A new class of parabolic problems connected with Newton s polygon

Abstract

are connected with the boundary conditions which do not fit into the classical theory of parabolic problems. The theory of parabolic (as well as elliptic and parameter-elliptic) problems even for scalar operators is deeply connected with the theory of mixed order system of pseudodierential operators acting on the boundary. The matrix-symbol of this system is the so-called Lopatinskii matrix. In standard parabolic problems, the corresponding system is parabolic in the sense of Solonnikov [8]. We introduce a more general class of parabolic boundary value problems replacing systems parabolic in the sense of [8] by a more general class of N-parabolic systems studied by the second author in [10]. In the definition of these systems the notion of Newton’s polygon plays a crucial role. It will be shown that the two mentioned examples from mathematical physics belong to this new class of boundary problems. We now come to the formulation of the two problems under consideration. For simplicity of presentation, we restrict ourselves to the model case of the half-space R n := {x = (x 0 ,xn) 2 R n : xn > 0} with boundary R n 1 . First, the linearized Stefan problem with Gibbs-Thompson correction is given by @tu(x,t) u(x,t) = f(x,t) (t > 0,x 2 R n ),