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Arttu Karppinen

Publications and source records attributed to Arttu Karppinen.

9 recordsLinked to original sources
arXiv2023

Local second order regularity of solutions to elliptic Orlicz-Laplace equation

We consider Orlicz--Laplace equation $-div(\frac{φ'(|\nabla u|)}{|\nabla u|}\nabla u)=f$ where $φ$ is an Orlicz function and either $f=0$ or $f\in L^\infty$. We prove local second order regularity results for the weak solutions $u$ of the Orlicz--Laplace equation. More precisely, we show that if $ψ$ is another Orlicz function that is close to $φ$ in a suitable sense, then $\frac{ψ'(|\nabla u|)}{|\nabla u|}\nabla u\in W^{1,2}_{loc}$. This work contributes to the building up of quantitative second order Sobolev regularity for solutions of nonlinear equations.

math.AP↗
arXiv2023

A direct proof of existence of weak solutions to fully anisotropic and inhomogeneous elliptic problems

We provide a direct proof of existence and uniqueness of weak solutions to a broad family of strongly nonlinear elliptic equations with lower order terms. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak--Orlicz spaces generated by an $N$-function $M:Ω\times\mathbb{R}^d\to[0,\infty)$. Neither $\nabla_2$ nor $Δ_2$ conditions are imposed on $M$. Our results cover among others problems with anisotropic polynomial, Orlicz, variable exponent, and double phase growth.

math.AP↗
arXiv2022

Stability of solutions to obstacle problems with generalized Orlicz growth

We consider nonlinear equations having generalized Orlicz growth (also known as Musielak--Orlicz growth). We prove that if differential operators $\mathcal{A}_i$ converge locally uniformly to an operator $\mathcal{A}$, then the sequence of solutions $(u_i)$ has a subsequence converging to solution $u$ of the limit operator in Sobolev and Hölder norms.

math.AP↗
arXiv2021

Sharp growth conditions for boundedness of maximal function in generalized Orlicz spaces

We study sharp growth conditions for the boundedness of the Hardy-Littlewood maximal function in the generalized Orlicz spaces. We assume that the generalized Orlicz function $ϕ(x, t)$ satisfies the standard continuity properties (A0), (A1) and (A2). We show that if the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space to itself then $ϕ(x,t)/ t^p$ is almost increasing for large $t$ for some $p>1$. Moreover we show that the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space $L^ϕ(\mathbb{R}^n)$ to itself if and only if $ϕ$ is weakly equivalent to a generalized Orlicz function $ψ$ satisfying (A0), (A1) and (A2) for which $ψ(x,t)/ t^p$ is almost increasing for all $t>0$ and some $p>1$.

math.FA↗
arXiv2021

Fractional operators and their commutators on generalized Orlicz spaces

In this paper we examine boundedness of fractional maximal operator. The main focus is on commutators and maximal commutators on generalized Orlicz spaces for fractional maximal functions and Riesz potentials. We prove their boundedness between generalized Orlicz spaces and give a characterization for functions of bounded mean oscillation. To best of our knowledge, these results are also new in the special case of double phase spaces.

math.FA↗
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↗