Related papers: A fully nonlinear transmission problem degeneratin…
A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove…
We find an explicit form of weak solutions to a Riemann problem for a degenerate semilinear parabolic equation with piecewise constant diffusion coefficient. It is demonstrated that the phase transition lines (free boundaries) correspond to…
We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions…
We develop a variational technique for some wide classes of nonlinear evolutions. The novelty here is that we derive the main information directly from the corresponding Euler-Lagrange equations. In particular, we prove that not only the…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
In this paper, we consider a kind of degenerate normalized $p$-Laplacian equation with general variable exponents. We establish local $C^{1,\alpha'}$ regularity of viscosity solutions by making use of the compactness argument, scaling…
In this paper, we present a problem involving fully nonlinear elliptic operators with Hamiltonian, which can present a singularity or degenerate as the gradient approaches the origin. The model studied here, allows the appearance of plateau…
In the present work, we derive functional upper bounds for the potential error arising from finite-element boundary-element coupling formulations for a nonlinear Poisson-type transmission problem. The proposed a posteriori error estimates…
For elliptic interface problems in two- and three-dimensions, this paper establishes a priori error estimates for Crouzeix-Raviart nonconforming, Raviart-Thomas mixed, and discontinuous Galerkin finite element approximations. These…
The benefits of transmission line switching are well-known in terms of reducing operational cost and improving system reliability of power systems. However, finding the optimal power network configuration is a challenging task due to the…
We propose finite difference methods for degenerate fully nonlinear elliptic equations and prove the convergence of the schemes. Our focus is on the pure equation and a related free boundary problem of transmission type. The cornerstone of…
We find analytical solutions to the Cahn-Hilliard equation for the dynamics of an interface in a system with a conserved order parameter (Model B). We show that, although steady-state solutions of Model B are unphysical in the far-field,…
Decentralized optimization is a powerful paradigm that finds applications in engineering and learning design. This work studies decentralized composite optimization problems with non-smooth regularization terms. Most existing gradient-based…
We consider cooperative multi-agent consensus optimization problems over an undirected network of agents, where only those agents connected by an edge can directly communicate. The objective is to minimize the sum of agent-specific…
The boundary value problems for linear and nonlinear singular degenerate differential-operator equations are studied. We prove a well-posedeness of linear problem and optimal regularity result for the nonlinear problem which occur in fluid…
This paper addresses the problems of spline interpolation on smooth Riemannian manifolds, with or without the inclusion of least-squares fitting. Our unified approach utilizes gradient flows for successively connected curves or networks,…
We present a method for the resolution of (oscillatory) nonlinear problems. It is based on the application of the Linear Delta Expansion to the Lindstedt-Poincar\'e method. By applying it to the Duffing equation, we show that our method…
Many problems in modern robotics can be addressed by modeling them as bilevel optimization problems. In this work, we leverage augmented Lagrangian methods and recent advances in automatic differentiation to develop a general-purpose…
We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface. Typically, the amplitude of the oscillations is of the order of $\epsilon$ while…
The aim of this paper is to study the obstacle problem with an elliptic operator having degenerate coercivity. We prove the existence of an entropy solution to the obstacle problem under the assumption of $L^{1}-$summability on the data.…