Solvability of uniformly elliptic fully nonlinear PDE

Fourth-order numerical solution of a fractional PDE with the nonlinear source term in the electroanalytical chemistry

The aim of this paper is to study the high order difference scheme for the solution of a fractional partial differential equation (PDE) in the electroanalytical chemistry. The space fractional derivative is described in the Riemann-Liouville sense. In the proposed scheme we discretize the space derivative with a fourth-order compact scheme and use the Grunwald- Letnikov discretization of the Ri...

Partial Differential Equations in the 20th Century

Contents 1. Introduction. 2. Models of PDE 's in the 18th and 19th century. 3. Methods of calculating solutions in the 19th century. 4. Developments of rigorous theories of solvability in the last decades of the 19th century. 5. The period 18901900: the beginning of modern PDE and the work of Poincare. 6. The Hilbert programs. 7. S. Bernstein and the beginning of a priori estimates. 8. Solvabil...

Perturbing Fully Nonlinear Second Order Elliptic Equations

We present two types of perturbations with reverse effects on some scalar fully nonlinear second order elliptic differential operators: on the other hand, first order perturbations which destroy the global solvability of the Dirichlet problem, in smooth bounded domains of Rn; on the other hand, an integral perturbation which restore the local solvability, on compact connected manifolds without ...

An eigenvalue problem for a fully nonlinear elliptic equation with gradient constraint

We consider the problem of finding λ ∈ R and a function u : Rn → R that satisfy the PDE max { λ+ F(D2u)− f (x), H(Du) = 0, x ∈ Rn . Here F is elliptic, positively homogeneous and superadditive, f is convex and superlinear, and H is typically assumed to be convex. Examples of this type of PDE arise in the theory of singular ergodic control. We show that there is a unique λ∗ for which the above e...

Solvability of monotone systems of fully nonlinear elliptic PDE's