Prof. Lavi Karp

A. Articles in Refereed Journals

1.L. Karp, Construction of quadrature domains R^⁴in from quadrature domains in R^² , Complex Variables and Elliptic Equations 17 (1992), 179-189.

2. L. Karp, Generalized Newton potential and its applications, Journal of Mathematical Analysis and Applications 174 (1993), 480-497.

3. L. Karp, On the Newtonian potential for ellipsoids, Complex Variables and Elliptic Equations 25 (1995), 367–372.

4. L. Karp, Liouville-type theorems for second order elliptic differential equations, Annali Della Scuola Normale Supriore Di Pisa 22 (1995), 275-298.

5. L. Karp and A. Margulis, Newtonian potential theory for unbounded sources and applications to free boundary problems, Journal d'Analyse Mathématique 70 (1996), 1-63.

6. L. Karp and H. Shahgholian, Regularity of a free boundary problem, Journal of Geometrical Analysis 9, No. 4 (1999), 653-669.

7. L. Karp and H. Shahgholian, On the optimal growth of functions with bounded Laplacian, Electronic Journal of Differential Equations 2000 No. 03(2000), 1-9.

8. L. A. Caffarelli, L. Karp and H. Shahgholian, Regularity of a free boundary problem with application to the Pompeiu problem, Annals of Mathematics 151, No.1 (2000), 269-292.

Annals of Mathematics was ranked first among the mathematical journals when the paper was published.

9. L. Karp, T. Kilpeläinen, A. Petrsyan and H. Shahgholian, On the porosity of Free boundaries in degenerate variational inequalities, Journal of Differential Equations 164, No. 1(2000), 110-117.

10. L. Karp and H. Shahgholian, Regularity of a free boundary problem near the infinity point, Communications in Partial Differential Equations 25, No.11–12 (2000), 2055-2086.

11. W. K. Hayman, L. Karp and H.S. Shapiro, Newtonian capacity and quasi balayage, Rendiconti di Matematica e delle sue Applicazioni. 20, No. 7(2000), 93-129.

12. L. Karp and H.S. Shapiro, Isolated singularities of harmonic functions, in Modern Developments in Multivariate Approximation, edit by W. Hausmann, K. Jetter, M. Reimer and J. Stöcker editors, Birkhäuser (2003), 165-174.

13. L. Karp, Multivalid analytic continuation of the Cauchy transform, Potential Analysis 24 (2006), 1-13.

14. U. Brauer and L. Karp, Local existence of classical solutions for the Einstein-Euler system using weighted Sobolev spaces of fractional order, Comptes Rendus Mathématique 345, Issue 1 (2007), 49-54.

15. L. Karp, On null quadrature domains, Computational Methods and Function Theory, (2008) No. 1, 57–72.

16. L. Karp, On the well-posedness of the vacuum Einstein's equations, Journal of Evolution equations 11 (2011), 641-673.

17. U. Brauer and L. Karp, Well-posedness of the Einstein-Euler system in asymptotically flat spacetimes: The constraint equations, Journal of Differential Equations 251 (2011), 1428-1446.

18. L. Karp, Global solutions to bubble growth in porous media, Journal of Mathematical Analysis and Applications 382 (2011), 132-139.

19. L. Karp and A. Margulis, Null quadrature domains and a free boundary problem for the Laplacian, Indiana University Mathematics Journal 61, No. 2 (2012), 859-882.

20. U. Brauer and L. Karp, Local existence of solutions of self gravitating relativistic perfect fluids, Communications in Mathematical Physics 325 (2014), 105-141.

21. L. Karp, Asymptotic properties of unbounded quadrature domains the plane, European Journal of Applied Mathematics 26 (2015), 175-191.

22. U. Brauer and L. Karp, Elliptic equations in weighted Besov spaces on asymptotically flat Riemannian manifolds, manuscripta mathematica, 148 (2015), 59-97.

23. A. Goldvard and L. Karp, On the composition of finite rotations in R^⁴, Journal Geometry and Symmetry in Physics, 39 (2015), 33-43.

24. L. Karp and E. Lundberg, A four-dimensional Neumann ovaloid, Arkiv för Matematik 55 (2017), 185-198.

25. U. Brauer and L. Karp, Local existence of solutions to the Euler-Poisson system, including densities without compact support, Journal of Differential Equations, 264 (2018), 755-785.

26. U. Brauer and L. Karp, Continuity of the flow map for symmetric hyperbolic systems and its application to the Euler-Poisson system, Accepted to Journal d'Analyse Mathématique, 42 pages.

27. U. Brauer and L. Karp, The non-isentropic Euler-Einstein system written in a symmetric hyperbolic form, Accepted to Springer INdAM Series.

28. U. Brauer and L. Karp, Local existence of solutions to the Euler-Poisson system, including densities without compact support}, Proceedings of the 12th ISAAC Congress in Aveiro, Portugal, Birkhäuser "Research Perspectives" series (accepted).

B. Edit Books

1. M. Agranovsky, L. Karp, D. Shoikhet and and L. Zalcman, Proceedings of an international conference, Complex Analysis and Dynamical Systems, Contemporary Mathematics Vol. 364, American Mathematical Society, 2004.

2. M. Agranovsky, L. Karp and D. Shoikhet, Proceedings of a international conference Complex Analysis and Dynamical System II, conference in honor of Professor Lawrence Zaclman 60th birthday, Contemporary Mathematics, Vol. 382, American Mathematical Society, 2005.

3. M. Agranovsky, D. Bshouty, L. Karp, S. Reich, D. Shoikhet and L. Zalcman, Proceedings of an international conference Complex Analysis and Dynamical System III, a conference in honor of the Retirement of Dov Aharonov, Lev Aizenberg, Samuel Krushkal, and Uri Srebro, Contemporary Mathematics, vol. 455, American Mathematical Society, 2008.

4. M. Agranovsky, M. Ben–Artzi, G. Galloway, L. Karp, S. Reich, D. Shoikhet, G. Weinstein and L. Zalcman, Complex Analysis and Dynamical Systems IV: Part 1. Function Theory and Optimization, Contemporary Mathematics vol. 553, American Mathematical Society, 2011.

5. M. Agranovsky, M. Ben-Artzi, G. Galloway, L. Karp, S. Reich, D. Shoikhet, G. Weinstein and L. Zalcman, Complex Analysis and Dynamical Systems IV: Part 2. General Relativity, Geometry, and PDE, Contemporary Mathematics vol. 554, American Mathematical Society, 2011.

6. M. Agranovsky, M. Ben-Artzi, G. Galloway, L. Karp, V. Maz'ya, S. Reich, D. Shoikhet, G. Weinstein and L. Zalcman, Complex Analysis and Dynamical Systems V, Contemporary Mathematics vol. 591, American Mathematical Society, 2013.

7.  M. Agranovsky, M. Ben-Artzi, G. Galloway, L. Karp, D. Khavinson, S. Reich, G. Weinstein and L. Zalcman, Complex Analysis and Dynamical Systems VI Part 1: PDE, Differential Geometry, Radon Transform, Contemporary Mathematics vol. 653, American Mathematical Society, 2015.

8. M. Agranovsky, M. Ben-Artzi, G. Galloway, L. Karp, D. Khavinson, S. Reich, G. Weinstein and L. Zalcman, Complex Analysis and Dynamical Systems VI Part 2: Complex analysis, quasiconformal mapping, complex dynamic, Contemporary Mathematics vol. 667, American Mathematical Society, 2016.

9. M. Agranovsky, M. Ben-Artzi, C. Bénéteau, L. Karp, D. Khavinson, S. Reich, D. Shoikhet, G. Weinstein,L. Zalcman, Complex Analysis and Dynamical Systems VII, Contemporary Mathematics vol. 699, American Mathematical Society, 2017.

C. Submitted

U. Brauer and L. Karp, Global existence of a nonlinear wave equation arising from Nordström's theory of gravitation, 26 pages.

D. In Preparation

1. L. Karp and M. Reissig, The Euler-Poisson equations in physical vacuum and with an adiabatic constant equals to 2
2. U. Brauer and L. Karp, Continuous dependence on the initial data for the Cauchy problem for Einstein equations.

3. U. Brauer and L. Karp, Global existence of the Nordström-Euler system.

E. In Arxiv and ResearchGate

1. U. Brauer and L. Karp, Well-posedness of the Einstein-Euler system in asymptotically flat spacetimes, https://arxiv.org/abs/0810.5045, 76 pages.

2. A. Goldvard and L. Karp, Applications of matrices multiplication to determinant rotations formulas in R^ⁿ, http://arxiv.org/abs/1010.3729, 10 pages.

3. U. Brauer and L. Karp, Interpolation of nonlinear contractive operators, Preprint, 4 pages.

Einstein equations are the cornerstone of General Relativity. They are non-linear partial differential
equations (PDE). The research in this field has a long history, and a longstanding interest in both
mathematics and physics. My own interest lies within the mathematical aspects, and in particular
with Einstein Equations that are coupled with a perfect fluid (relativistic flows). The main means
to analyze this type of flows is to transfer them into a system of PDE, which results in a convoluted
system consisting of both elliptic and hyperbolic types of equations. A major difficulty is that the
density is not strictly positive in many situations. This phenomenon is called a physical vacuum and
it causes substantial mathematical challenges that require novel approaches to investigate them.
I am also studying the topic of free boundary problems. The free refers here to an overdetermined
system of equations and therefore the boundary of a domain, or the shape of a body, cannot be
arbitrary. Such type problems arising from example in ice-melting, the evolution of oil spot surrounding by water and Hele-Shaw flows. These types of problems are related to potential theory and
Quadrature Domains. A challenging open problem is the classification of Null quadrature domains,
which means the classification of all domains such that the integral of all harmonic and integrable
functions over them is zero. This topic actually goes back to Newton era and it is related to a
classical theorem of Newton that an elliptical shell induces no gravitational force in the cavity. The
classification in the plane has been known since the early eighteens of the last century, but in higher
space dimensions it remains an open problem.

Einstein equations are the cornerstone of General Relativity. They are non-linear partial differential equations (PDE). The research in this field has a long history, and a longstanding interest in both mathematics and physics. My own interest lies within the mathematical aspects, and in particular with Einstein Equations that are coupled with a perfect fluid (relativistic flows). Tn means to analyze this type of flows is to transfer them into a system of PDE, which results in a convoluted system consisting of both elliptic and hyperbolic types of equations. A major difficulty is that the density is not strictly positive in many situations. This phenomenon is called a physical vacuum and it causes substantial mathematical challenges that require novel approaches to investigate them.

I am also studying the topic of free boundary problems. The free refers here to an overdetermined system of equations and therefore the boundary of a domain, or the shape of a body, cannot be arbitrary. Such type problems arising from example in ice-melting, the evolution of oil spot surrounding by water and Hele-Shaw flows. These types of problems are related to potential theory and Quadrature Domains. A challenging open problem is the classification of Null quadrature domains, which means the classification of all domains such that the integral of all harmonic and integrable functions over them is zero. This topic actually goes back to Newton era and it is related to a classical theorem of Newton that an elliptical shell induces no gravitational force in the cavity. The classification in the plane has been known since the early eighteens of the last century, but in higher space dimensions it remains an open problem.