Elliptic boundary value problem

Remarks

Example: the Dirichlet problem $-\Delta u = f$ in $\Omega$, $u = g$ on $\partial \Omega$ has $m = 1$, the Laplacian is properly elliptic, and the Dirichlet condition satisfies the Lopatinskii–Shapiro condition. Non-example: $-\Delta u = f$ with the two conditions $u = 0$ and $\partial_\nu u = 0$ on $\partial \Omega$ is not elliptic, since an operator of order $2$ takes exactly one boundary condition. Agmon, Douglis and Nirenberg proved an a priori estimate for elliptic problems. For each integer $s \ge 0$ there is $C > 0$ with $$\norm{u}_{H^{2m+s}(\Omega)} \le C \Big( \norm{Au}_{H^s(\Omega)} + \sum_{j=1}^m \norm{B_j u}_{H^{2m+s-m_j-1/2}(\partial \Omega)} + \norm{u}_{L^2(\Omega)} \Big),$$ where $H^k$ denotes Sobolev spaces. For a properly elliptic operator of order $2$, the Dirichlet condition always satisfies the Lopatinskii–Shapiro condition. Indeed, the bounded solutions of the ordinary differential equation form a line spanned by a function that does not vanish at $0$.