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Julio D. Rossi

Publications and source records attributed to Julio D. Rossi.

68 records · Page 4Linked to original sources
arXiv2015

Eigenvalues for a nonlocal pseudo $p-$Laplacian

In this paper we study the eigenvalue problems for a nonlocal operator of order $s$ that is analogous to the local pseudo $p-$Laplacian. We show that there is a sequence of eigenvalues $λ_n \to \infty$ and that the first one is positive, simple, isolated and has a positive and bounded associated eigenfunction. For the first eigenvalue we also analyze the limits as $p\to \infty$ (obtaining a limit nonlocal eigenvalue problem analogous to the pseudo infinity Laplacian) and as $s\to 1^-$ (obtaining the first eigenvalue for a local operator of $p-$Laplacian type). To perform this study we have to introduce anisotropic fractional Sobolev spaces and prove some of their properties.

math.AP↗
arXiv2015

The first nontrivial eigenvalue for a system of $p-$Laplacians with Neumann and Dirichlet boundary conditions

We deal with the first eigenvalue for a system of two $p-$Laplacians with Dirichlet and Neumann boundary conditions. If $Δ_{p}w=\mbox{div}(|\nabla w|^{p-2}w)$ stands for the $p-$Laplacian and $\fracα{p}+\fracβ{q}=1,$ we consider $$ \begin{cases} -Δ_pu= λα|u|^{α-2} u|v|^β &\text{ in }Ω,\\ -Δ_q v= λβ|u|^α|v|^{β-2}v &\text{ in }Ω,\\ \end{cases} $$ with mixed boundary conditions $$ u=0, \qquad |\nabla v|^{q-2}\dfrac{\partial v}{\partial ν}=0, \qquad \text{on }\partial Ω. $$ We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem $$ λ_{p,q}^{α,β} = \min \left\{\dfrac{\displaystyle\int_Ω\dfrac{|\nabla u|^p}{p}\, dx +\int_Ω\dfrac{|\nabla v|^q}{q}\, dx} {\displaystyle\int_Ω |u|^α|v|^β\, dx} \colon (u,v)\in \mathcal{A}_{p,q}^{α,β}\right\}, $$ where $$ \mathcal{A}_{p,q}^{α,β}=\left\{(u,v)\in W^{1,p}_0(Ω)\times W^{1,q}(Ω)\colon uv\not\equiv0\text{ and }\int_Ω|u|^α|v|^{β-2}v \, dx=0\right\}. $$ We also study the limit of $λ_{p,q}^{α,β} $ as $p,q\to \infty$ assuming that $\fracα{p} \to Γ\in (0,1)$, and $ \frac{q}{p} \to Q \in (0,\infty)$ as $p,q\to \infty.$ We find that this limit problem interpolates between the pure Dirichlet and Neumann cases for a single equation when we take $Q=1$ and the limits $Γ\to 1$ and $Γ\to 0$.

math.AP↗
arXiv2014

The first eigenvalue of the $p-$Laplacian on quantum graphs

We study the first eigenvalue of the $p-$Laplacian (with $1<p<\infty$) on a quantum graph with Dirichlet or Kirchoff boundary conditions on the nodes. We find lower and upper bounds for this eigenvalue when we prescribe the total sum of the lengths of the edges and the number of Dirichlet nodes of the graph. Also we find a formula for the shape derivative of the first eigenvalue (assuming that it is simple) when we perturb the graph by changing the length of an edge. Finally, we study in detail the limit cases $p\to \infty$ and $p\to 1$.

math-ph↗
arXiv2014

Estimates for Nonlinear Harmonic Measures on Trees

In this paper we give some estimates for nonlinear harmonic measures on trees. In particular, we estimate in terms of the size of a set $D$ the value at the origin of the solution to $ u(x)=F((x,0),\dots,(x,m-1))$ for every $x\in\mathbb{T}_m, $ a directed tree with $m$ branches with initial datum $f+χ_D$. Here $F$ is an averaging operator on $\mathbb{R}^m$, $x$ is a vertex of a directed tree $\mathbb{T}_m$ with regular $m$-branching and $(x,i)$ denotes a successor of that vertex for $0\le i\le m-1$.

math.AP↗
arXiv2014

Existence, uniqueness and decay rates for evolution equations on trees

We study evolution equations governed by an averaging operator on a directed tree, showing existence and uniqueness of solutions. In addition we find conditions of the initial condition that allows us to find the asymptotic decay rate of the solutions as $t\to \infty$. It turns out that this decay rate is not uniform, it strongly depends on how the initial condition goes to zero as one goes down in the tree.

math.AP↗
arXiv2014

An existence result for the infinity laplacian with non-homogeneous Neumann boundary conditions using Tug-of-War games

In this paper we show how to use a Tug-of-War game to obtain existence of a viscosity solution to the infinity laplacian with non-homogeneous mixed boundary conditions. For a Lipschitz and positive function $g$ there exists a viscosity solution of the mixed boundary value problem, $$ \{\begin{array}{ll} \displaystyle -Δ_{\infty}u(x)=0\quad & \text{in} Ω, \displaystyle \frac{\partial u}{\partial n}(x)= g (x)\quad & \text{on} Γ_N, \displaystyle u(x)= 0 \quad & \text{on} Γ_D. \end{array}. $$

math.AP↗
arXiv2014

The unique continuation property for a nonlinear equation on trees

In this paper we study the game $p-$Laplacian on a tree, that is, $$ u(x)=\fracα2\left\{\max_{y\in §(x)}u(y) + \min_{y\in §(x)}u(y)\right\} + \fracβ{m}\sum_{y\in §(x)} u(y), $$ here $x$ is a vertex of the tree and $S(x)$ is the set of successors of $x$. We study the family of the subsets of the tree that enjoy the unique continuation property, that is, subsets $U$ such that $u\mid_U=0$ implies $u \equiv 0$.

math.AP↗
arXiv2013

An obstacle problem for Tug-of-War games

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.

math.AP↗
arXiv2013

Decay estimates for nonlinear nonlocal diffusion problems in the whole space

In this paper we obtain bounds for the decay rate in the $L^r (\rr^d)$-norm for the solutions to a nonlocal and nolinear evolution equation, namely, $$u_t(x,t) = \int_{\rr^d} K(x,y) |u(y,t)- u(x,t)|^{p-2} (u(y,t)- u(x,t)) \, dy, $$ with $ x \in \rr^d$, $ t>0$. Here we consider a kernel $K(x,y)$ of the form $K(x,y)=ψ(y-a(x))+ψ(x-a(y))$, where $ψ$ is a bounded, nonnegative function supported in the unit ball and $a$ is a linear function $a(x)= Ax$. To obtain the decay rates we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $ T(u) = - \int_{\rr^d} K(x,y) |u(y)-u(x)|^{p-2} (u(y)-u(x)) \, dy$, with $1 \leq p < \infty$. The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole $\rr^d$: $$ λ_{1,p} (\rr^d) = 2(\int_{\rr^d} ψ(z) \, dz)|\frac{1}{|\det{A}|^{1/p}} -1|^p. $$ Moreover, we deal with the $p=\infty$ eigenvalue problem studying the limit as $p \to \infty$ of $λ_{1,p}^{1/p}$.

math.AP↗
arXiv2012

Discontinuous gradient constraints and the infinity Laplacian

Motivated by tug-of-war games and asymptotic analysis of certain variational problems, we consider a gradient constraint problem involving the infinity Laplace operator. We prove that this problem always has a solution that is unique if a certain regularity condition on the constraint is satisfied. If this regularity condition fails, then solutions obtained from game theory and $L^p$-approximation need not coincide.

math.AP↗
arXiv2009

p(x)-Harmonic functions with unbounded exponent in a subdomain

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 $Ω$.

math.AP↗
arXiv2009

A mixed problem for the infinity laplacian via Tug-of-War games

In this paper we prove that a function $ u\in\mathcal{C}(\barΩ)$ is the continuous value of the Tug-of-War game described in \cite{PSSW} if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions {-Δ_{\infty}u(x)=0\quad & \text{in} Ω, \frac{\partial u}{\partial n}(x)=0\quad & \text{on} Γ_N, u(x)=F(x)\quad & \text{on} Γ_D. By using the results in \cite{PSSW}, it follows that this viscous PDE problem has a unique solution, which is the unique {\it absolutely minimizing Lipschitz extension} to the whole $\barΩ$ (in the sense of \cite{Aronsson} and \cite{PSSW}) of the boundary data $ F:Γ_D\to\R $.

math.AP↗