Related papers: Infinity-Harmonic Potentials and Their Streamlines
We study the streamlines of $\infty$-harmonic functions in planar convex rings. We include convex polygons. The points where streamlines can meet are characterized: they lie on certain curves. The gradient has constant norm along…
We study parameterized elliptic systems on symmetric domains with additional system symmetries. We prove the existence of continua of nontrivial solutions bifurcating from the constant branch determined by a critical point of the potential,…
This paper deals with the Landesman-Lazer type problem of elliptic equations associated with homogeneous Dirichlet boundary conditions. By using some dynamical arguments we derive some new results on bifurcation from infinity and…
We give a generalization of a theorem of B\^ocher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a…
In the paper the asymptotic bifurcation of solutions to a parameterized stationary semilinear Schr\"odinger equation involving a potential of the Kato-Rellich type is studied. It is shown that the bifurcation from infinity occurs if the…
An equilibrium of a planar, piecewise-$C^1$, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here…
We prove the uniqueness for viscosity solutions of a differential equation involving the infinity-Laplacian with a variable exponent. A version of the Harnack's inequality is derived for this minimax problem.
We study the global structure of the set of radial solutions of a nonlinear Dirichlet problem involving the p-Laplacian with p>2, in the unit ball of $R^N$, $N \ges 1$. We show that all non-trivial radial solutions lie on smooth curves of…
The main goal of this paper is to address an important conjecture in the field of differential equations in the presence of a harmonic potential. While in the subcritical case, the uniqueness of positive solution has been addressed by…
We study a singularly perturbed problem related to infinity Laplacian operator with prescribed boundary values in a region. We prove that solutions are locally (uniformly) Lipschitz continuous, they grow as a linear function, are strongly…
It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem…
The goal of this paper is to discuss some of the results in [31] and [32] and expand upon the work there by proving a global weak existence result as well as a first bubbling analysis in finite time. In addition, an alternative local…
We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to…
In this paper, we consider the Laplace equation with a class of indefinite superlinear boundary conditions and study the uniqueness of positive solutions that this problem possesses. Superlinear elliptic problems can be expected to have…
We provide a new and simple proof based on Harnack's inequality to the Lipschitz continuity of the solutions of a class of free boundary problems.
In this paper, we analyze an eigenvalue problem for nonlinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We prove bifurcation results from trivial solutions and from infinity for…
We consider in this paper elliptic equations which are perturbations of Laplace's equation by a compactly supported potential. We show that in dimension greater than three for a wide class of potentials all the solutions are globally…
We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order PDEs. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the…
We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \begin{align*} \begin{aligned} -\Delta u &= f_\lambda(|x|,u,|\nabla u|) &&\text{in }\Omega, u &= 0 &&\text{on }\partial\Omega, \end{aligned}…
We propose a monotone, and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized Infinity Laplacian, which could be related to the family of so--called two--scale methods. We show that this method is…