The Sharpiro–Lopatinskij Condition for Elliptic Boundary Value Problems

Abstract

Elliptic boundary value problems are well posed in suitable Sobolev spaces, if the boundary conditions satisfy the Shapiro–Lopatinskij condition. We propose here a criterion (which also covers over-determined elliptic systems) for checking this condition. We present a constructive method for computing the compatibility operator for the given boundary value problem operator, which is also necessary when checking the criterion. In the case of two independent variables we give a formulation of the criterion for the Shapiro–Lopatinskij condition which can be checked in a finite number of steps. Our approach is based on formal theory of PDEs, and we use constructive module theory and polynomial factorisation in our test. Actual computations were carried out with computer algebra systems Singular and MuPad.

Symbolic-numeric computation of implicit riquier bases for PDE

“…Such bases only give local, and sometimes, unnatural boundary and initial conditions. We direct the reader to Krupchyk et al [10,11] for very interesting work on linking formal properties (such as formal integrability and involutivity) to elliptic BVP.…”

“…Applying (10) to the leading derivatives of R, we obtain an × m matrix (σ i,j ) which is called the signature matrix of R (see Pryce [13] for the ode case):…”

Regular and Singular Boundary Problems in Maple

“…A first approach along these lines, for the very simple case of linear coordinate changes, was presented in [2] and is currently being refined. Studying singular boundary problems for LPDEs from a symbolic point of view is also very interesting; see for example [21] for a Gröbner bases approach to compute the (hierarchy of) compatibility conditions for elliptic boundary problems. It would be tempting to combine the tools of involutive systems used there with the setting of operator rings used here.…”

Compatibility Complexes for Overdetermined Boundary Problems

“…The construction of compatibility complexes is useful and even necessary when investigating the well-posedness of overdetermined boundary value problems. In [8] and [7] we have used compatibility complexes to study well-posedness of elliptic problems and moreover, in [9] compatibility complexes are even used in the numerical solution of PDEs. Note that constructions given in this paper are also essential in the theory of overdetermined parabolic and hyperbolic systems of PDEs.…”

“…01 − f 7 10 + f 10 f 5 01 − f 8 10 + f 11 f 6 01 − f 9 10 + f 12 −f 1 10 − f 2 01 + f 3 + f 10 01 − f 12 10     …”