RESEARCH

Research areas that I am interested in:

  • Geometric Function Theory
  • Real and Harmonic Analysis
  • Optimal Mass Transportation
  • Nonlinear elliptic PDE
  • Geometric Measure Theory
  • Complex Analysis
  • Transport equation
  • Non-smooth vector fields
  • Euler equation
  • Inverse Boundary Value Problems
  • Geometric Analysis
  • Gradient flows in Banach spaces

Some keywords:

  • quasiconformal mappings and quasiregular mappings;
  • Calderón-Zygmund theory;
  • apriori estimates for elliptic equations; existence and uniqueness;
  • degenerate elliptic operators; commutators;
  • function spaces; interpolation; characterizations;
  • optimal mass transportation; continuity equation;
  • Hamilton-Jacobi equations; viscosity solutions;
  • Wardrop equilibrium; transport density; Monge problem;
  • Muckenhoupt weights and reverse Hölder inequalities;
  • nonlinear elliptic and parabolic PDEs; flows of solutions;
  • Fréchet manifolds; curves in the Wasserstein space; surfaces in the Sobolev space;
  • calculus of variations; smoothness of minimizers;
  • linear transport equation; well posedness; admissible vector fields
  • extremal transport problems and nonlinear transport equations
  • non-smooth flows; DiPerna-Lions flows
  • Bergman projections; truncated spaces; stability;

Some research projects:

  • 2 – dimensional manifolds of quasiconformal mappings
  • Muckenhoupt weights, quasiconformal mappings, and elliptic operators
  • Besov-Lipschitz and Triebel-Lizorkin estimates for nonlinear elliptic equations
  • traffic problems in optimal mass transportation; Wardrop equilibrium
  • Bergman kernels, Beltrami equations and Schwarz domains
  • regularity of  viscosity solutions of Hamilton-Jacobi equations
  • apriori L^2 bounds for degenerate elliptic equations in divergence form
  • well-posedness for the transport equation in non-reflexive Banach spaces
  • flows of non-smooth vector fields with exponentially integrable divergence
  • vorticity estimates for the Euler equation