In this note we study the periodic homogenization problem for a particular bidimensional selfadjoint elliptic operator of the second order. Theoretical and numerical considerations allow us to conjecture explicit formulae for the coefficients of the homogenized operator.

In this paper, we consider second order neutrons diffusion problem with coefficients in L∞(Ω). Nodal method of the lowest order is applied to approximate the problem's solution. The approximation uses special basis functions [1] in which the coefficients appear. The rate of convergence obtained is O(h2) in L2(Ω), with a free rectangular triangulation.

We consider a conducting body which presents some (unknown) perfectly insulating defects, such as cracks or cavities, for instance. We perform measurements of current and voltage type on a (known) part of the boundary of the conductor. We prove that, even if the defects are unknown, the current and voltage measurements at the boundary uniquely determine the corresponding electrostatic potential inside the conductor. A corresponding stability result, related to the stability of Neumann problems with...

We consider a class of stationary viscous Hamilton-Jacobi equations aswhere $\lambda \ge 0$, $A\left(x\right)$ is a bounded and uniformly elliptic matrix and $H(x,\xi )$ is convex in $\xi$ and grows at most like ${\left|\xi \right|}^q+f\left(x\right)$, with $1\<q\<2$ and $f\in L^{N/q^{\text{'}}}\left(\Omega \right)$. Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate,i.e.${(1+|u\left|\right)}^{\overline{q}-1}u\in H_0^1\left(\Omega \right)$, for a certain (optimal) exponent $\overline{q}$. This completes the recent results in [15],...

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is$$\left\{\begin{array}{cc}-div\left(a\left(x\right)\right(1+{\left|\nabla u\right|}^2{)}^{\frac{p-2}{2}}{\nabla u)+b\left(x\right)(1+\left|\nabla u\right|}^2{)}^{\frac{\lambda }{2}}=f\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$where $\Omega$ is a bounded open subset of ${ℝ}^N$, $N\ge 2$, $2-1/N\<p\<N$, $a$ belongs to $L^{\infty }\left(\Omega \right)$, $a\left(x\right)\ge {\alpha }_0\>0$, $f$ is a function in $L^1\left(\Omega \right)$, $b$ is a function in $L^r\left(\Omega \right)$ and $0\le \lambda \<{\lambda }^{*}(N,p,r),$ for some $r$ and ${\lambda }^{*}(N,p,r)$.

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is
$$\left\{-\text{div}\left(a\left(x\right)\right(1+{\left|\nabla u\right|}^2{)}^{\frac{p-2}{2}}{\nabla u)+b\left(x\right)(1+\left|\nabla u\right|}^2{)}^{\frac{\lambda }{2}}=f\text{in}\Omega ,u=0\text{on}\partial \Omega ,\right.$$
where Ω is a bounded open subset of ${ℝ}^N$, N > 2, 2-1/N < p < N , a belongs to L∞(Ω), $a\left(x\right)\ge {\alpha }_0>0$, f is a function in L1(Ω), b is a function in $L^r\left(\Omega \right)$ and 0 ≤ λ < λ *(N,p,r), for some r and λ *(N,p,r).

We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by $-\Delta u+\lambda \frac{{\left|\nabla u\right|}^2}{u^r}=f\left(x\right),\lambda ,r>0.$ The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they...