## 3 The Method
### Preliminaries and Results for the Minimising Problem Associated with (7)  
Before dealing with the results of the insurance of the minimisation problem, we recall some preliminary tools. Let \(\Omega\) be an open-bounded domain in \(\mathbb{R}^{N}\) with a sufficient regular boundary. Throughout this section, we assume that \(p\) is a measurable function from \(\Omega\) to \(\lvert 1\rvert;+\infty[\): We write  
\[p^{+}:=\sup_{x\in\Omega}p(x),\qquad p^{-}:=\inf_{x\in\Omega}p(x).\]  
\(L^{p(\cdot)}(\Omega)\), the Lebesgue space with a variable exponent \(p(\cdot)\), which is the so-called Nakano space and a special sort of Musielak-Orlicz spaces (see [9]). It is defined as follows:  
\[L^{p(\cdot)}(\Omega):=\{u:\Omega\to\mathbb{R};\text{ measurable in }\Omega\text{ and }\int_{\Omega}|u(x)|^{p(x)}dx<+\infty\}\]This space is equipped with the Luxemburg-type norm  
\[\left\|u\right\|_{p(x)}=\inf\Biggl{\{}s>0;\,\int_{\Omega}\left|\frac{u(x)}{s} \right|^{p(x)}dx\leq 1\Biggr{\}}. \tag{9}\]  
\(W^{\mathrm{L},p(\cdot)}(\Omega)\) designates the variable exponent Sobolev spaces on the open-bounded domain \(\Omega\). It is defined as follows:  
\[W^{\mathrm{L},p(\cdot)}(\Omega):=\left\{u\in L^{p(\cdot)}(\Omega);\frac{ \partial u}{\partial x_{i}}\in L^{p(\cdot)}(\Omega)\text{ for all }i=1,2,\cdots,N\right\}\]  
with the norm  
\[\left\|u\right\|_{W^{\mathrm{L},p(\cdot)}(\Omega)}:=\left\|u\right\|_{p(x)}+ \left\|\
abla u\right\|_{p(x)} \tag{10}\]  
where \(\left\|\
abla u\right\|_{p(x)}\) defines the norm of \(\left|\
abla u\right|\) on the Lebesgue space \(L^{p(\cdot)}(\Omega)\). The complete and full results on the Lebesgue space and Sobolev spaces with a variable exponent can be found in the monograph in [10]. Now, let us consider the energy function \(E(f,f_{0},\lambda)\) defined as in (7), and then the smoothing process corresponds to the optimal problem. Below, we give the existence and uniqueness results. **Proposition 1**. : _Under the assumptions \(1<p^{\leq p(x)<p^{+}<2}\) and \(f_{0}\in L^{2}(\Omega)\), the minimising problem_  
\[\inf_{f\in W^{\mathrm{L},p}(\Omega)}\biggl{[}\int_{\Omega}\left|\
abla f \right|^{p(x)}dx+\frac{\lambda}{2}\int_{\Omega}\left|f-f_{0}\right|^{2}dx\biggr{]}. \tag{11}\]  
_has a unique solution \(f^{\star}\) in \(W^{\mathrm{L},p}(\Omega)\)._  
Proof. : We present the most important points to ensure that the desired result (11) is achieved. Throughout the proof, we denote by \(\Phi\) the application defined for any \(f\) in \(W^{\mathrm{L},p(x)}(\Omega)\) by  
\[\Phi(f)=\int_{\Omega}\frac{1}{p(x)}|\
abla f(x)|^{p(x)}dx+\frac{\lambda}{2} \int_{\Omega}|f(x)-f_{0}(x)|^{2}dx. \tag{12}\]  
The proof will be divided into two steps. 1. Step 1 In the first step, we establish the existence of an element \(f^{\star}\) of \(W^{\mathrm{L},p(x)}(\Omega)\) such that \(\Phi(f^{\star})=\inf_{f\in W^{\mathrm{L},p(x)}(\Omega)}\Phi(f)\). For this, we can already notice that, for any \(f\in W^{\mathrm{L},p(x)}(\Omega)\), \(\Phi(f)\) is strictly positive. This implies that there exists a real number \(\alpha\), such that \[\inf_{f\in BV(\Omega)}\Phi(f)=\alpha.\] (13)  
To do this, we consider a minimising sequence \((f_{n})_{n}\) of \(\Phi\).