arXiv subjects

Pablo Ochoa

Publications and source records attributed to Pablo Ochoa.

24 records · Page 2Linked to original sources

arXiv2020

Effect of Non-linear Lower Order Terms in Quasilinear Equations Involving the $p(x)$-Laplacian

In this work, we study the existence of $W_0^{1, p(\cdot)}$-solutions to the following boundary value problem involving the $p(\cdot)$-Laplacian operator: \begin{equation*} \left\lbrace \begin{array}{l} -Δ_{p(x)}u+|\nabla u|^{q(x)}=λg(x)u^{η(x)}+f(x), \quad\textnormal{ in } Ω, \\\qquad \,\,\,\,\,\quad \quad\qquad\quad u\geq 0, \quad\textnormal{ in } Ω \qquad \,\,\,\,\,\quad \quad\qquad\quad u= 0, \,\,\quad \text{on } \partialΩ. \end{array} \right. \end{equation*}under appropriate ranges on the variable exponents. We give assumptions on $f$ and $g$ in terms of the growth exponents $q$ and $η$ under which the above problem has a non-negative solution for all $λ> 0$.

math.AP↗

arXiv2020

Solving singular evolution problems in sub-Riemannian groups via deterministic games

In this manuscript, we prove existence of viscosity solutions to singular parabolic equations in Carnot groups. We develop the analysis by constructing appropriate deterministic games adapted to the algebraic and differential structures of Carnot groups. We point out that the proof of existence does not require comparison and it is based on an Arzela-Ascoli-type theorem.

math.AP↗

arXiv2020

A capacity-based condition for existence of solutions to fractional elliptic equations with first-order terms and measures

In this manuscript, we appeal to Potential Theory to provide a sufficient condition for existence of distributional solutions to fractional elliptic problems with non-linear first-order terms and measure data $ω$: $$ \left\{ \begin{array}{rcll} (-Δ)^su&=&|\nabla u|^q + ω\quad \text{in }\mathbb{R}^n,\, \,\,s \in (1/2, 1)\\u & > &0 \quad \text{in } \mathbb{R}^{n}\\\lim_{|x|\to \infty}u(x) & =& 0, \end{array} \right. $$under suitable assumptions on $q$ and $ω$. Roughly speaking, the condition for exis\-tence states that if the measure data is locally controlled by the Riesz fractional capacity, then there is a global solution for the equation. We also show that if a positive solution exists, necessarily the measure $ω$ will be absolutely continuous with respect to the associated Riesz capacity, which gives a partial reciprocal of the main result of this work. Finally, estimates of $u$ in terms of $ω$ are also given in different function spaces.

math.AP↗

arXiv2017

On viscosity and weak solutions for non-homogeneous p-Laplace equations

In this manuscript we study the relation between viscosity and weak solutions for non-homogeneous p-Laplace equations with lower order term depending on $x$, $u$ and $\nabla u$. More precisely, we prove that any locally bounded viscosity solution constitutes a weak solution, extending previous results by Juutinen, Lindqvist and Manfredi on the homogeneous case, and Julin and Juutinen for a linear right hand side. Moreover, we provide a converse statement in the full case under extra assumptions on the regularity of the solutions.

math.AP↗

arXiv2014

On the variational limits of lattice energies on prestrained elastic bodies

We study the asymptotic behaviour of the discrete elastic energies in presence of the prestrain metric $G$, assigned on the continuum reference configuration $Ω$. When the mesh size of the discrete lattice in $Ω$ goes to zero, we obtain the variational bounds on the limiting (in the sense of $Γ$-limit) energy. In case of the nearest-neighbour and next-to-nearest-neibghour interactions, we derive a precise asymptotic formula, and compare it with the non-Euclidean model energy relative to $G$.

math.NA↗

arXiv2014

Variational models for prestrained plates with Monge-Ampère constraint

We derive a new model for pre-strained thin films, which consists of minimizing a biharmonic energy of deformations $v\in W^{2,2}$ satisfying the Monge-Ampère constraint $\det\nabla^2v = f$. We further discuss multiplicity properties of the minimizers of this model, in some special cases.

math.AP↗