Per ogni soluzione della (1) nel dominio limitato $\mathrm{\Omega }$,, appartenente a $H_0^2(\mathrm{\Omega })$ e soddisfacente le condizioni (2), si dimostra la maggiorazione (5), valida nell'intorno di ogni punto $x^0$ del contorno; si consente a $\partial \mathrm{\Omega }$ di essere singolare in $x^0$.

For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an $O\left(ϵ\right)$ estimate in $H^1$ for solutions with Dirichlet condition.

We prove global pointwise estimates for the Green function of a parabolic operator with potential in the parabolic Kato class on a $C^{1,1}$ cylindrical domain Ω. We apply these estimates to obtain a new and shorter proof of the Harnack inequality [16], and to study the boundary behavior of nonnegative solutions.

Consider the problem of time-periodic strong solutions of the Stokes system modelling viscous incompressible fluid flow past a rotating obstacle in the whole space ℝ³. Introducing a rotating coordinate system attached to the body yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In a recent paper [2] the author proved $L^q$-estimates of second order derivatives uniformly in the angular and translational velocities, ω and...

A theorem on estimates of solutions of impulsive parabolic equations by means of solutions of impulsive ordinary differential equations is proved. An application to the population dynamics is given.

Whenever nonlinear problems have to be solved through approximation methods by solving related linear problems a priori estimates are very useful. In the following this kind of estimates are presented for a variety of equations related to generalized first order Beltrami systems in the plane and for second order elliptic equations in ${ℝ}^m$. Different types of boundary value problems are considered. For Beltrami systems these are the Riemann-Hilbert, the Riemann and the Poincaré problem, while for elliptic...

We consider a class of perturbations of the degenerate Ornstein-Uhlenbeck operator in ${ℝ}^N$. Using a revised version of Bernstein’s method we provide several uniform estimates for the semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ associated with the realization of the operator $𝒜$ in the space of all the bounded and continuous functions in ${ℝ}^N$

We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$-Laplacian and the Navier $p$-biharmonic operator on a ball of radius $R$ in ${ℝ}^N$ and its asymptotics for $p$ approaching $1$ and $\infty$. Let $p$ tend to $\infty$. There is a critical radius $R_C$ of the ball such that the principal eigenvalue goes to $\infty$ for $0<R\le R_C$ and to $0$ for $R>R_C$. The critical radius is $R_C=1$ for any $N\in \mathbb{N}$ for the $p$-Laplacian and $R_C=\sqrt{2N}$ in the case of the $p$-biharmonic operator. When $p$ approaches $1$, the principal eigenvalue of the Dirichlet...

$L^{\infty }$-estimates of weak solutions are established for a quasilinear non-diagonal parabolic system with a special structure whose leading terms are modelled by p-Laplacians. A generalization of the weak maximum principle to systems of equations is employed.

A-priori estimates in weighted Hölder norms are obtained for the solutions of a one- dimensional boundary value problem for the heat equation in a domain degenerating at time t = 0 and with boundary data involving simultaneously the first order time derivative and the spatial gradient.

We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the Hölder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.

For the Dirichlet Laplacian in the exterior of a strictly convex obstacle, we show that the number of scattering poles of modulus $\le r$ in a small angle $\theta$ near the real axis, can be estimated by Const ${\theta }^{3/2}{r}^n$ for $r$ sufficiently large depending on $\theta$. Here $n$ is the dimension.