On some improperly posed problem for degenerate quasilinear elliptic equations
Kazuya Hayasida · Journal of the Mathematical Society of Japan · 1994
We consider the Cauchy problem for degenerate quasilinear elliptic equations, and we give an estimate for their solutions with a prescribed bound.For linear elliptic equations such an estimate is known (see $e$ .$g.,$ $[1],$ $[3]$ and [5] $)$ .In general the Cauchy problem is not well-posed for elliptic equations, that is, it is improperly posed for these equations.We give the following example due to Lavrentiev's book [5, p. 19]: Let $\Omega$ be a bounded domain in the plane.Let its boundary $\partial\Omega$ be smooth, and let $n$ be the outer normal of $\partial\Omega$ .Let $\Gamma_{1}$ be an open subset of $\partial\Omega$ and $\Gamma_{2}=\partial\Omega-\Gamma_{1}$ .The part $\Gamma_{1}$ is said to be an initial surface.Let $u$ be in $C^{1}(\overline{\Omega})$ and harmonic in $\Omega$ .We assume that $|u(x)|+| \frac{\partial}{\partial n}u(x)|\leqq\epsilon$ $x\in\Gamma_{1}$ , $|u(x)|+| \frac{\partial}{\partial n}u(x)|$ $ $M$ $x\in\Gamma_{2}$ .