arXiv subjects
Publications and source records attributed to Petteri Harjulehto.
We study integrability properties of the Wolff potential in context of Choquet integrals with respect to the Hausdorff content. As an application we give integrability results to the solutions of the $p$-Laplace equation in this context.
We study Poincare-Sobolev type inequalities for compactly supported smooth functions which are defined in the Euclidean $n$-space and whose absolute value of gradient are Choquet $δ/n$-integrable with respect to the $δ$-dimensional Hausdorff content, $n\geq 2$, $δ\in (0,n]$. In particular, our results imply a new Sobolev inequality for quasicontinuous functions defined in the Sobolev space $W^{1,1}_0(\mathbb{R}^n)$. As an application we extend a recently introduced superlevel Sobolev inequality into a context of the Hausdorff content.
We prove a relaxation result for a quasi-convex bulk integral functional with variable exponent growth in a suitable space of bounded variation type. A key tool is a decomposition under mild assumptions of the energy into absolutely continuous and singular parts weighted via a recession function.
We consider Choquet integrals with respect to dyadic Hausdorff content of non-negative functions which are not necessarily Lebesgue measurable. We study the theory of Lebesgue points. The studies yield convergence results and also a density result between function spaces. We provide examples which show sharpness of the main convergence theorem. These examples give additional information about the convergence in the norm also, namely the difference of the functions in this setting and continuous functions.
We study minimizers of non-autonomous energies with minimal growth and coercivity assumptions on the energy. We show that the minimizer is nevertheless the solution of the relevant Euler--Lagrange equation or inequality. The main tool is an extension result for convex $C^1$-energies.
We consider integrals in the sense of Choquet with respect to the $δ$-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean $n$-space, $n\geq 2$, $0<δ\le n$. In particular, for these functions we prove Sobolev inequalities in the limiting case $p=δ/n$ and in the case $p>δ$, here $p$ is the integrability exponent of the absolute value of the gradient of any given function. The results complement previously known Poincaré-Sobolev and Morrey inequalities.
We introduce Riesz potentials for non-Lebesgue measurable functions by taking the integrals in the sense of Choquet with respect to Hausdorff content and prove boundedness results for these operators. Some earlier results are recovered or extended now using integrals taken in the sense of Choquet with respect to Hausdorff content. Some earlier results also for maximal operators are considered, but now for non-measurable functions.
We consider inequalities where integrals are defined in the sense of Choquet with respect to Hausdorff content. We study cases where continuously differentiable functions are defined on open, connected sets with so much regularity that there exists a pointwise estimate between the values of a function and its gradient under the maximal operator or the Riesz potential, at every point of the set. We show that certain Hardy inequalities and Poincare-Sobolev inequalities are valid in this context.
Conditions for harmonic analysis in generalized Orlicz spaces have been studied over the past decade. One approach involves the generalized inverse of so-called weak $Φ$-functions. It featured prominently in the monograph Orlicz Spaces and Generalized Orlicz Spaces [P. Harjulehto and P. Hästö, Lecture Notes in Mathematics, vol. 2236, Springer, Cham, 2019]. While generally successful, the inverse function formulation of the decay condition (A2) in the monograph contains a flaw, which we explain and correct in this note. We also present some new results related to the conditions, including a more general result for the density of smooth functions.
We study convergence of generalized Orlicz energies when the lower growth-rate tends to infinity. We generalize results by Bocea--Mihăilescu (Orlicz case) and Eleuteri--Prinari (variable exponent case) and allow weaker assumptions: we are also able to handle unbounded domains with irregular boundary and non-doubling energies.
Motivated by the image denoising problem and the undesirable stair-casing effect of the total variation method, we introduce bounded variation spaces with generalized Orlicz growth. Our setup covers earlier variable exponent and double phase models. We study the norm and modular of the new space and derive a formula for the modular in terms of the Lebesgue decomposition of the derivative measure and a location dependent recession function. We also show that the modular can be obtained as the $Γ$-limit of uniformly convex approximating energies.
We study minimizers of non-autonomous functionals \begin{align*} \inf_u \int_Ωφ(x,|\nabla u|) \, dx \end{align*} when $φ$ has generalized Orlicz growth. We consider the case where the upper growth rate of $φ$ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on "truncating" the function $φ$ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.
We show local Hölder continuity of quasiminimizers of functionals with non-standard (Musielak--Orlicz) growth. Compared with previous results, we cover more general minimizing functionals and need fewer assumptions. We prove Harnack's inequality and a Morrey type estimate for quasiminimizers. Combining this with Ekeland's variational principle, we obtain local Hölder continuity for $ω$-minimizers.
In this paper we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We prove certain properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property with respect to the modular) and the properties of the new double phase operator (continuity, strict monotonicity, (S$_+$)-property). In contrast to the known constant exponent case we are able to weaken the assumptions on the data. Finally we show the existence and uniqueness of corresponding elliptic equations with right-hand sides that have gradient dependence (so-called convection terms) under very general assumptions on the data. As a result of independent interest, we also show the density of smooth functions in the new Musielak-Orlicz Sobolev space even when the domain is unbounded.
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.
A way to measure the lower growth rate of $φ:Ω\times [0,\infty) \to [0,\infty)$ is to require $t \mapsto φ(x,t)t^{-r}$ to be increasing in $(0,\infty)$. If this condition holds with $r=1$, then \[ \inf_{u\in f+W^{1, φ}_0(Ω)}\int_Ωφ(x, |\nabla u|) \, dx \] with boundary values $f\in W^{1,φ}(Ω)$ does not necessary have a minimizer. However, if $φ$ is replaced by $φ^p$, then the growth condition holds with $r=p > 1$ and thus (under some additional conditions) the corresponding energy integral has a minimizer. We show that a sequence $(u_p)$ of such minimizers convergences when $p \to 1^+$ in a suitable $\mathrm{BV}$-type space involving generalized Orlicz growth and obtain the $Γ$-convergence of functionals with fixed boundary values and of functionals with fidelity terms. %We complement our results by showing that some previous papers by some of the authors are included in our analysis.
We study weak-type estimates and exponential integrability for the variable order Riesz potential. As an application we prove an exponential integrability result with respect to the Hausdorff content for functions from variable exponent Sobolev spaces. In particular, the earlier exponential integrability results are improved to a corresponding one with respect to the Choque integral whenever John domains are considered. Moreover, new exponential integrability results also for domains with outward cusps are obtained.
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 $p>1$.