Contributions in nonlocal PDEs, optimal Lipschitz extensions, metric slopes and nonlinear optimization

Abstract

This thesis addresses questions at the intersection of: (I) partial integro–differential equations, (II) analysis inmetric spaces and (III) nonconvex optimization.In Part (I), we study partial integro–differential equations including degenerate nonlocal operators associated with state–dependent Lévy measures coupled with superlinear Hamiltonians. Chapter 2 establishes the comparison principle for such equations with a combined Wasserstein/Total Variation-continuity assumption, which is one of the weakest conditions employed in the context of viscosity approach. Chapter 3 deals with the regularity of viscosity solutions. We improve the Hölder exponent in the general case and we show that if the Lévy measures are weakly elliptic, then the solutions are Lipschitz continuous.Part (II) concerns metric slopes and absolutely minimal Lipschitz extensions. Chapter 4 provides a proof of absolutely minimal Lipschitz extensions in compact metric space and introduces a notion of such extensions in quasi–metric spaces. Chapter 5 provides a determination result for continuous bounded from below functions related to metric slopes and a condition that controls asymptotic behavior. The result can be extended to a large class of so–called descent moduli in complete metric spaces. Chapter 6 studies the well–posedness of Eikonal equations in metric spaces associated with the global slope operator. Consequently, we establish a new integration formula for lower semicontinuous functions via global slopes.Part (III) includes results related to growth condition of nonconvex functionals and the solvability of absolute value equations. Chapter 7 provides the equivalence between the growth condition at strict local minimizer in reflexive spaces with a variant of the so-called tilt stability property and an analog of the classical Polyak–Łojasiewicz condition, where the gradient is replaced by linear perturbations. Chapter 8 establishes the equivalence between nonlinear absolute value equations and an implicit nonlinear complementarity problem. This enables smoothing techniques to approximate the solutions of such equations.