## 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.

### MSC:

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