Research Interests

Partial Differential Equations

Continuum Physics

Functional Analysis

Degrees

• PhD, 2008,  Faculty of Natural Science, Department of Mathematics, University of Zagreb (thesis: Contributions to the theory of Friedrichs’ and hyperbolic systems, in Croatian, adviser: N. Antonić)
• MSc, 2004,  Faculty of Natural Science, Department of Mathematics, University of Zagreb
• BSc, 1999,  Faculty of Natural Science, Department of Mathematics, University of Zagreb
• 1995, Gymnasium, Vinkovci

Publications

1. K. Burazin, I. Crnjac, Application of explicit energy bounds in optimization of 3D elastic structures, Optimization and Engineering 25 (2024), 1179-1202
   Abstract

   We present a novel numerical method for calculating optimal design in topology optimization problems for 3D linear elastic structures. The algorithm is based on necessary conditions of optimality for problem which was obtained by relaxing the original one via the homogenization method in the sense of operators (G- or H-convergence), and can be implemented for self-adjoint problems. The method relies on recently obtained explicit expressions for the lower Hashin–Shtrikman bound on complementary energy and information on the microstructure that saturates the bound. We tested the algorithm on two benchmark examples, namely the cantilever and the bridge problem. The algorithm provides the solution in a first few iterations and the true composites appear in the optimal design. We also implement a penalization procedure to obtain classical design with slight increase of the cost functional.
2. K. Burazin, M. Erceg, M. Waurick, Evolutionary Equations are G-compact, Journal of Evolution Equations 1 (2024)
   Abstract

   We prove a compactness result related to G-convergence for autonomous evolutionary equations in the sense of Picard. Compared to previous work related to applications, we do not require any boundedness or regularity of the underlying spatial domain; nor do we assume any periodicity or ergodicity assumption on the potentially oscillatory part. In terms of abstract evolutionary equations, we remove any compactness assumptions of the resolvent modulo kernel of the spatial operator. To achieve the results, we introduced a slightly more general class of material laws. As a by-product, we also provide a criterion for G-convergence for time-dependent equations solely in terms of static equations.
3. K. Burazin, I. Crnjac, M. Vrdoljak, Explicit Hashin-Shtrikman bounds in 3D linearized elasticity, Filomat 38/17 (2024), 6033-6048
   Abstract

   In this paper, we are dealing with Hashin-Shtrikman bounds in the context of linearized elasticity. We give an explicit computation of the upper bound on the primal energy in three space dimensions and the corresponding microstructure that saturates this bound. Due to equivalency between the upper Hashin-Shtrikman bound on the primal energy and the lower Hashin-Shtrikman bound on the complementary energy, the explicit calculation of the latter arises. These calculations have straightforward applications in evaluation of experimental results and numerical schemes regarding composite elastic materials, as well as structural optimization problems.
4. K. Burazin, M. Erceg, M. Waurick, G-convergence of Friedrichs systems revisited, Mathematical Methods in the Applied Sciencies 2024 (2024)
   Abstract

   We revisit homogenisation theory for Friedrichs systems. In particular, we show that G-compactness can be obtained under severely weaker assumptions than in the original work of Burazin and Vrdoljak (2014). In this way we extend the applicability of G-compactness results for Friedrichs systems to equations that yield memory effects in the homogenised limit and detour any usage of compactness techniques previously employed.
5. K. Burazin, S. Gutiérrez, J. Jankov Pavlović, Optimal design of Kirchhoff-Love plates under the low contrast assumption, Optimization and Engineering 25 (2024), 821-839
6. N. Antonić, K. Burazin, J. Jankov Pavlović, Small-amplitude homogenization of elastic plates via H-measures, Journal of Applied Mathematics and Mechanics 102/5 (2022), 1-9
7. K. Burazin, J. Jankov Pavlović, On the effective properties of composite elastic plate, Journal of Mathematical Analysis and Applications 495/1 (2021)
8. K. Burazin, I. Crnjac, M. Vrdoljak, Optimality criteria method in 2D linearized elasticity problems, Applied Numerical Mathematics 160 (2021), 192-204
9. K. Burazin, J. Jankov Pavlović, Small-amplitude homogenization of elastic plate equation, Applicable Analysis 100/5 (2021), 1039-1050
10. K. Burazin, I. Crnjac, Convergence of the optimality criteria method for multiple state optimal design problems, Computers & Mathematics with Applications 79/5 (2020), 1382-1392
11. K. Burazin, D. Mitrović, Apriori estimates for fractional diffusion equation, Optimization Letters 13/8 (2019), 1793-1801
12. K. Burazin, M. Vrdoljak, Exact Solutions in Optimal Design Problems for Stationary Diffusion Equation, Acta Applicandae Mathematicae 161/1 (2019), 71-88
13. K. Burazin, J. Jankov Pavlović, M. Vrdoljak, Homogenization of elastic plate equation, Mathematical Modelling and Analysis 23/2 (2018), 190-204
14. K. Burazin, On Unique Solutions of Multiple-State Optimal Design Problems on an Annulus, Journal of optimization theory and applications 177 (2018), 329-344
15. K. Burazin, I. Crnjac, M. Vrdoljak, Variant of optimality criteria method for multiple state optimal design problems, Communications in Mathematical Sciences 16/6 (2018), 1597-1614
16. N. Antonić, K. Burazin, I. Crnjac, M. Erceg, Complex Friedrichs systems and applications, Journal of Mathematical Physics 58/10 (2017), 1-22
17. K. Burazin, M. Erceg, Non-stationary abstract Friedrichs systems, Mediterranean Journal of Mathematics 13/6 (2016), 3777-3796
18. K. Burazin, M. Erceg, Estimates for mild solutions to semilinear Cauchy problems, Electronic Journal of Differential Equations 2014/194 (2014), 1-10
19. K. Burazin, M. Vrdoljak, Homogenisation theory for Friedrichs systems, Communications on Pure and Applied Analysis 13/3 (2014), 1017-1044
20. N. Antonić, K. Burazin, M. Vrdoljak, Second-order equations as Friedrichs systems, Nonlinear analysis: real world applications 14/1 (2014), 290-305
21. N. Antonić, K. Burazin, M. Vrdoljak, Heat equation as a Friedrichs system , Journal of Mathematical Analysis and Applications 404 (2013), 537-553
22. N. Antonić, K. Burazin, Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems, Journal of Differential Equations 250 (2011), 3630-3651
23. N. Antonić, K. Burazin, M. Vrdoljak, Connecting classical and abstract theory of Friedrichs systems via trace operator , ISRN Mathematical Analysis 2011 (2011), 1-14
24. N. Antonić, K. Burazin, Intrinsic boundary conditions for Friedrichs systems, Communications in Partial Differential Equations 35/9 (2010), 1690-1715
25. N. Antonić, K. Burazin, Graph spaces of first-order linear partial differential operators, Mathematical Communications 14/1 (2009), 135-156
26. N. Antonić, K. Burazin, On equivalent descriptions of boundary conditions for Friedrichs systems, Mathematica Montisnigri 22 (2009), 5-13
27. K. Burazin, Estimates on the weak solution of semilinear hyperbolic systems, Annali dell'Universita' di Ferrara 54 (2008), 229-243
28. N. Antonić, K. Burazin, Graph representation for asymptotic expansion in homogenisation of nonlinear first-order equations, Annali dell'Universita' di Ferrara 53 (2007), 149-176

1. N. Antonić, K. Burazin, On certain properties of spaces of locally Sobolev functions, Conference on Applied Mathematics and Scientific Computing, Brijuni, 2005, 109-120

1. K. Burazin, U. Radojičić, Uvod u varijacijski račun i njegova povijest, Osječki matematički list 16/2 (2016), 111-133
   Abstract

   U radu su prikazane osnove varijacijskog računa u slučaju kada traženi ekstrem ovisi samo o jednoj realnoj varijabli te je dan osvrt na njegov povijesni razvoj. Izvedena je Euler-Lagrangeova jednadžba kao nužan uvjet optimalnosti, te je pokazano da je ona i dovoljan uvjet ekstrema u slučaju konveksnog lagrangiana. Teorija je ilustrirana na povijesno važnim primjerima najkraćeg puta, geodetske krivulje na sferi, te problema brahistokrone.
2. K. Burazin, J. Jankov Pavlović, Glazba titrajuće žice, Osječki matematički list 14/1 (2014)
3. K. Burazin, Z. Tomljanović, I. Vuksanović, Prigušivanje mehaničkih vibracija, Math.e : hrvatski matematički elektronski časopis 24 (2014)
4. K. Burazin, Nelinearne diofantske jednadžbe, Osječki matematički list 7 (2007)

1. K. Burazin, J. Jankov Pavlović, I. Kuzmanović Ivičić, I. Soldo, Primjene diferencijalnog i integralnog računa funkcija jedne varijable, Sveučilište Josipa Jurja Strossmayera u Osijeku - Odjel za matematiku, Osijek, 2017.

Projects

• Homogenization and optimization in mechanics of solids

The project is funded by Croatian Science Foundation (December 2023 – December 2027). Project leader: Krešimir Burazin.

• Homogenization, dimension reduction and structural optimization in continuum mechanics

The project was funded by Croatian Science Foundation (October 2018 – April 2023). Project leader: prof. Igor Velčić from University of Zagreb.

• Real-time measurements and forecasting for successful prevention and management of seasonal allergies in Croatia-Serbia cross-border region

The project is funded within Interreg IPA CBC Croatia – Serbia 2014-2020. Partners: Institut BioSens Novi Sad (Lead Beneficiary), University of Novi Sadu – Faculty of Natural Sciences, City of Osijek, University of Osijek – Department of Mathematics. Coordinator for University of Osijek – Department of Mathematics: prof. Kristian Sabo

• Weak convergence methods and applications

The project was funded by Croatian Science Foundation (July 2014 – June 2018). Project leader: prof. Nenad Antonić from University of Zagreb.

• Calculus of variation, optimization and applications
  

This was bilateral project with Serbia (funded by Croatian Ministry of Science and Serbian Ministry of Science 2016 – 2017). Project leaders: N. Krejić (serbian side) and K. Burazin (croatian side)

• Cost Action Mathematics for Industry Network, MI-NET, TD1409

The project is funded within programme TD COST Action TD1409. Member since 2015 as Management Committee substitut

• Center of Excellence for Applications of Mathematics

The project was part of DAAD (German Academic Exchange Service) Programme Academic Reconstruction of South Eastern Europe which was established in late 1999, and it is funded by the German Federal Foreign Office until 2016. Eighteen universities from Germany and south-east Europe were participating in this project. Coordinators from German side were prof. Margareta Heilmann from University of Wuppertal, prof Heiner Gonska from University of Duisburg-Essen and Dozent Dr. Daniela Kacsó from University of Bochum. Coordinator for University of Osijek: K. Burazin.

• Evolution Friedrichs systems

This project run in academin year 2013/14 and was funded by J.J. Strossmayer University of Osijek. Project leader: K. Burazin

Participation (as researcher) in work of following projects founded by Croatian Ministry of Science:

• Titrajuća rješenja parcijalnih diferencijalnih jednadžbi (MZT 037 101)

Project run from July 2002 until June 2006 under leadership of prof. Nenad Antonić from University of Zagreb.

• Titrajuća rješenja parcijalnih diferencijalnih jednadžbi (MZOS 037-0372787-2795)

Project run from 2007 until June 2013 under leadership of prof. Nenad Antonić from University of Zagreb.

Professional Activities

Professional Society Membership

  HMD – Croatian Mathematical Society, Department Osijek

  GAMM – Gesellschaft für Angewandte Mathematik und Mechanik

Refereeing/Reviewing

Glasnik matematički

Journal of Functional Analysis

Mathematical Communications

Osječki matematički list

Editorial Boards

Mathematical Communications

Invited Lectures

Primjena kompaktnosti kompenzacijom na zakone sačuvanja, Matematički kolokvij, Osijek, March 2003

On equivalence of two descriptions of boundary conditions for Friedrichs systems, The fifth international conference of applied mathematics and computing, Plovdiv, August 2008

Friedrichsovi sustavi, Matematički kolokvij, Osijek, April 2009

Friedrichsovi sustavi, Znanstveni kolokvij HMDa, Zagreb, February 2012

Non-stationary Friedrichs systems, ERC-NUMERIWAVES Seminar, Basque Center for Applied Mathematics, Bilbao, November 2013

Non-stationary Friedrichs systems, Karlsruhe Institute of Technology, Karlsruhe, July, 2014

Explicit solutions of multiple state optimal design problems, Departman za matematiku i informatiku, Novi Sad, May 2016

Homogenization of Kirchhoff-Love plate equation and composite plates, International Workshop on PDEs: analysis and modelling, Zagreb, Croatia, June 17-20, 2018

Workshops and Conferences

with talk:

Nonlocal effects in homogenisation: nonlinear first-order equation, Bosnian-Croatian Analysis Meeting, Bihać, May 2001

Graphs and asymptotic expansion in homogenisation of nonlinear equations, Applied Mathematics in Changing World, Berlin, September 2001

On certain properties of spaces of locally Sobolev functions, Applied Mathematics and Scientific computing, Brijuni, June 2003

Graph representation for asymptotic expansion in homogenisation of nonlinear first-order equations, Applied Mathematics and Scientific Computing, Brijuni, June 2005

Estimates on weak solutions of semilinear hyperbolic systems, Applied Mathematics and Scientific Computing, Brijuni, July 2007

On equivalence of two descriptions of boundary conditions for Friedrichs systems, The fifth international conference of applied mathematics and computing, Plovdiv, August 2008

Krein spaces applied to Friedrichs’ systems, The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, May 2009

How to pose boundary conditions for Friedrichs systems?, Applied Mathematics and Scientific Computing, Zadar, September 2009

Boundary conditions for Friedrichs’ systems,Multiscale Problems in Science and Technology. Challenges to Mathematical Analysis and Perspectives 3, Dubrovnik, June 2010

Heat equation as a Friedrichs system,Topics in PDE, Microlocal and Time-frequency Analysis, Novi Sad,  September 2012

Homogenisation theory for Friedrichs systems, 8th Conference on Applied Mathematics and Scientific Computing (Dedicated to the memory of Professor Mladen Rogina), Šibenik, June 2013

Non-stationary Friedrichs systems,  PDEs, Continuum Mechanics and Numerical Analysis – A Conference in Honor of the 80th Anniversary of professor Ibrahim Aganovic, Dubrovnik, May 2014

Non-stationary Friedrichs systems, Chemnitz-Zagreb Workshop on Harmonic Analysis for PDE, Applications, and related topics, Chemnitz, July 2014

Explicit solutions in optimal design problems for stationary diffusion equation, Partial differential equations, optimal design and numerics, Benasque, August-September 2015

Exact solutions in optimal design problems for stationary diffusion equation, The Fourth Najman Conference on Spectral Problems for Operators and Matrices, Opatija, September 2015

Exact solutions in multiple state optimal design problems, Calculus of variations and its applications (on the occasion of Luisa Mascarenhas’ 65th birthday), Lisbon, December 2015

Exact solutions of multiple state optimal design problems, 6th  Croatian Mathematical Congress, Zagreb, June 2016

Exact solutions of multiple state optimal design problems,  International Workshop on PDEs: analysis and  modelling (Celebrating 80th anniversary of professor Nedžad Limić), Zagreb, June 2016

Exact solutions of multiple state optimal design problems,  Mini-workshop on PDEs: analysis and modelling, Zagreb, January 2017

Homogenization of Kirchhoff-Love plate equation and composite plates, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Munich, March 19 – 23, 2018

On composite elastic plate and Hashin-Shtrikman bounds, VIII Partial differential equations, optimal design and numerics, Benasque, August 18 – 30, 2019

Explicit Hashin-Shtrikman bounds in 3D linearized elasticity and applications, Workshop on Control of Dynamical Systems, Dubrovnik, June 14 – 16, 2021

Explicit Hashin-Shtrikman energy bounds in 3D linearized elasticity and applications, Analysis, PDEs and Applications, Celebrating 60th anniversary of professor Nenad Antonić, Dubrovnik, June 19 – 25, 2022

Numerical method for compliance minimization in 3D linearized elasticity, 11th Conference on Applied Mathematics and Scientific Computing, Brijuni, September 5 – 9, 2022

without talk

Phase transitions in crystalline solids, MPI–MIS Leipzig, April 2000

Second Croatian Mathematical Congress, Zagreb, June 2000

Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives, Dubrovnik, September 2000

Selected issues in the mechanics of crystalline solids, Padova, October 2000

Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives 2, Dubrovnik, October 2007

Forth Croatian Mathematical Congress, Osijek, June 2008

International Conference on Generalised  Functions – GF2016, Dubrovnik, September 2016

Applications of Generalized Functions in Harmonic Analysis, Mechanics, Stochastics and PDE, Novi Sad, Serbia, October 25 – 27, 2017

International Workshop on Optimal Control of Dynamical Systems and Applications, Osijek, Croatia, June 20-22, 2018

Ninth Conference on Applied Mathematics and Scientific Computing, Šibenik, Croatia, September 17-20, 2018

Study Visits Abroad and Professional Improvements

Scuola Matematica Interuniversitaria, Perugia, Italy, July-August 1999

Max Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany, April 2000

Carnegie Mellon University, Pittsburgh, USA, June 2001

University of Novi Sad, Novi Sad, Serbia, September 2010, August-September 2011

Bulgarian Academy of Science , Sofia, Bulgaria, October 2011, June 2012

Basque Center for Applied Mathematics, Bilbao, November 2013

Babes-Bolyai University, Cluj-Napoca, September 2014

Johann Radon Institute for Computational and Applied Mathematics, Linz, October 2014

Service Activities

Member of the Scientific Committee for the Field of Mathematics,  2021 –

Chairman of the Working seminar 2014 –

Chairman of Seminar for Optimization and Applications 2017 –

Vice chair for education and students at Department of Mathematics, J. J. Strossmayer University of Osijek, October 2013 – September 2017

Secretary of the Mathematical Colloquium,  2003 – 2013

Member of the organizing  committee  of:

• International Workshop on PDEs: analysis and  modelling (Celebrating 80th anniversary of professor Nedžad Limić), Zagreb, June 19 – 22 2016
• International Conference on Generalised  Functions – GF2016, Dubrovnik, September 4 – 9 2016
• Mini-workshop on PDEs: analysis and modelling, Zagreb, January 4 – 6 2017
• Analysis, PDEs and Applications, Celebrating 60th anniversary of professor Nenad Antonić, Dubrovnik, June 19 – 25, 2022
• 8th Croatian Mathematical Congress, Osijek, July 2 – 5, 2024

Teaching

Nastavne aktivnosti u zimskom semestru Akademske 2023/2024

Funkcije više varijabli (Multidimensional Calculus)

Obične diferencijalne jednadžbe (Ordinary differential equations)

Nastavne aktivnosti u ljetnom semestru Akademske 2023/2024

Kompleksna analiza (Complex analysis)

Primjene diferencijalnog i integralnog računa I (Applications of Calculus I)

Normirani prostori (Normed spaces)

Metode matematičke fizike (Methods of Mathematical Physics)

Konzultacije (Office Hours): Srijeda (Wed) 11:00. Konzultacije su moguće i po dogovoru.

Teme diplomskih i završnih radova

Sljedeće teme dostupne su za diplomske i završne radove. Moguće su i druge teme prema dogovoru. Zainteresirani studenti mogu se javiti radi detaljnijih informacija.

Sustavi običnih diferencijalnih jednadžbi

Fourierovi redovi

Varijacijski račun

Banachovi i Hilbertovi prostori

L^p prostori (samo diplomski)

Soboljevljevi prostori (samo diplomski)

Jednadžba provođenja topline

Jednadžba stacionarnog provođenja topline

Valna jednadžba

Linearizirana elastičnost (samo diplomski)

Distribucije (samo diplomski)

Teaching experience

List of courses taught: Metric Spaces, Ordinary Differential Equations, Methods of Mathematical Physics, Multivariable Calculus, Complex Analysis, Calculus 3, Calculus 4, Vector Spaces, Algebra, Elementary Mathematics 2,  Differential Equations (Department of Physics), Mathematics 3, Mathematics 4 (Faculty of Civil Engineering)

Courses taught at other universities:

University of Zagreb, Faculty of Electrical Engineering and Computing: Calculus 1, Calculus 2, Calculus 3, Calculus 4, Linear Algebra

University of Bihać: Differential Geometry

College Lavoslav Ružička, Vukovar: Statistics

Postgraduate programmes: Hyperbolic and Friedrichs systems, Real and Functional Analysis, Optimal Control of Linear Dynamical Systems, Homogenization and Applications in Optimal Design

Personal

Married, two children

Member od Croatian Mountain Rescue Service

Sport interests: climbing, mountaineering, skiing, running, cycling; past: martial arts (Karate, Jiu-jitsu, Brasilian Jiu-Jitsu).