248 research outputs found
Optimal Constants in the Theory of Sobolev Spaces and PDEs
Recent research activities on sharp constants and optimal inequalities have shown their impact on a deeper understanding of geometric, analytical and other phenomena in the context of partial differential equations and mathematical physics. These intrinsic questions have applications not only to a-priori estimates or spectral theory but also to numerics, economics, optimization, etc
International Conference on Nonlinear Differential Equations and Applications
Dear Participants, Colleagues and Friends
It is a great honour and a privilege to give you all a warmest welcome to the first Portugal-Italy Conference on Nonlinear Differential Equations and Applications (PICNDEA).
This conference takes place at the Colégio Espírito Santo, University of Évora, located in the beautiful city of Évora, Portugal. The host institution, as well the associated scientific research centres, are committed to the event, hoping that it will be a benchmark for scientific collaboration between the two countries in the area of mathematics.
The main scientific topics of the conference are Ordinary and Partial Differential Equations, with particular regard to non-linear problems originating in applications, and its treatment with the methods of Numerical Analysis. The fundamental main purpose is to bring together Italian and Portuguese researchers in the above fields, to create new, and amplify previous collaboration, and to follow and discuss new topics in the area
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher
This review is dedicated to recent results on the 2d parabolic-elliptic
Patlak-Keller-Segel model, and on its variant in higher dimensions where the
diffusion is of critical porous medium type. Both of these models have a
critical mass such that the solutions exist globally in time if the mass
is less than and above which there are solutions which blowup in finite
time. The main tools, in particular the free energy, and the idea of the
methods are set out. A number of open questions are also stated.Comment: 26 page
Resultados de existência para equações elípticas com termos singulares
Doutoramento em Matemática e AplicaçõesEsta dissertação estuda em detalhe três problemas elípticos: (I) uma classe de
equações que envolve o operador Laplaciano, um termo singular e nãolinearidade
com o exponente crítico de Sobolev, (II) uma classe de equações
com singularidade dupla, o expoente crítico de Hardy-Sobolev e um termo
côncavo e (III) uma classe de equações em forma divergente, que envolve um
termo singular, um operador do tipo Leray-Lions, e uma função definida nos
espaços de Lorentz.
As não-linearidades consideradas nos problemas (I) e (II), apresentam
dificuldades adicionais, tais como uma singularidade forte no ponto zero (de
modo que um "blow-up" pode ocorrer) e a falta de compacidade, devido à
presença do exponente crítico de Sobolev (problema (I)) e Hardy-Sobolev
(problema (II)). Pela singularidade existente no problema (III), a definição
padrão de solução fraca pode não fazer sentido, por isso, é introduzida uma
noção especial de solução fraca em subconjuntos abertos do domínio.
Métodos variacionais e técnicas da Teoria de Pontos Críticos são usados para
provar a existência de soluções nos dois primeiros problemas. No problema (I),
são usadas uma combinação adequada de técnicas de Nehari, o princípio
variacional de Ekeland, métodos de minimax, um argumento de translação e
estimativas integrais do nível de energia. Neste caso, demonstramos a
existência de (pelo menos) quatro soluções não triviais onde pelo menos uma
delas muda de sinal. No problema (II), usando o método de concentração de
compacidade e o teorema de passagem de montanha, demostramos a
existência de pelo menos duas soluções positivas e pelo menos um par de
soluções com mudança de sinal. A abordagem do problema (III) combina um
resultado de surjectividade para operadores monótonos, coercivos e
radialmente contínuos com propriedades especiais do operador de tipo Leray-
Lions. Demonstramos assim a existência de pelo menos, uma solução no
espaço de Lorentz e obtemos uma estimativa para esta solução.This dissertation study mainly three elliptical problems: (I) a class of equations,
which involves the Laplacian operator, a singular term and a nonlinearity with
the critical Sobolev exponent, (II) a class of equations with double singularity,
the critical Hardy-Sobolev exponent and a concave term and (III) a class of
equations in divergent form, which involves a singular term, a Leray-Lions
operator, and a function defined on Lorentz spaces.
The nonlinearities considered in problems (I) and (II), bring additional difficulties
which, as the strong singularity at zero (so blow-up may occur) and the lack of
compactness due to the presence of a Sobolev critical exponent (problem (I))
and a Hardy-Sobolev critical exponent (problem (II)). In problem (III), the
singularity implies that the standard definition of weak solution may not make
sense. Therefore is necessary to introduce a special notion of weak solution on
open subsets of the domain.
Variational methods and Critical Point Theory techniques are used to prove the
existence of solutions in the two first problems. In problem (I), our method
combines Nehari's techniques, Ekeland's variational principle, minimax
methods, a translation argument and integral estimates of the energy level. In
this case, we prove the existence of (at least) four nontrivial solutions where at
least one of them is sign-changing. In problem (II), we prove the existence of at
least two positive solutions and a pair of sign-changing solutions, using the
concentration-compactness method and the mountain pass theorem. The
approach in problem (III) combines a surjectivity result for monotone, coercive
and radially continuous operators with special properties of Leray-Lions
operators. We prove the existence of at least one solution in a Lorentz space
and obtain an estimative for the solution
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