This book contains papers that engage a wide set of classical and modern topics in partial differential equations, including linear and nonlinear equations, variational problems, the Navier-Stokes system, and the Boltzmann equation. The results include existence and uniqueness theorems, qualitative properties of solutions, a priori estimates, and nonexistence theorems. Table of Contents: J. Andersson, H. Shahgholian, and G. S. Weiss -- Regularity below the $C^2$ threshold for a torsion problem, based on regularity for Hamilton-Jacobi equations; A. Arkhipova -- Signorini-type problem in $\mathbb{R}^N$ for a class of quadratic functional; M. Bildhauer and M. Fuchs -- A 2D-invariant of a theorem of Uraltseva and Urdaletova for higher order variational problems; M. Bostan, I. M. Gamba, and T. Goudon -- The linear Boltzmann equation with space periodic electric field; L. Caffarelli and L. Silvestre -- Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations; P. Constantin and G. Seregin -- Holder continuity of solutions of 2D Navier-Stokes equations with singular forcing; M. Giaquinta, P. M. Mariano, G. Modica, and D. Mucci -- Currents and curvature varifolds in continuum mechanics; N. M. Ivochkina -- On classic solvability of the $m$-Hessian evolution equation; N. V. Krylov -- About an example of N. N. Ural'tseva and weak uniqueness for elliptic operators; V. Maz'ya and R. McOwen -- On the fundamental solution of an elliptic equation in nondivergence form; G. Mingione -- Boundary regularity for vectorial problems; A. Nazarov and A. Reznikov -- Attainability of infima in the critical Sobolev trace embedding theorem on manifolds; M. V. Safonov -- Non-divergence elliptic equations of second order with unbounded drift; V. V. Zhikov and S. E. Pastukhova -- Global solvability of Navier-Stokes equations for a nonhomogeneous non-Newtonian fluid. (TRANS2/229)