arXiv subjects
Publications and source records attributed to Juan J. Manfredi.
We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war.
In this article we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit of the theory of viscosity solutions for PDEs. The equations include the graph Laplacian, the $p$-Laplacian, the Infinity Laplacian, the Mean Curvature equation, and the Eikonal operator on the graph.
We characterize $p-$harmonic functions in the Heisenberg group in terms of an asymptotic mean value property, where $1<p<\infty$, following the scheme described in Manfredi et al. (2009) for the Euclidean case. The new tool that allows us to consider the subelliptic case is a geometric lemma, Lemma 3.2 below, that relates the directions of the points of maxima and minima of a function on a small subelliptic ball with the unit horizontal gradient of that function.
We study the horizontal mean curvature flow in the Heisenberg group by using the level-set method. We prove the uniqueness, existence and stability of axisymmetric viscosity solutions of the level-set equation. An explicit solution is given for the motion starting from a subelliptic sphere. We also give several properties of the level-set method and the mean curvature flow in the Heisenberg group.
We show that quasi-minimizers of non-homogeneous energy functionals on metric measure spaces are locally Hölder continuous and satisfy the Harnack inequality. We assume that the spaces are doubling and support a Poincaré inequality. The proof is based on the De Giorgi method, combined with the "expansion of positivity" technique.
We study the Dirichlet problem $-÷(|\nabla u|^{p(x)-2} \nabla u) =0 $ in $Ω$, with $u=f$ on $\partial Ω$ and $p(x) = \infty$ in $D$, a subdomain of the reference domain $Ω$. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as $n \to \infty$ of the solutions $u_n$ to the corresponding problem when $p_n(x) =p(x) \wedge n$, in particular, with $p_n = n$ in $D$. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, $\infty$-harmonic within $D$. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of $Ω$.
We find sharp bounds for the norm inequality on a Pseudo-hermitian manifold, where the L^2 norm of all second derivatives of the function involving horizontal derivatives is controlled by the L^2 norm of the sub-Laplacian. Perturbation allows us to get a-priori bounds for solutions to sub-elliptic PDE in non-divergence form with bounded measurable coefficients. The method of proof is through a Bochner technique. The Heisenberg group is seen to be en extremal manifold for our inequality in the class of manifolds whose Ricci curvature is non-negative.
Let $F\in W^{1,n}_{\text{loc}}(Ω; \Bbb R^n)$ be a mapping with nonnegative Jacobian $J_F(x)=\det DF(x)\ge 0$ for a.e. $x$ in a domain $Ω\subset\Bbb R^n$. The {\it dilatation} of $F$ is defined (almost everywhere in $Ω$) by the formula $$K(x)=\frac{|DF(x)|^n}{J_F(x)}\cdot$$ Iwaniec and \v Sver\' ak \ncite{IS} have conjectured that if $p\ge n-1$ and $K\in L^{p}_{\text{loc}}(Ω)$ then $F$ must be continuous, discrete and open. Moreover, they have confirmed this conjecture in the two-dimensional case $n=2$. In this article, we verify it in the higher- dimensional case $n\ge 2$ whenever $p>n-1$.