Non-Homogeneous Boundary Value Problems and Applications

Sobolev Regularity fort > 0 in Quasilinear Parabolic Equations

“…In [6], we have studied the global solvability of (1.4) in the Sobolev spaces H m,m/2 (Q) introduced by Lions and Magenes in [4]: setting…”

The boundary element solution of the Cauchy steady heat conduction problem in an anisotropic medium

“…We denote by H 1=2 ( i ) and H 1 ( i ) the spaces of the functions belonging to H 1=2 ( ) and H 1 ( ) that are bounded on i and H 1=2 ( i ) * and H 1 ( i ) * the dual spaces of H 1=2 ( i ) and H 1 ( i ), respectively, for i ∈ {0; 1} (see, e.g. Reference [8]). If is smooth then, according to Kozlov et al [4], the above algorithm produces two sequences of approximate solutions {T (2k) (x)} k¿0 and {T (2k+1) (x)} k¿0 which both converge in H 1 ( ) to the solution T of the Cauchy problem given by Equations (1), (3) and (4) for any initial guess u 0 ∈ H 1=2 ( 0 ).…”

“…We note that provided the initial guess u 0 is in H 1=2 ( 0 ) and the boundary data f and q are in H 1=2 ( 1 ) and H 1=2 ( 1 ) * , respectively, the problems given at step 3 of the algorithm are both wellposed and uniquely solvable in H 1 ( ) (see Reference [8]). These intermediate mixed well-posed problems given by Equations (11) and (13) are solved using the BEM described in Section 2.…”

A field-based finite element method for magnetostatics derived from an error minimization approach

“…Equation (1a) expresses the constitutive magnetic relationship, while (1b) and (1c) are known as Maxwell's equations. We assume, beÿtting the physics of the situation, that f B is the normal trace of some ÿeld B ∈ H (div; ) on B , that is f B belongs to the dual space of H 1=2 00 ( B ) [10]. Similarly, f H is the tangential trace of some ÿeldH ∈ H (curl; ) on H .…”