On the homogenization of partial integro-differential-algebraic equations

We present a Hilbert space perspective to homogenization of standard linear evolutionary boundary value problems in mathematical physics and provide a unified treatment for (non-)periodic homogenization problems in thermodynamics, elasticity, electro-magnetism and coupled systems thereof. The approach permits the consideration of memory problems as well as differential-algebraic equations. We show that the limit equation is well-posed and causal. We rely on techniques from functional analysis and operator theory only.Comment: Thoroughly revised, changed title, reviewer's comments incorporated, added some references. 41 page

Stabilization via Homogenization

In this short note we treat a 1+1-dimensional system of changing type. On different spatial domains the system is of hyperbolic and elliptic type, that is, formally, $\partial_t^2 u_n-\partial_x^2 u_n = \partial_t f$ and $u_n-\partial_x^2 u_n= f$ on the respective spatial domains $\bigcup_{j\in \{1,\ldots,n\}} \big(\frac{j-1}{n},\frac{2j-1}{2n}\big)$ and $\bigcup_{j\in \{1,\ldots,n\}} \big(\frac{2j-1}{2n},\frac{j}{n}\big)$. We show that $(u_n)_n$ converges weakly to $u$, which solves the exponentially stable limit equation $\partial_t^2 u +2\partial_t u + u -4\partial_x^2 u = 2(f+\partial_t f)$ on $[0,1]$. If the elliptic equation is replaced by a parabolic one, the limit equation is \emph{not} exponentially stable.Comment: 8 pages; some typos corrected; referee's comments incorporate

On abstract grad-div systems

For a large class of dynamical problems from mathematical physics the skew-selfadjointness of a spatial operator of the form ⁎A=(0−C⁎C0), where C:D(C)⊆H0→H1 is a closed densely defined linear operator between Hilbert spaces H0,H1, is a typical property. Guided by the standard example, where C=grad=(∂1⋮∂n) (and ⁎−C⁎=div, subject to suitable boundary constraints), an abstract class of operators C=(C1⋮Cn) is introduced (hence the title). As a particular application we consider a non-standard coupling mechanism and the incorporation of diffusive boundary conditions both modeled by setting associated with a skew-selfadjoint spatial operator A

Well-posedness via Monotonicity. An Overview

The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a comprehensive class of such problems. We elaborate the applicability of our scheme with a number examples. A brief discussion of stability and homogenization issues is also provided.Comment: Thoroughly revised version. Examples correcte