Theory and Background Material for PDEs

Abstract

This chapter surveys the principal theoretical issues concerning the solving of partial differential equations (PDEs). In the first section we recall the notions of elliptic, parabolic, and hyperbolic PDEs, since we are going to study the regularity of solutions for all these types of equations in Chaps. 4– 6. In the subsequent sections we study weak formulations of our PDEs. In a weak formulation an equation is no longer required to hold absolutely and has instead weak solutions only with respect to certain ‘test functions’. We present the Lax-Milgram Theorem theorem which guarantees the existence and uniqueness of such a weak solution. In particular, it turns out that for elliptic problems with zero Dirichlet boundary conditions, the Sobolev spaces Open image in new window (and appropriate generalizations for parabolic and hyperbolic problems) are ‘good’ spaces among which we can look for weak solutions to a given PDE. Furthermore, we provide sufficient conditions on the coefficients of the involved differential operators and the functions on the right hand side of our equations, which guarantee the existence and uniqueness of a weak solution.Moreover, for elliptic problems we also present regularity results of the weak solution in the fractional Sobolev scale Hs( Ω). For parabolic problems this will be investigated later on (and compared with the results presented here) in Chap. 5.Finally, we discuss the concept of operator pencils generated by (elliptic) boundary value problems, since singularities of solutions on non-smooth domains can be described in terms of spectral properties of certain pencils. We use these pencils in Chap. 5 when proving some of the regularity results in Kondratiev spaces.