### Krešimir Burazin

Josip Juraj Strossmayer University of Osijek

### 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

- K. Burazin, M. Erceg, M. Waurick, Evolutionary Equations are G-compact, Journal of Evolution Equations
**1**(2024) - K. Burazin, I. Crnjac, M. Vrdoljak, Explicit Hashin-Shtrikman bounds in 3D linearized elasticity, Filomat
**1**(2024)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. - K. Burazin, I. Crnjac, Application of explicit energy bounds in optimization of 3D elastic structures, Optimization and Engineering
**2023**(2023)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. - K. Burazin, S. Gutiérrez, J. Jankov Pavlović, Optimal design of Kirchhoff-Love plates under the low contrast assumption, Optimization and Engineering
**2023**(2023) - 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 - K. Burazin, J. Jankov Pavlović, On the effective properties of composite elastic plate, Journal of Mathematical Analysis and Applications
**495**/1 (2021) - K. Burazin, I. Crnjac, M. Vrdoljak, Optimality criteria method in 2D linearized elasticity problems, Applied Numerical Mathematics
**160**(2021), 192-204 - K. Burazin, J. Jankov Pavlović, Small-amplitude homogenization of elastic plate equation, Applicable Analysis
**100**/5 (2021), 1039-1050 - 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 - K. Burazin, D. Mitrović, Apriori estimates for fractional diffusion equation, Optimization Letters
**13**/8 (2019), 1793-1801 - K. Burazin, M. Vrdoljak, Exact Solutions in Optimal Design Problems for Stationary Diffusion Equation, Acta Applicandae Mathematicae
**161**/1 (2019), 71-88 - K. Burazin, J. Jankov Pavlović, M. Vrdoljak, Homogenization of elastic plate equation, Mathematical Modelling and Analysis
**23**/2 (2018), 190-204 - K. Burazin, On Unique Solutions of Multiple-State Optimal Design Problems on an Annulus, Journal of optimization theory and applications
**177**(2018), 329-344 - 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 - N. Antonić, K. Burazin, I. Crnjac, M. Erceg, Complex Friedrichs systems and applications, Journal of Mathematical Physics
**58**/10 (2017), 1-22 - K. Burazin, M. Erceg, Non-stationary abstract Friedrichs systems, Mediterranean Journal of Mathematics
**13**/6 (2016), 3777-3796 - K. Burazin, M. Erceg, Estimates for mild solutions to semilinear Cauchy problems, Electronic Journal of Differential Equations
**2014**/194 (2014), 1-10 - K. Burazin, M. Vrdoljak, Homogenisation theory for Friedrichs systems, Communications on Pure and Applied Analysis
**13**/3 (2014), 1017-1044 - N. Antonić, K. Burazin, M. Vrdoljak, Second-order equations as Friedrichs systems, Nonlinear analysis: real world applications
**14**/1 (2014), 290-305 - N. Antonić, K. Burazin, M. Vrdoljak, Heat equation as a Friedrichs system , Journal of Mathematical Analysis and Applications
**404**(2013), 537-553 - N. Antonić, K. Burazin, Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems, Journal of Differential Equations
**250**(2011), 3630-3651 - N. Antonić, K. Burazin, M. Vrdoljak, Connecting classical and abstract theory of Friedrichs systems via trace operator , ISRN Mathematical Analysis
**2011**(2011), 1-14 - N. Antonić, K. Burazin, Intrinsic boundary conditions for Friedrichs systems, Communications in Partial Differential Equations
**35**/9 (2010), 1690-1715 - N. Antonić, K. Burazin, Graph spaces of first-order linear partial differential operators, Mathematical Communications
**14**/1 (2009), 135-156 - N. Antonić, K. Burazin, On equivalent descriptions of boundary conditions for Friedrichs systems, Mathematica Montisnigri
**22**(2009), 5-13 - K. Burazin, Estimates on the weak solution of semilinear hyperbolic systems, Annali dell'Universita' di Ferrara
**54**(2008), 229-243 - 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

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

- K. Burazin, U. Radojičić, Uvod u varijacijski račun i njegova povijest, Osječki matematički list
**16**/2 (2016), 111-133U 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. - K. Burazin, J. Jankov Pavlović, Glazba titrajuće žice, Osječki matematički list
**14**/1 (2014) - K. Burazin, Z. Tomljanović, I. Vuksanović, Prigušivanje mehaničkih vibracija, Math.e : hrvatski matematički elektronski časopis
**24**(2014) - K. Burazin, Nelinearne diofantske jednadžbe, Osječki matematički list
**7**(2007)

- 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.

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

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)

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

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:

*Oscilatorna rješenja parcijalnih diferencijalnih jednadžbi (MZT 037 015)*

*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.

*Functional analysis methods in mathematical modelling*

This is bilateral project with serbia (cofinanced by Serbian Ministry of Science) that run from 2011 until 2012. Project leaders: prof. Marko Vrdoljak from University of Zagreb (croatian side) and prof. Jelena Aleksić from University of Novi Sad (serbian side).

### Professional Activities

**Professional Society Membership**

HMD – Croatian Mathematical Society, Department Osijek

GAMM – Gesellschaft für Angewandte Mathematik und Mechanik

**Refereeing/Reviewing**

Journal of Functional Analysis

**Editorial Boards**

**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).