Local uniqueness in the Cauchy problem for second order elliptic equations with non-Lipschitzian coefficients
The author considers second-order elliptic operators of the form: \[ E(x,D_x)u(x)= \sum^d_{j,k=0} a_{j,k}(x)D_{xj}D_{xk}u(x)+ \sum^d_{j=0} a_j(x)D_{xj}u(x)+ c(x)u(x). \] If the coefficients \(a_j(x)\), \(c(x)\) are locally essentially bounded and \(a_{j,k}(x)\) are real-valued, satisfy \[ \sum^d_{j,k=0} a_{j,k}(x)\xi_j\xi_k\geq C_K|\xi|^2,\quad K\text{ compact},\;K\subset\Omega\subset\mathbb{R}^{d+1} \] and belong to a certain space of continuous functions \(C^\chi\), the local uniqueness of the Cauchy problem is proved. The special space \(C^\chi(\Omega)\) is defined as the space of all continuous functions \(f\) defined on \(\Omega\), such that for any compact \(K\subset\Omega\) the following inequality holds \[ |f(x)- f(y)|\leq C|x-y|\chi\Biggl({1\over|x-y|^\varepsilon}\Biggr),\quad x,y\in K, \] where \(\varepsilon\), \(C\) are two constants, \(\varepsilon\in(0,1]\), \(C>0\), \(|x-y|<\varepsilon\), and \(\chi\) is a positive nondecreasing continuous function satisfying some boundedness conditions. Examples of such functions are: \(\log(C_1s)\), \(\log(C_2s)\log(\log(C_3s))\). These function spaces \(C^\chi\) have been used previously in the theory of fluid dynamics.
scientific article; zbMATH DE number 1051637