Serge Vlăduţ

Nonlinear Elliptic Equations and Nonassociative Algebras

This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equation ...

Singular solution to Special Lagrangian Equations

Abstract We prove the existence of non-smooth solutions to three-dimensional Special Lagrangian Equations in the non-convex case.

The John equation for tensor tomography in three-dimensions*

John proved that a function on the manifold of lines in belongs to the range of the x-ray transform if and only if satisfies some second order differential equation and obeys some smoothness and decay conditions. We generalize the John equation to the case of the x-ray transform on arbitrary rank symmetric tensor fields: a function on the manifold of lines in belongs to the range of the x-ray transform on rank m symmetric tensor fields if and only if satisfies some differential equation of order and obeys some smoothness and decay conditions.

Weak Solutions of Nonvariational Elliptic Equations

We discuss basic properties (uniqueness and regularity) of viscosity solutions to fully nonlinear elliptic equations of the form F (x,Du) = 0, which includes also linear elliptic equations of nondivergent form. In the linear case we consider equations with discontinuous coefficients. Mathematics Subject Classification (2010). Primary 35J15, 35D30, 35D40, 35J60 Secondary 17A35, 20G41, 53C38, 60G46 .