Open Access

Orlicz regularity of the gradient of solutions to quasilinear elliptic equations in the plane

  • Linda Maria De Cave1,
  • Luigi D’Onofrio2Email author and
  • Roberta Schiattarella3
Boundary Value Problems20162016:103

https://doi.org/10.1186/s13661-016-0607-6

Received: 29 July 2015

Accepted: 10 May 2016

Published: 21 May 2016

Abstract

Given a planar domain Ω, we study the Dirichlet problem
$$\textstyle\begin{cases} {-}\operatorname {div}A(x, \nabla v)= f & \mbox{in } \Omega,\\ v=0 & \mbox{on } \partial\Omega, \end{cases} $$
where the higher-order term is a quasilinear elliptic operator, and f belongs to the Zygmund space \(L (\log L)^{\delta} (\log\log\log L)^{\frac{\beta}{2}}(\Omega)\) with \(\beta\geq0\) and \(\delta\geq \frac{1}{2}\).

We prove that the gradient of the variational solution \(v \in W^{1,2}_{0}(\Omega)\) belongs to the space \(L^{2} (\log L)^{2\delta-1} (\log \log\log L)^{\beta}(\Omega)\).

Keywords

gradient regularity quasilinear elliptic equations Zygmund spaces

MSC

35J62 35B65

1 Introduction

In this paper we consider the following Dirichlet problem on a bounded open set \(\Omega\subset \mathbb {R}^{2}\) with \(\mathcal{C}^{1}\) boundary:
$$ \textstyle\begin{cases} {-}\operatorname {div}A(x, \nabla v) = f & \mbox{in } \Omega,\\ v=0 & \mbox{on } \partial\Omega, \end{cases} $$
(1.1)
where f belongs to the Zygmund space \(L(\log L)^{\delta} (\log\log\log L)^{\frac{\beta}{2}}(\Omega)\) with \(\beta\geq0\) and \(\delta\geq\frac{1}{2}\). We prove that the distributional gradient of the unique solution \(v\in W^{1,2}_{0}(\Omega)\) to (1.1) satisfies \(|\nabla v|\in L^{2} (\log L)^{2\delta-1} (\log\log\log L)^{\beta }(\Omega)\).
Here \(A: \Omega\times \mathbb {R}^{2} \longrightarrow \mathbb {R}^{2}\) is a mapping of Leray-Lions type [1], that is,
$$ \begin{aligned} &A( \cdot, \xi) \mbox{ is measurable for all } \xi\in \mathbb {R}^{2}, \mbox{ and}\\ &A(x, \cdot) \mbox{ is continuous for almost every } x\in\Omega. \end{aligned} $$
(1.2)
Moreover, we assume that there exists \(K\geq1\) such that, for almost every \(x\in\Omega\) and for any \(\xi, \eta\in \mathbb {R}^{2}\),
$$ \begin{aligned} (\mathrm{i}) &\quad \bigl|A(x, \xi) - A(x, \eta)\bigr| \leq K|\xi- \eta|,\\ (\mathrm{ii}) &\quad |\xi- \eta|^{2} \leq K \bigl\langle A(x, \xi) - A(x, \eta), \xi -\eta\bigr\rangle ,\\ (\mathrm{iii}) &\quad A(x,0)=0. \end{aligned} $$
(1.3)
In [2], under assumptions (1.2) and (1.3), the authors proved the existence and uniqueness of the solution to the Dirichlet problem with \(f\in L^{1}(\Omega)\) in the grand Sobolev space \(W^{1,2)}_{0}(\Omega)\). Precisely, \(W^{1,2)}_{0}(\Omega)\) is the space of functions \(v\in W^{1,1}_{0}(\Omega)\) whose gradients belong to the grand Lebesgue space \(L^{2)}(\Omega)\) (see Section 2 for a definition).

Nowadays, a vast literature is available dealing with several types of a priori estimates on the gradients of solutions to equations of this kind; see, for example, [35].

We are interested in cases where the solution is the variational \(W^{1,2}(\Omega)\) solution. The minimal assumption on f that guarantees this is \(f\in L (\log L)^{\frac{1}{2}}(\Omega)\). This follows by the embedding in the plane (see [6, 7], and [8])
$$W^{1, 2}_{0} (\Omega) \hookrightarrow\exp_{2}( \Omega) $$
and by the duality relation (see [9])
$$\bigl((\exp_{2}) (\Omega) \bigr)'=L(\log L)^{\frac{1}{2}} (\Omega) . $$
In [10], the authors interpolate between the data spaces
$$L(\log L)^{\frac{1}{2}} (\Omega) \quad\mbox{and}\quad L(\log L) (\Omega). $$
To this aim, the following estimate was proved for \(0 \leq\beta\leq1\):
$$ \| \nabla v\|_{L^{2} (\log L)^{\beta}(\Omega)} \leq C(K, \beta) \| f \|_{L (\log L)^{\frac{(\beta+1)}{2}}(\Omega)}. $$
(1.4)
When f belongs to the Zygmund space \(L(\log L)^{\frac{1}{2}} (\log \log L)^{\frac{\beta}{2}}(\Omega)\) for \(0\leq\beta<2\), the unique solution v to the Dirichlet problem (1.1) satisfies \(|\nabla v|\in L^{2}(\log\log L)^{\beta}(\Omega)\) with the estimate
$$ \| \nabla v\|_{L^{2} (\log\log L)^{\beta}(\Omega)} \leq C(K, \beta) \| f\|_{L (\log L)^{\frac{1}{2}} (\log\log L)^{\frac{\beta}{2}}(\Omega)} $$
(1.5)
(see [11]). This generalizes a result of [12] obtained for \(\beta=1\).
Starting from the results of [11], in [13], the authors of the present paper prove an analogue of the previous result when the critical Zygmund class \(L(\log L)^{\frac{1}{2}}(\Omega)\) is perturbed in a weaker way, namely with perturbations of order \(\log \log \log L\). Precisely, in [13], it is proved that if \(\beta\geq 0\), then
$$ \| \nabla v\|_{L^{2} (\log\log\log L)^{\beta}(\Omega)} \leq C(K, \beta) \| f \|_{L (\log L)^{\frac{1}{2}} (\log\log\log L)^{\frac{\beta }{2}}(\Omega)}. $$
(1.6)
The aim of this paper is to extend the results of [13] to the case \(f\in L(\log L)^{\delta} (\log\log \log L)^{\frac {\beta}{2}}(\Omega)\) with \(\beta\geq0\) and \(\delta\geq\frac{1}{2}\), that is, to prove the following:

Theorem 1.1

Let \(A= A(x, \xi)\) satisfy (1.2) and (1.3), and let \(\beta \geq0\), \(\delta\geq\frac{1}{2}\). Then, if \(f\in L (\log L)^{\delta }(\log \log\log L)^{\frac{\beta}{2}}(\Omega)\), the gradient of the unique finite energy solution \(v\in W^{1,2}_{0}(\Omega)\) to the Dirichlet problem (1.1) belongs to the Orlicz space \(L^{2} (\log L)^{2\delta -1} (\log\log \log L)^{\beta}(\Omega, \mathbb {R}^{2})\), and the following estimate holds:
$$\| \nabla v\|_{L^{2} (\log L)^{2\delta-1}(\log\log\log L)^{\beta }(\Omega ;\mathbb {R}^{2})} \leq C(K, \beta, \delta) \| f\|_{L (\log L)^{\delta} (\log \log\log L)^{\frac{\beta}{2}}(\Omega)}. $$

In order to prove this theorem, we will find an integral expression equivalent to the Luxemburg norm in the Zygmund class (see Theorem 3.1), which is based on a method recently introduced in [14, 15].

We note that our method allows us to prove estimates (1.4) and (1.6) for any \(\beta\geq0\) (in particular, see Lemmas 2.3 and 2.4).

2 Preliminaries

Let Ω be a bounded domain in \(\mathbb {R}^{n}\), \(n\geq2\). A function u belongs to the Lebesgue space \(L^{p}(\Omega)\) with \(1\leq p < \infty\) if and only if
$$\| u\|_{L^{p}(\Omega)}= \biggl( \fint_{\Omega}|u|^{p} \,dx \biggr)^{\frac {1}{p}}< + \infty, $$
where \(\fint_{\Omega}=\frac{1}{|\Omega|} \int_{\Omega}\).

Now we recall some useful function spaces slightly larger than the classical Lebesgue spaces.

2.1 Grand Lebesgue spaces

For \(1< p<\infty\), let us consider the class, denoted by \(L^{p)}(\Omega )\), consisting of all measurable functions \(u\in\bigcap_{1\leq q < p} L^{q}(\Omega)\) such that
$$\sup_{0< \varepsilon\leq p-1} \biggl\{ \varepsilon \fint_{\Omega} \bigl|u(x)\bigr|^{p- \varepsilon} \biggr\} ^{\frac{1}{p-\varepsilon}} < + \infty $$
which was introduced in [16]; \(L^{p)}(\Omega)\) becomes a Banach space, the grand Lebesgue space \(L^{p)}(\Omega)\), equipped with the norm
$$\| u \|_{L^{p)}(\Omega)}= \sup_{0< \varepsilon\leq p-1} \varepsilon ^{\frac{1}{p}} \biggl\{ \fint_{\Omega} \bigl|u(x)\bigr|^{p- \varepsilon} \biggr\} ^{\frac{1}{p-\varepsilon}}. $$
Moreover, \(\| u \|_{L^{p)}(\Omega)} \) is equivalent to
$$\sup_{0< \varepsilon\leq p-1} \biggl\{ \varepsilon \fint_{\Omega} \bigl|u(x)\bigr|^{p- \varepsilon} \biggr\} ^{\frac{1}{p-\varepsilon}}. $$
In general, if \(0<\alpha<\infty\), then we can define the space \(L^{\alpha, p)}(\Omega)\) as the space of all measurable functions \(u\in \bigcap_{1\leq q < p} L^{q}(\Omega)\) such that
$$\|u\|_{L^{\alpha, p)}(\Omega)}=\sup_{0< \varepsilon\leq p-1} \bigl\{ \varepsilon^{\frac{\alpha}{p}}\|u\|_{p- \varepsilon} \bigr\} < + \infty . $$

2.2 Orlicz spaces

Let Ω be an open set in \(\mathbb {R}^{n}\) with \(n\geq2\). A function \(\Phi: [0, + \infty) \rightarrow[0, + \infty)\) is called a Young function if it is convex, left-continuous, and vanishes at 0; thus, any Young function Φ admits the representation
$$\Phi(t)= \int_{0}^{t} \phi(s)\,ds \quad\mbox{for } t \geq0, $$
where \(\phi: [0, + \infty) \rightarrow[0, + \infty)\) is a nondecreasing left-continuous function that is neither identically equal to 0 nor to ∞.
The Orlicz space associated to Φ, named \(L^{\Phi }(\Omega)\), consists of all Lebesgue-measurable functions \(f: \Omega \rightarrow \mathbb {R}\) such that
$$\int_{\Omega} \Phi\bigl(\lambda|f|\bigr) < \infty\quad\mbox{for some } \lambda= \lambda(f)>0. $$
\(L^{\Phi}(\Omega)\) is a Banach space equipped with the Luxemburg norm
$$\| f \|_{L^{\Phi}(\Omega)} = \inf \biggl\{ \frac{1}{\lambda}: \int _{\Omega} \Phi\bigl(\lambda|f|\bigr) \leq1 \biggr\} . $$
Examples of Orlicz spaces:
  1. (1)

    If \(\Phi(t)= t^{p}\) for \(1 \leq p < \infty\), then \(L^{\Phi }(\Omega)\) is the classical Lebesgue space \(L^{p}(\Omega)\).

     
  2. (2)

    If \(\Phi(t)= t^{p} ( \log(a+ t) )^{q}\) with either \(p>1\) and \(q\in \mathbb {R}\) or \(p=1\) and \(q\geq0\) and where \(a \geq e\), then \(L^{\Phi}(\Omega)\) is the Zygmund space denoted by \(L^{p} (\log L)^{q}(\Omega)\).

     
  3. (3)

    If \(\Phi(t)= t^{p} (\log(a+t) )^{q_{1}} ( \log\log \log(a + t) )^{q_{2}}\) with either \(p>1\) and \(q_{1},q_{2}\in \mathbb {R}\) or \(p=1\) and \(q_{1},q_{2}\geq0\) and where \(a \geq e^{e^{e}}\), then \(L^{\Phi }(\Omega)\) is the space \(L^{p} (\log L)^{q_{1}}(\log\log\log L)^{q_{2}} (\Omega)\).

     
  4. (4)

    If \(\Phi(s)= e^{t^{a}}-1\) and \(a>0\), then \(L^{\Phi}(\Omega )\) is the space of a-exponentially integrable functions \(\operatorname {EXP}_{a}(\Omega)\).

    We denote by \(\exp_{a}(\Omega)\) the closure of \(L^{\infty}(\Omega )\) in \(\operatorname {EXP}_{a}(\Omega)\).

     
The Young complementary function is given by
$$\tilde{\Phi}(t)= \int_{0}^{t} \phi^{-1}(s)\,ds, $$
where
$$\phi^{-1}(s)= \sup\bigl\{ r: \phi(r) \leq s \bigr\} . $$
Moreover, the following Hölder-type inequality holds:
$$\biggl\vert \int_{\Omega} f(x) g(x)\,dx \biggr\vert \leq C(\Phi) \| f\| _{L^{\Phi}(\Omega)}\| g\|_{L^{\tilde{\Phi}}(\Omega)} $$
for \(f\in L^{\Phi}(\Omega)\) and \(g\in L^{\tilde{\Phi }}(\Omega)\).

Definition 2.1

A Young function Φ satisfies the \(\Delta_{2}\)-condition (\(\Phi\in \Delta_{2}\)) if
$$\Phi(2s) \leq C \Phi(s) $$
for some constant \(C\geq2\) and all \(s>0\).

By the Riesz representation theorem, if Φ and Φ̃ belong to the class \(\Delta_{2}\), then the dual space of \(L^{\Phi }(\Omega)\) is \(L^{\tilde{\Phi}}(\Omega)\).

Now we recall the explicit expression of the duals of some Orlicz spaces (see [1719]).

Theorem 2.1

Let \(\Omega\subset \mathbb {R}^{n}\) be an open set. If \(1 < p < \infty\) and \(q, q_{1}, q_{2}\in \mathbb {R}\), then
  • \((L^{p} (\log L)^{q}(\Omega) )' \cong L^{p'} (\log L)^{-\frac{q}{p-1}}(\Omega)\),

  • \((L^{p} (\log\log\log L)^{q} (\Omega) )' \cong L^{p'} (\log\log\log L)^{-\frac{q}{p-1}}(\Omega)\),

  • \((L^{p} (\log L)^{q_{1}}(\log\log\log L)^{q_{2}} (\Omega ) )' \cong L^{p'} (\log L)^{-\frac{q_{1}}{p-1}}(\log\log\log L)^{-\frac {q_{2}}{p-1}}(\Omega)\),

where \(p'\) is the conjugate exponent of p, that is, \(\frac{1}{p}+ \frac{1}{p'}=1\).
If \(p=1\) and \(q>0\), then
  • \((L(\log L)^{q}(\Omega) )' \cong \operatorname {EXP}_{\frac{1}{q}} (\Omega)\).

Given two Young functions Φ and Ψ, we say that Ψ dominates Φ globally (respectively near infinity) if there exists a constant \(k>0\) such that
$$\Phi(t)\leq\Psi(kt) \quad\mbox{for all } t\geq0 \mbox{ (respectively for all } t \geq t_{0} \mbox{ for some } t_{0}>0) ; $$
moreover, Φ and Ψ are equivalent globally (respectively near infinity, \(\Phi\cong\Psi\)) if each dominates the other globally (respectively near infinity). If Φ̃ and Ψ̃ are the complementary Young functions of, respectively, Φ and Ψ, then Ψ dominates Φ globally (or near infinity) if and only if Φ̃ dominates Ψ̃ globally (or near infinity). Similarly, Φ and Ψ are equivalent if and only if Φ̃ and Ψ̃ are equivalent. We have the following result.

Theorem 2.2

The continuous embedding \(L^{\Psi}(\Omega)\hookrightarrow L^{\Phi }(\Omega)\) holds if and only if either Ψ dominates Φ globally or Ψ dominates Φ near infinity and Ω has finite measure.

Finally, we recall the definition of the Orlicz-Sobolev spaces \(W^{1,\Psi}(\Omega)\) and \(W^{1,\Psi }_{0}(\Omega)\) (see [2023]). The space \(W^{1,\Psi }(\Omega)\) consists of the equivalence classes of functions u in \(L^{\Psi}(\Omega)\) whose distributional gradients u belong to \(L^{\Psi}\). This is a Banach space with respect to the norm given by
$$\|u\|_{W^{1,\Psi}(\Omega)}=\|u\|_{L^{\Psi}(\Omega)}+\|\nabla u\| _{L^{\Psi}(\Omega)} . $$
As in the case of the ordinary Sobolev space, \(W^{1,\Psi }_{0}(\Omega)\) coincides with the closure of \(C^{\infty}_{0}(\Omega)\) in \(W^{1,\Psi}(\Omega)\).

2.3 Orlicz-Sobolev imbeddings

Lemma 2.3

Let \(\Phi(t)=\exp \{\frac{t^{\frac{1}{\delta}}}{ (\log (e + \log (e+ t)) )^{\frac{\beta}{2\delta}}} \}-1\) with \(\beta\in \mathbb {R}\) and \(\delta>0\). Then
$$ \tilde{\Phi}(t)\cong t (\log t )^{\delta} ( \log \log \log t )^{\frac{\beta}{2}}. $$
(2.1)

Proof

Since Φ is a Young function, by definition we have
$$\Phi(t)= \int_{0}^{t} \phi(s)\,ds, $$
where ϕ is equivalent near infinity to
$$\Phi(s) \cdot \biggl[\frac{s^{\frac{1}{\delta}-1}}{\delta (\log\log s )^{\frac{\beta}{2\delta}}}- \frac{\beta s^{\frac{1}{\delta }-1}}{2\delta(\log s) \cdot (\log\log s )^{\frac{\beta }{2\delta } +1}} \biggr]. $$
For large s, we have
$$\phi(s)\cong\Phi(s) \frac{s^{\frac{1}{\delta}-1}}{\delta (\log\log s )^{\frac{\beta}{2\delta}}}, $$
and we will prove that, near infinity,
$$ \phi(s)\cong\Phi(s). $$
(2.2)
We begin with the case \(\delta\leq1\). Then we can state that there exists \(c>1\) such that
$$\begin{aligned} \exp \biggl\{ \frac{s^{\frac{1}{\delta}}}{ (\log\log s )^{\frac {\beta}{2\delta}} } \biggr\} &\leq\exp \biggl\{ \frac{s^{\frac {1}{\delta }}}{ (\log\log s )^{\frac{\beta}{2\delta}} } \biggr\} \cdot \frac{s^{\frac{1}{\delta}-1}}{\delta (\log\log s )^{\frac{\beta }{2\delta}}} \\ &\leq\exp \biggl\{ \frac{(cs)^{\frac{1}{\delta }}}{ (\log\log(cs) )^{\frac{\beta}{2\delta}} } \biggr\} . \end{aligned}$$
Similarly, in the case \(\delta>1\), there exists \(c\in(0,1)\) such that
$$\begin{aligned} \exp \biggl\{ \frac{(cs)^{\frac{1}{\delta}}}{ (\log\log (cs) )^{\frac{\beta}{2\delta}} } \biggr\} &\leq\exp \biggl\{ \frac {s^{\frac {1}{\delta}}}{ (\log\log s )^{\frac{\beta}{2\delta}} } \biggr\} \cdot \frac{s^{\frac{1}{\delta}-1}}{\delta (\log\log s )^{\frac{\beta }{2\delta}}}\\ &\leq\exp \biggl\{ \frac{s^{\frac{1}{\delta }}}{ (\log \log s )^{\frac{\beta}{2\delta}} } \biggr\} . \end{aligned}$$
Hence, (2.2) is proved, and then it is not difficult to check that
$$\phi^{-1}(r) \cong(\log r)^{\delta} (\log\log\log r )^{\frac {\beta}{2}}. $$
By the definition of a complementary Young function, for large y, we obtain that
$$\tilde{\Phi}(y)= \int_{0}^{y} \phi^{-1} (r)\,dr\cong y ( \log y)^{\delta } (\log\log\log y )^{\frac{\beta}{2}}. $$
 □
Given a Young function Ψ such that
$$\int_{0} \biggl(\frac{r}{\Psi(r)} \biggr)\,dr< \infty, $$
we define \(\Phi: [0, + \infty) \rightarrow[0, + \infty)\) as
$$ \Phi(s)= \Psi\circ H^{-1}_{2}(s) \quad\mbox{for } s \geq0 , $$
(2.3)
where \(H^{-1}_{2}(s)\) is the (generalized) left-continuous inverse of the function \(H_{2}: [0, + \infty) \rightarrow[0, + \infty)\) given by
$$ H_{2}(r)= \biggl( \int_{0}^{r} \biggl(\frac{t}{\Psi(t)} \biggr)\,dt \biggr)^{\frac{1}{2}} \quad\mbox{for } r \geq0 . $$
(2.4)
In [24] and in [25], the author showed that Φ is a Young function and that the following Sobolev-Orlicz embedding theorem holds:
$$\| u\|_{L^{\Phi}(\Omega)} \leq C \| \nabla u \|_{L^{\Psi}(\Omega)} $$
for every function u in the Orlicz-Sobolev space \(W^{1, \Psi}(\Omega )\). As an application, we prove an embedding theorem, which can be regarded as an extension of Lemma 2.4 in [13].

Lemma 2.4

Let \(\Omega\subset \mathbb {R}^{2}\) be an open bounded set with \(\mathcal{C}^{1}\) boundary. Consider the Young function
$$\Psi(t)= t^{2} (\log t )^{1-2\delta} ( \log\log\log t )^{-\beta} $$
with \(\beta\in \mathbb {R}\) and \({\delta\geq\frac{1}{2}}\). Then
$$W^{1, \Psi}(\Omega) \hookrightarrow L^{\Phi}(\Omega), $$
where
$$ \Phi(s)\cong e^{s^{\frac{1}{\delta}} ( \log\log s )^{-\frac {\beta}{2\delta}}} . $$
(2.5)

Proof

By (2.4) we have that
$$H_{2}(r)= \biggl( \int_{0}^{r} \frac{(\log t)^{2\delta-1}(\log\log\log t)^{\beta}}{t}\,dt \biggr)^{\frac{1}{2}} \cong ( \log r )^{\delta} (\log\log\log r)^{\frac{\beta}{2}}. $$
Moreover, as shown in the proof of Lemma 2.3, the inverse function \(H^{-1}_{2}(s)\) is equivalent near infinity to
$$e^{s^{\frac{1}{\delta}} ( \log\log s )^{-\frac{\beta }{2\delta}}}. $$
By (2.3) we obtain that
$$\begin{aligned} \Phi(s) &\cong e^{2s^{\frac{1}{\delta}} ( \log\log s )^{-\frac{\beta}{2\delta}}} \bigl(s^{\frac{1}{\delta}}( \log\log s)^{-\frac {\beta}{2\delta}} \bigr)^{1-2\delta} \bigl(\log\log s^{\frac {1}{\delta }}( \log\log s)^{-\frac{\beta}{2\delta}} \bigr)^{-\beta}\\ &\cong e^{s^{\frac{1}{\delta}} ( \log\log s )^{-\frac{\beta }{2\delta}}}, \end{aligned}$$
and we conclude that
$$W^{1, \Psi}(\Omega) \hookrightarrow L^{\Phi} (\Omega) . $$
 □

Remark 2.5

The previous lemma for \(\delta=\frac{1}{2}\) and \(\beta=0\) was proved in [6, 7], and [8]. The case \(\beta=0\) and \(\delta> \frac{1}{2}\) is proved in [26].

3 Equivalent norm on the Zygmund spaces \(L^{q}(\log L)^{-\gamma }(\log\log\log L)^{-\beta}(\Omega)\)

The main tool of this section is to obtain an integral expression equivalent to the Luxemburg norm in \(L^{q}(\log L)^{-\gamma} ( \log \log\log L )^{-\beta}(\Omega)\) with \(1< q<\infty\), \(\beta\geq0\) and \(\gamma>0\).

If f is a measurable function on Ω, we set
$$ |\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} ( \log\log\log L )^{-\beta }(\Omega)}= \biggl\{ \int_{0}^{\varepsilon_{0}} \varepsilon^{\gamma-1} \| f\| _{L^{q- \varepsilon} ( \log\log\log L )^{-\beta}(\Omega )}^{q} \,d\varepsilon \biggr\} ^{\frac{1}{q}} . $$
(3.1)
Here \(\varepsilon_{0} \in \,] 0, q-1]\) is fixed.
For \(\beta=0\), (3.1) becomes
$$|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma}(\Omega)}= \biggl\{ \int _{0}^{\varepsilon_{0}} \varepsilon^{\gamma-1} \| f \|_{L^{q- \varepsilon}(\Omega)}^{q} \,d\varepsilon \biggr\} ^{\frac{1}{q}} $$
as in [15].

Theorem 3.1

We have \(f\in L^{q} (\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)\) if and only if
$$|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)} < + \infty. $$
Moreover, \(|\!|\!|\cdot |\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta}(\Omega)}\) is a norm equivalent to the Luxemburg one, that is, there exist constants \(C_{i}= C_{i}(q, \beta,\gamma,\varepsilon _{0})\), \(i=1, 2\), such that, for all \(f\in L^{q}(\log L)^{-\gamma} (\log \log \log L )^{-\beta}(\Omega)\),
$$\begin{aligned} C_{1} \| f \|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)} &\leq|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log \log\log L )^{-\beta}(\Omega)}\\ &\leq C_{2} \| f \|_{L^{q}(\log L)^{-\gamma } (\log\log\log L )^{-\beta}(\Omega)}. \end{aligned}$$

Proof

It is easy to check that \(|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log \log L )^{-\beta}(\Omega)}\), defined by (3.1), is a norm on \(L^{q}(\log L)^{-\gamma}(\log\log\log L)^{-\beta}(\Omega)\).

Moreover, for any measurable function f and for a.e. \(x\in\Omega\), if \(a\geq e^{e^{e}}\), then we have
$$|f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon}\leq|f|^{q-\varepsilon} \leq2^{q-1} \bigl[a^{q}+|f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon} \bigr], $$
and so we deduce
$$\begin{aligned} |f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon}\bigl(\log\log\log\bigl(a+|f|\bigr) \bigr)^{-\beta}\leq{}& |f|^{q-\varepsilon}\bigl(\log\log\log\bigl(a+|f|\bigr) \bigr)^{-\beta}\\ \leq{}&2^{q-1} \bigl[a^{q}+|f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon} \bigr]\\ &{}\times\bigl(\log\log \log \bigl(a+|f|\bigr)\bigr)^{-\beta} . \end{aligned}$$
Integrating over Ω, we get
$$\begin{aligned} &\fint_{\Omega} |f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon}\bigl(\log \log\log \bigl(a+|f|\bigr)\bigr)^{-\beta}\,dx\\ &\quad\leq\|f\|_{L^{q-\varepsilon}(\log\log\log L)^{-\beta}(\Omega)} ^{q-\varepsilon} \\ &\quad\leq2^{q-1}a^{q} +2^{q-1} \fint_{\Omega}|f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon }\bigl(\log \log\log\bigl(a+|f|\bigr)\bigr)^{-\beta}\,dx . \end{aligned}$$
Then we multiply for \(\varepsilon^{\gamma-1}\) and integrate between 0 and \(\varepsilon_{0}\) to obtain:
$$\begin{aligned} & \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma-1} \biggl[ \fint _{\Omega} |f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon}\bigl(\log \log\log\bigl(a+|f|\bigr)\bigr)^{-\beta}\,dx \biggr]\,d\varepsilon \\ &\quad \leq \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma -1}\|f\|_{L^{q-\varepsilon}(\log\log\log L)^{-\beta}(\Omega)} ^{q-\varepsilon}\,d\varepsilon\\ &\quad\leq2^{q-1}a^{q} \frac{\varepsilon_{0}^{\gamma}}{\gamma}+ 2^{q-1} \int _{0}^{\varepsilon_{0}}\varepsilon^{\gamma-1} \biggl[ \fint_{\Omega} |f|^{q}\bigl(a+|f|\bigr)^{-\varepsilon}\bigl(\log \log\log\bigl(a+|f|\bigr)\bigr)^{-\beta}\,dx \biggr]\,d\varepsilon. \end{aligned}$$
Thanks to Lemma 4.3 of [11], used with the choice \(b=a+|f|\), we obtain that there exist two constant \(C_{1}\), \(C_{2}\), depending only on γ and \(\varepsilon_{0}\), such that
$$\begin{aligned} C_{1} \fint_{\Omega} & |f|^{q}\bigl(\log\bigl(a+|f|\bigr) \bigr)^{-\gamma} \bigl(\log \log\log \bigl(a+|f|\bigr) \bigr)^{-\beta}\,dx \\ & \leq \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma -1}\|f\|_{L^{q-\varepsilon}(\log\log\log L)^{-\beta}(\Omega )}^{q-\varepsilon}\,d\varepsilon \\ & \leq C_{2} \biggl[1+ \fint_{\Omega} |f|^{q}\bigl(\log \bigl(a+|f|\bigr) \bigr)^{-\gamma} \bigl(\log\log\log\bigl(a+|f|\bigr) \bigr)^{-\beta}\,dx \biggr]. \end{aligned}$$
(3.2)
If \(|\!|\!|f|\!|\!|_{L^{q} (\log\log\log L )^{-\beta}(\Omega )}\) is finite, then since
$$ \|f\|^{q-\varepsilon} _{L^{q-\varepsilon}(\log\log\log L)^{-\beta }(\Omega)}\leq\|f\|^{q} _{L^{q-\varepsilon}(\log\log\log L)^{-\beta }(\Omega)}+1, $$
by the first inequality in (3.2) we get that \(f\in L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta}(\Omega)\). Moreover, if \(|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log \log L )^{-\beta}(\Omega)}=1\), then
$$\fint_{\Omega} |f|^{q}\bigl(\log(a+|f|) \bigr)^{-\gamma} \bigl(\log\log\log (a+|f|) \bigr)^{-\beta}\,dx \leq C_{3} , $$
where \(C_{3}\) is a constant independent on f. By homogeneity, for any measurable f, we get
$$\|f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega )}\leq C_{3} |\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log \log L )^{-\beta}(\Omega)} . $$
Before proving the converse, we recall that
$$ \sup_{0 < \sigma\leq q-1} \sigma^{\frac{\gamma}{q-\sigma}} \| f\| _{L^{q-\sigma} (\log\log\log L )^{-\beta}(\Omega)} \leq C_{4} \| f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)}. $$
(3.3)
Indeed, if we fix \(a\geq e^{e^{e}}\) and proceed as in Lemma 1.2 in [16], using the Hölder inequality and the inequality
$$\log^{\lambda}(a+t)\leq\lambda^{\lambda}(a+t), $$
we obtain
$$\begin{aligned} &\int_{\Omega}|f|^{q-\sigma} \bigl(\log\log\log \bigl(a+|f| \bigr) \bigr)^{-\beta}\\ &\quad= \int_{\Omega}\frac{|f|^{q-\sigma} (\log\log\log (a+|f|) )^{-\beta+\frac{\beta(q-\sigma)}{q}-\frac{\beta(q-\sigma )}{q}} (\log (a+|f|) )^{\frac{\gamma(q-\sigma)}{q}}}{ (\log(a+|f|) )^{\frac{\gamma(q-\sigma)}{q}}}\\ &\quad\leq \biggl[ \int_{\Omega}\frac{|f|^{q} (\log\log\log (a+|f|) )^{-\beta}}{ (\log(a+|f|) )^{\gamma}} \biggr]^{\frac {q-\sigma }{q}} \\ &\qquad{}\times\biggl[ \int_{\Omega}\bigl(\log\log\log \bigl(a+|f| \bigr) \bigr)^{(-\beta+\frac {\beta(q-\sigma)}{q})\frac{q}{\sigma}} \bigl(\log\bigl(a+|f|\bigr) \bigr)^{\frac {\gamma(q-\sigma)}{\sigma}} \biggr]^{\frac{\sigma}{q}} \\ &\quad\leq \biggl[ \int_{\Omega}\frac{|f|^{q} (\log\log\log (a+|f|) )^{-\beta}}{ (\log(a+|f|) )^{\gamma}} \biggr]^{\frac {q-\sigma }{q}} \\ &\qquad{}\times\biggl[ \biggl(\frac{\gamma(q-\sigma)}{\sigma} \biggr)^{\frac {\gamma (q-\sigma)}{\sigma}} \int_{\Omega}\bigl(\log\log\log \bigl(a+|f| \bigr) \bigr)^{-\beta}\bigl(a+|f|\bigr) \biggr]^{\frac{\sigma}{q}}\\ &\quad\leq \biggl[ \int_{\Omega}\frac{|f|^{q} (\log\log\log (a+|f| ) )^{-\beta}}{ (\log(a+|f|) )^{\gamma}} \biggr]^{\frac {q-\sigma }{q}} \biggl[ \biggl(\frac{\gamma(q-\sigma)}{\sigma} \biggr)^{\frac {\gamma (q-\sigma)}{\sigma}} \int_{\Omega}\bigl(a+|f|\bigr) \biggr]^{\frac{\sigma}{q}}. \end{aligned}$$
Hence, elevating both sides of this inequality to the power \(\frac {1}{q-\sigma}\) and then multiplying both of them by \(\sigma^{\frac {\gamma}{q-\sigma}}\), we deduce
$$\begin{aligned} \biggl[\sigma^{\gamma}&\int_{\Omega}|f|^{q-\sigma} \bigl(\log\log\log \bigl(a+|f| \bigr) \bigr)^{-\beta} \biggr]^{\frac{1}{q-\sigma}}\\ &\quad\leq \biggl[ \int_{\Omega}\frac{|f|^{q} (\log\log\log (a+|f| ) )^{-\beta}}{ (\log(a+|f|) )^{\gamma}} \biggr]^{\frac {1}{q}} \bigl(a|\Omega|+ \|f\|_{L^{1}(\Omega)} \bigr)^{\frac{\sigma}{q(q-\sigma )}}\gamma ^{\frac{\gamma}{q}}(q-\sigma)^{\frac{\gamma}{q}}\sigma^{\frac {\gamma \sigma}{q(q-\sigma)}}, \end{aligned}$$
and passing to the supremum with respect to \(\sigma\in(0,q-1]\), we get formula (3.3) with
$$C_{4}=\gamma^{\frac{\gamma}{q}}\sup_{0< \sigma\leq q-1} \bigl\{ \bigl(a| \Omega|+\|f\|_{L^{1}(\Omega)} \bigr)^{\frac{\sigma}{q(q-\sigma )}}(q- \sigma)^{\frac{\gamma}{q}}\sigma^{\frac{\gamma\sigma }{q(q-\sigma )}} \bigr\} . $$
If \(f\in L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)\), that is, if
$$ \|f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega )}< \infty $$
(3.4)
by (3.3), then there exists a constant \(C_{5}\) independent on f such that
$$ \|f\|_{L^{q-\varepsilon} (\log\log\log L )^{-\beta }(\Omega )}\leq C_{5} \varepsilon^{-\frac{\gamma}{q-\varepsilon}} \|f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)}. $$
(3.5)
By (3.5) we get
$$\begin{aligned} \|f\|^{q}_{L^{q-\varepsilon}(\log\log\log L)^{-\beta}(\Omega)}&= \|f\|^{q-\varepsilon}_{L^{q-\varepsilon}(\log\log\log L)^{-\beta }(\Omega )} \|f\|^{\varepsilon}_{L^{q-\varepsilon}(\log\log\log L)^{-\beta }(\Omega )} \\ &\leq C_{6} \|f\|^{q-\varepsilon}_{L^{q-\varepsilon}(\log\log\log L)^{-\beta}(\Omega)} \|f\|^{\varepsilon}_{L^{q}(\log L)^{-\gamma } (\log\log\log L )^{-\beta}(\Omega)}. \end{aligned}$$
(3.6)
Hence, by (3.2) we obtain that \(|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma } (\log\log\log L )^{-\beta}(\Omega)} <+\infty\). Indeed, if
$$\|f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega )}=1, $$
by (3.6) and (3.2) we get
$$|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)}< C_{7} , $$
where the constant \(C_{7}\) is independent on f. By homogeneity we conclude the proof, obtaining
$$|\!|\!|f|\!|\!|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta }(\Omega)}< C_{7} \|f\|_{L^{q}(\log L)^{-\gamma} (\log\log\log L )^{-\beta}(\Omega)} . $$
 □

4 Proof of Theorem 1.1

In this section, before proving Theorem 1.1, we state a regularity result for elliptic equations with right-hand side in divergence form. For convenience of the reader, we recall Theorem 3.1 of [2].

Theorem 4.1

Let \(A=A(x,\xi)\) be a Leray-Lions mapping that satisfies (1.3). Then there exists \(\sigma_{0}= \sigma_{0}(K)>0\) such that, for \(|\sigma |\leq\sigma_{0}\) and \(\underline{\chi}_{1}\), \(\underline{\chi}_{2}\in L^{2-\sigma}( \Omega; \mathbb {R}^{2})\), each of the two problems
$$\begin{aligned}& \textstyle\begin{cases} \operatorname {div}A(x,\nabla\varphi_{1})=\operatorname {div}\underline{\chi}_{1} & \textit{in } \Omega,\\ \varphi_{1}\in W^{1,2-\sigma}_{0}(\Omega) , \end{cases}\displaystyle \end{aligned}$$
(4.1)
$$\begin{aligned}& \textstyle\begin{cases} \operatorname {div}A(x,\nabla\varphi_{2})=\operatorname {div}\underline{\chi}_{2} & \textit{in } \Omega,\\ \varphi_{2}\in W^{1,2-\sigma}_{0}(\Omega) , \end{cases}\displaystyle \end{aligned}$$
(4.2)
has a unique solution and
$$\| \nabla\varphi_{1}-\nabla\varphi_{2} \|_{L^{2-\sigma}(\Omega)} \leq C(K) \| \underline{\chi}_{1} - \underline{\chi}_{2} \|_{L^{2-\sigma }(\Omega)}, $$
where \(C(K)>0\) depends only on K.

Theorem 4.1 allows us to prove the following:

Theorem 4.2

Let \(A=A(x,\xi)\) be a Leray-Lions mapping that satisfies (1.3). Then, if \(\gamma>0\) and \(\beta\geq0\), for \(i=1,2\) and for any \(\underline{\chi}_{i}\in L^{2}(\log L)^{-\gamma}(\log\log\log L)^{-\beta }(\Omega; \mathbb {R}^{2})\), there exists a unique solution \(\varphi_{i}\) to the Dirichlet problem
$$\begin{aligned} \textstyle\begin{cases} \operatorname {div}A(x,\nabla\varphi_{i})=\operatorname {div}\underline{\chi}_{i} & \textit{in } \Omega,\\ \varphi_{i}\in W^{1,1}_{0}(\Omega) . \end{cases}\displaystyle \end{aligned}$$
(4.3)
Moreover,
$$ \|\nabla\varphi_{1}-\nabla\varphi_{2} \|_{L^{2}(\log L)^{-\gamma}(\log \log\log L)^{-\beta}(\Omega)}\leq C\|\underline{\chi}_{1}-\underline{\chi }_{2}\| _{L^{2}(\log L)^{-\gamma}(\log\log\log L)^{-\beta}(\Omega)}, $$
(4.4)
where \(C=C(\beta,\gamma,K)>0\) is a positive constant that depends on the parameters K, β, and γ.

Proof

By Theorem 4.1 there exists a positive constant \(\sigma_{0}=\sigma (K)\) such that if \(|\sigma|\leq\sigma_{0}\), then for \(i=1,2\) and for any \(\underline{\chi}_{i}\in L^{2-\sigma}(\Omega; \mathbb {R}^{2})\), problem (4.3) admits a unique solution \(\varphi_{i} \in W^{1, 2-\sigma}_{0}\), and
$$ \|\nabla\varphi_{1}-\nabla\varphi_{2} \|_{L^{2-\sigma}(\Omega)}\leq C\| \underline{\chi}_{1}-\underline{ \chi}_{2}\|_{L^{2-\sigma}(\Omega)} , $$
(4.5)
where \(C=C(K)>0\) is a positive constant that depends only on the parameter K.
If \(\gamma>0\) and \(\beta\geq0\) are fixed, using Theorem 3.1, we obtain
$$\begin{aligned} &\|\nabla\varphi_{1}-\nabla \varphi_{2}\|^{2}_{L^{2}(\log L)^{-\gamma}(\log \log \log L)^{-\beta}(\Omega)} \\ &\quad\leq C_{1}(\beta, \gamma)|\!|\!|\nabla\varphi_{1}-\nabla\varphi _{2} |\!|\!|^{2}_{L^{2}(\log L)^{-\gamma}(\log\log\log L)^{-\beta}(\Omega )}\\ &\quad=C_{1}(\beta,\gamma) \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma-1} \| \nabla \varphi_{1}-\nabla\varphi_{2}\|_{L^{2- \varepsilon} ( \log \log \log L )^{-\beta}(\Omega)}^{2}\,d\varepsilon. \end{aligned}$$
For \(\beta=0\), by Theorem 4.1 we get
$$ \|\nabla\varphi_{1}-\nabla\varphi_{2}\|^{2}_{L^{2}(\log L)^{-\gamma }(\Omega )} \leq C_{2}(\gamma, K) \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma -1} \| \underline{\chi}_{1}-\underline{\chi}_{2}\|_{L^{2- \varepsilon}(\Omega)}^{2}\,d\varepsilon. $$
If \(\beta>0\), then with a suitable choice of \(\lambda_{0}\), by Theorem 3 in [13] and Theorem 4.1, we get
$$\begin{aligned} &\|\nabla\varphi_{1}-\nabla \varphi_{2}\|^{2}_{L^{2}(\log L)^{-\gamma}(\log \log \log L)^{-\beta}(\Omega)}\\ &\quad\leq C_{3}(\beta,\gamma) \int_{0}^{\varepsilon_{0}}\varepsilon^{\gamma-1} \biggl[ \int_{0}^{\lambda_{0}}\bigl(1+\log|\log\lambda|\bigr)^{-\beta -1}\bigl(\lambda|\log \lambda|\bigr)^{-1}\\ &\qquad{}\times\| \nabla\varphi_{1}-\nabla \varphi_{2}\| _{L^{2-\varepsilon -\lambda}(\Omega)}^{2-\varepsilon}\,d\lambda \biggr]^{\frac {2}{2-\varepsilon }}\,d\varepsilon \\ &\quad \leq C_{4}(\beta,\gamma,K) \int_{0}^{\varepsilon_{0}}\varepsilon ^{\gamma-1} \biggl[ \int_{0}^{\lambda_{0}}\bigl(1+\log|\log\lambda|\bigr)^{-\beta -1}\bigl(\lambda|\log \lambda|\bigr)^{-1}\\ &\qquad{}\times\| \underline{\chi}_{1}-\underline{ \chi}_{2}\| _{L^{2-\varepsilon-\lambda}(\Omega)}^{2-\varepsilon}\,d\lambda \biggr] ^{\frac{2}{2-\varepsilon}}\,d\varepsilon\\ &\quad\leq C_{5}(\beta,\gamma,K) \int_{0}^{\varepsilon_{0}}\varepsilon ^{\gamma-1} \| \underline{\chi}_{1}-\underline{\chi}_{2}\|_{L^{2- \varepsilon} ( \log \log\log L )^{-\beta}(\Omega)}^{2}\,d\varepsilon. \end{aligned}$$
Using again Theorem 3.1 in the last term, we have
$$\begin{aligned} &\|\nabla\varphi_{1}-\nabla \varphi_{2}\|^{2}_{L^{2}(\log L)^{-\gamma}(\log \log \log L)^{-\beta}(\Omega)}\\ &\quad\leq C_{5}( \beta,\gamma,K)|\!|\!|\underline{\chi}_{1}-\underline{\chi }_{2}|\!|\!|^{2}_{L^{2}(\log L)^{-\gamma}(\log\log\log L)^{-\beta}(\Omega )}\\ &\quad\leq C_{6}(\beta,\gamma,K)\|\underline{\chi}_{1}- \underline{\chi }_{2}\| ^{2}_{L^{2}(\log L)^{-\gamma} (\log\log\log L)^{-\beta}(\Omega)} . \end{aligned}$$
 □

Now we are in position to prove the main theorem.

Proof of Theorem 1.1

Since \(L^{\widetilde{\Phi}}(\Omega)=L (\log L)^{\delta}(\log\log \log L)^{\frac{\beta}{2}}(\Omega)\) is a subspace of \(L(\log L)^{\frac {1}{2}}(\Omega)\) if \(\beta\geq0\) and \(\delta\geq\frac{1}{2}\), we can ensure (as already observed) that (1.1) has a unique finite energy solution \(v\in W^{1,2}_{0}(\Omega)\).

In order to prove Theorem 1.1, we want to apply the regularity result given by Theorem 4.2. To do this, as already showed in the papers [10, 11, 13], and [12], we need to linearize problem (1.1). We will use a linearization procedure introduced in [27] that preserves the ellipticity bounds.

For shortness, we do not give all the details of the linearization procedure, and we refer, for example, to proof of Theorem 1.1 in [11]. So we know that there exists a symmetric, definite positive, and measurable matrix-valued function \(B=B(x)\) such that
$$A(x, \nabla v)= B(x) \nabla v. $$
Then, the unique finite energy solution \(v\in W^{1,2}_{0}(\Omega)\) of (1.1) with \(f\in L^{\widetilde{\Phi}}(\Omega)\) solves also the following linear problem:
$$\begin{aligned} \textstyle\begin{cases} -\operatorname {div}B(x)\nabla v=f & \mbox{in } \Omega, \\ v=0 & \mbox{on } \partial\Omega, \end{cases}\displaystyle \end{aligned}$$
(4.6)
that is,
$$ \int_{\Omega}B(x)\nabla v\nabla\varphi= \int_{\Omega}f\varphi, \quad\forall\varphi\in W^{1,2}_{0}( \Omega) . $$
(4.7)

The case \(\boldsymbol{\beta= 0}\) and \(\boldsymbol{\frac{1}{2}\leq \delta\leq1}\) has been proved in [10].

The case \(\boldsymbol{\beta> 0}\) and \(\boldsymbol{\delta=\frac {1}{2}}\) has been proved in [13].

Now, if \(\boldsymbol{\beta\geq0}\) and \(\boldsymbol{\delta>\frac {1}{2}}\), then we fix \(\underline{\chi}\in C^{1}(\overline{\Omega})\) such that
$$\|\underline{\chi}\|_{L^{2}(\log L)^{-(2\delta-1)}(\log\log\log L)^{-\beta }(\Omega; \mathbb {R}^{2})}\leq1, $$
and we consider the unique finite energy solution φ to the linear Dirichlet problem
$$\begin{aligned} \textstyle\begin{cases} -\operatorname {div}B(x)\nabla\varphi=\operatorname {div}\underline{\chi} & \mbox{in } \Omega, \\ \varphi=0 & \mbox{on } \partial\Omega, \end{cases}\displaystyle \end{aligned}$$
where \(B(x)\) is the matrix given by the linearization procedure. By Theorem 4.2 we have
$$\begin{aligned} &\|\nabla\varphi\|_{L^{2}(\log L)^{-(2\delta-1)}(\log\log\log L)^{-\beta }(\Omega)}\\ &\quad\leq C(\beta,\delta,K)\|\underline{\chi}\|_{L^{2}(\log L)^{-(2\delta -1)}(\log\log\log L)^{-\beta}(\Omega)}\leq C(\beta,\delta,K), \end{aligned}$$
and so, using Lemma 2.4, we obtain
$$ \|\varphi\|_{L^{\Phi}(\Omega)}\leq C_{1}(\beta,\delta,K) , $$
(4.8)
where \(\Phi(s)\cong e^{s^{\frac{1}{\delta}} ( \log \log s )^{-\frac{\beta}{2\delta}}}\), and \(C_{1}(\beta,K)\) is another constant depending only on β, δ, and K.
Thanks to the fact that v satisfies the linear problem (4.6) and that \(B(x)\) is a symmetric matrix, using Lemma 2.3 and the Hölder inequality between the complementary spaces \(L^{\Phi}(\Omega)\) and \(L^{\widetilde{\Phi}}(\Omega)\), by (4.8) we obtain that, for any \(\underline{\chi}\in C^{1}(\overline{\Omega}; \mathbb {R}^{2})\) such that \({\|\underline {\chi }\|_{L^{2}(\log L)^{-(2\delta-1)}(\log\log\log L)^{-\beta}(\Omega )}\leq 1}\), we have
$$\begin{aligned} \biggl|\int_{\Omega}\nabla v\cdot\underline{\chi} \biggr|&= \biggl| \int _{\Omega}v \operatorname {div}\underline {\chi} \biggr| \\ &= \biggl| \int_{\Omega}v \operatorname {div}\bigl(B(x)\nabla\varphi\bigr) \biggr|= \biggl| \int_{\Omega}B(x)\nabla v\cdot\nabla\varphi \biggr| \\ &= \biggl|\int_{\Omega}f\varphi \biggr|\leq C_{2}(\beta,\delta)\| \varphi\| _{L^{\Phi }(\Omega)}\|f\|_{L(\log L)^{\delta}(\log\log\log L)^{\frac{\beta }{2}}(\Omega)} \\ &\leq C_{2}(\beta,\delta,K)\|f\|_{L(\log L)^{\delta}(\log\log\log L)^{\frac{\beta}{2}}(\Omega)} , \end{aligned}$$
(4.9)
where \(C_{2}(\beta,\delta,K)\) is a constant that depends only on β, δ, and K.

By Theorem 2.1 the dual space of \(L^{2}(\log L)^{-(2\delta -1)}(\log \log\log L)^{-\beta}(\Omega)\) is \(L^{2}(\log L)^{2\delta-1} (\log\log \log L)^{\beta}(\Omega)\).

Now, since \(C^{1}(\overline{\Omega}; \mathbb {R}^{2})\) is dense in \(L^{2}(\log L)^{-(2\delta-1)}(\log\log\log L)^{-\beta}(\Omega)\) (see [20], Theorem 8.20 and [23], Corollary 5), passing to the supremum in (4.9) under the conditions \(\underline{\chi}\in C^{1}(\overline {\Omega}; \mathbb {R}^{2})\), \({\|\underline{\chi}\|_{L^{2}(\log L)^{-(2\delta-1)}(\log\log\log L)^{-\beta}(\Omega; \mathbb {R}^{2})}\leq 1}\), we obtain
$$\|\nabla v\|_{L^{2}(\log L)^{2\delta-1}(\log\log\log L)^{\beta }(\Omega )}\leq c(\beta,\delta,K)\|f\|_{L(\log L)^{\delta}(\log\log\log L)^{\frac {\beta}{2}}(\Omega)}, $$
as desired. □

Remark 4.3

In [27], it was proved that the linearization procedure holds in any dimension with the following ellipticity bounds:
$$|\xi|^{2}+ \bigl|A(x, \xi)\bigr|^{2} \leq \biggl( K+ \frac{1}{K} \biggr) \bigl\langle A(x, \xi), \xi\bigr\rangle ,\quad\xi\in \mathbb {R}^{n}, \mbox{ a.e. }x\in\Omega. $$
We would like to point out that the linear growth of \(A(x, \xi)\) with respect to ξ is absolutely essential for the previous results. The main difficulty with the n-harmonic-type equations (\(n \neq2\)) is due to the lack of uniqueness for very weak solutions.

Declarations

Acknowledgements

This research was partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilita’ e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second author has been partially supported by projects Regione Campania Legge 5/2007 (‘Spazi pesati ed applicazioni al calcolo delle variazioni’) and ‘Sostegno alla Ricerca Individuale’ of Università degli Studi di Napoli ‘Parthenope’.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Section de Mathématiques, École Polytechnique Fédérale de Lausanne
(2)
Università degli Studi di Napoli ‘Parthenope’
(3)
Dipartimento di Matematica e Applicazioni ‘R. Caccioppoli’, Università degli Studi di Napoli ‘Federico II’

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