aligned}$$

because, by the range Q(1, 1) of $(s, \, y)$ and the condition (3.5) of $r_0$,

$$\begin{aligned}&\displaystyle { - 1 \le s \le 0; \quad |y| \le 1; }\\&\displaystyle { \rho _0 = \frac{ (t_0 + R^{\lambda _0})^{1/\lambda _0} - |x_0| }{4}; \quad r_0 \le \rho _0/2; \quad L^{2 - p} (r_0)^2 \le (\rho _0)^{\lambda _0} } \end{aligned}$$

and so, we have the estimations

$$\begin{aligned}&\displaystyle { r_0/(\rho _0/2) \le 1; }\\&\displaystyle { (t_0 + L^{2 - p} (r_0)^2 s + R^{\lambda _0})^{1/\lambda _0} - |x_0 + r_0 y| \ge \left( t_0 + R^{\lambda _0} - (\rho _0)^{\lambda _0} \right) ^{1/\lambda _0} - (|x_0| + r_0) }\\&\displaystyle { \ge \frac{(t_0 + R^{\lambda _0})^{1/\lambda _0} + |x_0|}{2} - |x_0| - r_0 \ge \rho _0 - \frac{\rho _0}{2} = \frac{\rho _0}{2}. } \end{aligned}$$

We arrange some terms in an appropriate way to have

$$\begin{aligned}&\displaystyle { \frac{1}{|Q (r^2, \, r)|} \, \int \limits _{Q (r^2, \, r)} w^{ \alpha + 2 - \frac{2}{p} + \frac{2 (\alpha + 1)}{m} } \, \mathcal{C}^{\frac{q (m + 2)}{m}} \, d z } \nonumber \\&\quad \displaystyle { \le \, \frac{ C \, (\alpha + p)^{3 (1 + 2/m)} \, |Q (\rho ^2, \, \rho )|^{1 + 2/m} }{ (\rho - r)^{2 (1 + 2/m)} } \left( \frac{1}{ |Q (\rho ^2, \, \rho )| } \, \int \limits _{Q (\rho ^2, \, \rho )} w^{\alpha + 2 - \frac{2}{p}} \, \mathcal{C}^q \, d z \, + 1 \right) ^{1 + \frac{2}{m}}. } \nonumber \\ \end{aligned}$$

(6.27)

Here let $\{\rho _k\}$ be a sequence of radii, defined as

$$\begin{aligned} \displaystyle { \rho _k = 2^{- 1} \, \left( 1 + 2^{- k}\right) ; \quad 1 \ge \rho _k \searrow 1/2; \quad Q_k = Q ((\rho _k)^2, \, \rho _k) (0) } \end{aligned}$$

(6.28)

and $\{\alpha _k\}$ be a sequence of exponents

$$\begin{aligned}&\displaystyle { \theta = 1 + \frac{2}{m}; \quad q > 1; \quad q_k = q \, \theta ^k; \quad 0< q< q_k \nearrow \infty \quad ; } \nonumber \\&\alpha _k = \theta ^k + 1 - \frac{2}{p}; \quad 2 - \frac{2}{p} = : \alpha _0 < \alpha _k \nearrow \infty ; \quad \alpha _{k + 1} = \alpha _k + \frac{ 2 (\alpha _k - 1 + 2 \, p^{- 1}) }{m}\nonumber \\&\quad = \alpha _k \, \theta - \frac{2 (p - 2)}{m p}. \end{aligned}$$

(6.29)

We choose $r = \rho _{k + 1}$, $\rho = \rho _k$ and $\alpha = \alpha _k$ in (6.27) and make routine computation to have

$$\begin{aligned}&\displaystyle { \frac{1}{|Q_{k + 1}|} \, \int \limits _{Q_{k + 1}} w^{ \alpha _{k + 1} } \, \mathcal{C}^{q_{k + 1}} \, d z \le C^k \, (\alpha _k + p)^{3 \theta } \, \left( \frac{1}{ |Q_k| } \, \int _{Q_k} w^{\alpha _k} \, \mathcal{C}^{q_k} \, d z \, + 1 \right) ^\theta ; } \nonumber \\&\displaystyle { \frac{1}{|Q_{k + 1}|} \, \int \limits _{Q_{k + 1}} w^{ \alpha _{k + 1} } \, \mathcal{C}^{q_{k + 1}} \, d z + 1 \le 2 \, C^k \, (\alpha _k + p)^{3 \theta } \, \left( \frac{1}{ |Q_k| } \, \int _{Q_k} w^{\alpha _k} \, \mathcal{C}^{q_k} \, d z \, + 1 \right) ^\theta }\nonumber \\ \end{aligned}$$

(6.30)

which is computed by sequences (6.28) and (6.29) as

$$\begin{aligned} \displaystyle { \left( \frac{1}{|Q_{k + 1}|} \, \int \limits _{Q_{k + 1}} w^{ \alpha _{k + 1} } \, \mathcal{C}^{q_{k + 1}} \, d z + 1 \right) ^{ \frac{1}{\theta ^{k + 1}} } \le C^{\frac{k}{\theta ^k}} \, \left( \frac{1}{ |Q_k| } \, \int _{Q_k} w^{\alpha _k} \, \mathcal{C}^{q_k} \, d z \, + 1 \right) ^{ \frac{1}{\theta ^k} }. } \qquad \end{aligned}$$

(6.31)

An iterative application of (6.31) yields, as $k \rightarrow \infty$,

(6.32)

where we use the relation of exponents

$$\begin{aligned}&\displaystyle { q_{k + 1} = q_0 \, \theta ^{k + 1} < q_0 \alpha _{k + 1} \Longleftrightarrow \alpha _{k + 1} = \theta ^{k + 1} + 1 - \frac{2}{p} > \theta ^{k + 1}; }\\&\displaystyle { 0 \le \mathcal{C} (s, \, y) \le 1, \quad (s, \, y) \in Q (1, 1) (0) } \end{aligned}$$

and the limit as $k \rightarrow \infty$

$$\begin{aligned} \frac{ \alpha _{k + 1} }{ \theta ^{k + 1} } = 1 + \frac{p - 2}{p \, \theta ^{k + 1}} \rightarrow 1. \end{aligned}$$

Finally, scaling back in (6.32) yields the desired estimate (3.6). $\square$