Existence and uniqueness theorems for periodic solutions of an elliptic equation with discontinuous and non-differentiable coefficients
The equation under study is the elliptic equation in n-dimensions, \[ (5.2)\quad -\sum_{i,j}a_{ij}D_{x_ i}D_{x_ j}\nu +\lambda \nu =g, \] with coefficient functions periodic in each variable. The author's purpose is to establish the existence, and in dimension \(n=3\) uniqueness of periodic solutions in the space \(W_{loc}^{5/2,2}(R^ n)\). The main result states that under the hypothesis \[ a_{ij}\in W_{\#}^{n/(n+1),n+1}(R^ n)\cap L^{\infty}(R^ n),\quad \sum_{ij}a_{ij}\xi_ i\xi_ i\geq \nu \sum_{i}\xi^ 2_ i,\quad \forall \xi \in R^ n,\quad \nu =cons\tan t\quad >\quad 0,\quad g\in W_{\#}^{,2}(R^ n), \] there exists a \(\lambda_ 0\) such that for \(\lambda \leq \lambda_ 0\) problem (5.2) has at least one solution in \(W_{\#}^{5/2,2}(R^ n)\). In case \(n=3\) the solution is unique. The space \(C_{\#}^{\infty}(R^ n)\) denotes the functions in \(C^{\infty}(R^ n)\), of period 1 in each variable, \(W_{\#}^{s,p}(R^ n)\) denotes the closure of \(C_{\#}^{\infty}(R^ n)\) in the Sobolev norm. This is therefore the class of functions belonging to \(W^{s,p}\) on every bounded subset of \(R^ n\), and having the desired periodicity. A major first step is to establish a priori bounds for the related problem \[ (1.5)\quad \sum_{ij}A_{ij}D_{x_ i}D_{x_ j}u-\nu D^ 2_ tu+\lambda u=f\in W_{\#}^{1,2}(S);\quad u\in W_{\#}^{3,2}(S),\quad S=R^ n\times (0,1), \] for \(\lambda\) a sufficiently large constant. The functions \(A_{ij}(x,t)\) satisfy the conditions \[ A_{ij}=A_{ji}\in W_{\#}^{1,n+1}(S)\cap L^{\infty}(S);\quad \sum A_{ij}\xi_ i\xi_ j\geq \nu \sum \xi^ 2_ i,\quad \forall \xi \in R^ n, \] where \(\nu\) is a positive constant. It will follow that \(D^ 2_ tu\) has zero trace for \(t=0\) and \(t=1\).
scientific article; zbMATH DE number 4025934