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We extend the Gibbs conditioning principle to an abstract setting combining infinitely many linear equality constraints and non-linear inequality constraints, which need not be convex. A conditional large large deviation principle (LDP) is…
We prove existence of a value for two-player zero-sum stopper vs. singular-controller games on finite-time horizon, when the underlying dynamics is one-dimensional, diffusive and bound to evolve in $[0,\infty)$. We show that the value is…
We consider a PDE-constrained optimization problem governed by a free boundary problem. The state system is based on coupling the Laplace equation in the bulk with a Young-Laplace equation on the free boundary to account for surface…
We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical $p$-Laplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide…
In this paper, we study the following singular problem associated with mixed operators (the combination of the classical Laplace operator and the fractional Laplace operator) under mixed boundary conditions \begin{equation*} \label{1}…
We study the consequences of the equivalence between the least gradient problem and a boundary-to-boundary optimal transport problem in two dimensions. We extend the relationship between the two problems to their respective dual problems,…
We first study Clarke's tangent cones at infinity to unbounded subsets of $\mathbb{R}^n.$ We prove that these cones are closed convex and show a characterization of their interiors. We then study subgradients at infinity for extended real…
We prove local Lipschitz continuity and Harnack's inequality for value functions of the stochastic game tug-of-war with noise and running payoff. As a consequence, we obtain game-theoretic proofs for the same regularity properties for…
We consider a stationary variational inequality with gradient constraint and obstacle. We prove that this problem can be described by an equation using a Lagrange multiplier and a characteristic function. The Lagrange multiplier contains…
We prove several results for the Dirichlet, Neumann and Regularity problems for the Laplace equation in graph Lipschitz domains in the plane, considering $A_{\infty}$-measures on the boundary. More specifically, we study the…
Under non-exponential discounting, we develop a dynamic theory for stopping problems in continuous time. Our framework covers discount functions that induce decreasing impatience. Due to the inherent time inconsistency, we look for…
We study nonlinear parabolic PDEs with Orlicz-type growth conditions. The main result gives the existence of a unique solution to the obstacle problem related to these equations. To achieve this we show the boundedness of weak solutions and…
In equality-constrained optimization, a standard regularity assumption is often associated with feasible point methods, namely the gradients of constraints are linearly independent. In practice, the regularity assumption may be violated. To…
The main difficulty in solving the discrete constrained problem is its poor and even ill condition. In this paper, we transform the discrete constrained problems on de Rham complex to Laplace-like problems. This transformation not only make…
We study the relationship between the Dirichlet and Regularity problem for parabolic operators of the form $ L = \mbox{div}(A\nabla\cdot) - \partial_t $ on cylindrical domains $ \Omega = \mathcal O \times \mathbb R $, where the base $…
We give a sufficient condition for the essential self-adjointness of a perturbation of the square of the magnetic Laplacian on an infinite weighted graph. The main result is applicable to graphs whose degree function is not necessarily…
We characterize regular boundary points in terms of a barrier family for a general form of a parabolic equation that generalizes both the standard parabolic $p$-Laplace equation and the normalized version arising from stochastic game…
We find that to the dynamics of a given dissipative system a $p=1$ differential form can be associated with a general decomposition into a potential term and a non-potential residual part. If the residual part is absent the form is closed…
Cardinality constraints in optimization are commonly of $L^0$-type, and they lead to sparsely supported optimizers. An efficient way of dealing with these constraints algorithmically, when the objective functional is convex, is…
The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue…