Let Snf be the n-th partial sum of the Vilenkin-Fourier series of f∈ L1(G). For 1<p- ≤ p+ < ∞, we characterize all exponent p(⋅) such that if f∈ Lp(⋅)(G), Snf converges to f in Lp(⋅)(G).
Excluding the Lavrentiev's phenomenon is crucial in studying regularity of minimizers of variational functionals. It can be excluded, provided that smooth functions are dense in a suitable Musielak-Orlicz-Sobolev space, in modular topology. I will present new results concerning both density of smooth functions and its consequences for the absence of Lavrentiev's phenomenon. Moreover, density results in Musielak-Orlicz-Sobolev spaces may be applied to the theory of existence of PDEs of so-called non-standard growth. This is based on [B, Chlebicka, JFA, 2022] and [B, Chlebicka, Miasojedow, arXiv:2210.15217].
We study local weighted Poincaré and Poincaré-Sobolev type inequalities with Lorentz quasi-norm Lp,q on the left hand side, where 1< p <∞ and 0< q< ∞. We obtain those inequalities as a consequence of a general self-improving property shared by functions satisfying the generalized Poincaré inequality 1/|Q| ∫Q |f-fQ|dx ≤ a(Q) for all cubes Q, where fQ=1/|Q| ∫Q f and a is some functional over cubes that satisfy the SDps(w) condition.
We consider some variational functionals F satisfying a (p,q)-growth condition. Under suitable assumptions, we prove that the lower semicontinuous envelope of F with respect to W1,q coincides with F. In other words, the Lavrentiev term is equal to zero for any admissible function u in W1,p.
The main goal of this poster is to present a method to obtain inequalities related to the classical Poincaré-Sobolev inequalities. This method is mainly based around the self-improving properties of generalized Poincaré inequalities, first appearing in [PR19] and later improved in [LLO22]. For this poster we will use this method to obtain fractional Poincaré-Sobolev type inequalities, with the particularity that in one of the sides of the inequality the measure we will consider is a non-doubling weight. In the following we will state the main results used for this case, followed by the main steps for the proof of this result. Regarding the application of this kind of methods in the context of fractional Poicaré-Sobolev inequalities more result can be found in [HS+22]. [HS+22] Ritva Hurri-Syrjänen, Javier C. Martı́nez-Perales, Carlos Pérez, and Antti V Vähäkangas. “On the BBM-Phenomenon in Fractional Poincaré–Sobolev Inequalities with Weights”. In: International Mathematics Research Notices (2022). doi: 10.1093/imrn/rnac246. [LLO22] Andrei K. Lerner, Emiel Lorist, and Sheldy Ombrosi. “Operator-free sparse domination”. In: Forum of Mathematics, Sigma 10 (2022), Paper No. e15, 28. [PR19] Carlos Pérez and Ezequiel Rela. “Degenerate Poincaré-Sobolev inequalities”. In: Transactions of the American Mathematical Society 372 (2019), pp. 6087–6133.
We present a regularity result on a class of nonstandard variational problems involving perimeter and a non-linear version of Dirichlet energy motivated by a model for instability for a ferromagnetic fluid.
We consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show, that every slowly varying function of this type is equivalent to a slowly varying function that has continuous classical derivatives of all orders.
The steady motion of a viscous incompressible fluid in an obstructed finite pipe is modeled through the Navier-Stokes equations with mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while a transversal flux rate F is prescribed along the pipe. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument based on Bernoulli's law. Through variational techniques and with the use of an exact flux carrier, an explicit upper bound on F (in terms of the viscosity, diameter and length of the tube) ensuring the uniqueness of such weak solution is given. This upper bound is shown to converge to zero at a given rate as the length of the pipe goes to infinity. In an axially symmetric framework, we also prove the existence of a weak solution displaying rotational symmetry.