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Author: Liero, Matthias × Subject: p(x)-Laplacian ×

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Now showing 1 - 3 of 3
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    Systems describing electrothermal effects with p(x)-Laplacian like structure for discontinuous variable exponents
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Bulíc̆ek, Miroslav; Glitzky, Annegret; Liero, Matthias
    We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the p(x)-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a priori L1 term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles
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    Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Bulicek, Miroslav; Glitzky, Annegret; Liero, Matthias
    We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian-type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.
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    Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Glitzky, Annegret; Liero, Matthias
    We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current-voltage laws and selfheating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a p(x)-Laplace-type problem with piecewise constant exponent. We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincaré inequalities, and a Gehring-type Lemma for the p(x)-Laplacian. Finally, Schauders fixed-point theorem is used to show the existence of solutions.