Jose Manuel Mazón

This paper deals with evolution problem for the $1$-Laplacian with mixed boundary conditions on a bounded open set $Ω$ of $\R^N$. We prove existence and uniqueness of strong solutions for data in $L^2(Ω)$ by mean of the theory of maximal monotone operator. We also see that if the flux on the boundary is~$1$ (that is, the maximum possible) then these strong solutions can be seen as the large solutions introduced in \cite{MP}. We give explicit examples of solutions.

This paper deals with Gelfand-type problems $$\begin{aligned} \qquad \qquad \left\{ \begin{array}{ll} - \Delta _m u = \lambda f(u), \quad & \hbox {in} \ \Omega , \ \lambda >0, \\ u =0, \quad & \hbox {on} \ \partial _m\Omega , \end{array} \right. \end{aligned}$$ - Δ m u = λ f ( u ) , in Ω , λ > 0 , u = 0 , on ∂ m Ω , in the framework of Random Walk Spaces, which includes as particular cases: Gelfand-type problems posed on locally finite weighted connected graphs and Gelfand-type problems driven by convolution integrable kernels. Under the same assumption on the nonlinearity f as in the local case, we show there exists an extremal parameter $$\lambda ^* \in (0, \infty )$$ λ ∗ ∈ ( 0 , ∞ ) such that, for $$0 \le \lambda < \lambda ^*$$ 0 ≤ λ < λ ∗ , problem (0.1) admits a minimal bounded solution $$u_\lambda $$ u λ and there are not solution for $$\lambda > \lambda ^*$$ λ > λ ∗ . Moreover, assuming f is convex, we show that Problem (0.1) admits a minimal bounded solution for $$\lambda = \lambda ^*$$ λ = λ ∗ . We also show that $$u_\lambda $$ u λ are stable, and, for f strictly convex, we show that they are the unique stable solutions. We give simple examples that illustrate the many situations that can occur when solving Gelfand-type problems on weighted graphs.

Random walk spaces are a general framework for the study of PDEs. They include as particular cases locally finite weighted connected graphs and nonlocal settings involving symmetric integrable kernels on $${\mathbb {R}}^N$$ R N . We are interested in the study of evolution problems involving two random walk structures so that the associated functionals have different growth on each structure. We also deal with the case of a functional with different growth on a partition of the random walk.

This paper deals with Gelfand-type problems \begin{equation}\label{Gelfand10} \qquad\qquad\left\{\begin{array}{ll} - Δ_m u = λf(u), \quad&\hbox{in} \ Ω, \ λ>0, \\[10pt] u =0, \quad&\hbox{on} \ \partial_mΩ, \end{array} \right. \end{equation} in the framework of Random Walk Spaces, which includes as particular cases: Gelfand-type problems posed on locally finite weighted connected graphs and Gelfand-type problems driven by convolution integrable kernels. Under the same assumption on the nonlinearity $f$ as in the local case, we show there exists an extremal parameter $λ^* \in (0, \infty)$ such that, for $0 \leq λ< λ^*$, problem \eqref{Gelfand10} admits a minimal bounded solution $u_λ$ and there are not solution for $λ> λ^*$. Moreover, assuming $f$ is convex, we show that Problem \eqref{Gelfand10} admits a minimal bounded solution for $λ= λ^*$. We also show that $u_λ$ are stable, and, for $f$ strictly convex, we show that they are the unique stable solutions. We give simple examples that illustrate the many situations that can occur when solving Gelfand-type problems on weighted graphs.