Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects

In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321--359]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques

Complete damage in linear elastic materials - Modeling, weak formulation and existence results

In this work, we introduce a degenerating PDE system with a time-depending domain for complete damage processes under time-varying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a quasi-static balance equation for the displacement field which are strongly nonlinearly coupled. In our proposed model, the material may completely disintegrate which is indispensable for a realistic modeling of damage processes in elastic materials. Complete damage theories lead to several mathematical problems since for instance coercivity properties of the free energy are lost and, therefore, several difficulties arise. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an $SBV$-framework. The main aim is to prove existence of weak solutions for the introduced degenerating model. In addition, we show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions which is a novelty in the theory of complete damage models of this type

A compressible mixture model with phase transition

We introduce a new thermodynamically consistent diffuse interface model of Allen--Cahn/Navier--Stokes type for multi-component flows with phase transitions and chemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e.~a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized Allen-Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a Young--Laplace and a Stefan type law

A solution of Braess' approximaiton problem on powers of the distance function

The polynomial approximation behaviour of the class of functions Fs:R2(x0,y0)−>R,Fs(x,y)=((x−x0)2+(y−y0)2)(−s),sin(0,infty), is studied in [Bra01]. There it is claimed that the obtained results can be embedded in a more general setting. This conjecture will be confirmed and complemented by a different approach than in [Bra01]. The key is to connect the approximation rate of F_s with its holomorphic continuability for which the classical Bernstein approximation theorem is linked with the convexity of best approximants. Approximation results of this kind also play a vital role in the numerical treatment of elliptic differential equations [Sau]

Maximal convergence theorems for functions of squared modulus holomorphic type and various applications

In this paper we extend the theory of maximal convergence introduced by Walsh to functions of squared modulus holomorphic type. We introduce in accordance to the well-known complex maximal convergence number for holomorphic functions a real maximal convergence number for functions of squared modulus holomorphic type and prove several maximal convergence theorems. We achieve that the real maximal convergence number for F is always greater or equal than the complex maximal convergence number for g and equality occurs if L is a closed disk in R^2. Among other various applications of the resulting approximation estimates we show that for functions F of squared holomorphic type which have no zeros in a closed disk B_r the relation limsupntoinftysqrt[n]En(Br,F)=limsupntoinftysqrt[n]En(partialBr,F) is valid, where E_n is the polynomial approximation error

The degenerate and non-degenerate Stefan problem with inhomogeneous and anisotropic Gibbs--Thomson law

The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end approximate solutions are constructed by means of variational functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law in a weak generalized BV-formulation

Bernstein--Walsh type theorems for real analytic functions in several variables

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in R^N. In particular, we investigate the polynomial approximation behavior for functions $F: L \to C, L= { (Re z, Im z ) : z \in K}$, of the type $F= g \overline{ h}$, where g and h are holomorphic in a neighborhood of a compact set $K \subset C^N$. To this end the maximal convergence number $rho(S_c,f)$ for continuous functions f defined on a compact set S_c \subset \C^N is connected to a maximal convergence number $\rho(S_r,F)$ for continuous functions F defined on a compact set $S_r \subset \R^N$

A solution of Braess' approximation problem on powers of the distance function

The polynomial approximation behaviour of the class of functions F_s: R^2\(x_0, y_0 ) -> R, F_s(x,y) = ( (x-x_0)^2 + (y-y_0)^2 )^(-s), s \in (0, \infty), is studied in [Bra01]. There it is claimed that the obtained results can be embedded in a more general setting. This conjecture will be confirmed and complemented by a different approach than in [Bra01]. The key is to connect the approximation rate of F_s with its holomorphic continuability for which the classical Bernstein approximation theorem is linked with the convexity of best approximants. Approximation results of this kind also play a vital role in the numerical treatment of elliptic differential equations [Sau]