Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u=0u=0 in a domain with many small holes

We perform the homogenization of the semilinear elliptic problem \begin{equation*} \begin{cases} u^\varepsilon \geq 0 & \mbox{in} \; \Omega^\varepsilon,\\ \displaystyle - div \,A(x) D u^\varepsilon = F(x,u^\varepsilon) & \mbox{in} \; \Omega^\varepsilon,\\ u^\varepsilon = 0 & \mbox{on} \; \partial \Omega^\varepsilon.\\ \end{cases} \end{equation*} In this problem $F(x,s)$ is a Carath\'eodory function such that $0 \leq F(x,s) \leq h(x)/\Gamma(s)$ a.e. $x\in\Omega$ for every $s > 0$, with $h$ in some $L^r(\Omega)$ and $\Gamma$ a $C^1([0, +\infty[)$ function such that $\Gamma(0) = 0$ and $\Gamma'(s) > 0$ for every $s > 0$. On the other hand the open sets $\Omega^\varepsilon$ are obtained by removing many small holes from a fixed open set $\Omega$ in such a way that a "strange term" $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on $x$. We already treated this problem in the case of a "mild singularity", namely in the case where the function $F(x,s)$ satisfies $0 \leq F(x,s) \leq h(x) (\frac 1s + 1)$. In this case the solution $u^\varepsilon$ to the problem belongs to $H^1_0 (\Omega^\varepsilon)$ and its definition is a "natural" and rather usual one. In the general case where $F(x,s)$ exhibits a "strong singularity" at $u = 0$, which is the purpose of the present paper, the solution $u^\varepsilon$ to the problem only belongs to $H_{\tiny loc}^1(\Omega^\varepsilon)$ but in general does not belongs to $H^1_0 (\Omega^\varepsilon)$ any more, even if $u^\varepsilon$ vanishes on $\partial\Omega^\varepsilon$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results

A semilinear elliptic equation with a mild singularity at u=0u=0: existence and homogenization

In this paper we consider semilinear elliptic equations with singularities, whose prototype is the following \begin{equation*} \begin{cases} \displaystyle - div \,A(x) D u = f(x)g(u)+l(x)& \mbox{in} \; \Omega,\\ u = 0 & \mbox{on} \; \partial \Omega,\\ \end{cases} \end{equation*} where $\Omega$ is an open bounded set of $\mathbb{R}^N,\, N\geq 1$, $A\in L^\infty(\Omega)^{N\times N}$ is a coercive matrix, $g:[0,+\infty)\rightarrow [0,+\infty]$ is continuous, and $0\leq g(s)\leq {{1}\over{s^\gamma}}+1$ $\forall s>0$, with $0<\gamma\leq 1$ and $f,l \in L^r(\Omega)$, $r={{2N}\over{N+2}}$ if $N\geq 3$, $r>1$ if $N=2$, $r=1$ if $N=1$, $f(x), l(x)\geq 0$ a.e. $x \in \Omega$. We prove the existence of at least one nonnegative solution and a stability result; moreover uniqueness is also proved if $g(s)$ is nonincreasing or "almost nonincreasing". Finally, we study the homogenization of these equations posed in a sequence of domains $\Omega^\epsilon$ obtained by removing many small holes from a fixed domain $\Omega$

Bounded solutions for non-parametric mean curvature problems with nonlinear terms

In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain $\Omega$ of $\mathbb{R}^N$. The mean curvature, that depends on the location of the solution $u$ itself, is asked to be of the form $f(x)h(u)$, where $f$ is a nonnegative function in $L^{N,\infty}(\Omega)$ and $h:\mathbb{R}^+\mapsto \mathbb{R}^+$ is merely continuous and possibly unbounded near zero. As a preparatory tool for our analysis we propose a purely PDE approach to the prescribed mean curvature problem not depending on the solution, i.e. $h\equiv 1$. This part, which has its own independent interest, aims to represent a modern and up-to-date account on the subject. Uniqueness is also handled in presence of a decreasing nonlinearity. The sharpness of the results is highlighted by mean of explicit examples

Results on parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities

We study existence and non existence results for a class of parabolic problems related to some Caffarelli Kohn Nirenberg inequalities, which include the classical Hardy inequality. The paper starts by the results of Baras and Goldstein concerning existence and blow-up for the solution to the heat equation with a critical potential, results which use as essential tool the Hardy inequality

Asymptotic analysis of singular problems in perforated cylinders.

In this paper, we deal with elliptic problems having terms singular in the variable uu which represents the solution. The problems are posed in cylinders Ωnε of height 2n and perforated according to a parameter ε. We study existence, uniqueness and asymptotic behavior of the solutions uεn as the cylinders become infinite (n→+∞) and the size of the holes decreases while the number of the holes increases (ε→0)

Quali-quantitative analysis of eight Rosmarinus officinalis essential oils of different origin. First report.

Aim. It is well known that the pharmacological activity of essential oils depends on their major components, which may vary enormously. The aim of the present study was to determine the chemical composition of samples of essential oil of rosemary of different origins, in order to identify the main therapeutic constituents, according to European Pharmacopoeian (EP). Material and Methods. Analytical GC/MS was carried out on a total of eight samples of essential oil of rosemary: seven samples were commercial products from producers located in different geographical areas; the last sample was prepared in our labo- ratory from fresh flowering terminal sprigs of rosemary collected in Siena&rsquo;s Province. results. The most representative constituents of the essential oils tested, were 1,8-cineole and camphor. Other components also occurred in significant quantities in some samples, for example and &alpha;- and &beta;-pinene, limonene and caryophyllene, in- dicating clear phytochemical differences among samples. discussion. The high quantity of eucalyptol and camphor detected in the samples made them particularly suited for treating minor respiratory disorders. Eucalyptol is expectorant and liquefies bronchial secretions; camphor increases the interval bet- ween inspiration and expiration and increases the activity of the parasympathetic nervous system, facilitating respiration. On the other hand, the essential oils analyzed by us were not suitable for perfume production, because they contained little or no positive aromatic components

Elliptic equations having a singular quadratic gradient term and a changing sign datum

In this paper we study a singular elliptic problem whose model is \begin{eqnarray*} - \Delta u= \frac{|\nabla u|^2}{|u|^\theta}+f(x), in \Omega\\ u = 0, on \partial \Omega; \end{eqnarray*} where $\theta\in (0,1)$ and $f \in L^m (\Omega)$, with $m\geq \frac{N}{2}$. We do not assume any sign condition on the lower order term, nor assume the datum $f$ has a constant sign. We carefully define the meaning of solution to this problem giving sense to the gradient term where $u=0$, and prove the existence of such a solution. We also discuss related questions as the existence of solutions when the datum $f$ is less regular or the boundedness of the solutions when the datum $f \in L^m (\Omega)$ with $m> \frac{N}{2}$

The Dirichlet problem for singular elliptic equations with general nonlinearities

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline u\geq 0& \text{in}\ \Omega, \newline u=0 & \text{on}\ \partial \Omega, \end{cases}$$ where, $\Delta_{1}$ is the $1$-laplace operator, $\Omega$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary, $h(s)$ is a continuous function which may become singular at $s=0^{+}$, and $f$ is a nonnegative datum in $L^{N,\infty}(\Omega)$ with suitable small norm. Uniqueness of solutions is also shown provided $h$ is decreasing and $f>0$. As a by-product of our method a general theory for the same problem involving the $p$-laplacian as principal part, which is missed in the literature, is established. The main assumptions we use are also further discussed in order to show their optimality

Optimization of extraction of drugs containing polyphenols using an innovative technique

The role of polyphenols in human health nowadays is well established and these natural products, found in many plant species, are the active ingredients of drugs, food supplements and cosmetics. Extraction procedure is pivotal to obtain high quality herbal products but paradoxically this factor is often underrated and obsolete techniques are used. In this work we compared the classic and most used method of maceration and an innovative and standardized technique of extraction, Estrattore Naviglio((R)), processing ten common medicinal plants containing polyphenols and for each analysing specific biological markers such as flavonoids, anthocyanosides and caffeic derivatives in addition to total polyphenols content. Estrattore Naviglio((R)) guaranteed a significant improvement of the chemical quality of extracts combining effectiveness with rapidity and reproducibility. In this work we further investigated the optimization of drug extractions by replicating operations varying parameters setting on Estrattore Naviglio((R)) instrument

Quasi linear parabolic equations with degenerate coercivity having a quadratic gradient term

We study existence and regularity of distributional solutions for possibly degenerate quasi-linear parabolic problems having non linear lower order terms which depends on the solution and on its gradient