2008, Denk, Robert, Volevič, Leonid R.

A new class of boundary value problems for parabolic operators is introduced which is based on the Newton polygon method. We show unique solvability and a priori estimates in corresponding L₂-Sobolev spaces. As an application, we discuss some linearized free boundary problems arising in crystallization theory which do not satisfy the classical parabolicity condition. It is shown that these belong to the new class of parabolic boundary value problems, and two-sided estimates for their solutions are obtained.

2002, Denk, Robert, Volevič, Leonid R.

In this paper boundary value problems are studied for systems with large parameter, elliptic in the sense of Douglis-Nirenberg. We restrict ourselves on model problems acting in the half-space. It is possible to define parameter-ellipticity for such problems, in particular we formulate Shapiro-Lopatinskii type conditions on the boundary operators. It can be shown that parameter-elliptic boundary value problems are uniquely solvable and that their solutions satisfy uniform a priori estimates in parameter-dependent norms. We essentially use ideas from Newton's polygon method and of Vishik-Lyusternik boundary layer theory.

2001, Denk, Robert, Volevič, Leonid R.

In this paper we discuss ellipticity conditions for some parameter-dependent boundary value problems which do not satisfy the Agmon-Agranovich-Vishik condition of ellipticity with parameter. The appropriate definition of ellipticity uses the concept of the Newton polygon. For the corresponding boundary value problems with small parameter we construct the formal asymptotic solution, thus explaining the nature of the Shapiro-Lopatinskii condition for these problems.

2000, Denk, Robert, Volevič, Leonid R.

A large class of parameter-dependent elliptic problems was introduced in the sixties by Agmon and Agranovich-Vishik. Some important examples, however, cannot be treated with the classical methods, for instance, singularly perturbed problems and the resolvent of Douglis-Nirenberg systems. This paper gives a short survey on the results of the so-called Newton polygon approach applied to singularly perturbed boundary value problems and to systems of mixed order depending in a specific way on a complex parameter.

2007, Denk, Robert, Volevič, Leonid R.

A new class of boundary value problems for parabolic operators is introduced. We discuss some linearized free boundary problems not satisfying the classical parabolicity condition. It is shown that they belong to this class and by means of the Newton polygon method the nontrivial two-sided estimates of these problems are found.

2002, Denk, Robert, Volevič, Leonid R.

In this paper pencils of partial differential operators depending polynomially on a complex parameter and corresponding boundary value problems with general boundary conditions are studied. We define a concept of ellipticity for such problems (for which the parameter-dependent symbol in general is not quasi-homogeneous) in terms of the Newton polygon and introduce related parameter-dependent norms. It is shown that this type of ellipticity leads to unique solvability of the boundary value problem and to two-sided a priori estimates for the solution.

2000, Denk, Robert, Volevič, Leonid R.

In this paper we study mixed-order (Douglis-Nirenberg) boundary value problems which depend on a real parameter but which are not elliptic with parameter in the sense of Agmon-Agranovich-Vishik. Using the method of the Newton polygon, we are able to prove a priori estimates for the solutions of such problems in corresponding Sobolev spaces. For the related singularly perturbed problem the boundary layer structure of the solutions is described. As an application of the a priori estimate, we obtain new estimates for a transmission problem studied by Faierman.

2002, Denk, Robert, Volevič, Leonid R.

This paper gives a survey on the concept of parameter-ellipticity and parabolicity for mixed order systems depending on a complex parameter. We consider both systems of operators acting on closed manifolds and on manifolds with boundary. The main results include unique solvability of the corresponding equations or boundary value problems and uniform estimates on the solutions in terms of parameter-dependent norms.

2001, Denk, Robert, Mennicken, Reinhard, Volevič, Leonid R.

In this paper parameter-dependent partial differential operators are investigated which satisfy the condition of N-ellipticity with parameter, an ellipticity condition formulated with the use of the Newton polygon. For boundary value problems with general boundary operators we define N-ellipticity including an analogue of the Shapiro Lopatinskii condition. It is shown that the boundary value problem is N-elliptic if and only if an a priori estimate with respect to certain parameter-dependent norms holds. These results are closely connected with singular perturbation theory and lead to uniform estimates for problems of Vishik Lyusternik type containing a small parameter.

2000, Denk, Robert, Mennicken, Reinhard, Volevič, Leonid R.

An operator family of densely defined closed linear operators and the In this paper operator pencils depending polynomially on the spectral parameter are studied which act on a manifold with boundary and satisfy the condition of N -ellipticity with parameter, a generalization of the notion of ellipticity with parameter as introduced by Agmon and Agranovich-Vishik. Sobolev spaces corresponding to a Newton polygon are defined and investigated; in particular it is possible to describe their trace spaces. With respect to these spaces, an a priori estimate holds for the Dirichlet boundary value problem connected with an N-elliptic pencil, and a right parametrix is constructed.