Nikos Katzourakis

An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L

1 History, Examples, Motivation and First Definitions.- 2 Second Definitions and Basic Analytic Properties of the Notions.- 3 Stability Properties of the Notions and Existence via Approximation.- 4 Mollification of Viscosity Solutions and Semi convexity.- 5 Existence of Solution to the Dirichlet Problem via Perrons Method.- 6 Comparison results and Uniqueness of Solution to the Dirichlet Problem.- 7 Minimisers of Convex Functionals and Viscosity Solutions of the Euler-Lagrange PDE.- 8 Existence of Viscosity Solutions to the Dirichlet Problem for the Laplacian.- 9 Miscellaneous topics and some extensions of the theory.

Absolutely minimising generalised solutions to the equations of vectorial calculus of variations in \(L^\infty \)

Consider the supremal functional 1

Existence and uniqueness of global strong solutions to fully nonlinear second order elliptic systems

\begin{aligned} E_\infty (u,A) := \Vert \mathscr {L}(\cdot ,u,\mathrm {D}u)\Vert _{L^\infty (A)},\quad A\subseteq \Omega , \end{aligned}

On the Dirichlet problem for fully nonlinear elliptic Hessian systems

E∞(u,A):=‖L(·,u,Du)‖L∞(A),A⊆Ω,applied to

On the well-posedness of global fully nonlinear first order elliptic systems

W^{1,\infty }

Remarks on the validity on the maximum principle for the ∞-Laplacian

W1,∞ maps