prof. RNDr. Jan Franců, CSc.
E-mail:   francu@fme.vutbr.cz  WWW:   http://www.mat.fme.vutbr.cz/home/francu Dept.:   Institute of Mathematics Dept. of Mathematical Analysis Position:   Professor Room:   A1/1839

Supervised courses:

• Equations of Mathematical Physics I (9RF1)
• Functional Analysis and Function Spaces (9FAP)
• Modern Methods of Solving Differential Equations (SDR)
• Modern Methods of Solving Differential Equations (SDR-A)
• Partial Differential Equations (SPD)

Publications:

• FRANCŮ, J.:
  O řešení problémů slabé konvergence,
  Kvaternion, Vol.2013, (2013), No.1, pp.27-44, ISSN 1805-1324
  journal article - other
• FRANCŮ, J.:
  Outline of Nguetseng's approach to non-periodic homogenization,
  Mathematics for applications, Vol.1, (2012), No.2, pp.117-128, ISSN 1805-3610
  journal article - other
• FRANCŮ, J.; SVANSTEDT, N.:
  Some remarks on two-scale convergence and periodic unfolding,
  Application of Mathematics, Vol.57, (2012), No.4, pp.359-375, ISSN 0373-6725, Matematický ústav AVČR
  journal article - other
• FRANCŮ, J.; NOVÁČKOVÁ, P.; JANÍČEK, P.:
  Torsion of a Non-circular Bar,
  Engineering Mechanics, Vol.19, (2012), No.1, pp.45-60, ISSN 1802-1484
  journal article - other
• FRANCŮ, J.; NECHVÁTAL, L.:
  Homogenization of Monotone Problems with Uncertain Coefficients,
  Mathematical Modelling and Analysis, Vol.16, (2011), No.3, pp.432-441, ISSN 1392-6292, Taylor&Francis a VGTU
  journal article - other
• FRANCŮ, J.:
  On two-scale convergence and periodic unfolding,
  Tatra Mountains Mathematical Publications, Vol.48, (2011), No.1, pp.73-81, ISSN 1210-3195
  journal article - other
• FRANCŮ, J.:
  Modification of unfolding approach to two-scale convergence,
  Mathematica Bohemica, Vol.135, (2010), No.4, pp.403-412, ISSN 0862-7959
  journal article - other

List of publications at Portal BUT

Abstracts of most important papers:

• FRANCŮ, J.; SVANSTEDT, N.:
  Some remarks on two-scale convergence and periodic unfolding,
  Application of Mathematics, Vol.57, (2012), No.4, pp.359-375, ISSN 0373-6725, Matematický ústav AVČR
  journal article - other
  The paper discusses some aspects of the adjoint definition of two-scale convergence based on periodic unfolding. As is known this approach removes problems concerning choice of the appropriate space for admissible test functions. The paper proposes a modified unfolding which conserves integral of the unfolded function and hence simplifies the proofs and its application in homogenization theory. The article provides also a self-contained introduction to two-scale convergence and gives ideas for generalization to non-periodic homogenization.
• FRANCŮ, J.; NECHVÁTAL, L.:
  Homogenization of Monotone Problems with Uncertain Coefficients,
  Mathematical Modelling and Analysis, Vol.16, (2011), No.3, pp.432-441, ISSN 1392-6292, Taylor&Francis a VGTU
  journal article - other
  The homogenization problem for a nonlinear elliptic equation modelling some physical phenomena set in a periodically heterogeneous medium is studied. Contrary to the usual approach, the coefficients in the equation are supposed to be uncertain functions from a given set of admissible data satisfying suitable monotonicity and continuity conditions. The problem with uncertainties is treated by means of the worst scenario method.
• FRANCŮ, J.; KREJČÍ, P.:
  Homogenization of scalar wave equations with hysteresis,
  Continuum Mech Therm, Vol.11, (1999), No.6, pp.371-390, ISSN 0935-1175
  journal article - other
  The paper deals with a scalar wave equation of the form $\rho u_{tt} = (F[u_x])_x + f$ where $F$ is a Prandtl-Ishlinskii operator and $\rho, f$ are given functions. This equation describes longitudinal vibrations of an elastoplastic rod. The mass density $\rho$ and the Prandtl-Ishlinskii distribution function $\eta$ are allowed to depend on the space variable $x$. We prove existence, uniqueness and regularity of solution to a corresponding initial-boundary value problem. The system is then homogenized by considering a sequence of equations of the above type with spatially periodic data $\rho^\eps$ and $\eta^\eps$, where the spatial period $\eps$ tends to $0$. We identify the homogenized limits $\rho^*$ and $\eta^*$ and prove the convergence of solutions $u^\e$ to the solution $u^*$ of the homogenized equation.
• FRANCŮ, J.:
  Modelling of the Czochralski flow,
  Abstract and Applied Analysis, Vol.3, (1998), No.1-2, pp.1-40, ISSN 1085-3375
  journal article - other
  Czochralski method of industrial production of silicon single crystal consists in pulling up the single crystal from Silicon melt. The flow of the melt during this production is called Czochralski flow. Its character determines concentration of desired oxygen impurity in the crystal. The mathematical description of the Czochralski flow consists of a coupled system of six P.D.E. in cylindrical coordinates containing Navier-Stokes equations (with the stream function), heat convection-conduction equation, convection-diffusion equation for oxygen impurity and an equation describing magnetic field effect. The paper contains derivation of the model, its weak and operator formulation, its justification and proof of existence of solution to the both stationary and evolutionary problems.
• FRANCŮ, J.:
  Monotone operators. A survey directed to applications to differential equations,
  APPLICATIONS OF MATHEMATICS, Vol.35, (1990), pp.257-301, ISSN 0862-7940, Academia
  journal article - other
  The paper deals with the existence of solutions to equations of the form Au=b with operators monotone in a broader sense, including pseudomonotone operators and operators satisfying conditions S and M. The first part of the paper which has a methodical character is concluded by the proof of an existence theorem for the equation on a reflexive separable Banach space with a bounded demicontinuous coercive operator satisfying condition (M)o. The second part which has a character of survey compares various types of continuity and monotony and introduces further results. In the third part application of this theory to proofs of existence theorems for boundary value problems for ordinary and partial differential equations is illustrated by examples.