Boundedness and higher integrability of minimizers to a class of two-phase free boundary problems under non-standard growth conditions

1.   Introduction

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}(n\geq 2)$. Let $\psi\in W^{1, p(\cdot)}(\Omega)\cap L^{\infty}(\Omega)$ and $g\in L^{q(\cdot)}(\Omega)$ with $p, q\in C(\Omega; (1, +\infty))$ being certain given functions. The aim of this paper is to study heterogeneous, two-phase free boundary problems

over the set $\mathcal {K} = \left\{ u\in W^{1, p(\cdot)}(\Omega):u-\psi\in W_0^{1, p(\cdot)}(\Omega) \right\}$ in the framework of Sobolev spaces with variable exponents, where $f:\Omega \times\mathbb{R}^{n}\rightarrow \mathbb{R}$ is a Carathéodory function having a form

with $L\geq 1$ being a constant. The non-differentiable potential $F_{\gamma}(\cdot)$ is given by

where $\gamma \in [0, 1]$ is a parameter, and $\lambda_{+}, \lambda_{-}\in \mathbb{R}$ are positive constants with $\lambda_{+} > \lambda_{-}$. As usual, $u^{\pm}: = \max\{\pm u, 0\}$, and by convention,

As is well known, the lower limiting case, i.e., $\gamma = 0$, relates to the jets and cavities problems. The upper case, i.e., $\gamma = 1$, relates to obstacle-type problems. The intermediary problem, i.e., $\gamma\in(0, 1)$, can be used to model the density of certain chemical species in reaction with a porous catalyst pellet, and has intrigued a number of mathematicians in the past decades.

It should be mentioned that a large class of functionals and identical obstacle problems under non-standard growth conditions have been studied in ^{[1,3,4,5,8,17]}, which provide the reference estimates, and suitable localization and freezing techniques, etc., to treat the non-standard growth exponents in the functional governed by (1.1). It is well known that the boundedness of minimizers plays a crucial role in getting regularity results. For more details about the history of free boundary problems of these types, we refer to the work ^[15], where the authors provided a complete description of regularity theory for the free boundary problems governed by (1.1) with $f(z)\equiv|z|^{p}$ and a constant $p\in[2, +\infty)$. Local and global higher integrability results for solutions or derivatives of the solutions to the obstacle problems, one may refer to ^{[9,10,11,12,19]} and the references therein. The existence and asymptotic analysis of nontrivial solutions for some related double-phase problems under unbalanced growth conditions may be referred to ^[16,18,21] and the references therein.

In this paper, we would like to extend several known results to a larger class of free boundary problems governed by (1.1). We shall establish the existence, boundedness, and integrability of minimizers of $\mathcal {J}_{\gamma}(u)$. The results obtained in this paper are not only extensions of the one in the one-phase obstacle problems under non-standard growth conditions (see, e.g., ^[3,4]), but also a supplement to the one in the degenerate free boundary problems studied in ^[15], as we also consider the singular case $p\in(1, 2)$.

In the rest of the paper, we first introduce some notations used in this paper. In Section 2, we state basic assumptions on the functions $f, p$, and $q$ and main results on the existence, boundedness, and higher integrability of minimizers, which are proved in Sections 3 and 4, respectively.

Notation. Denote by $B_{R}(x)$ the open ball in $\mathbb{R}^{n}$ with center $x$ and radius $R > 0$, and $|B_{R}(x)|$ is the Lebesgue measure of $B_{R}(x)$. For an integrable function $u$ defined on $B_{R}(x)$, let $(u)_{x, R}: = \frac{1}{|B_{R}(x)|}{\int_{B_{R}(x)}u(x)\text{d}x}$. Without confusion, for $R > 0$, we will write $B_{R}$ and $(u)_{R}$ instead of $B_{R}(x)$ and $(u)_{x, R}$ respectively. Let $C, c, C_1, C_2, C_3, ...$ denote constants that may be different from each other, but independent of $\gamma$.

The variable exponent of Lebesgue space $L^{p(\cdot)}(\Omega)$ is defined by

with the norm $\|u\|_{L^{p(\cdot)}(\Omega)}: = \inf\left\{\lambda > 0;\ \int_{\Omega}\left|\frac{u(x)}{\lambda}\right|^{p(x)}\text{d}x\leq 1\right\}.\notag$ The variable exponent Sobolev space $W^{1, p(\cdot)}(\Omega)$ is defined by

with the norm $\|u\|_{W^{1, p(\cdot)}(\Omega)}: = \|u\|_{L^{p(\cdot)}(\Omega)}+\|\nabla u\|_{L^{p(\cdot)}(\Omega)}.\notag$ Define $W^{1, p(\cdot)}_{0}(\Omega)$ as the closure of $C^{\infty}_{0}(\Omega)$ in $W^{1, p(\cdot)}(\Omega)$. If $\Omega$ is bounded and $p(\cdot)$ satisfies (2.5) specified in Section 2, then the spaces $L^{p(\cdot)}(\Omega), W^{1, p(\cdot)}(\Omega)$, and $W^{1, p(\cdot)}_{0}(\Omega)$ are all separable and reflexive Banach spaces. $\|\nabla u\|_{L^{p(\cdot)}(\Omega)}$ is an equivalent norm of $\|u\|_{W^{1, p(\cdot)}_0(\Omega)}$ defined for $W^{1, p(\cdot)}_{0}(\Omega)$. We refer to ^[6,7,13] for the elementary properties and more details of the space $W^{1, p(\cdot)}(\Omega)$.

2.   Main results

In this paper, we always propose the following growth, ellipticity, and continuity conditions on the function $f$:

where $L\geq 1$ and $\mu\in[0, 1]$ are constants. Let $\omega:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ be a nondecreasing continuous function, vanishing at zero, which represents the modulus of $p\in C(\Omega; (1, +\infty))$:

and satisfies $\limsup\limits_{R\rightarrow0}\omega(R)\log\left(\frac{1}{R}\right) < +\infty.$ Without loss of generality, we assume that

Assume further that

with

Let $q \in C(\Omega; (1, +\infty))$ satisfy the conditions of the types (2.3) and (2.4) and

A function $u\in \mathcal{K}$ is said to be a minimizer of the functional $\mathcal {J}_{\gamma}(u)$ governed by (1.1) if $\mathcal {J}_{\gamma}(u)\leq \mathcal {J}_{\gamma}(v)$ for all $v\in \mathcal{K}$.

The first result obtained in this paper is concerned with the existence and uniform (w.r.t. $\gamma$) boundedness of minimizers for the functional $\mathcal {J}_{\gamma}(u)$.

Theorem 2.1. Assume that (2.1)–(2.7) hold. Then, for each $\gamma\in[0, 1]$, there exists a minimizer $u_{\gamma}\in\mathcal {K}$ of the functional $\mathcal {J}_{\gamma}(u)$. Furthermore, $u_{\gamma}$ is bounded. More precisely,

The second result obtained in this paper is the following theorem, which indicates higher integrability of minimizers of the functional $\mathcal {J}_{\gamma}(u)$.

Theorem 2.2. Assume that (2.1)–(2.7) hold and $u_{\gamma}\in \mathcal {K}$ is a minimizer of the functional $\mathcal {J}_{\gamma}(u).$ Then there exist two positive constants $C_{0}$ and $\delta_{0} < q_{-}\left(1-\frac{1}{p_{-}}\right)-1$, both depending only on $n, p_{\pm}, \lambda_{\pm}, q_{-}, L, M$, and $\Omega$, such that f

3.   Existence and $L^{\infty}$-boundedness of minimizers

In this section, we prove Theorem 2.1 in a similar way as in ^[15].

Proof of Theorem 2.1. Firstly, we prove the existence of a minimizer of the functional $\mathcal {J}_{\gamma}(u)$.

Let $I_{0}: = \min\{\mathcal {J}_{\gamma}(u):u\in \mathcal {K}\}$. We claim that $I_{0} > -\infty$. Indeed, for any $u\in \mathcal {K}$, by Poincaré's inequality, there exists a positive constant $C = C(n, p_{\pm}, \Omega)$ such that

which implies

and

where $C_{1}$ and $C_{2}$ are positive constants depending only on $n, p_{\pm}$, and $\Omega$.

Due to $q(x)\geq q_{-}$, we deduce from (2.7), Hölder's inequality, and Young's inequality with $\epsilon > 0$ that

where $\varepsilon\in(0, 1)$ will be chosen later.

Now we consider two cases: $\|\nabla u\|_{L^{p(\cdot)}(\Omega)} > 1$ and $\|\nabla u\|_{L^{p(\cdot)}(\Omega)}\leq 1$.

Case 1: $\|\nabla u\|_{L^{p(\cdot)}(\Omega)} > 1$. It follows from (2.2), (3.2), and (3.5) that

Choose $\varepsilon\in (0, 1)$ such that $L^{-1}C_{1}-\varepsilon > 0$, then, (3.7) yields

Case 2: $\|\nabla u\|_{L^{p(\cdot)}(\Omega)}\leq 1$. We deduce from (2.2), (3.3), and (3.5) that

Choose $\varepsilon\in (0, 1)$ such that $L^{-1}C_{2}-\varepsilon > 0$, then, (3.8) gives

Now we prove the existence of a minimizer of $\mathcal {J}_{\gamma}(u)$. Let $u_{j}\in \mathcal{K}$ be a minimizing sequence. We will show that $\{u_{j}-\psi\}$ (up to a subsequence) is bounded in $W^{1, p(\cdot)}_{0}(\Omega)$. Without loss of generality, we assume that $\|\nabla u_{j}\|_{L^{p(\cdot)}(\Omega)} > 1$. For $j\gg 1$, we have $\mathcal {J}_{\gamma}(u_{j}) \leq I_{0}+1$.

From (3.1), (3.4), (3.6), and Young's inequality with $\varepsilon > 0$, we obtain

where $C_{8}$ and $C_{9}$ depend only on $L, I_{0}, p_{\pm}, \Omega$, and $\|g\|_{L^{q(\cdot)}(\Omega)}$. Then, we get

which, along with Poincaré's inequality, ensures that $\{u_{j}-\psi\}$ is bounded in $W^{1, p(\cdot)}_{0}(\Omega)$. Therefore, there is a function $u\in \mathcal{K}$ such that, up to a subsequence,

With a slight modification of the proof of [20, Theorem 1.6], we infer from (2.1) and (2.2) that

For $\gamma\in(0, 1]$, by the pointwise convergence, we have

For $\gamma = 0$, recalling that $\lambda_{+} > \lambda_{-} > 0$, we have

which implies

In addition, since $u_{j}\rightarrow u$ a.e. in $\Omega$, it follows from the Dominated Convergence Theorem that

Therefore, it holds that

From (3.9), (3.10), and (3.11), we conclude that

which indicates the existence of a minimizer in $\mathcal{K}$.

Secondly, we establish the $L^{\infty}$- boundedness of $u_{\gamma}$. Hereafter, in this proof, we will refer to $u_{\gamma}$ as $u$.

Let $j_{0}: = [\sup\limits_{\partial\Omega} |\psi|]$ be the smallest positive integer above $\sup\limits_{\partial\Omega} |\psi|$. For each $j\geq j_{0}$, we define the truncated function $u_{j}:\Omega\rightarrow \mathbb{R}$ by

where $\text{sgn}(u) = 1$ if $u > 0$ and $\text{sgn}(u) = -1$ if $u\leq 0$. Define the set $A_{j}: = \{|u| > j\}$.

For $\gamma\in(0, 1]$, in view of the minimality of $u$, we derive that

Now we estimate each term on the right-hand side of (3.13).

Then, we get

For the first term in the right-hand side of (3.13), we deduce that

For $\gamma = 0$, it suffices to notice that $u_{j} > 0$ and $u$ have the same sign. From the choice of the truncated function, we know that $(|u|-j)^{+}\in W^{1, p(\cdot)}_{0}(A_{j})$. Applying Hölder's inequality and the embedding theorem, we have

where $t\in C(\Omega; (1, +\infty))$ satisfies $\frac{1}{t(\cdot)} = 1-\frac{1}{p(\cdot)}-\frac{1}{q(\cdot)}$, we denote

and the constant $C$ in the last inequality depends only on $p_{\pm}, q_{-}, n, \Omega$, and $\|g\|_{L^{q(\cdot)}(\Omega)}$.

From (3.13) to (3.16), we infer that

where $C$ depends only on $p_{\pm}, q_{-}, n, \Omega$, and $\|g\|_{L^{q(\cdot)}(\Omega)}$.

Now we consider two cases: $\|\nabla u\|_{L^{p(\cdot)}(A_{j})} > 1$ and $\|\nabla u\|_{L^{p(\cdot)}(A_{j})}\leq 1$.

Case 1: $\|\nabla u\|_{L^{p(\cdot)}(A_{j})} > 1$. We deduce from (2.2), (3.17), and Young's inequality with $\epsilon > 0$ that

which implies

Therefore, we have

where $C$ depends only on $L, p_{\pm}, q_{-}, n, \Omega$, and $\|g\|_{L^{q(\cdot)}(\Omega)}$.

Analogous to (3.16), we deduce that

where in the last inequality we used (3.18), and the constant $C$ depends only on $L, p_{\pm}, q_{-}, n, \Omega$, and $\|g\|_{L^{q(\cdot)}(\Omega)}$.

Case 2: $\|\nabla u\|_{L^{p(\cdot)}(A_{j})}\leq 1$. Analogously, we may obtain

where the constant $C$ depends only on $L, p_{\pm}, q_{-}, n, \Omega$, and $\|g\|_{L^{q(\cdot)}(\Omega)}$.

Now, combining (3.19) and (3.20), we get

where $C$ depends only on $L, p_{\pm}, q_{-}, n, \Omega$, and $\|g\|_{L^{q(\cdot)}(\Omega)}$.

Notice that by (2.6) and (2.7) we have $\frac{1}{q_{-}} < \frac{1}{n}-\frac{1}{p_{-}}+\frac{1}{p_{+}}$ and $\frac{1}{p_{+}} -\frac{1}{p_{-}} +\frac{1}{n} > 0$, respectively, thus

Notice also that $\|u\|_{L^{1}(A_{j_{0}})}\leq \left(1+|A_{j_{0}}|^{\frac{p_{-}-1}{p_{-}}}\right)\|u\|_{L^{p(x)}(A_{j_{0}})}\leq C$. Then, applying [14, Lemma 5.1], we obtain the boundedness of minimizers. □

Remark 3.1. Note that in ^[5], the assumption that $\int_{\Omega} |\nabla u|^{p(x)}\mathit{\text{d}}x\leq M$ with some $M\geq 0$ is proposed for establishing local regularity of minimizers of functionals having a form $\int_{\Omega} f(x, u, \nabla u)\mathit{\text{d}}x$, while in this paper, we are able to show that any minimizer $u_{\gamma}$ of $\mathcal {J}_{\gamma}(u)$ is uniformly bounded w.r.t. $\gamma\in[0, 1]$ in $W^{1, p(\cdot)}(\Omega)$ by using the $L^{\infty}$- estimate of $u_{\gamma}$. Indeed, we have

where $M = M\left(L, n, q_{-}, p_{\pm}, \lambda_{\pm}, \Omega, \|\psi\|_{L^{\infty}(\partial\Omega)}, \|g\|_{L^{q(\cdot)}(\Omega)}\right)$ is a positive constant. Therefore, we conclude that $u_{\gamma}-\psi \in W_0^{1, p(\cdot)}(\Omega)$ with $\|u_{\gamma}\|_{W^{1, p(\cdot)}(\Omega)} \leq C$, where $C$ is independent of $\gamma$.

4.   Higher integrability

In this section, we prove the higher integrability of minimizers of $\mathcal {J}_{\gamma}(u)$. We first recall some important lemmas that will be used in the proof.

Lemma 4.1 (^[5]). Let $\theta\in(0, 1), A > 0$, and $B\geq 0$ be constants, and $h\in L^{p(\cdot)}(B_{R})$. If $k\geq 0$ is a bounded function on $(r, R)$ and satisfies

for all $r\leq t < s\leq R$, there exists a constant $C\equiv C(\theta, p_{+})$ such that

Lemma 4.2 (Gehring-type Lemma, ^[2]). Let $E$ be a closed subset of $\overline{\Omega}$. Consider two nonnegative functions $f, g\in L^1(\Omega)$ and $p\in(1, +\infty)$ such that there holds

for almost all $x\in \Omega\backslash E$ with $B_{\rho}\cap E = \emptyset$, for some constant $b$. Then, there exist constants $C = C(n, p, q, b)$ and $\epsilon = \epsilon(n, p, b)$ such that

holds true for all $q\in [p, p+\epsilon)$, where $\widetilde{g}(x): = \frac{|B_{d(x, E)}(x)\cap\Omega|}{| \Omega |} g(x)$.

Based on Lemma 4.2 and the technique of iteration, we can prove the higher integrability of minimizers of $\mathcal {J}_{\gamma}(u)$.

Proof of Theorem 2.2. Let $0 < R < R_{0}\leq 1$ and $x_{0}\in B_{R}$ with $\overline{B}_{R_{0}}(x_{0})\subset\Omega$. Let $t, s\in\mathbb{R}$ with $\frac{R}{2} < t < s < R$. Let $\eta\in C_{c}^{\infty}(B_{R}), 0\leq \eta\leq 1$, be a cut-off function with $\eta\equiv1$ on $B_{t}, \eta\equiv0$ outside $B_{s}$, and $|\nabla \eta|\leq \frac{2}{s-t}$.

In the sequel, we refer to $u_{\gamma}$ as $u$. Let $z: = u-\eta(u-(u)_{R})$. We deduce from (2.2) and the minimality of $u$ that

where as in the last but one inequality, we used the fact that the inequality

holds true for all $\varphi\in W^{1, p(\cdot)}_{0}(\Omega)$ with $\text{spt}\; \varphi\subset\subset\Omega$. Indeed, it follows from the minimality of $u$ that

Now we estimate each term at (4.1).

where $C = C(p_{+}, p_{-})$ is a positive constant.

A direct calculus shows that

where $C = C(\lambda_{+}, \lambda_{-})$ is a positive constant.

Then, by Young's inequality, we deduce that

where $C = C(p_{\pm}, \lambda_{\pm})$ is a positive constant.

The Young's inequality also gives

where $C = C(p_{+}, p_{-})$ is a positive constant.

Combining (4.1)–(4.6), we obtain

where the constant $C$ depends only on $L, p_{\pm}$, and $\lambda_{\pm}$.

Now, "filling the hole, " we get

which, along with Lemma 4.1, implies that

Let $p_{1}: = \min_{x\in \overline{B}_{R}}p(x)$ and $p_{2}: = \max_{x\in \overline{B}_{R}}p(x)$. By Sobolev–Poincaré's inequality, we deduce that there exists $\nu\in(0, 1)$ such that

where in the last inequality we used the result stated in Remark 3.1.

Combining (4.8) and (4.9), we get

where $C = C(n, p_{\pm}, \lambda_{\pm}, L, M, \Omega)$.

Now applying Lemma 4.2, we conclude that there exists $\delta_{0}\in\left(0, q_{1}\left(1-\frac{1}{p_{-}}\right)-1\right)$ such that (2.8) holds true. □

5.   Conclusions

In this paper, we proved the existence, uniform boundedness, and a higher integrability of minimizers of the functional $J_{\gamma}(u)$ under the framework of Sobolev spaces with variable exponents. Based on the obtained results, we will further study the regularity such as Hölder continuity of minimizers of the functional $J_{\gamma}(u)$.

Author contributions

Jiayin Liu and Jun Zheng: Writing-original draft preparation; Jiayin Liu and Jun Zheng: writing-review and editing; All authors equally contributed to this work. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

J. Liu was supported by the NSF of Ningxia Hui Autonomous Region (No. 2022AAC03237 and No. 2021AAC03184) and the NSF of Gansu Province (No. 23JRRA864). J. Zheng was supported by the NSF of China (No. 11901482).

Conflict of interest

All authors declare no conflicts of interest in this paper.