On a phase transition model
The authors study a phase transition model with a periodic and non-autonomous term. The problem is written as \(-\Delta u+AG^{\prime }(u)=0\) in \(\mathbb{R}^{n}\), where \(A\in C^{1}(\mathbb{R}^{n})\) is a 1-periodic function, and \(G\) is a double-well potential, \(G(u)=u^{2}(1-u)^{2}\). The authors look for a solution which takes its values in \((0,1)\), which is near 1 on \(A^{T}=\cup _{i\in T}(i+\Omega )\) (\(\Omega =\{x\in (0,1)^{n}\mid A(x)>0\}\)) and near 0 on \(B^{T}=\cup _{i\in \mathbb{Z}^{n}\setminus T}(i+\Omega )\) with \(T\subset \mathbb{Z}^{n}\). The authors first consider the perturbed equation \[ -\Delta u+A_{\varepsilon }G^{\prime }(u)=0 \] in \(\mathbb{R}^{n}\), where \(A_{\varepsilon }=1+\frac{1}{\varepsilon }A\). The main result of the paper proves that for \(\varepsilon \) small enough and for every \(T\subset \mathbb{Z}^{n}\) there exists a solution \(U_{\varepsilon ,T}\) of the above problem which converges uniformly to 1 on \(A^{T}\) and to 0 on \( B^{T}\) and which satisfies further properties. The proof distinguishes between the cases \(T\) finite and \(T\) general. The authors define the energy functional associated to the perturbed problem as \[ J_{\varepsilon }(u)=\int_{ \mathbb{R}^{n}}(\frac{1}{2}\left| \nabla u\right| ^{2}+A_{\varepsilon }G(u))dx. \] They define the set of admissible functions as \(\Gamma (T)=\{u\in \mathcal{W}\mid u\geq a>1/2\) on \(A_{T}\) and \(au\leq b<1/2\) on \(B_{T}\}\), where \(A_{T}=\cup _{i\in T}(i+\Omega _{d})\) (\(\Omega _{d}=\{x\in \Omega \mid \left| x-\partial \Omega \right| >d\}\)), \( B_{T}=\cup _{i\in \mathbb{Z}^{n}\setminus T}(i+\Omega _{d})\) and \(\mathcal{W} \) is the closure of \(C_{0}^{\infty }(\mathbb{R}^{n})\) with respect to the norm \[ \left( \int_{\mathbb{R}^{n}}\left| \nabla u\right| ^{2}dx+\int_{\left[ -1,1\right] ^{n}}u^{2}\right) ^{1/2}. \] The proof consists of verifying that \(\mathcal{M}_{\varepsilon }(T)=\{u\in \Gamma (T)\mid J_{\varepsilon }(u)=\inf_{\Gamma (T)}J_{\varepsilon }\}\) is non void, establishing the properties of the elements of this set of minimizers and passing to the limit when \(\varepsilon \) goes to \(0\).
scientific article; zbMATH DE number 6161251
scientific article