Splitting-type variational problems with asymmetrical growth conditions

Abstract

Splitting-type variational problems

$$\begin{aligned} \int _{\Omega }\sum _{i=1}^n f_i(\partial _i w) \, \textrm{d}x\rightarrow \min \end{aligned}$$

with superlinear growth conditions are studied by assuming

$$\begin{aligned} h_i(t) \le f''_i(t) \le H_i(t) \qquad (*) \end{aligned}$$

with suitable functions $h_i$, $H_i$: $\mathbb {R}\rightarrow \mathbb {R}^+$, $i=1$, ..., n, measuring the growth and ellipticity of the energy density. Here, as the main feature, we do not impose a symmetric behaviour like $h_i(t)\approx h_i(-t)$ and $H_i(t) \approx H_i(-t)$ for large |t|. Assuming quite weak hypotheses on the functions appearing in $(*)$, we establish higher integrability of $|\nabla u|$ for local minimizers $u\in L^\infty (\Omega )$ by using a Caccioppoli-type inequality with some power weights of negative exponent.

1 Introduction

Suppose that $\Omega \subset \mathbb {R}^n$ is a bounded Lipschitz domain and consider the variational integral

$$\begin{aligned} J[w]:= \int _{\Omega }f\left( \nabla w\right) \, \textrm{d}x\end{aligned}$$

(1.1)

of splitting-type, i.e.

$$\begin{aligned} f:\, \mathbb {R}^n \rightarrow \mathbb {R}\,, \quad f(Z) = \sum _{i=1}^n f_i(Z_i) \end{aligned}$$

(1.2)

with strictly convex functions $f_i$: $\mathbb {R}\rightarrow \mathbb {R}$ of class $C^2(\mathbb {R})$, $i=1$, ..., n, satisfying in addition some suitable superlinear growth and ellipticity conditions.

Problem (1.1), (1.2) serves as a prototype for non-uniformly elliptic variational problems. After Giaquinta’s counterexample [1] and the pioneering work of, e.g., Marcellini [2, 3], Acerbi and Fusco [4], Fusco and Sbordone [5] and many others it is well understood that the ratio of the highest and the lowest eigenvalue of $D^2f$ is the crucial quantity for proving the regularity of solutions. The reader will find an extensive overview including different settings of non-uniformly elliptic variational problems in the recent paper [6]. Without going into further details we refer to the series of references given in this paper. We just like to finish this short considerations with the remark, that the unbounded counterexamples constructed by Giaquinta et al. are complemented, e.g., by the work of Fusco and Sbordone [7].

In Section 1.3 of [6], the authors consider general growth conditions which, roughly speaking, means that the energy density f is controlled in the sense of

$$\begin{aligned} g(|Z|)|\xi |^2 \le D^2 f(Z)(\xi ,\xi )\,, \quad \left| D^2f(Z)\right| \le G(|Z|)\,, \end{aligned}$$

(1.3)

with suitable functions g, G: $\mathbb {R}^+_0 \rightarrow \mathbb {R}^+$. Then, under appropriate assumptions on g, G, a general approach to regularity theory is given in [6].

Our note is motivated by the observation, that in (1.2) there is no obvious reason to assume some kind of symmetry for the functions $f_i$, i.e. in general we have $f_i(t) \not = f_i(-t)$ and, as one model case, we just consider ($q_i^{\pm } > 1$, $i=1,\dots , n$)

$$\begin{aligned} f_i(t) \approx |t|^{q_i^-}\quad \text{ if }\quad t \ll -1\,, \qquad f_i(t) \approx |t|^{q_i^+}\quad \text{ if }\quad t \gg 1 \,. \end{aligned}$$

(1.4)

Then, both for $t\ll -1$ and for $t \gg 1$, the functions $f_i$ just behave like a uniform power of |t|. Nevertheless, the power $q_i^-$ enters the left-hand side of (1.3) and $q^+_i$ is needed on the right-hand side of (1.3).

This motivates to study the model case (1.2) and to establish regularity results for solutions under the weaker assumption

$$\begin{aligned} h_i(t) \le f_i''(t) \le H_i(t) \quad t \in \mathbb {R}\,, \end{aligned}$$

(1.5)

with suitable functions $h_i$, $H_i$: $\mathbb {R}\rightarrow \mathbb {R}^+$, $i=1,\dots n$.

There is another quite subtle difficulty in studying regularity of solutions to splitting-type variational problems: in [8] the authors consider variational integrals of the form ($1 \le k < n$)

$$\begin{aligned} I[w,\Omega ] = \int _{\Omega }\left[ f\left( \partial _1 w,\dots ,\partial _{k} w\right) + g\left( \partial _{k+1}w,\dots , \partial _n w\right) \right] \, \textrm{d}x, \end{aligned}$$

(1.6)

where f and g are of p and q-growth, respectively (p, $q >1$). Then the regularity of bounded solutions follows in the sense of [8], Theorem 1.1, without any further condition relating p and q. The proof argues step by step and works since the energy density splits into two parts. If, as supposed in (1.2), the energy density splits in more than two components, then one has to be more careful dealing with the exponents and some more restrictive (but still quite weak) assumptions have to be made. In this sense Remark 1.3 of [8] might be a little bit misleading. We note that a splitting structure into two components as supposed in (1.6) is also assumed, e.g., in [9] and related papers.

In the following we consider the variational integral (1.1), (1.2) defined on the energy class

$$\begin{aligned} E_f(\Omega ):= \left\{ w\in W^{1,1}(\Omega ):\, \int _{\Omega }f(\nabla w) \, \textrm{d}x< \infty \right\} \,. \end{aligned}$$

We are interested in local minimizers u: $\Omega \rightarrow \mathbb {R}$ of class $E_f(\Omega )$, i.e. it holds that

$$\begin{aligned} \int _{\Omega }f(\nabla u) \, \textrm{d}x\le \int _{\Omega }f(\nabla w) \, \textrm{d}x\end{aligned}$$

(1.7)

for all $w\in E_f(\Omega )$ such that ${\text {spt}}(u-w) \Subset \Omega$.

Notation. We will always denote by $q_i^+ >1$, $q_i^- >1$, $1\le i \le n$, real exponents and we let for fixed $1 \le i \le n$

$$\begin{aligned} \underline{q}_i:= \min \left\{ q_i^{\pm }\right\} \,,\quad \overline{q}_i:= \max \left\{ q_i^{\pm }\right\} . \end{aligned}$$

(1.8)

Moreover, we let

$$\begin{aligned} \Gamma :\; [0,\infty ) \rightarrow \mathbb {R}\,,\quad \Gamma (t) = 1 + t^2. \end{aligned}$$

Recalling the idea sketched in (1.4), (1.5) we denote by $h_i$ and $H_i$, $i=1$, ..., n, functions $\mathbb {R}\rightarrow \mathbb {R}^+$ such that with positive constants $\underline{a}_i$, $\overline{a}_i$

$$\begin{aligned} \left. \begin{array}{l} \displaystyle \underline{a}_i \Gamma ^{\frac{q_i^{-}-2}{2}}(|t|)\quad \text{ if } \,\,t < -1\\ \displaystyle \underline{a}_i \Gamma ^{\frac{q_i^{+}-2}{2}}(|t|)\quad \text{ if } \,\, t > 1\\ \end{array}\right\} \le h_i(t) \end{aligned}$$

(1.9)

and

$$\begin{aligned} H_i(t) \le \left\{ \begin{array}{l} \displaystyle \overline{a}_i \Gamma ^{\frac{q_i^{-}-2}{2}}(|t|)\quad \text{ if } \,\,t < -1\\ \displaystyle \overline{a}_i \Gamma ^{\frac{q_i^{+}-2}{2}}(|t|)\quad \text{ if } \,\, t > 1\\ \end{array}\right. \,. \end{aligned}$$

(1.10)

We consider functions $f_i$: $\mathbb {R}\rightarrow [0,\infty )$ of class $C^2(\mathbb {R})$, $i=1$, ..., n, such that for all $t\in \mathbb {R}$

$$\begin{aligned} h_i(t) \le f''_i(t) \le H_i(t) \end{aligned}$$

(1.11)

and note that (1.11) immediately implies for all $i \in \{1, \dots , n\}$ with constants $b_i>0$

$$\begin{aligned} |f_i'(t)| \le b_i \left\{ \begin{array}{l} \displaystyle \Gamma ^{\frac{q_i^{-}-1}{2}}(|t|)\;\text{ if } \,\,t < -1\\ \displaystyle \Gamma ^{\frac{q_i^{+}-1}{2}}(|t|)\;\text{ if } \,\, t > 1\\ \end{array}\right\} \,. \end{aligned}$$

(1.12)

Moreover we obtain (maybe up to additive constants) for all $i=1$, ..., n with constants $\underline{c}_i$, $\overline{c}_i > 0$

$$\begin{aligned} \underline{c}_i \left\{ \begin{array}{l} \displaystyle \Gamma ^{\frac{q_i^{-}}{2}}(|t|)\;\text{ if } \,\,t< -1 \\ \displaystyle \Gamma ^{\frac{q_i^{+}}{2}}(|t|)\;\text{ if }\,\, t> 1 \\ \end{array}\right\} \le f_i(t) \le \overline{c}_i \left\{ \begin{array}{rcl} \displaystyle \Gamma ^{\frac{q_i^{-}}{2}}(|t|);\text{ if } \,\,t < -1\\ \displaystyle \Gamma ^{\frac{q_i^{+}}{2}}(|t|)\;\text{ if } \,\,t > 1\\ \end{array}\right\} \,. \end{aligned}$$

(1.13)

With this notation our main result reads as follows.

Theorem 1.1

Suppose that for $i=1$, ..., n the functions $f_i$: $\mathbb {R}\rightarrow [0,\infty )$ are of class $C^2(\mathbb {R})$ and satisfy (1.11) with $h_i$, $H_i$ given in (1.9), (1.10) for $q_i^{\pm } > 1$.

With the notation (1.8) we assume in addition:

• (i) in the case $n=2$ and $\overline{q}_2 >2$ we suppose that

  $$\begin{aligned} \overline{q}_2 < 2 \underline{q_1} + 2. \end{aligned}$$

  (1.14)

  By (1.14) we may choose $\rho _1 >1$ such that

  $$\begin{aligned} \rho _1 < 2 \frac{\underline{q}_1}{\overline{q}_2-2} \end{aligned}$$

  and we further suppose that

  $$\begin{aligned} \overline{q}_1 < \underline{q}_2[2+\rho _1]+2. \end{aligned}$$

  (1.15)

  In the case $n=2$ and $1 \le \overline{q}_2 \le 2$ we may take any $\rho _1 < \infty$ which follows from (5.1) with $\theta _j = 0$.

• (ii) in the case $n \ge 3$ suppose that we have for every fixed $1 \le i \le n$

  $$\begin{aligned} \overline{q}_j< & {} 2 \underline{q}_i + 2 \qquad \text{ for } \text{ all }\qquad i < j \le n, \end{aligned}$$

  (1.16)
  $$\begin{aligned} \overline{q}_j< & {} 3 \underline{q}_i +2 \qquad \text{ for } \text{ all }\qquad 1 \le j < i. \end{aligned}$$

  (1.17)

If $u \in L^\infty (\Omega ) \cap E_f(\Omega )$ denotes a local minimizer of (1.1), (1.2), i.e. of

$$\begin{aligned} J[w] = \int _{\Omega }\left[ \sum _{i=1}^n f_i\left( \partial _i w\right) \right] \, \textrm{d}x, \end{aligned}$$

then for every $1\le i \le n$, for some $\delta >1/2$ and for any ball $B_{2r}(x_0) \Subset \Omega$

$$\begin{aligned} \int _{B_r(x_0)} f_i (\partial _i u) \Gamma ^{\delta }(|\partial _i u|) \, \textrm{d}x\le c \end{aligned}$$

(1.18)

with a finite local constant c.

Remark 1.1

• i) In the two dimensional case as discussed in [8] we have $p=q_2 \le q_1 =q$ and $\overline{q}_j = \underline{q}_j = q_j$, $j=1$, 2. In this case (1.14) gives no restriction on the exponents since we have chosen w.l.o.g. $p \le q$. In an analogous way it is possible to renumber the exponents in the general form (1.16), (1.17) (or in the same spirit (2.5), (2.6) of Theorem 2.1 below) in order to have these conditions after reordering. For (1.15) we observe

  $$\begin{aligned} p \left[ 2+ \frac{2q}{p-2}\right] + 2 = \frac{2pq}{p-2}+2(p+1) > 2q, \end{aligned}$$

  hence (1.15) as well does not restrict the class of admissible exponents p and q. Here and in the following we always suppose w.l.o.g. that we have (5.1) (see Section 2 for the precise notation) since otherwise no further hypotheses on the exponents have to be made.

• ii) Of course with an explicit choice of $\rho _1$ and the parameter $\rho _2$ (see (5.11)) the choice of $\delta$ in (1.18) can be made precise. Moreover, given (1.18) one may iterate the arguments in order to improve the integrability results. We leave the details to the reader.

• iii) In the case $n \ge 3$ we could choose parameters $\rho _{i,j\not =i}$ as outlined by the choice of $\rho _1$ and $\rho _2$ in the case $n=2$. We prefer the much simpler formulation of (1.16) and (1.17).

Theorem 1.1 describes the typical situation we have in mind. The proof however is not limited to this particular case which leads to the generalized version stated in Theorem 2.1 below.

Although the algebraic choice of parameters in the general form appears somehow involved, we prefer the general formulation since it clearly indicates the idea of the proof. We start with an Ansatz involving both $f_i(\partial _i u)$ and $\Gamma (|\partial _i u|)$ where the asymmetric structure enters by exploiting the structure of $f_i$ combined with the relation to its derivatives. This Ansatz leads to the first general inequalities and we end up with some mixed terms which have to be discussed in the last section. There we combine a careful pointwise analysis with an iteration procedure which generalizes the arguments given in [8]. We note that even in the symmetric case splitting into more than 2 groups the known results are generalized by our Theorems.

In Sect. 3 we shortly sketch a regularization procedure via Hilbert-Haar solutions while Sect. 4 presents the main inequalities for the iteration procedure of Sect. 5. This completes the proof of Theorem 2.1 and hence Theorem 1.1.

2 Precise assumptions on f

The suitable larger class of admissible energy densities is given by the following assumption.

Assumption 2.1

The energy density f,

$$\begin{aligned} f:\, \mathbb {R}^n \rightarrow \mathbb {R}\,, \quad f(Z) = \sum _{i=1}^n f_i\left( Z_i\right) , \end{aligned}$$

introduced in (1.2) is supposed to satisfy the following hypotheses.

• (i) The function $f_i$: $\mathbb {R}\rightarrow [0,\infty )$, $i=1$, ..., n, is of class $C^2(\mathbb {R})$ and for all $t \in \mathbb {R}$ we have $f_i''(t) >0$. For $1 \le i \le n$ we suppose superlinear growth in the sense of

  $$\begin{aligned} \lim _{t\rightarrow \pm \infty } \left| f_i'(t)\right| = \infty \end{aligned}$$

  and at most of polynomial growth in the sense that for some $s> 0$ we have for |t| sufficiently large

  $$\begin{aligned} f_i(t) \le c |t|^s\quad \text{ with } \text{ a } \text{ finite } \text{ constant } \,\, c \,. \end{aligned}$$

• (ii) For $i\in \{1,\dots ,n\}$ and for

  $$\begin{aligned} 0 \le \theta _i < \frac{1}{2} \end{aligned}$$

  (2.1)

  we suppose that for all |t| sufficiently large

  $$\begin{aligned} c_1 \Gamma ^{1-\theta _i}(|t|) f''_i (t)\le & {} f_i (t) \le c_2 f''_i(t) \Gamma ^{1+ \theta _i}(|t|)\,, \end{aligned}$$

  (2.2)
  $$\begin{aligned} \left| f'_i(t)\right| ^2\le & {} c_3 f''_i(t) f_i(t) \Gamma ^{\theta _i}(|t|), \end{aligned}$$

  (2.3)

  where $c_1$, $c_2$ and $c_3$ denote positive constants.

• (iii) We let

  $$\begin{aligned} \Gamma ^{\frac{q_i^{\pm }}{2}}(t) = \left\{ \begin{array}{l} \displaystyle \Gamma ^{\frac{q_i^{-}}{2}}(|t|)\;\text{ if } \,\,t < 0 \\ \displaystyle \Gamma ^{\frac{q_i^{+}}{2}}(|t|)\;\text{ if } \,\,t \ge 0 \end{array}\right\} \,. \end{aligned}$$

  and suppose that $f_i$, $i=1$, ...,n, satisfies with $q_i^{\pm } >1$, with positive constants $c_4$, $c_5$ and for |t| sufficiently large

  $$\begin{aligned} c_4 \Gamma ^{\frac{q_i^\pm }{2}}(|t|) \le f_i(t) \le c_5 \Gamma ^{\frac{q_i^\pm }{2}}(|t|). \end{aligned}$$

  (2.4)

Remark 2.1

• 1. If $f_i$ is a power growth function like, e.g., $f_i(t) = (1+t^2)^{p_i/2}$, $p_i > 1$ fixed, then we have

  $$\begin{aligned} c \Gamma (|t|) f_i''(t) \le f_i(t) \le c\Gamma (|t|) f_i''(t) \,, \end{aligned}$$

  i.e. (2.2) with $\theta _i=0$. Our asymmetric model case given by (1.9)–(1.13) as well is an admissible choice satisfying (2.2).

• 2. By convexity it is well known (see, e.g., [10], exercise 1.5.9, p. 53) that if a convex function is at most of growth rate s, then we have at most the growth rate $s-1$ for its derivative. Hence, (2.4) together with the right-hand side of (2.2) imply (2.3).

• 3. The condition (2.2) with $\theta _i < 1/2$ formally corresponds with the condition $q < p+2$ in the standard (p, q)-case (see, e.g., [11], Chapter 5, and the references quoted therein).

Theorem 2.1

Suppose that we have Assumption 2.1 with $q_i^{\pm }>1$, $i=1$, ..., n. With the above notation we assume in addition that we have for every fixed $1 \le i \le n$

$$\begin{aligned} \overline{q}_j< & {} \left( 1-\theta _j\right) \left[ 2 \left( \underline{q}_i-2 \theta _i\right) +2\right] \nonumber \\{} & {} \qquad \text{ for } \text{ all }\qquad i < j \le n\,, \end{aligned}$$

(2.5)

$$\begin{aligned} \overline{q}_j< & {} \left( 1-\theta _j\right) \left[ (2+\tau ) \left( \underline{q}_i-2\theta _i\right) +2\right] , \quad \tau := \frac{1-2\theta _j}{1-\theta _j}, \nonumber \\{} & {} \qquad \text{ for } \text{ all }\qquad 1 \le j < i. \end{aligned}$$

(2.6)

If $u \in L^\infty (\Omega ) \cap E_f(\Omega )$ denotes a local minimizer of (1.1), (1.2), i.e. of

$$\begin{aligned} J[w] = \int _{\Omega }\left[ \sum _{i=1}^n f_i(\partial _i w)\right] \, \textrm{d}x, \end{aligned}$$

then for every $1\le i \le n$, for some $\delta > 1/2$ and for any ball $B_{2r}(x_0) \Subset \Omega$ we have

$$\begin{aligned} \int _{B_r(x_0)} f_i (\partial _i u) \Gamma ^{\delta -\theta _i}(|\partial _i u|) \, \textrm{d}x\le c \end{aligned}$$

(2.7)

with a finite local constant c.

Remark 2.2

In particular we note that (2.5), (2.6) reduce to (1.16), (1.17) if $\theta _i$ is equal to zero.

3 Some remarks on regularization

We have to start with a regularization procedure such that the expressions given below are well defined. We follow Section 2 of [8] and fix a ball $D \Subset \Omega$. If u denotes the local minimizer in the sense of (1.7) and if $\varepsilon > 0$ is sufficiently small, we consider the mollification $(u)_\varepsilon$ of u w.r.t. the radius $\varepsilon$. We consider the Dirichlet-problem

$$\begin{aligned} \int _D \sum _{i=1}^n f_i(\partial _i w) \, \textrm{d}x\rightarrow \min \end{aligned}$$

among all Lipschitz mappings $\overline{D}\rightarrow \mathbb {R}$ with boundary data $(u)_\varepsilon$. According to, e.g., [12], there exists a unique (Hilbert-Haar) solution $u_\varepsilon$ to this problem.

Exactly as outlined in [8], Lemma 2.1 and Lemma 2.2, we obtain:

Lemma 3.1

Let $\underline{q}:= \min _{1\le i \le n} \underline{q}_i$

• i) We have as $\varepsilon \rightarrow 0$

  $$\begin{aligned} u_\varepsilon \rightharpoondown u \quad \text{ in }\, W^{1,\underline{q}}(D)\,, \qquad \int _D \sum _{i=1}^n f_i(\partial _i u_\varepsilon ) \, \textrm{d}x\rightarrow \int _D \sum _{i=1}^n f_i(\partial _i u) \, \textrm{d}x\,. \end{aligned}$$

• ii) There is a finite constant $c >0$ not depending on $\varepsilon$ such that

  $$\begin{aligned} \Vert u_\varepsilon \Vert _{L^{\infty }(D)} \le c \,. \end{aligned}$$

• iii) For any $\alpha < 1$ we have $u_\varepsilon \in C^{1,\alpha }(D)\cap W^{2,2}_{{\text {loc}}}(D)$.

We then argue as follows: consider a local minimizer u of (1.1), (1.2) and the approximating sequence $\{u_\varepsilon \}$ minimizing the functional

$$\begin{aligned} J[w,D]:= \int _D \sum _{i=1}^n f_i\left( \partial _i w_i\right) \, \textrm{d}x\end{aligned}$$

(3.1)

w.r.t. the data $(u)_\varepsilon$. In particular we have a sequence of local J[w, D]-minimizers. We apply the a priori results of the next section to $u_\varepsilon$ and Theorem 1.1 follows from Lemma 3.1 passing to the limit $\varepsilon \rightarrow 0$.

4 General inequalities

The main result of this section is Proposition 4.2 which is not depending on the hypotheses made in Assumption 2.1, iii).

We will rely on the following variant of Caccioppoli’s inequality which was first introduced in [13]. We also refer to Section 6 of [14] on Caccioppoli-type inequalities involving powers with negative exponents, in particular we refer to Proposition 6.1.

Lemma 4.1

Fix $l\in \mathbb {N}$ and suppose that $\eta \in C^\infty _0(D)$, $0 \le \eta \le 1$. If we consider a local minimizer $u \in W^{1,\infty }_{{\text {loc}}}(D) \cap W^{2,2}_{{\text {loc}}}(D)$ of the variational functional

$$\begin{aligned} I[w] = \int _{D}g(\nabla w) \, \textrm{d}x\end{aligned}$$

with energy density g: $\mathbb {R}^n \rightarrow \mathbb {R}$ of class $C^2$ satisfying $D^2 g(Z)(Y,Y) > 0$ for all Y, $Z\in \mathbb {R}^n$, then for any fixed $i \in \{1,\dots , n\}$ we have

$$\begin{aligned}{} & {} \int _{D}D^2 g (\nabla u)\left( \nabla \partial _i u, \nabla \partial _i u\right) \eta ^{2l} \Gamma ^{\beta }(|\partial _i u|) \, \textrm{d}x\\{} & {} \quad \le c \int _{D}D^2 g (\nabla u) (\nabla \eta ,\nabla \eta )\eta ^{2l-2} \Gamma ^{1+\beta }(|\partial _i u|) \, \textrm{d}x\end{aligned}$$

for any $\beta > - 1/2$.

To the end of our note we always consider a fixed ball

$$\begin{aligned} B=B_{2r}(x_0) \Subset D\,. \end{aligned}$$

With this notation we have the following auxiliary proposition.

Proposition 4.1

Suppose that we have i) of Assumption 2.1 and let $\eta \in C^{\infty }_0(B)$, $0 \le \eta \le 1$, $\eta \equiv 1$ on $B_r(x_0)$, $|\nabla \eta | \le c/r$. Moreover, we assume that $u \in L^\infty (D) \cap W^{1,\infty }_{{\text {loc}}}(D) \cap W^{2,2}_{{\text {loc}}}(D)$.

Then we have for fixed $\gamma \in \mathbb {R}$, for all $k>0$ sufficiently large and for $i=1$, ..., n the starting inequalities (no summation w.r.t. i)

$$\begin{aligned}{} & {} \int _{B}f_i(\partial _i u) \Gamma ^{1+\gamma }(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\le c \left[ 1 + \int _{B}|\partial _i \partial _i u| \Gamma ^{\gamma }(|\partial _i u|) f_i(\partial _i u) \eta ^{2k}\, \textrm{d}x\right. \nonumber \\{} & {} \quad \left. + \int _{B}|\partial _i\partial _i u| \, |f_i'|(\partial _i u) \, \Gamma ^{\frac{1}{2}+\gamma }(|\partial _i u|) \eta ^{2k} \, \textrm{d}x\right] \,, \end{aligned}$$

(4.1)

where the constant may depend on $\Vert u\Vert _{L^\infty }$ and on $r>0$.

Remark 4.1

• (i) The idea of the proof of Proposition 4.1 is based on an integration by parts using the boundedness of u. An Ansatz of this kind was already made by Choe [15], where all relevant quantities are depending on $|\nabla u|$. Here the main new feature is to work with the energy density f which is not depending on the modulus of $\nabla u$.

• (ii) We note that for the proof of Proposition 4.1 no minimizing property of u is needed.

Proof of Proposition 4.1

With $i\in \{1,\dots , n\}$ fixed we obtain using an integration by parts

$$\begin{aligned} \int _{B}f_i(\partial _i u){} & {} \Gamma ^{1+\gamma }(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} = \int _{B}|\partial _i u|^2 f_i(\partial _i u) \Gamma ^{\gamma }(|\partial _i u|)\eta ^{2k}\, \textrm{d}x+ \int _{B}f_i(\partial _i u) \Gamma ^{\gamma }(|\partial _i u|)\eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} = - \int _{B}u \partial _i \Big [\partial _i u f_i(\partial _i u) \Gamma ^{\gamma }(|\partial _i u|) \eta ^{2k}\Big ]\, \textrm{d}x+ \int _{B}f_i(\partial _i u) \Gamma ^{\gamma }(|\partial _i u|)\eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \le c \int _{B}|\partial _i \partial _i u| \Gamma ^{\gamma }(|\partial _i u|) f_i(\partial _i u) \eta ^{2k} \, \textrm{d}x\nonumber \\{} & {} \quad + c \int _{B}|\partial _i \partial _i u|\, |\partial _i u| \, |f'_i|(\partial _i u)\, \Gamma ^{\gamma }(|\partial _i u|) \eta ^{2k} \, \textrm{d}x\nonumber \\{} & {} \quad + c \int _{B}|\partial _i u| f_i(\partial _i u) \Gamma ^{\gamma }(|\partial _i u|) \eta ^{2k-1} |\partial _i \eta | \, \textrm{d}x+ \int _{B}f_i(\partial _i u) \Gamma ^{\gamma }(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} = I_{1,i} + I_{2,i} + I_{3,i} + I_{4,i}\,. \end{aligned}$$

(4.2)

In (4.2) we discuss $I_{3,i}$: for $\varepsilon >0$ sufficiently small we estimate

$$\begin{aligned} I_{3,i}\le & {} \int _{B}|\partial _i u| f_i^{\frac{1}{2}}(\partial _i u) \Gamma ^{\frac{\gamma }{2}}(|\partial _i u|) \eta ^{k} f_i^{\frac{1}{2}}(\partial _i u) \Gamma ^{\frac{\gamma }{2}}(|\partial _i u|)\eta ^{k-1} |\nabla \eta | \, \textrm{d}x\nonumber \\\le & {} \varepsilon \int _{B}|\partial _i u|^2 f_i(\partial _i u) \Gamma ^{\gamma }(|\partial _i u|) \eta ^{2k} dx\nonumber \\{} & {} +c(\varepsilon ,r) \int _{B}f_i(\partial _i u) \Gamma ^{\gamma }(|\partial _i u|) \eta ^{2k-2}\, \textrm{d}x\,. \end{aligned}$$

(4.3)

The first integral on the right-hand side of (4.3) is absorbed in the left-hand side of (4.2), i.e.

$$\begin{aligned}{} & {} \int _{B}f_i (\partial _i u) \Gamma ^{1+\gamma }(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \le I_{1,i} + I_{2,i} + c(\varepsilon ,r) \int _{B}f_i(\partial _i u) \Gamma ^{\gamma }(|\partial _i u|)\eta ^{2k-2}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \quad + \int _{B}f_i (\partial _i u) \Gamma ^{\gamma }(|\partial _i u|)\eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \le I_{1,i} + I_{2,i} + c(\varepsilon ,r) \int _{B}f_i(\partial _i u) \Gamma ^{\gamma }(|\partial _i u|)\eta ^{2k-2}\, \textrm{d}x\,. \end{aligned}$$

(4.4)

Discussing the remaining integral we recall that the function $f_i(t) \Gamma ^{1+\gamma }(|t|)$ is at most of polynomial growth, hence we may apply the auxiliary Lemma 4.2 below to the functions $\varphi (t) = f_i(t) \Gamma ^{\gamma }(|t|)$ and $\psi (t):= f_i(t) \Gamma ^{1+\gamma }(|t|)$ with the result that for some $\rho >1$ and for all $t \in \mathbb {R}$

$$\begin{aligned} f_i(t) \Gamma ^{\gamma }(|t|) \le c \big [f_i(t) \Gamma ^{1+\gamma }(|t|)\big ]^{\frac{1}{\rho }} + c \end{aligned}$$

(4.5)

with a suitable finite constant c.

With (4.5) we estimate for $\tilde{\varepsilon } >0$ sufficiently small and for $k > \rho ^* = \rho /(\rho -1)$

$$\begin{aligned} c(\varepsilon ,r) \int _{B}f_i(\partial _i u){} & {} \Gamma ^{\gamma }(|\partial _i u|) \eta ^{2k-2}\, \textrm{d}x\nonumber \\{} & {} \le c(\varepsilon ,r) \int _{B}\big [f_i(\partial _i u) \Gamma ^{1+\gamma }(|\partial _i u|)\big ]^{\frac{1}{\rho }} \eta ^{\frac{2k}{\rho }} \eta ^{\frac{2k}{\rho ^*}-2}\, \textrm{d}x+c\nonumber \\{} & {} \le \tilde{\varepsilon } \int _{B}f_i (\partial _i u) \Gamma ^{1+\gamma }(|\partial _i u|)\eta ^{2k}\, \textrm{d}x+ c(\tilde{\varepsilon },\varepsilon ,r) \int _{B}\eta ^{2(k-\rho ^*)} \, \textrm{d}x+c \,.\nonumber \\ \end{aligned}$$

(4.6)

The inequalities (4.4) and (4.6) complete the proof of the proposition by absorbing the first integral on the right-hand side of (4.6) in the left-hand side of (4.4). $\square$

It remains to give an elementary proof of the following auxiliary Lemma.

Lemma 4.2

For $m\in \mathbb {N}$ we consider functions $\varphi$, $\psi$: $\mathbb {R}^m \rightarrow [0,\infty )$ such that $\psi (X) \le c \Gamma ^\tau (|X|)$ for some $\tau >0$ and for all $X\in \mathbb {R}^m$. Suppose that we have for some $\varepsilon >0$ and for all $X \in \mathbb {R}^n$

$$\begin{aligned} \varphi (X) \le c \Gamma ^{-\varepsilon }(|X|) \psi (X)\,. \end{aligned}$$

Then there exists a real number $\rho > 1$ and a constant $C >0$ such that

$$\begin{aligned} \varphi (X) \le \big [\psi (X)\big ]^{\frac{1}{\rho }} + C\,. \end{aligned}$$

Proof

Let $\delta := \varepsilon /\tau$, w.l.o.g. $\delta < 1$, i.e. for all $X \in \mathbb {R}^m$

$$\begin{aligned} 1+ \psi ^\delta (X) \le 1+ c\Gamma ^\varepsilon (|X|) \le (1+c) \Gamma ^\varepsilon (|X|)\,, \end{aligned}$$

hence we have by assumption

$$\begin{aligned} \varphi (X)\le & {} c \big [ 1 + \psi ^{\delta }(X)\big ]^{-1} \psi (X)\\\le & {} \left\{ \begin{array}{ccl} c&{} \text{ if } &{} \psi ^\delta (X) \le 1\\ c \psi ^{1-\delta }(X) &{}\text{ if }&{} \psi ^{\delta }(X) > 1 \end{array}\right\} \,. \end{aligned}$$

The lemma follows with the choice $\rho = 1/(1- \delta )$. $\square$

With the help of Proposition 4.1 we now establish the main inequality of this section.

Proposition 4.2

Suppose that we have Assumption 2.1 and let $\eta \in C^{\infty }_0(B)$, $0 \le \eta \le 1$, $\eta \equiv 1$ on $B_r(x_0)$, $|\nabla \eta | \le c/r$. Moreover, we assume that $u \in L^\infty (D) \cap W^{1,\infty }_{{\text {loc}}}(D) \cap W^{2,2}_{{\text {loc}}}(D)$ is a local minimizer of (3.1).

For $i\in \{1,\dots ,n\}$ we choose ${\sigma _i}$ satisfying (recall $0< \theta _i < 1/2$)

$$\begin{aligned} \theta _i< {\sigma _i}< 1/2 \,. \end{aligned}$$

Moreover, again for $i=1$, ..., n we choose arbitrary real numbers $\gamma _i >-1$, $\beta _i>-1/2$ subject to the condition

$$\begin{aligned} \gamma _i+ {\sigma _i}=: \beta _i > - \frac{1}{2}\,. \end{aligned}$$

(4.7)

Then we have for any sufficiently large real number $k>0$

$$\begin{aligned} \int _{B}f_i (\partial _i u){} & {} \Gamma ^{1+\gamma _{i}}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \le c \Bigg [ 1+ \sum _{j\not = i} \int _{B}f''_{j}(\partial _j u) \Gamma ^{1+\beta _i}(|\partial _i u|) \eta ^{2k-2}\, \textrm{d}x\Bigg ] \,. \end{aligned}$$

(4.8)

Proof

We recall the starting inequality (4.1),

$$\begin{aligned} \int _{B}f_i (\partial _i u) \Gamma ^{1+\gamma _{i}}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\le c \Big [1+ I_{1,i} + I_{2,i}\Big ] \,, \end{aligned}$$

(4.9)

where we fix $i \in \{1,\dots , n\}$. We estimate for fixed $\beta _i$ as above

$$\begin{aligned} I_{1,i}= & {} \int _{B}|\partial _i \partial _i u| \big [f''_i(\partial _i u)\big ]^{\frac{1}{2}} \Gamma ^{\frac{\beta _i}{2}}(|\partial _i u|) \big [f''_i (\partial _i u)\big ]^{-\frac{1}{2}} \Gamma ^{-\frac{\beta _i}{2}}(|\partial _i u|)\nonumber \\{} & {} \qquad \cdot \Gamma ^{\gamma _{i}}(|\partial _i u|)f_i(\partial _i u)\eta ^{2k}\, \textrm{d}x\nonumber \\\le & {} c \int _{B}f''_i(\partial _i u) |\partial _i\partial _i u|^2 \Gamma ^{\beta _i}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} + c \int _{B}\big [f''_i (\partial _i u)\big ]^{-1} \Gamma ^{\gamma _{i}-{\sigma _i}}(|\partial _i u|) f_i^2(\partial _i u) \eta ^{2k}\, \textrm{d}x\,. \end{aligned}$$

(4.10)

The second integral on the right-hand side of (4.10) is handled with the help of the right-hand side of (2.2) using in addition Lemma 4.2 (recalling ${\sigma _i}> \theta _i$)

$$\begin{aligned} \int _{B}\big [f''_i (\partial _i u)\big ]^{-1}{} & {} \Gamma ^{\gamma _{i}-{\sigma _i}}(|\partial _i u|) f_i^{2}(\partial _i u) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \le \int _{B}\Big [f_i(\partial _i u) \Gamma ^{1+\gamma _{i} - ({\sigma _i}-\theta _i)}(|\partial _i u|) \Big ] \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \le \int _{B}\Big [f_i(\partial _i u) \Gamma ^{1+\gamma _{i}}(|\partial _i u|) \Big ]^{\frac{1}{\rho }}\eta ^{\frac{2k}{\rho }} \eta ^{\frac{2k}{\rho *}}\, \textrm{d}x+ c \nonumber \\{} & {} \le \varepsilon \int _{B}f_i(\partial _i u) \Gamma ^{1+\gamma _{i}}(|\partial _i u|)\eta ^{2k}\, \textrm{d}x+ c(\varepsilon ,r) \,. \end{aligned}$$

(4.11)

Absorbing terms it is shown up to now (using (4.9)– (4.11))

$$\begin{aligned} \int _{B}f_i (\partial _i u){} & {} \Gamma ^{1+\gamma _{i}}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \le c \Bigg [1+ \int _{B}{f''_i}(\partial _i u) |\partial _i\partial _i u|^2 \Gamma ^{\beta _i}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x+ I_{2,i}\Bigg ] \,. \end{aligned}$$

(4.12)

Let us consider $I_{2,i}$, $i\in \{1, \dots ,n\}$. With $\beta _i > - 1/2$ as above we have

$$\begin{aligned} I_{2,i}= & {} \int _{B}|\partial _i\partial _i u| \big [f''_i(\partial _i u)\big ]^{\frac{1}{2}} \Gamma ^{\frac{\beta _i}{2}}(|\partial _i u) \big [f''_i(\partial _i u) \big ]^{-\frac{1}{2}} \Gamma ^{-\frac{\beta _i}{2}}(|\partial _i u|)\nonumber \\{} & {} \quad \cdot \Gamma ^{\frac{1}{2}+\gamma _{i}}(|\partial _i u|) |f_i'|(\partial _i u)\eta ^{2k}\, \textrm{d}x\nonumber \\\le & {} c \int _{B}{f''_i} (\partial _i u) |\partial _i \partial _i u|^2 \Gamma ^{\beta _i}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} + c \int _{B}\big [f''_i (\partial _i u) \big ]^{-1} \Gamma ^{1+\gamma _{i}-{\sigma _i}}(|\partial _i u|) |f_i'|^2(\partial _i u) \eta ^{2k}\, \textrm{d}x\,. \end{aligned}$$

(4.13)

The first integral on the right-hand side of (4.13) already occurs in (4.12) and the second one is handled with (2.3) and Lemma 4.2 (recalling ${\sigma _i}> \theta _i$)

$$\begin{aligned}{} & {} \int _{B}\big [f''_i (\partial _i u) \big ]^{-1}\Gamma ^{1+\gamma _{i}-{\sigma _i}}(|\partial _i u|) |f_i'|^2(\partial _i u) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \qquad \quad \qquad \le \int _{B}f_i (\partial _i u) \Gamma ^{1+\gamma _{i}-({\sigma _i}-\theta _i)}(|\partial _i u|)\eta ^{2k} \, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \qquad \quad \qquad \le \int _{B}\Big [f_i (\partial _i u) \Gamma ^{1+\gamma _{i}}(|\partial _i u|)\Big ]^{\frac{1}{\rho }}\eta ^{\frac{2k}{\rho }} \eta ^{\frac{2k}{\rho *}}\, \textrm{d}x+ c \nonumber \\{} & {} \qquad \qquad \qquad \quad \qquad \le \varepsilon \int _{B}f_i (\partial _i u) \Gamma ^{1+\gamma _{i}}(|\partial _i u|)\eta ^{2k}\, \textrm{d}x+ c(\varepsilon ,r) \end{aligned}$$

(4.14)

and once more the integral on the right-hand side is absorbed.

To sum up, (4.12) implies with the help of (4.13) and (4.14) for $i=1$, ..., n

$$\begin{aligned}{} & {} \int _{B}f_i (\partial _i u)\Gamma ^{1+\gamma _{i}}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \quad \le c \Bigg [1+ \int _{B}f''_i(\partial _i u) |\partial _i\partial _i u|^2 \Gamma ^{\beta _i}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\Bigg ] \,. \end{aligned}$$

(4.15)

Discussing the right-hand side of (4.15) we apply Lemma 4.1, where we let $f(Z) = \sum _{j=1}^n f_j(Z_j)$ and fix $i\in \{1,\dots , n\}$:

$$\begin{aligned}{} & {} \int _{B}f''_i(\partial _i u) |\partial _i\partial _i u| ^2 \Gamma ^{\beta _i}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad \le c \int _{B}D^2 f(\nabla u)\big (\partial _i \nabla u, \partial _i \nabla u\big ) \Gamma ^{\beta _i}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad \le c \int _{B}D^2 f(\nabla u) \big (\nabla \eta ,\nabla \eta ) \Gamma ^{1+\beta _i}(|\partial _i u|)\eta ^{2k-2}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad \le c(r) \sum _{j=1}^n \int _{B}f''_j(\partial _j u) \Gamma ^{1+\beta _i}(|\partial _i u|) \eta ^{2k-2}\, \textrm{d}x\,. \end{aligned}$$

(4.16)

For $j=i$ on the right-hand side of (4.16) we now apply the left-hand side of (2.2) and again Lemma 4.2 with the result (recall $\theta _i$, ${\sigma _i}<1/2$)

$$\begin{aligned}{} & {} \int _{B}f''_i(\partial _i u)\Gamma ^{1+\beta _i} (|\partial _i u|) \eta ^{2k-2}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \qquad \le \int _{B}f_i(\partial _i u) \Gamma ^{\gamma _{i}+\theta _i+{\sigma _i}}(|\partial _i u|)\eta ^{2k-2}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \qquad \le \int _{B}\Big [f_i(\partial _i u) \Gamma ^{1+\gamma _{i}}(|\partial _i u|))\Big ]^{\frac{1}{\rho }} \eta ^{\frac{2k}{\rho }} \eta ^{\frac{2k}{\rho ^*}-2}\, \textrm{d}x+c\nonumber \\{} & {} \qquad \qquad \qquad \le \varepsilon \int _{B}f_i(\partial _i u) \Gamma ^{1+\gamma _{i}}(|\partial _i u|)\eta ^{2k}\, \textrm{d}x+ c(\varepsilon ,r) \,. \end{aligned}$$

(4.17)

Note that the integral on the right-hand side of (4.17) can be absorbed in the left-hand side of (4.15). This proves Proposition 4.2. $\square$

5 Iteration

Step 1 - preliminaries. We start with an elementary proposition recalling and relating the relevant parameters of the problem.

Proposition 5.1

With $q_i^\pm$, $\underline{q}_i$, $\overline{q}_i$, $\theta _i$, $-1/2 < \beta _i = \gamma _i +{\sigma _i}$, $i=1$, ..., n, as above we further let

$$\begin{aligned} \omega _i^\pm := \frac{q_i^{\pm }}{2} + \gamma _i \,, \quad i \in \{1,\dots , n\}\,. \end{aligned}$$

W.l.o.g. (since otherwise the claim (5.3) trivially holds on account of $\sigma _j < 1/2$) we assume that we have for $j \in \{1,\dots ,n\}$

$$\begin{aligned} \overline{q}_j > 2 (1-\theta _j) \,. \end{aligned}$$

(5.1)

We fix $\tau \ge 0$, i, $j \in \{1,\dots ,n\}$ and suppose in addition to (4.7) that $\gamma _i$ (and $\beta _i$) are given such that

$$\begin{aligned} 1 + \gamma _i < \frac{\underline{q}_i\left( 1-\theta _{j}\right) }{2} \frac{2+\tau }{\overline{q}_j -2\left( 1-\theta _{j}\right) } - {\sigma _i}\left( 1-\theta _{j}\right) \frac{\tau +\frac{\overline{q}_j}{\left( 1-\theta _{j}\right) }}{\overline{q}_j -2\left( 1-\theta _{j}\right) } \end{aligned}$$

(5.2)

This yields (for any combination of $q_j^{\pm }$ and $q_i^{\pm }$, recall $\sigma _i < 1/2$, hence $\omega _i^{\pm }-\beta _i >0$)

$$\begin{aligned} q^{\pm }_j \frac{1+\beta _i}{\omega _i^{\pm } - \beta _i} < 2 \left( 1-\theta _{j}\right) \frac{1+\frac{q_i^{\pm }}{2} + \gamma _i}{\omega _i^{\pm }-\beta _i} + \tau \left( 1-\theta _{j}\right) \,. \end{aligned}$$

(5.3)

Proof

We note that

$$\begin{aligned} 1+ \gamma _i < \frac{\underline{q}_i\left( 1-\theta _{j}\right) }{2}\frac{2+\tau }{\overline{q}_j-2\left( 1-\theta _{j}\right) } - {\sigma _i}\left( 1-\theta _{j}\right) \frac{\tau +\overline{q}_j/\left( 1-\theta _{j}\right) }{\overline{q}_j -2\left( 1-\theta _{j}\right) } \,, \end{aligned}$$

is equivalent to

$$\begin{aligned} (1+ \gamma _i) \big [\overline{q}_j - 2 \left( 1-\theta _{j}\right) \big ] < \underline{q}_i\left( 1-\theta _{j}\right) + \tau \left[ \frac{\underline{q}_i\left( 1-\theta _{j}\right) }{2}-{\sigma _i}\left( 1-\theta _{j}\right) \right] - {\sigma _i}\overline{q}_j\,. \end{aligned}$$

Writing this in the form

$$\begin{aligned} \overline{q}_j(1+\beta _i) < 2 \left( 1-\theta _{j}\right) \left[ 1+ \gamma _i + \frac{\underline{q}_i}{2}\right] + \tau \left( 1-\theta _{j}\right) \left[ \frac{\underline{q}_i}{2}-{\sigma _i}\right] \end{aligned}$$

and recalling that we have by definition $\omega ^{\pm }_i - \beta _i = (q_i^\pm /2) - {\sigma _i}$ we obtain as an equivalent inequality

$$\begin{aligned} \overline{q}_j \frac{1+\beta _i}{\omega _i^{\pm } - \beta _i} < 2 \left( 1-\theta _{j}\right) \frac{1+\frac{\underline{q}_i}{2} + \gamma _i}{\omega _i^{\pm }-\beta _i} + \tau \left( 1-\theta _{j}\right) \frac{\underline{q}_i-2{\sigma _i}}{q_i^{\pm }-2{\sigma _i}}. \square \end{aligned}$$

Up to now no relation between $q_i^+$ and $q_i^-$ was needed due to our particular Ansatz depending on t instead of |t|.

Step 2 - main inequality. To complete the proofs of Theorem 1.1 and Theorem 2.1 it remains to handle the mixed terms on the right-hand side of (4.8). Here, of course, it is no longer possible to argue with the structure conditions for fixed i, i.e. to argue with $q_i^\pm$ separated from each other in disjoint regions.

Throughout the rest of this section we suppose that the assumptions of Theorem 2.1 are satisfied.

Consider a set $U \subset \Omega$ and a $C^1$-function v: $\Omega \rightarrow \mathbb {R}$. We let for any $i \in \{1, \dots , n\}$

$$\begin{aligned} U \cap [\partial _i v \ge 0] =: U_i^+[v] =: U_i^+\,, \qquad U \cap [\partial _i v < 0] =: U_i^-[v] =: U_i^-\,, \end{aligned}$$

in particular U can be written as the disjoint union

$$\begin{aligned} U = U_i^+ \cup U_i^- \end{aligned}$$

for every $1 \le i \le n$.

Using this notation, recalling Proposition 4.2 and the left-hand side of (2.2) we have for every $1 \le i \le n$

$$\begin{aligned}{} & {} \int _{B}f_i (\partial _i u) \Gamma ^{1+\gamma _{i}}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \quad \le c \Bigg [1+\sum _{j\not = i} \int _{B}f''_{j}(\partial _j u) \Gamma ^{1+\beta _i}(|\partial _i u|) \eta ^{2k-2}\, \textrm{d}x\Bigg ]\nonumber \\{} & {} \qquad \qquad \quad \le c\Bigg [1+ \sum _{j\not = i}\int _{B}f_j(\partial _j u) \Gamma ^{\theta _j -1}(|\partial _j u|) \Gamma ^{1+\beta _i}(|\partial _i u|)\eta ^{2k-2}\, \textrm{d}x\Bigg ] \,. \end{aligned}$$

(5.4)

Fix $i \in \{1, \dots , n\}$ and let

$$\begin{aligned} \kappa _i^{\pm } = \frac{1+\omega _i^\pm }{1+\beta _i}\,,\qquad \hat{\kappa }_i^{\pm } = \frac{1+\omega _i^{\pm }}{\omega _i^{\pm } - \beta _i}\,. \end{aligned}$$

For the choice of exponents we observe

$$\begin{aligned} 1 + \omega _i^{\pm }> 1+ \beta _i \quad \Leftrightarrow \quad \frac{q_i^\pm }{2} > {\sigma _i}\end{aligned}$$

which follows from $\sigma _i < 1/2$.

We obtain for fixed $1 \le i \le n$ and for $\varepsilon > 0$ sufficiently small (note that the ball B is divided into two parts w.r.t. the function $\partial _i u$)

$$\begin{aligned}{} & {} \sum _{j\not = i}\int _{B}f_j(\partial _j u) \Gamma ^{\theta _j-1}(|\partial _j u|)\Gamma ^{1+\beta _i}(|\partial _i u|) \eta ^{2k-2}\, \textrm{d}x\nonumber \\{} & {} \quad \le c \sum _{j\not =i}\sum _{\pm } \int _{B_{i}^{\pm }} \big (1+f_j(\partial _j u)\big ) \Gamma ^{\theta _j -1}(|\partial _j u|) \Gamma ^{1+\beta _i}(|\partial _i u|) \eta ^{2k-2}\, \textrm{d}x\nonumber \\{} & {} \quad \le \sum _{j\not = i}\sum _{\pm } \Bigg [\varepsilon \int _{B_{i}^{\pm }} \Gamma (|\partial _i u|)^{1+\omega _i^\pm } \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \quad + c(\varepsilon ) \int _{B_{i}^{\pm }} \big (1+f_j(\partial _j u)\big )^{\frac{1+\omega _i^\pm }{\omega _i^{\pm }-\beta _i}} \Gamma ^{(\theta _j -1)\frac{1+\omega _i^\pm }{\omega _i^\pm - \beta _i}}(|\partial _j u|)\, \textrm{d}x\Bigg ] \,. \end{aligned}$$

(5.5)

By (2.4) we have on $B_{i}^{\pm }$ for $|\partial _i u|$ sufficiently large $\Gamma (|\partial _i u|)^{q_i^{\pm }/2} \le c f_i(\partial _i u)$, hence by the definition of $\omega _i^{\pm }$

$$\begin{aligned} \varepsilon \sum _{j\not = i}\sum _{\pm } \int _{B_{i}^{\pm }} \Gamma (|\partial _i u|)^{1+\omega _i^\pm } \eta ^{2k}\, \textrm{d}x\le c (n-1) \varepsilon \int _{B}f_i(\partial _i u) \Gamma ^{1+\gamma _i}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x+ c\nonumber \\ \end{aligned}$$

(5.6)

and, as usual, the integral on the right-hand side can be absorbed in (5.4).

We will finally show with the help of an iteration procedure that for every $1 \le i \le n$

$$\begin{aligned}{} & {} \sum _{j\not = i} \sum _{\pm }\int _{B_{i}^{\pm }} \big (1+f_j(\partial _j u)\big )^{\frac{1+\omega _i^\pm }{\omega _i^{\pm }-\beta _i}} \Gamma ^{(\theta _j -1) \frac{1+\omega _i^\pm }{\omega _i^\pm - \beta _i}}(|\partial _j u|) \, \textrm{d}x\le c\,, \end{aligned}$$

(5.7)

which completes the proof of Theorem 2.1 since we have by (5.4), (5.5) and (5.6) for every $1 \le i \le n$

$$\begin{aligned}{} & {} \int _{B}f_i (\partial _i u) \Gamma ^{1+\gamma _{i}}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\nonumber \\{} & {} \qquad \qquad \le c \Bigg [1+ \sum _{j\not =i}\sum _{\pm } \int _{B_{i}^{\pm }} \big (1+f_j(\partial _j u)\big )^{\frac{1+\omega _i^\pm }{\omega _i^{\pm }-\beta _i}} \Gamma ^{(\theta _j -1)\frac{1+\omega _i^\pm }{\omega _i^\pm - \beta _i}}(|\partial _j u|)\, \textrm{d}x\Bigg ] \,. \end{aligned}$$

(5.8)

In order to establish (5.7) let us suppose that (5.2) is true with a real number $\tau \ge 0$. Then we may apply Proposition 5.1 and (5.3) implies in the case $\overline{q}_j> 2 \left( 1-\theta _{j}\right)$ (recall $f_j \approx c \Gamma ^{q_j^\pm /2}$)

$$\begin{aligned} \Gamma ^{-(1-\theta _j) \frac{1+\omega _i^\pm }{\omega _i^\pm - \beta _i} - (1-\theta _j) \frac{\tau }{2} }(|\partial _j u|)\le & {} c \big (1+f_j(\partial _j u)\big )^{-\frac{1+\beta _i}{\omega _i^\pm - \beta _i}}\nonumber \\= & {} c \big (1+f_j(\partial _j u)\big )^{1-\frac{1+\omega _i^{\pm }}{\omega _i^\pm - \beta _i}} \,. \end{aligned}$$

Thus we obtain

$$\begin{aligned} \big (1+f_j(\partial _j u)\big )^{\frac{1+\omega _i^\pm }{\omega _i^{\pm }-\beta _i}} \Gamma ^{(\theta _j -1) \frac{1+\omega _i^\pm }{\omega _i^\pm - \beta _i}}(|\partial _j u|)\le c \big (1+f_j(\partial _j u)\big ) \Gamma ^{(1-\theta _j) \frac{\tau }{2}}(|\partial _j u|) \,. \end{aligned}$$

(5.9)

We note that (5.9) is formulated uniformly w.r.t. the index j and the symbol ± is just related to $\partial _i u$.

In the case $\overline{q}_j \le 2 \left( 1-\theta _{j}\right)$ we have

$$\begin{aligned} (1+f_j(\partial _j u)) \le c \Gamma ^{1-\theta _j}(|\partial _j u|) \,, \end{aligned}$$

hence

$$\begin{aligned} \big (1+f_j(\partial _j u)\big )^{\frac{1+\omega _i^\pm }{\omega _i^{\pm }-\beta _i}} \Gamma ^{(\theta _j - 1) \frac{1+\omega _i^\pm }{\omega _i^\pm - \beta _i}}(|\partial _j u|) \le c \end{aligned}$$

and (5.9) holds for any $\tau \ge 0$.

Inequality (5.9) is the main tool for the iteration procedure leading to the claim (5.7).

The strategy is the following: we start with $i=1$ and use (5.2) with an appropriate real number $\gamma _1$ which gives (5.9) for $i=1$ and $2 \le j \le n$. In this first step $\tau =0$ is chosen such that we have a priori integrability on the right-hand side.

For $i=2$ and $3\le j \le n$ the same is done in the next step. We note that for $j=1$ we may benefit from the integrability obtained before with $i=1$, i.e. we may choose an appropriate $\tau >0$ in (5.2).

The iteration is done with the case $i=n$.

Step 3 - proof of Theorem 1.1, (i). Let us start with the easiest case i) of Theorem 1.1, i.e. $n=2$, $\theta _i =0$, $i=1$, 2.

On account of $\theta _1=0$ we may choose $\sigma _1$ arbitrarily small and (5.2) for $\overline{q}_2 > 2$ and $\tau =0$ becomes with $\rho _1 >1$

$$\begin{aligned} 1+ \gamma _1 =: \frac{1}{2}\rho _1 < \frac{\underline{q}_1}{\overline{q}_2 -2}\,. \end{aligned}$$

(5.10)

We note that (5.10) is satisfied with some $\rho _1 >1$ if we have

$$\begin{aligned} \overline{q}_2 < 2 \underline{q}_1 + 2 \,, \end{aligned}$$

which corresponds to our assumption (1.14).

This implies by (5.8) and (5.9)

$$\begin{aligned} \int _{B}f_1 (\partial _1 u) \Gamma ^{\frac{\rho _1}{2}}(|\partial _1 u|) \eta ^{2k}\, \textrm{d}x\le c \,. \end{aligned}$$

As a consequence, in the second step we consider (5.2) with

$$\begin{aligned} \tau< \rho _1 < 2 \frac{\underline{q}_1}{\overline{q}_2 -2}\,. \end{aligned}$$

This leads to the condition ($\rho _2 > 1$)

$$\begin{aligned} \frac{1}{2}< 1+ \gamma _2 =: \frac{1}{2}\rho _2< \frac{\underline{q}_2}{2} \frac{2 + \rho _1}{\overline{q}_1-2}\,. \end{aligned}$$

(5.11)

This condition is satisfied for an appropriate $\rho _2$ if we suppose that

$$\begin{aligned} \overline{q}_1 < \underline{q}_2 [2+\rho _1]+ 2 \end{aligned}$$

with $\rho _1$ satisfying (5.10). This proves Theorem 1.1i).

Step 4 - iteration and proofs of the theorems.

We first note that we have (5.2) with

$$\begin{aligned} \gamma _i + {\sigma _i}= \beta _i \end{aligned}$$

for $\beta _i$ sufficiently close to $-1/2$ if

$$\begin{aligned} \overline{q}_j < (2+\tau ) \underline{q}_i(1-\theta _j) +2(1-\theta _j)(1- (2+\tau ) {\sigma _i}) \,. \end{aligned}$$

and for ${\sigma _i}> \theta _i$ sufficiently close to $\theta _i$ we are led to

$$\begin{aligned} \overline{q}_j < (2+\tau ) \underline{q}_i(1-\theta _j) +2(1-\theta _j)(1- (2+\tau ) \theta _i) \,. \end{aligned}$$

(5.12)

$\underline{i=1.}$

Again with $\beta _i$ sufficiently close to $-1/2$ (here for $i=1$) and choosing ${\sigma _i}> \theta _i$ sufficiently close to $\theta _i$ ($i=1$), (5.2) is valid with the choice $\tau = 0$ if we have (recall (5.12))

$$\begin{aligned} \overline{q}_j < (1-\theta _j) \left[ 2 \left( \underline{q}_i-2 \theta _i\right) +2\right] \qquad \text{ for } \text{ all }\qquad 2 \le j \le n\,, \end{aligned}$$

(5.13)

i.e. in the particular case $i=1$ under consideration

$$\begin{aligned} \overline{q}_j < \left( 1-\theta _j\right) \left[ 2 \left( \underline{q}_1-2 \theta _1\right) +2\right] \qquad \text{ for } \text{ all }\qquad 2 \le j \le n\,, \end{aligned}$$

and this is just assumption (2.5) for $i=1$.

From (5.2) we deduce (5.9) for $i=1$, $\tau = 0$, and (5.7) follows from (5.9) for $i=1$ and for all $2 \le j \le n$ with the choice $\tau =0$

$$\begin{aligned}{} & {} \sum _{j\not =1}\sum _{\pm }\int _{{B_i}^{\pm }}\big (1+f_j(\partial _j u)\big )^{\frac{1+\omega _1^\pm }{\omega _1^{\pm }-\beta _1}} \Gamma ^{(\theta _j -1) \frac{1+\omega _1^\pm }{\omega _1^\pm - \beta _1}}(|\partial _j u|)\, \textrm{d}x\nonumber \\{} & {} \le c \int _{B}\big (1+f_j(\partial _j u)\big ) \, \textrm{d}x\le c\,, \end{aligned}$$

(5.14)

Returning to (5.8) we insert (5.14) which yields (w.l.o.g. $1+\gamma _1 = 1+\beta _1 - \sigma _1 > \delta - \theta _1$ for some $\delta > 1/2$)

$$\begin{aligned} \int _{B}f_1(\partial _1 u) \Gamma ^{\delta -\theta _1}(|\partial _1 u|) \eta ^{2k}\, \textrm{d}x\le c \end{aligned}$$

(5.15)

for some $\delta > 1/2$

$\underline{1 < i \le n.}$

Suppose that we have (5.13) (again compare (2.5)) in the sense

$$\begin{aligned} \overline{q}_j < (1-\theta _j) \left[ 2 \left( \underline{q}_i-2 \theta _i\right) +2\right] \qquad \text{ for }\qquad i+1 \le j \le n \,. \end{aligned}$$

(5.16)

With the same argument leading to (5.14) we have for all $i+1 \le j \le n$

$$\begin{aligned} \sum _{j> i} \sum _{\pm }\int _{B_i^\pm }\left( 1+f_j(\partial _j u)\right) ^{\frac{1+\omega _i^\pm }{\omega _i^{\pm }-\beta _i}} \Gamma ^{(\theta _j -1) \frac{1+\omega _i^\pm }{\omega _i^\pm - \beta _i}}(|\partial _j u|) \, \textrm{d}x\le c. \end{aligned}$$

(5.17)

Moreover, we suppose that by iteration we have (5.15) for $1 \le j < i$, i.e. by decreasing radii in this finite iteration, if necessary, we have w.l.o.g.

$$\begin{aligned} \int _{B}f_j(\partial _j u) \Gamma ^{\delta -\theta _j}(|\partial _j u|) \, \textrm{d}x\le c , \quad 1 \le j < i, \end{aligned}$$

(5.18)

for some $\delta > 1/2$.

Then we return to (5.2) with the choice $\tau = (1-2\theta _j)/(1-\theta _j)$. For $\beta _i$ sufficiently close to $-1/2$ and ${\sigma _i}$ sufficiently close to $\theta _i$ we are lead to the condition (recall (5.12))

$$\begin{aligned} \overline{q}_j < \left( 1-\theta _j\right) \left[ (2+\tau ) \left( \underline{q}_i-2\theta _i\right) +2\right] , \quad \tau = \frac{1-2\theta _j}{1-\theta _j}, \end{aligned}$$

(5.19)

$1 \le j < i$, and (5.19) is just the assumption (2.6).

With (5.2) we again have (5.9), now with $\tau = (1-2\theta _j)/(1-\theta _j)$, hence

$$\begin{aligned}{} & {} \sum _{j< i}\sum _{\pm } \int _{B_i^\pm } \big (1+f_j(\partial _j u)\big )^{\frac{1+\omega _i^\pm }{\omega _i^{\pm }-\beta _i}} \Gamma ^{(\theta _j -1)\frac{1+\omega _i^\pm }{\omega _i^\pm - \beta _i}}(|\partial _j u|)\, \textrm{d}x\nonumber \\{} & {} \quad \le c \sum _{j< i} \int _{B}\big (1+f_j(\partial _j u)\big ) \Gamma ^{\delta -\theta _j}(|\partial _j u|)\, \textrm{d}x\le c \,, \end{aligned}$$

(5.20)

where the last estimate follows from (5.18) for some $\delta > 1/2$.

With (5.17) and (5.20) one has

$$\begin{aligned} \sum _{j\not = i}\sum _{\pm }\int _{B_i^\pm }\big (1+f_j(\partial _j u)\big )^{\frac{1+\omega _i^\pm }{\omega _i^{\pm }-\beta _i}} \Gamma ^{(\theta _j -1)\frac{1+\omega _i^\pm }{\omega _i^\pm - \beta _i}}(|\partial _j u|) \, \textrm{d}x\le c\,, \end{aligned}$$

(5.21)

which exactly as in the case $i=1$ shows for some sufficiently small $\delta > 1/2$

$$\begin{aligned} \int _{B}f_i(\partial _i u) \Gamma ^{\delta -\theta _i}(|\partial _i u|) \eta ^{2k}\, \textrm{d}x\le c \,, \end{aligned}$$

(5.22)

hence with (5.22) we proceed one step in the iteration of (5.19). This completes the proof of Theorem 2.1. $\square$