Harnack inequality and regularity for degenerate quasilinear elliptic equations

We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,\alpha}$ local estimates for solutions of a degenerate equation in non divergence form

Monotonicity and symmetry of singular solutions to quasilinear problems

We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved moving plane procedure

Importance Sampling for a Markov Modulated Queuing Network with Customer Impatience until the End of Service

For more than two decades, there has been a growing of interest in fast simulation techniques for estimating probabilities of rare events in queuing networks. Importance sampling is a variance reduction method for simulating rare events. The present paper carries out strict deadlines to the paper by Dupuis et al for a two node tandem network with feedback whose arrival and service rates are modulated by an exogenous finite state Markov process. We derive a closed form solution for the probability of missing deadlines. Then we have employed the results to an importance sampling technique to estimate the probability of total population overflow which is a rare event. We have also shown that the probability of this rare event may be affected by various deadline values.Importance Sampling, Queuing Network, Rare Event, Markov Process, Deadline

Implicit equations involving the pp-Laplace operator

In this work we study the existence of solutions $u \in W^{1,p}_0(\Omega)$ to the implicit elliptic problem $f(x, u, \nabla u, \Delta_p u)= 0$ in $\Omega$, where $\Omega$ is a bounded domain in $\mathbb R^N$, $N \ge 2$, with smooth boundary $\partial \Omega$, $1< p< +\infty$, and $f\colon \Omega \times \mathbb R \times \mathbb R^N \times \R \to \R$. We choose the particular case when the function $f$ can be expressed in the form $f(x, z, w, y)= \varphi(x, z, w)- \psi(y)$, where the function $\psi$ depends only on the $p$-Laplacian $\Delta_p u$. We also present some applications of our results.Comment: 15 pages; comments are welcom