Related papers: Discrete comparison principles for quasilinear ell…
We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on $L^1$ comparison) between solutions to an elliptic partial differential equation on a smooth bounded set $\Omega$ with a…
In this paper we study a general class of nonlinear elliptic problems in divergence form. First, we prove that the solutions to these problems satisfy a convexity property when the given domain is strictly convex. Then, making use of this…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
In this paper, we propose and analyze a multiscale method for a class of quasilinear elliptic problems of nonmonotone type with spatially multiscale coefficient. The numerical approach is inspired by the Localized Orthogonal Decomposition…
In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not…
The last decades saw growing interest across multiple disciplines in nonlinear phenomena described by partial differential equations (PDE). Integrability of such equations is tightly related with the Painleve property - solutions being free…
We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…
We present an ordinary differential equations approach to the analysis of algorithms for constructing $l_1$ minimizing solutions to underdetermined linear systems of full rank. It involves a relaxed minimization problem whose minimum is…
The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation (PDE) with singular non-linearity is analyzed. The PDE is a recently derived variant of a canonical model used in the modeling of…
We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations (PDEs) in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and…
We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in \cite{DFMST}. We apply it to generalized $p$-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative…
This work introduces finite element methods for a class of elliptic fully nonlinear partial differential equations. They are based on a minimal residual principle that builds upon the Alexandrov--Bakelman--Pucci estimate. Under rather…
We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The…
We derive a comparison principle for a degenerate elliptic partial differential equation without boundary conditions which arises naturally in optimal learning strategies. Our argument is direct and exploits the degeneracy of the…
Nonlinear multi-scale problems are ubiquitous in materials science and biology. Complicated interactions between nonlinearities and (nonseparable) multiple scales pose a major challenge for analysis and simulation. In this paper, we study…
We adopt the integral definition of the fractional Laplace operator and analyze an optimal control problem for a fractional semilinear elliptic partial differential equation (PDE); control constraints are also considered. We establish the…
A theorem providing necessary conditions enabling one to map a nonlinear system of first order partial differential equations to an equivalent first order autonomous and homogeneous quasilinear system is given. The reduction to quasilinear…
This work is devoted to the study of a Liouville comparison principle for entire weak solutions of quasilinear differential inequalities of the form $A(u) + |u|^{q-1}u \leq A(v) + |v|^{q-1}v$ on ${\Bbb R}^n$, where $n\geq 1$, $q$ is…
In this paper we investigate an adaptive discretization strategy for ill-posed linear prob- lems combined with a regularization from a class of semiiterative methods. We show that such a discretization approach in combination with a…
Comparison estimates are an important technical device in the study of regularity problems for quasilinear possibly degenerate elliptic and parabolic equations. Such tools have been employed indispensably in many papers of Mingione,…