Regularizing effect in some Mingione’s double phase problems with very singular data

1.   Introduction

The topic of this paper is inspired by one of the recent scientific interests of Rosario Mingione, the so–called "double phase" elliptic problem.

The main example of a double phase integral functional is

where $\Omega$ is an open, bounded subset of ${{\mathbb{R}}}^N$ ($N\geq 2$),

and

Since it is not assumed that the weight $\rho(x)$ is bounded away from zero (that is, it is not assumed that $\exists\; \rho_0\in{{\mathbb{R}}}^+$ such that $\rho(x)\geq\rho_0 > 0$), it is not possible to say, even under the assumption $p < q$, that the term $\rho(x)|\nabla v|^q$ is dominant, so that the set $\{x:\, \rho(x) = 0\}$ plays an important role.

Few years ago, R. Mingione found a name for such a problem: double phase problems. Since then, these problems and this terminology have become very popular.

Note that, the functional $J$ exhibits unbalanced growth: the $(p, q)$-growth in the Marcellini terminology (see ^[16]).

Nowadays, there is a huge literature concerning double phase elliptic problems. Here we only recall the fundamental papers ^[2,3,12,13], and recently ^[14,17].

The main example of a double phase elliptic nonlinear differential operator is the derivative of $J$, that is

In this paper we study the existence of distributional solutions, belonging to some standard Sobolev spaces, of Dirichlet problems with very singular data, and associated to differential operators of double phase type like

with $g(0) = 0$.

Namely, we deal with the existence of solutions of the following boundary value problem

where $\Omega$ is an open, bounded subset of ${{\mathbb{R}}}^N$ ($N\geq 2$),

$\, a(x)$ is a measurable function such that

We point out that

● in (1.4) the parameters $\lambda, p$ play the role of $p, q$ in (1.1)

● the operator presented in (1.3) also depends on a power of $u$;

● the coefficient $a(x)$ does not need to be smooth.

Our existence results hinge on the presence of the additional term

which strongly helps, even if it has a growth (with respect to the gradient) $\lambda\leq p$ and despite of the degeneracy due to the factor $|u|^{(r-1)\lambda+1}$.

As a matter of fact, this term provides a strong regularizing property: roughly speaking, we prove that the solution $u$ of (1.3), under a suitable relationship between the parameters $p$ and $\lambda$, is more regular (and it even exists) than the solution $y$ of

studied in ^[4,7,8].

The regularizing effect of some lower orders terms, in the framework of boundary value problems with $L^1$-data, is already known since the paper ^[10] by H. Brezis and W. A. Strauss. We also refer to the paper ^[1,6,9,11], where some Dirichlet problems with lower order terms of order zero or of order one, with natural growth with respect to $\nabla u$ are studied.

2.   The case of bounded data

This section deals with the case

In the sequel, given $k > 0$, we denote by $G_k(s)$ and $T_k(s)$ the classical truncated functions defined by

Let us introduce the following sequence of boundary value problems

As a consequence of the classical result due to J. Leray, J. L. Lions (see ^[15]) there exists ${u_{n}}\in{W_0^{1, p}(\Omega)}$ which is a weak solution of the above problem in the sense that the following integral identity holds

Moreover, due to the boundedness of $f$ and adapting the well known method used in ^[18], each ${u_{n}}$ is a bounded function and there exists a positive contant $C_f$, independent on $n$, such that

Thus, for any $n > C_f$ it holds $T_{n}({u_{n}}) = {u_{n}}$ and ${u_{n}}$ is a weak solution of the following Dirichlet problem

that is

Taking $u_n$ as test function in (2.3) and using the assumption (1.5) we have

Dropping the second (positive) term in the left–hand side and using the boundedness of $f$ we obtain

Here, and in the sequel, we denote by $C_i$ positive constants only depending on the data (but not on $n$).

Thus, there exist a subsequence, not relabelled, and a function $u \in {W_0^{1, p}(\Omega)} \cap L^{\infty}(\Omega)$ such that

Moreover, using estimate (2.4) we obtain, since $1 < \lambda\leq p$,

Thus,

In order to have

it is enough to prove that

Let us take $v = {u_{n}}-u$ as test function in (2.3)

Due to the positivity of the second term we get

We note that the first and the second integral in the right–hand side converge to $0$. Moreover,

and (since $|\nabla u |^{\lambda-1}\in\, L^{p'}$)

Thus, by the Lebesgue Theorem we get

which in turn implies

Then, (2.9) easily follows taking the limit as $n \to +\infty$ in (2.10) and the strong convergence (2.8) is proved. Finally, we take the limit as $n \to +\infty$ in (2.3) (using (2.9) and (2.11)) and we obtain the following existence theorem.

Theorem 2.1. Let $1 < \lambda \leq p < N$. Assume that (1.5), (1.6) hold and let

Then there exists a weak solution $u \in {W_0^{1, p}(\Omega)} \cap L^{\infty}(\Omega)$ which solves the problem (1.3) in the following weak sense

for any $v \in {W_0^{1, p}(\Omega)}.$

3.   Non regular data

In this section we assume that

and we will prove the existence of a distributional solution of problem (1.3)

3.1.   Approximating problems

Let $\left\{f_n\right\}$ be a sequence of bounded functions such that

and

Classical examples are ${f_{n}} = T_n[f]$ and ${f_{n}} = \frac{f}{1+\frac1n|f|}$.

Let us introduce the following approximate boundary value problems

By Theorem 2.1, there exists $u_n \in {W_0^{1, p}(\Omega)} \cap L^{\infty}(\Omega)$ such that, for any $v \in {W_0^{1, p}(\Omega)}$,

Let $k > 0$; by taking $T_k(u_n)$ as test function in the weak formulation (3.3) of problem (3.2) and dropping the positive second term, we can proceed as in ^[4] and the following lemma holds.

Lemma 3.1. Let $1 < \lambda \leq p < N$. Assume that the hypotheses (1.5), (1.6), (3.1) are satisfied. Then, for any $k > 0$ it holds

Moreover, there exists $C_0 > 0$ such that

and

Next, we will prove the following lemma.

Lemma 3.2. Let $1 < \lambda \leq p < N$. Assume that the hypotheses (1.5), (1.6), (3.1) are satisfied. Then there exists a positive constant $R$, independent on n, such that

Proof. We set $\eta = (r-1)\lambda+1$ (note that $\eta > 1$ since $r > 1$) and we take

as test function in (3.3). Dropping the positive term resulting by the principal part, we obtain

We fix $k > 0$. By the above estimate we have

Thus, putting together estimates (3.4) and (3.8), it follows (3.7).

Further improvements on the boundedness of ${u_{n}}$ and $\nabla {u_{n}}$, depending on the relationship between the parameters $p, \lambda$ and $r$, can be derived from the following lemma

Lemma 3.3. Let $1 < \lambda \leq p < N$. Assume that the hypotheses (1.5), (1.6), (3.1) are satisfied. Then there exist two positive constants $R_1, R_2$ independent of $n$ such that

and

with

Proof. By taking $v = \log(1+|{u_{n}}|)\frac{{u_{n}}}{|{u_{n}}|}$ as test function in the weak formulation (3.3) of problem (3.2) (see ^[8]), it is easy to see that

Using in the right–hand side the inequality

and the boundedness of $\left\{{u_{n}}\right\}$ in $L^1(\Omega)$ and, taking into account the positivity of each of the two integrals in the left-hand side and (3.7), the following two estimates hold

and

From estimate (3.11) we also deduce

which, together with inequality (3.4), implies

Now, we can use Sobolev inequality

and the estimate (3.9) follows.

Next, let us prove (3.10). We follow the outline of ^[8]. Note that since $\sigma < p$, by Hölder inequality with exponents $\frac{p}\sigma, \, \frac{p}{p-\sigma}$ and inequality (3.12), we have

and the proof is concluded, since, by the choice of $\sigma$, it follows $\frac\sigma{p-\sigma} = r\, \lambda^*$.

Remark 3.4. Note that in Lemmas 3.1 and 3.7 we only use the assumption $f \in L^1(\Omega)$, while Lemma 3.3 requires the additional hypothesis $f \in L^1 \log L^1 (\Omega)$. However, if $f$ is merely summable, the proof of Lemma 3.3 can be repeated in order to obtain the boundedness of $\{\nabla {u_{n}}\}$ in $W^{1, \sigma}_0(\Omega)$, for any $1 \leq \sigma < \frac{r\, p\, \lambda^*}{1+r\, \lambda^*}$.

Remark 3.5. We point out that

Moreover

Thus, Lemma 3.3 improves Lemma 3.7 if $1\leq \lambda < N\frac{pr-1}{Nr-1}$ and $p > 1+\frac{N-1}{r}$. (Note that $1+\frac{N-1}{r} \in \big]1, 2-\frac{1}{N}\big[$ since $r > 1$).

Remark 3.6. Let $2-\frac{1}{N} < p < N$. Taking into account only the contribution of the principal part and applying the results of ^[7] we deduce that the sequence $\left\{{u_{n}} \right\}$ is bounded in $W^{1, \frac{N(p-1)}{N-1}}_0(\Omega)$.

Thus the term

has a regularizing effect in the following two cases

i) $2-\frac1N < p < N \hbox{ and } \frac{(p-1)N}{N-1} < \lambda \leq p,$

ii) $1 < p\leq 2-\frac1N$ and $1 < \lambda \leq p$.

As a consequence of previous lemmas we prove the following two existence results.

Theorem 3.7. Let $1 < \lambda \leq p < N$. Assume that hypotheses (1.5), (1.6) and (3.1) hold.

Then there exists $u \in W^{1, \lambda}_0(\Omega)$, such that $g(u)|\nabla u|^{\lambda-1} \in L^1(\Omega)$, which solves the problem (1.3) in the following distributional sense

for any $v \in C^{\infty}_0(\Omega).$

Theorem 3.8. Let $1 < \lambda \leq p < N$. Assume that hypotheses (1.5), (1.6) and (3.1) hold.

Then there exists $u \in W^{1, \sigma}_0(\Omega)$, such that $g(u)|\nabla u|^{\lambda-1} \in L^1(\Omega)$, which solves the problem (1.3) in the distributional sense (3.14).

Remark 3.9. We explicitly remark that, in the case $\lambda = p$, Theorem 3.7 gives the existence of at least one solution with finite energy without any additional assumption on the summability of $f$. A similar regularizing effect occurs for the solution of the Dirichlet problem associated to the equation

where $f \in L^m(\Omega)$ with $1 < m < (p^*)'$, when a suitable balance between $m$ and $s$ holds, (see ^[11]) or to the equation

with $f \in L^1(\Omega)$ (see ^[9]).

The following Figure 1 summarizes the different regularity results in dependence of $p$ and $\lambda$.

If $(p, \lambda)$ belongs to the region $A$, the better regularity is the one obtained in ^[7], i.e., $u \in W^{1, \frac{N(p-1)}{N-1}}_0(\Omega)$; otherwise the better regularity is the one proved here.

If $(p, \lambda)$ belongs to the region $B$, Theorem 3.8 gives the existence of a distributional solution $u \in W^{1, \sigma}_0(\Omega)$; while in the region $C$ the better regularity is the one stated in Theorem 3.7, i.e., $u \in W^{1, \lambda}_0(\Omega)$.

At last, if $(p, \lambda)$ belongs to the colored region the result stated in Theorem 3.7 is new.

3.2.   Proof of Theorems 3.7 and 3.8

We begin with the proof of Theorem 3.7.

As a consequence of Lemma 3.1 and Lemma 3.7 there exist a subsequence, not relabelled, and a function $u \in W^{1, \lambda}_0(\Omega)$ such that

In order to take the limit as $n \to +\infty$ in (3.2) we have to prove that

We follow some techniques of ^[5]. For any $\xi \in {\mathbb{R}}^N$, we set

Let $j, k > 0$; using $v = T_j[u_n-T_k(u)]$ as test function in (3.3) we have

We note that

Moreover (since $A(x, \nabla T_k(u)) \nabla T_j[u-T_k(u)] = 0$)

and

since $B_n(0) = 0$. Thus, from (3.16) we deduce

where we have denoted by $\epsilon_n^1(k)$ and $\epsilon_n^2(k)$ two functions which go to $0$ as $n \to +\infty$, for any $k > 0$ and

Now, we use the above inequality in order to prove the $L^1$ compactness of the sequence $\{\nabla u_n\}$.

Let $0 < \theta < \dfrac{\lambda}{p}\,$ ($0 < \theta < 1$) and $k > 0$. Let us define

and let us prove that the previous integral converges to zero.

Indeed, it holds

where

and

with

We observe that

Using the Hölder inequality, with exponents $\frac{\lambda}{p\theta}$ and $\frac{\lambda}{\lambda-p\theta}$, and the a priori estimate (3.10), we have

Here and in the sequel, for any measurable set $E \subset {\mathbb{R}}^N,$ $|E|$ denotes its $N-$ dimensional measure. Moreover, by $\omega^i (k)$ we denote some quantities such that $\lim_{k \to \infty } \omega^i (k) = 0$. Now, we have to study the behavior of $J_{n, \Omega}$; it can be splitted as ($j \in {{\mathbb{N}}}$)

We estimate $J^1_{n, \Omega}$ and $J^2_{n, \Omega}$ by means of Hölder inequality with exponents $\frac1\theta$, $\frac{1}{1-\theta}$ and $\frac{\lambda}{p\theta}$, $\frac{\lambda}{\lambda-p\theta}$, respectively and we use inequalities (3.17) and (3.7), getting

Since

and

we obtain

Summing up the above inequality and (3.18) we have

Therefore,

which gives (for a suitable subsequence, still denoted by $u_n$)

and also (since $\theta$ is positive)

In ^[15], it is proved that, under our assumptions on the function $A(x, {\xi})$, the previous limit implies that

Thus

and, thanks to (3.6) we have

Next, we will prove that

Thanks to (3.19) we also deduce

Moreover, for any measurable set $E \subset \Omega$ we have

and the right–hand side goes to $0$ as $|E| \to 0$ uniformly w.r.t. $n$, since estimates (3.9) and (3.13) hold. Thus

Thanks to Vitali Theorem the convergence (3.22) is proved and

The above limit and the limit (3.21), allow us to take the limit as $n \to +\infty$ in (3.3) and the proof of Theorem 3.7 follows.

In order to prove Theorem 3.8 we note that as a consequence of Lemma 3.1 and Lemma 3.3. there exist a subsequence, not relabelled, and a function $u \in W^{1, \sigma}_0(\Omega)$ such that

and the proof can be performed as above. Precisely, in order to obtain the a.e. convergence of $\{\nabla {u_{n}}\}$ we just have to replace $\lambda$ with $\sigma$.

Acknowledgements

This work has been supported by Project EEEP & DLaD – Piano della Ricerca di Ateneo 2020-2022–PIACERI.

G. R. Cirmi is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Conflict of interest

The authors declare no conflict of interest.