Local and global behaviour of nonlinear equations with natural growth terms

This paper concerns a study of the pointwise behaviour of positive solutions to certain quasi-linear elliptic equations with natural growth terms, under minimal regularity assumptions on the underlying coefficients. Our primary results consist of optimal pointwise estimates for positive solutions of such equations in terms of two local Wolff's potentials.Comment: In memory of Professor Nigel Kalto

Fractional differentiability for solutions of nonlinear elliptic equations

We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,\xi)$ enjoys $B^\alpha_{\frac{n}\alpha, q}$ smoothness, local well-posedness is found in $B^\alpha_{p,q}$ for certain values of $p\in[2,\frac{n}{\alpha})$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,\xi)=A(x)\xi$, $G=0$ and $A\in B^\alpha_{\frac{n}\alpha,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^\alpha_{p,q}$ for each $p<\frac{n}\alpha$. Our main tool in the proof is a more general result, that holds also if ${\mathcal A}$ has growth $s-1$ in $Du$, $2\leq s\leq n$, and asserts local well-posedness in $L^q$ for each $q>s$, provided that $x\mapsto{\mathcal A}(x,\xi)$ satisfies a locally uniform $VMO$ condition

Second-order L2L^2-regularity in nonlinear elliptic problems

A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the existence and square integrability of the weak derivatives of the nonlinear expression of the gradient under the divergence operator. This provides a nonlinear counterpart of the classical $L^2$-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are established. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required. If the domain is convex, no regularity of its boundary is needed at all

Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces

We prove optimal integrability results for solutions of the p(x)-Laplace equation in the scale of (weak) Lebesgue spaces. To obtain this, we show that variable exponent Riesz and Wolff potentials map L1 to variable exponent weak Lebesgue spaces

Adjuvant or radical fractionated stereotactic radiotherapy for patients with pituitary functional and nonfunctional macroadenoma

Purpose: To evaluate the efficacy and toxicity of stereotactic fractionated radiotherapy (SFRT) for patients with pituitary macroadenoma (PMA).Methods and Materials: Between March 2000 and March 2009, 27 patients (male to female ratio, 1.25) with PMA underwent SFRT (median dose, 50.4 Gy). Mean age of the patients was 56.5 years (range, 20.3 - 77.4). In all but one patient, SFRT was administered for salvage treatment after surgical resection (transphenoidal resection in 23, transphenoidal resection followed by craniotomy in 2 and multiple transphenoidal resections in another patient). In 10 (37%) patients, the PMAs were functional (3 ACTH-secreting, 3 prolactinomas, 2 growth hormone-secreting and 2 multiple hormone-secretion). Three (11.1%) and 9 (33.3%) patients had PMA abutting and compressing the optic chiasm, respectively. Mean tumor volume was 2.9 +/- 4.6 cm(3). Eighteen (66.7%) patients had hypopituitarism prior to SFRT. The mean follow-up period after SFRT was 72.4 +/- 37.2 months.Results: Tumor size decreased for 6 (22.2%) patients and remained unchanged for 19 (70.4%) other patients. Two (7.4%) patients had tumor growth inside the prescribed treatment volume. The estimated 5-year tumor growth control was 95.5% after SFRT. Biochemical remission occurred in 3 (30%) patients with functional PMA. Two patients with normal anterior pituitary function before SFRT developed new deficits 25 and 65 months after treatment. The 5-year survival without new anterior pituitary deficit was thus 95.8%. Five patients with visual field defect had improved visual function and 1 patient with no visual defect prior to SFRT, but an optic chiasm abutting tumor, had a decline in visual function. The estimated 5-year vision and pituitary function preservation rates were 93.2% and 95.8%, respectively.Conclusions: SFRT is a safe and effective treatment for patients with PMA, although longer follow-up is needed to evaluate long-term outcomes. In this study, approximately 1 patient with visual field defect out of two had an improved visual

Mobility and Migrations in the Rural Areas of Mediterranean EU Countries

AbstractThis chapter focuses on the ambivalent nature of contemporary migrations in European rural areas. The growing presence of immigrants in these areas is a direct result of the restructuring of agriculture and global agri-food chains. Evidence indicates that while agricultural work and rural settings are decreasingly attractive to local populations, they represent a favourable environment to international newcomers, due to the higher chances to access livelihood resources. The non-visibility and informality that characterise rural settings and agricultural work arrangements provide on the one side opportunities for employment, while also fostering illegal labour practices and situations of harsh exploitation