arXiv subjects

Iwona Chlebicka

Publications and source records attributed to Iwona Chlebicka.

35 records · Page 2Linked to original sources

arXiv2020

Measure data elliptic problems with generalized Orlicz growth

We study nonlinear measure data elliptic problems involving the operator exposing generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. Approximable and renormalized solutions are proven to exist and coincide for arbitrary measure datum and to be unique when the datum is diffuse with respect to a relevant nonstandard capacity. For justifying that the class of measures is natural, a capacitary characterization of diffuse measures is provided.

math.AP↗

arXiv2020

Essentially fully anisotropic Orlicz functions and uniqueness to measure data problem

Studying elliptic measure data problem with strongly nonlinear operator whose growth is described by the means of fully anisotropic $N$-function, we prove the uniqueness for a broad class of measures. In order to provide it, the framework of capacities in fully anisotropic Orlicz-Sobolev spaces is developed and the~capacitary characterization of a~bounded measure is given. Moreover, we give an example of an anisotropic Young function $Φ$, such that $|ξ|^p \lesssimΦ(ξ)\lesssim |ξ|^p\log^α(1+|ξ|)$, with arbitrary $p\geq 1$, $α>0$, but so irregularly growing that % we call it essentially fully anisotropic. In fact, the Orlicz--Sobolev--type space generated by $Φ$ indispensably requires fully anisotropic tools to be handled.

math.AP↗

arXiv2020

Generalized superharmonic functions with strongly nonlinear operator

We study properties of $\mathcal{A}$-harmonic and $\mathcal{A}$-superharmonic functions involving an operator having generalized Orlicz-growth embracing besides Orlicz case also natural ranges of variable exponent and double-phase cases. In particular, Harnack's Principle and Minimum Principle are provided for $\mathcal{A}$-superharmonic functions and boundary Harnack inequality is proven for $\mathcal{A}$-harmonic functions.

math.AP↗

arXiv2020

Wolff potentials and local behaviour of solutions to measure data elliptic problems with Orlicz growth

We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for $A$-superharmonic functions with nonlinear operator $A:Ω\times\mathbb{R}^n\to\mathbb{R}^n$ having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls bounds from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfies conditions expressed in the natural scales. Finally, we give a variant of Hedberg--Wolff theorem on characterization of the dual of the Orlicz space.

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arXiv2020

Boundary regularity for manifold constrained $p(x)$-harmonic maps

We prove partial and full boundary regularity for manifold constrained $p(x)$-harmonic maps.

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arXiv2020

Removable sets in elliptic equations with Musielak-Orlicz growth

We characterize, in the terms of intrinsic Hausdorff measures, the size of~removable sets for Hölder continuous solutions to elliptic equations with Musielak-Orlicz growth. In the general case we provide an elegant form of the measure that captures -- as special cases -- the classical results, slightly refines the ones provided for problems stated in the variable exponent and double phase spaces and essentially improves the known one in the Orlicz case.

math.AP↗

arXiv2019

Fully anisotropic elliptic problems with minimally integrable data

We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic $N$-function, which is not necessarily of power type and need not satisfy the $Δ_2$ nor the $\nabla _2$-condition. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutions - in the approximable sense - is established. Their maximal regularity in Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of $L^1$-data.

math.AP↗

arXiv2019

Regularizing effect of the lower-order terms in elliptic problems with Orlicz growth

Under various conditions on the data we analyse how appearence of lower order terms affects the gradient estimates on solutions to a general nonlinear elliptic equation of the form \[-{\rm div}\, a(x,Du)+b(x,u)=μ\] with data $μ$ not belonging to the dual of the natural energy space but to Lorentz/Morrey-type spaces. The growth of the leading part of the operator is governed by a function of Orlicz-type, whereas the lower-order term satisfies the sign condition and is minorized with some convex function, whose speed of growth modulates the regularization of the solutions.

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arXiv2019

Removable sets in non-uniformly elliptic problems

We analyze fine properties of solutions to quasilinear elliptic equations with double phase structure and characterize, in the terms of intrinsic Hausdorff measures, the removable sets for Hölder continuous solutions.

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arXiv2018

Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or $L^1$ data

We investigate solutions to nonlinear elliptic Dirichlet problems of the type \[ \left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= μ&\qquad \mathrm{ in}\qquad Ω, u=0 &\qquad \mathrm{ on}\qquad \partialΩ, \end{array}\right. \] where $Ω$ is a bounded Lipschitz domain in $\mathbb{R}^n$ and $A(x,z,ξ)$ is a Carathéodory's function. The growth of~the~monotone vector field $A$ with respect to the $(z,ξ)$ variables is expressed through some $N$-functions $B$ and $P$. We do not require any particular type of growth condition of such functions, so we deal with problems in nonreflexive spaces. When the problem involves measure data and weakly monotone operator, we prove existence. For $L^1$-data problems with strongly monotone operator we infer also uniqueness and regularity of~solutions and their gradients in the scale of Orlicz-Marcinkiewicz spaces.

math.AP↗

arXiv2018

Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon

We provide existence and uniqueness of renomalized solutions to a general nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in $\mathbb{R}^n$. Namely we study \begin{equation*} \left\{\begin{array}{l } \partial_t u-{\rm div} A(t,x,\nabla u)= f(t,x) \in L^1(Ω_T),\\ u(0,x)=u_0(x)\in L^1(Ω). \end{array}\right. \end{equation*} The growth of the monotone vector field $A$ is assumed to be controlled by a generalized nonhomogeneous and anisotropic $N$-function $M:[0,T)\times Ω\times\mathbb{R}^n \to[0,\infty)$. Existence and uniqueness of renormalized solutions are proven in absence of~Lavrentiev's phenomenon. The condition we impose to ensure approximation properties of the space is a certain type of balance of interplay between the behaviour of $M$ for large $|ξ|$ and small changes of time and space variables. Its instances are log-Hölder continuity of variable exponent (inhomogeneous in time and space) or optimal closeness condition for powers in double phase spaces (changing in time). The noticeable challenge of this paper is considering the problem in non-reflexive and inhomogeneous fully anisotropic space that changes along time. New delicate approximation-in-time result is proven and applied in the construction of renormalized solutions.

math.AP↗

arXiv2018

Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon

We study a general nonlinear parabolic equation on a Lipschitz bounded domain in $\mathbb{R}^N$, \begin{equation*} \left\{\begin{array}{l l} \partial_t u-\mathrm{div} A(t,x,\nabla u)= f(t,x)&\text{in}\ \ Ω_T,\\ u(t,x)=0 &\ \mathrm{ on} \ (0,T)\times\partialΩ,\\ u(0,x)=u_0(x)&\text{in}\ Ω, \end{array}\right. \end{equation*} with $f\in L^\infty(Ω_T)$ and $u_0\in L^\infty(Ω)$. The growth of the monotone vector field $A$ is controlled by a generalized fully anisotropic $N$-function $M:[0,T)\timesΩ\times\mathbb{R}^N\to[0,\infty)$ inhomogeneous in time and space, and under no growth restrictions on the last variable. It results in the need of the integration by parts formula which has to be formulated in an advanced way. Existence and uniqueness of solutions are proven when the Musielak-Orlicz space is reflexive OR in absence of Lavrentiev's phenomenon. To ensure approximation properties of the space we impose natural assumption that the asymptotic behaviour of the modular function is sufficiently balanced. Its instances are log-Hölder continuity of variable exponent or optimal closeness condition for powers in double phase spaces. The noticeable challenge of this paper is cosidering the problem in non-reflexive and inhomogeneous fully anisotropic space that changes along time.

math.AP↗

arXiv2018

Gradient estimates for problems with Orlicz growth

We study a general nonlinear elliptic equation in the Orlicz setting with data not belonging to the dual of the energy space. We provide several Lorentz-type and Morrey-type estimates for the gradients of solutions under various conditions on the data.

math.AP↗

arXiv2018

A pocket guide to nonlinear differential equations in the Musielak--Orlicz spaces

The Musielak--Orlicz setting unifies the variable exponent, Orlicz, weighted Sobolev, and double-phase spaces. They inherit technical difficulties resulting from general growth and inhomogeneity. In this survey we present an overview of developments of the theory of existence of~PDEs in the setting including reflexive and non-reflexive cases, as well as isotropic and anisotropic ones. Particular attention is paid to problems with data below natural duality in absence of Lavrentiev's phenomenon.

math.AP↗

arXiv2018

Existence to nonlinear parabolic problems with unbounded weights

We consider the weighted parabolic problem of the type \begin{equation*} \begin{split} \left\{\begin{array}{ll} u_t-\mathrm{div}(ω_2(x)|\nabla u|^{p-2} \nabla u )= λω_1(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{split} \end{equation*} for quite a general class of possibly unbounded weights $ ω_1,ω_2$ satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that $λ$ is smaller than the optimal constant in the inequality.

math.AP↗

arXiv2018

Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions

We prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \[\partial_t u-\mathrm{div} A(x,\nabla u)= f\in L^1(Ω_T),\] on a Lipschitz bounded domain in $\mathbb{R}^n$. The growth of the weakly 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 on $M$ is continuity of log-Hölder-type, which results in good approximation properties of the space. However, the requirement of regularity can be skipped in the case of reflexive spaces. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments. Uniqueness results from the comparison principle.

math.AP↗

arXiv2018

Gossez's approximation theorems in the Musielak-Orlicz-Sobolev spaces

We prove the density of smooth functions in the modular topology in the Musielak-Orlicz-Sobolev spaces essentially extending the results of Gossez \cite{GJP2} obtained in the Orlicz-Sobolev setting. We impose new systematic regularity assumption on $M$ which allows to study the problem of density unifying and improving the known results in the Orlicz-Sobolev spaces, as well as the variable exponent Sobolev spaces. We confirm the precision of the method by showing the lack of the Lavrentiev phenomenon in the double-phase case. Indeed, we get the modular approximation of $W^{1,p}_0(Ω)$ functions by smooth functions in the double-phase space governed by the modular function $H(x,s)=s^p+a(x)s^q$ with $a\in C^{0,α}(Ω)$ excluding the Lavrentiev phenomenon within the sharp range $q/p\leq 1+α/N$. See \cite[Theorem~4.1]{min-double-reg1} for the sharpness of the result.

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