# Aram KarakhanyanLecturer in Mathematics The University of Edinburgh

1. Lecture notes on elliptic equations, pdf
2. New trends in free boundary problems, Joint with S. Dipierro and E. Valdinoci, submitted, pdf
3. Remarks on the thin obstacle problem and constrained Ginibre ensembles, submitted, pdf
4. Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients, Joint with S. Dipierro and E. Valdinoci, submitted, pdf
5. A geometric approach to regularity for nonlinear free boundary problems, submitted, pdf
6. A nonlinear free boundary problem with a self-driven Bernoulli condition, Joint with S. Dipierro and E. Valdinoci, submitted, pdf
7. Embedded $K$-surfaces with free boundaries, Joint with H. Aleksanyan
8. Capillary surfaces arising in singular perturbation problems, pdf
9. A class of unstable free boundary problems, Joint with S. Dipierro and E. Valdinoci, submitted, pdf
10. Blaschke's rolling ball theorem and the Trudinger-Wang monotone bending, Journal of Differential Equations, in press, pdf
11. Boundary behavior for a singular perturbation problem, Joint with H.Shahgholian, Nonlinear Analysis Series A, TMA, in press, pdf
12. A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two, Joint with S. Dipierro, pdf
13. Stratification of free boundary points for a two-phase variational problem, Joint with S. Dipierro, pdf
14. Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization, Joint with M.Stromqvist, Cal. of Variations and PDE, in press, pdf
15. Regularity for a quasilinear continuous casting problem, submitted, pdf
16. Potential theoretic approach to Schauder estimates for the fractional Laplacian, Joint with C. Bucur, Proceedings of the AMS, in press, pdf
17. On a conjecture of De Giorgi related to homogenization, Joint with H.Shahgholian, submitted, pdf
18. Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation, Discrete and Cont. Dynamical Systems, 36 (2016), no. 1, 261--277 pdf
19. Regularity for energy-minimizing area-preserving deformations, Journal of Elasticity, Volume 114, Issue 2 (2014), pp 213−223, pdf
20. Application of uniform distribution to homogenization of a thin obstacle problem with $p$-Laplacian, Joint with M.Stromqvist, Communications in PDEs, 39 (2014), no. 10 1870--1897, pdf
21. An inverse problem for the refractive surfaces with parallel lighting, SIAM Journal of Mathematical Analysis, in press, pdf
22. Existence and regularity of the reflector surfaces in $\mathbb R^{n+1}$, Archive for Rational Mechanics and Analysis, 213 (2014), no. 3, 833--885, pdf
23. Optimal Regularity for phase transition problems with convection, Annales de l'Institut Henri Poincare, Analyse non lineaire, 32 (2015), no. 4, 715--740, pdf
24. The Stefan problem with constant convection, Joint with J.Rodrigues, preprint, pdf
25. Analysis of a free boundary at contact point with Lipschitz data, Joint with H.Shahgholian, Transactions of the AMS,  367 (2015), no. 7, 5141--5175, pdf
26. Sufficient conditions for regularity of area-preserving deformations, Manuscripta Math, 138, 463--476, 2012, pdf
27. The geometry of solutions to a segregation problem for non-divergence systems, Joint with L.A. Caffarelli and Fang-Hua Lin, J. of Fixed Point Theory and Appl.,5 (2009), no. 2, 319--351, pdf
28. On the reflector shape design, Joint with Xu-Jia Wang, Journal of Differential Geometry, 84 (2010), no. 3, 561--610, pdf
29. On the regularity of weak solutions to refractor problem, Arm. J. Math. 2, Issue 1, pp. 28-37, 2009, pdf
30. On Derivation of Euler-Lagrange Equations for incompressible energy-minimizers, Joint with N.Chaudhuri, Cal. of Variations and PDE, 36 (2009), no. 4, 627--645, pdf
31. Lectures on gas flow in porous media, Joint with L.A. Caffarelli, Appl. Numer. Harmon. Anal., 133--157, Birkhauser Boston, Inc., Boston, MA, 2010, pdf
32. The reflector design problem, Joint with Xu-Jia Wang, Proceedings of Inter. Congress of Chinese Mathematicians, vol 2, 1-4, 1--24, 2007, pdf
33. On the Lipschitz regularity of solutions of minimum problem with free boundary, Interfaces and Free Boundaries, 10 (2008), no. 1, 79--86, pdf
34. The behavior of the free boundary near the fixed boundary for a minimization problem. Joint with C. Kenig and H. Shahgholian, Calc. Var. Partial Differential Equations 28 (2007), no. 1, 15--31, pdf
35. Up-to boundary regularity for a singular perturbation problem of $p$-Laplacian type. J. Differential Equations 226 (2006), no. 2, 558--571, pdf
36. Nonlinear free boundary problems with singular source terms. Joint with A. Hakobyan, Monatsh. Math. 142 (2004), no. 1-2, 7-16, pdf