Well-posedness for a transmission problem connecting first and second-order operators [PDF]
arXiv, 2023 We establish the existence and uniqueness of viscosity solutions within a domain $\Omega\subseteq\mathbb R^n$ for a class of equations governed by elliptic and eikonal type equations in disjoint regions. Our primary motivation stems from the Hamilton-Jacobi equation that arises in the context of a stochastic optimal control problem.
arxiv
On two notions of solutions to the obstacle problem for the singular porous medium equation [PDF]
arXiv, 2023 We show that two different notions of solutions to the obstacle problem for the porous medium equation, a potential theoretic notion and a notion based on a variational inequality, coincide for regular enough compactly supported obstacles.
arxiv
Large self-similar solutions of the parabolic-elliptic Keller--Segel model [PDF]
arXiv, 2020 We construct radial self-similar solutions of the, so called, minimal parabolic-elliptic Keller--Segel model in several space dimensions with radial, nonnegative initial conditions with are below the Chandrasekhar solution -- the singular stationary solution of this system.
arxiv
On the Gibbons' conjecture for equations involving the $p$-Laplacian [PDF]
arXiv, 2020 In this paper we prove the validity of Gibbons' conjecture for the quasilinear elliptic equation $ -\Delta_p u = f(u) $ on $\mathbb{R}^N.$ The result holds true for $(2N+2)/(N+2) < p < 2$ and for a very general class of nonlinearity $f$.
arxiv
Viscosity solutions of general viscous Hamilton-Jacobi equations [PDF]
arXiv, 2013 We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton-Jacobi equations.
arxiv
Symmetry and monotonicity of singular solutions of double phase problems [PDF]
arXiv, 2020 We consider positive singular solutions of PDEs arising from double phase functionals. Exploiting a rather new version of the moving plane method originally developed by Sciunzi, we prove symmetry and monotonicity properties of such solutions.
arxiv
Stochastic ordering by g-expectations [PDF]
arXiv, 2020 We derive sufficient conditions for the convex and monotonic g-stochastic ordering of diffusion processes under nonlinear g-expectations and g-evaluations. Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity, monotonicity and continuous dependence properties for the ...
arxiv
A weak comparison principle for solutions of very degenerate elliptic equations [PDF]
arXiv, 2012 We prove a comparison principle for weak solutions of elliptic quasilinear equations in divergence form whose ellipticity constants degenerate at every point where $\nabla u \in K$, where $K\subset \mathbb{R}^N$ is a Borel set containing the origin.
arxiv
Travelling wave solutions of the density-suppressed motility model [PDF]
arXiv, 2020 In this paper, we study the traveling wave solutions to the density-suppressed motility model describing the ``self-trapping'' mechanism that induces spatio-temporal pattern formations observed in the experiment. We establish the existence of traveling wavefronts with a minimal wave speed and discuss the selection of wave profiles supplemented with ...
arxiv
A level set crystalline mean curvature flow of surfaces [PDF]
arXiv, 2016 We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by regularized problems, and we also show the uniqueness and existence of a level set flow for bounded crystals.
arxiv