Krešimir Burazin

Intrinsic boundary conditions for Friedrichs systems

The admissible boundary conditions for symmetric positive systems of first-order linear partial differential equations, originally introduced by Friedrichs [11], were recently related to three different sets of intrinsic geometric conditions in graph spaces [10]. We rewrite their cone formalism in terms of an indefinite inner product space, which in a quotient by its isotropic part gives a Kreǐn space. This new viewpoint allows us to show that the three sets of intrinsic boundary conditions are actually equivalent, which will hopefully facilitate further investigation of their precise relation to the original Friedrichs boundary conditions.

Connecting Classical and Abstract Theory of Friedrichs Systems via Trace Operator

Based on recent progress in understanding the abstract setting for Friedrichs symmetric positive systems by Ern et al. (2007), as well as Antonic and Burazin (2010), we continue our efforts to relate these results to the classical Friedrichs theory. Following the approach via the trace operator, we extend the results of Antonic and Burazin (2011) to situations where the important boundary field does not consist only of projections, allowing the treatment of hyperbolic equations, besides the elliptic ones.

On Certain Properties of Spaces of Locally Sobolev Functions

In recent years the locally Sobolev functions got quite popular in works on applications of partial differential equations. However, the properties of those spaces have not been systematically studied and proved in the literature, resulting in many particular proofs by reduction to classical Sobolev spaces.

On Unique Solutions of Multiple-State Optimal Design Problems on an Annulus

We study the uniqueness and explicit derivation of the relaxed optimal solutions, corresponding to the minimization of weighted sum of potential energies for a mixture of two isotropic conductive materials on an annulus. Recently, it has been shown by Burazin and Vrdoljak that even for multiple-state problems, if the domain is spherically symmetric, then the proper relaxation of the problem by the homogenization method is equivalent to a simpler relaxed problem, stated only in terms of local proportions of given materials. This enabled explicit calculation of a solution on a ball, while problems on an annulus appeared to be more tedious. In this paper, we discuss the uniqueness of a solution of this simpler relaxed problem, when the domain is an annulus and we use the necessary and sufficient conditions of optimality to present a method for explicit calculation of the unique solution of this simpler proper relaxation, which is demonstrated on an example.

Complex Friedrichs systems and applications

Recently, there has been a significant development of the abstract theory of Friedrichs systems in Hilbert spaces [Ern, A., Guermond, J.-L., and Caplain, G., Commun. Partial Differ. Equations 32, 317–341 (2007) and Antonic, N. and Burazin, K., Commun. Partial Differ. Equations 35, 1690–1715 (2010)] and its applications to specific problems in mathematical physics. However, these applications were essentially restricted to real systems. We check that the already developed theory of abstract Friedrichs systems can be adjusted to the complex setting, with some necessary modifications, which allows for applications to partial differential equations with complex coefficients. We also provide examples where the involved Hilbert space is not the space of square integrable functions, as it was the case in previous studies, but rather its closed subspace or the space Hs(Rd;Cr), for real s. This setting appears to be suitable for particular systems of partial differential equations, such as the Dirac system, the Di...