Parameter-elliptic boundary value problems connected with the Newton polygon. (English) Zbl 1008.35020

From the text: Pencils of partial differential operators of the form \[ P(D,\lambda)= \sum_{\alpha,k} a_{\alpha k}\lambda^kD^\alpha \tag{1} \] of order \(2M\) depending polynomially on the complex parameter \(\lambda\) and acting in the half-space \(\mathbb{R}^n_+: =\{x=(x',x_n) \in\mathbb{R}^n: x_n>0\}\) and corresponding boundary value problems \(P(D,\lambda) u=f\) with general boundary conditions \(B_j(D) u=g_j\) \((1\leq j\leq M)\) are studied. The authors define the concept of \(N\)-ellipticity for pencils of the form (1) in terms of the Newton polygon and study equivalent conditions for this type of ellipticity. Furthermore, they introduce related parameter-dependent norms and show that this type of ellipticity leads to unique solvability of the boundary value problems and to two-sided a priori estimates for the solutions.


35J40 Boundary value problems for higher-order elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems