Related papers: Semismoothness for Solution Operators of Obstacle-…
In this paper we give a comprehensive treatment of a two-penalty boundary obstacle problem for a divergence form elliptic operator, motivated by applications to fluid dynamics and thermics. Specifically, we prove existence, uniqueness and…
We consider an optimal control problem for the steady-state Kirchhoff equation, a prototype for nonlocal partial differential equations, different from fractional powers of closed operators. Existence and uniqueness of solutions of the…
The paper starts with a description of the SCD (subspace containing derivative) mappings and the SCD semismooth* Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the…
We investigate the monotonicity method for fractional semilinear elliptic equations with power type nonlinearities. We prove that if-and-only-if monotonicity relations between coefficients and the derivative of the Dirichlet-to-Neumann map…
We consider the obstacle problem with irregular barriers for semilinear elliptic equation involving measure data and operator corresponding to a general quasi-regular Dirichlet form. We prove existence and uniqueness of a solution as well…
We consider optimal control problems of elliptic PDEs on hypersurfaces in 2- or 3-dimensional Euclidean space. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral…
We investigate full Lipschitzian and full H\"olderian stability for a class of control problems governed by semilinear elliptic partial differential equations, where all the cost functional, the state equation, and the admissible control…
Proximal methods are known to identify the underlying substructure of nonsmooth optimization problems. Even more, in many interesting situations, the output of a proximity operator comes with its structure at no additional cost, and…
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition…
In this paper, we consider a class of systems of nonlinear equations, which arise in discretized mixed formulations of problems in solid mechanics by $hp$-finite elements. We introduce a semismooth Newton solver for this specific class and…
For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary…
In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on…
We propose a novel linesearch variant of the trust region normal map-based semismooth Newton method developed in [Ouyang and Milzarek, Math. Program. 212(1-2), 389--435 (2025)] for solving a class of nonsmooth, nonconvex composite-type…
We develop an implementable stochastic proximal point (SPP) method for a class of weakly convex, composite optimization problems. The proposed stochastic proximal point algorithm incorporates a variance reduction mechanism and the resulting…
We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with…
Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea…
Fixed-point equations with Lipschitz operators have been studied for more than a century, and are central to problems in mathematical optimization, game theory, economics, and dynamical systems, among others. When the Lipschitz constant of…
This work investigates a dynamical system functioning as a nonsmooth adaptation of the continuous Newton method, aimed at minimizing the sum of a primal lower-regular and a locally Lipschitz function, both potentially nonsmooth. The…
Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the…
The paper investigates stability properties of solutions of optimal control problems for semilinear parabolic partial differential equations. H\"older or Lipschitz dependence of the optimal solution on perturbations are obtained for…