Unessential sets in the Cauchy problem for parabolic equations of second order (Q923824)

The author considers parabolic equations that satisfy uniqueness in the sense of Täcklind. A set E is called nonessential for the Cauchy problem if changing the initial data on E does not change the values of the solution. The author gives a sufficient condition in order that a set be nonessential for the Cauchy problem for a second order parabolic equation with continuous coefficients. Consider the operator \[ L=\sum^{n}_{i,j=1}a_{ij}(x,t)\frac{\partial^ 2}{\partial x_ ix_ j}+\sum^{n}_{i=1}b_ i(x,t)\frac{\partial}{\partial x_ i}+c(x,t), \] with continuous coefficients about which we assume \(\inf_{(t,x)}\sum^{n}_{i=1}a_{ii}(t,x)=M\) and \(\sup \max_{(t,x)| \xi | =1}\sum^{n}_{i,k=1}a_{ik}(x,t)\xi_ i\xi_ k=\alpha\) and that \(\sum^{n}_{i=1}| b_ i(x,t)| \leq B\), where \(B<1/2\). This last can be achieved by a change of coordinates. Let u be the solution of the Cauchy problem, continuous up to the hypersurface \(t=0\), with \(u(x,0)=0\) outside E. Let \[ \mu_{\sigma}(E)=\lim_{\epsilon \to 0}(\inf_{r_ i<\epsilon}\sum_{i}r_ i^{\sigma}) \] where the infimum is taken over balls of radius \(r_ i\) that cover E, denote the Hausdorff \(\sigma\) measure of E. The author's main result is the following Theorem. In order that a compact set E be nonessential for the Cauchy problem, it is sufficient that \(\mu_{\sigma}(E)=0\) for \(\sigma <M/\alpha\) where M and \(\alpha\) are defined above. If \(b_ i\equiv 0\), it is sufficient for the matrix \(\| a_{ij}\|\) to be nonnegative definite and then \(\sigma =M/\alpha.\) She gives an example of an operator L and a set S that fails to be nonessential although \(\mu_{M/\alpha +\epsilon}(S)=0\) for every \(\epsilon >0\) and \(\mu_{M/\alpha}(S)\neq 0\). This shows that the above theorem would be false for \(\sigma >M/\alpha\).