Abstract
Abstract We study integral operators L u ( χ ) = ∫ ℝ ℕ ψ ( u ( x ) − u ( y ) ) J ( x − y ) d y $\mathcal{L}u\left( \chi \right)=\int{_{_{\mathbb{R}}\mathbb{N}}\psi \left( u\left( x \right)-u\left( y \right) \right)J\left( x-y \right)dy}$ of the type of the fractional p-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel J considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem L u = f $\mathcal{L}u=f$ in a bounded domain Ω , $\Omega ,$ and boundary condition u ≡ 0 on Ω c ; ${{\Omega }^{c}};$ both cases f = f(x) and f = f(u) are considred, including the generalized eigenvalue problem f ( u ) = λ ψ ( u ) . $f\left( u \right)=\lambda \psi \left( u \right).$
Highlights
Un modelo de poblacionConsideremos una poblacion de individuos cuya densidad en cualquier punto x ∈ RN y tiempo t ≥ 0 estarepresentada por u(x, t)
The present thesis is dedicated to the study of certain integro-differential operators with differential order close to zero, which are interesting both, from the theoretical point of view and for the applications
In the second part we study nonlinear operators of the fractional p-Laplacian type, including weakly non-integrable kernels
Summary
Consideremos una poblacion de individuos cuya densidad en cualquier punto x ∈ RN y tiempo t ≥ 0 estarepresentada por u(x, t). Cuando el nucleo de Levy es una potencia no integrable, J(y) = |y|−N−α con 0 < α < 2, entonces el operador L es un multiplo del llamado laplaciano fraccionario (−∆)α/2u(x) = CN,αV.P. donde CN,α es una constante de normalizacion. Nuestro interes en esta parte de la tesis es estudiar operadores integrales del tipo p-laplaciano fraccionario para funciones Φ mas generales que solo potencias, ademas de considerar tambien nucleos en el lımite de integrabilidad. El Capıtulo 2 esta dedicada a estudiar el operador no local definido y la forma bilineal asociada, describiendo sus propiedades, incluyendo la accion del operador sobre diferentes funciones, dos desigualdades de Hardy y un resultado de simetrizacion para xx. Finalmente el Capıtulo 7 esta dedicado al estudio de los problemas elıpticos asociados para las diferentes reacciones comentadas anteriormente
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