arXiv subjects

Iwona Skrzypczak

Publications and source records attributed to Iwona Skrzypczak.

5 recordsLinked to original sources

arXiv2017

Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \begin{equation*} -{\rm div} A(x,\nabla u)= f\in L^1(Ω), \end{equation*} on a Lipschitz bounded domain in $\mathbb{R}^N$. The growth of the monotone vector field $A$ is controlled by a generalized nonhomogeneous and anisotropic $N$-function $M $. The approach does not require any particular type of growth condition of $M$ or its conjugate $M^*$ (neither $Δ_2$, nor $\nabla_2$). The condition we impose is log-Holder continuity of $M$, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.

math.AP↗

arXiv2016

Existence of solutions to degenerate parabolic problems with two weights via the Hardy inequality

The paper concentrates on the application of the following Hardy inequality \begin{equation*} \int_Ω\ |ξ(x)|^p ω_{1 }(x)dx\le \int_Ω|\nabla ξ(x)|^pω_{2 }(x)dx, \end{equation*} to the proof of existence of weak solutions to degenerate parabolic problems of the type \begin{equation*} \left\{\begin{array}{ll} u_t-div(ω_2(x)|\nabla u|^{p-2} \nabla u )= λW(x) |u|^{p-2}u& x\inΩ, u(x,0)=f(x)& x\inΩ, u(x,t)=0& x\in\partialΩ,\ t>0,\\ \end{array}\right. \end{equation*} on an open subset $Ω\subseteq\mathbb{R}^n$, not necessarily bounded, where \[W(x)\leq \min\{m,ω_1(x)\},\qquad m\in\mathbb{R}_+.\]

math.AP↗

arXiv2016

Caccioppoli--type estimates and Hardy--type inequalities derived from degenerated $p$--harmonic problems

We obtain Caccioppoli--type estimates for nontrivial and nonnegative solutions to the anticoercive partial differential inequalities of elliptic type involving degenerated $p$--Laplacian: $-Δ_{p,a} u:= -\mathrm{div}(a(x)|\na u|^{p-2}\na u)\ge b(x)Φ(u)$, where $u$ is defined in a domain $Ω$. Using Caccioppoli--type estimates, we obtain several variants of Hardy--type inequalities in weighted Sobolev spaces.

math.AP↗

arXiv2015

Hardy-type inequality in variable exponent Lebesgue spaces derived from nonlinear problem

We derive a family of weighted Hardy-type inequalities in the variable exponent Lebesgue space with an additional term of the form \[ \int_Ω |ξ|^{p(x)} μ_{1,β}(dx)\leqslant \int_Ω|\nabla ξ|^{p(x)}μ_{2,β}(dx)+\int_Ω\left|ξ{\log ξ} \right|^{p(x)} μ_{3,β}(dx), \] where $ξ$ is any compactly supported Lipschitz function. The involved measures depend on a certain solution to the partial differential inequality involving $p(x)$-Laplacian ${-}Δ_{p(x)} u\geqslant Φ$, where $Φ$ is a given locally integrable function, and $u$ is defined on an open and not necessarily bounded subset $Ω\subseteq\mathbb{R}^n $, and a certain parameter $β$. We derive new Caccioppoli-type inequality for the solution $u$. As its consequence we get Hardy-type inequality. We illustrate the result by several one-dimensional examples.

math.AP↗

arXiv2015

Variable exponent Hardy-type inequalities in $\mathbb{R}^n$

In this paper, we investigate further the weighted $p(x)$-Hardy inequality with the additional term of the form \[ \int_Ω|ξ|^{p(x)}μ_{1,β} (dx) \leqslant \int_Ω|\nabla ξ|^{p(x)}μ_{2,β} (dx)+\int_Ω\left|ξ{\log ξ} \right|^{p(x)} μ_{3,β} (dx), \] holding for Lipschitz functions compactly supported in $Ω\subseteq\mathbb{R}^n$. The involved measures depend on a certain solution to the partial differential inequality involving $p(x)$-Laplacian ${-}Δ_{p(x)} u\geqslant Φ$, where $Φ$ is a given locally integrable function, and $u$ is defined on an open and not necessarily bounded subset $Ω\subseteq\mathbb{R}^n $, and a certain parameter $β$. We focus on the $n$-dimensional case giving some examples. Moreover, we compare our inequalities with the existing in the literature.

math.AP↗