Comparison results for a nonlocal singular elliptic problem

We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in (Arch. Ration. Mech. Anal. 239 (2021 ) 1733–1770, Theorem 31). Some interesting consequences are L p regularity results and nonlocal energy estimates for solutions

Nonnegative solutions for a class of singular parabolic problems involving p-laplacian

We deal with the existence of nonnegative solutions u to parabolic problems with p-laplacian principal part and lower order terms which are singular in the u variabl

Existence of weak solutions to a nonlinear reaction-diffusion system with singular sources

We discuss the existence of a class of weak solutions to a nonlinear parabolic system of reaction-diffusion type endowed with singular production terms by reaction. The singularity is due to a potential occurrence of quenching localized to the domain boundary. The kind of quenching we have in mind is due to a twofold contribution: (i) the choice of boundary conditions, modeling in our case the contact with an infinite reservoir filled with ready-to-react chemicals and (ii) the use of a particular nonlinear, non-Lipschitz structure of the reaction kinetics. Our working techniques use fine energy estimates for approximating non-singular problems and uniform control on the set where singularities are localizing

Existence of weak solutions to a nonlinear reaction-diffusion system with singular sources

We discuss the existence of a class of weak solutions to a nonlinear parabolic system of reaction-diffusion type endowed with singular production terms by reaction. The singularity is due to a potential occurrence of quenching localized to the domain boundary. The kind of quenching we have in mind is due to a twofold contribution: (i) the choice of boundary conditions, modeling in our case the contact with an infinite reservoir filled with ready-to-react chemicals and (ii) the use of a particular nonlinear, non-Lipschitz structure of the reaction kinetics. Our working techniques use fine energy estimates for approximating non-singular problems and uniform control on the set where singularities are localizing

Complementi di Analisi Matematica II

Questo volume è rivolto agli studenti dei corsi di Analisi Matematica II che i tre autori tengono presso la Facoltà di Ingegneria Civile ed Industriale dell’Università “La Sapienza” di Roma per diversi corsi di laurea. Esso nasce dalla necessità di esporre i contenuti della prima parte di tali corsi e precisamente le funzioni di più variabili, il calcolo differenziale in più variabili, le curve algebriche piane, il calcolo integrale in due e tre variabili, il principio degli indivisibili di Cavalieri, gli integrali curvilinei, le superfici, le aree di superfici e gli integrali superficiali. Tali argomenti sono esposti in modo essenziale e sintetico, sono presentati molti esempi che illustrano la teoria e le dimostrazioni sono omesse. I testi indicati in bibliografia possono essere utili per gli approfondimenti necessari. La seconda parte del libro raccoglie molti testi d’esame degli ultimi anni accademici (dal 2011). Sono tutti svolti tranne alcuni esercizi che contengono domande di teoria e per essi si rimanda alla prima parte di questo libro o al testo di Metodi Matematici consigliato in bibliografia

Homogenization of elliptic problems involving interfaces and singular data

We prove existence and homogenization results for a family of elliptic problems involving interfaces and a singular lower order term. These problems model heat or electrical conduction in composite media

Derivation of macroscopic equilibrium models for heat conduction in finely mixed composite media with singular sources

We prove existence and homogenization results for a family (depending on a small parameter and on a parameter 2 f1; 0; 1g) of elliptic problems involving a singular lower order term and representing the Euler equations of energy functionals, which can be used to describe the equilibrium for the heat conduction in composite materials with two finely mixed phases having a microscopic periodic structure (for details on the related physical models see for instance [3, 4] and the reference quoted there). The same kind of energies can be also useful to study the electrical conduction in biological tissues (see for instance [1, 2], where the related parabolic problems without singular source are studied)