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Peter Hästö

Publications and source records attributed to Peter Hästö.

At least 19 recordsLinked to original sources
arXiv2026

Higher integrability for parabolic PDEs with generalized Orlicz growth

We prove higher integrability of the gradient of weak solutions to nonlinear parabolic systems whose prototype is \[ \partial_t u-\mathrm{div}\Big(\frac{φ'(z, |\nabla u|)}{|\nabla u|}\nabla u\Big) =0, \qquad u=(u^1,\dots,u^N), \] where $φ$ is a generalized Young function. Special cases of our main theorem include previously known results for the $p$-growth, the variable exponent and the double phase growth. Also included are previously unknown borderline double phase growth and perturbed variable exponent growth, among others. The problem is controlled by a natural requirement of comparison of $φ$ between points in intrinsic parabolic cylinders via an (A1)-condition, which unifies disparate conditions from the special cases. Moreover, we handle both the singular and degenerate cases at the same time, providing a unified proof of of a reverse Hölder type inequality.

math.AP↗
arXiv2026

Intrinsic Schauder estimates at nearly linear growth

We develop a nonlinear potential theoretic framework for Schauder estimates for vector-valued solutions of a broad class of nonautonomous variational problems at nearly linear growth. Our approach naturally embraces the variable exponent as well as the Double and Multi phase setting, yielding new regularity results in basic models and recovering optimal regularity recently established in specific cases.

math.AP↗
arXiv2026

Mean oscillation conditions for nonlinear equation and regularity results

We consider general nonlinear elliptic equations of the form \[ \operatorname{div}\, A(x,Du) = 0 \quad \text{in } Ω, \] where $A:Ω\times \mathbb R^n \to \mathbb R^n$ satisfies a quasi-isotropic $(p,q)$-growth condition, which is equivalent to the point-wise uniform ellipticity of $A$. We establish sharp and comprehensive mean oscillation conditions on $A(x,ξ)$ with respect to the $x$ variable to obtain $C^1$- and $W^{1,s}$-regularity results. The results provide new conditions even in the standard $p$-growth case with coefficient $\operatorname{div}(a(x)|Du|^{p-2}Du)=0$. Also included are variable exponent growth with and without perturbation as well as borderline double-phase growth and double-phase growth with a coefficient.

math.AP↗
arXiv2026

Relaxation of quasi-convex functionals with variable exponent growth

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.

math.AP↗
arXiv2024

Strategy flexibility in university mathematics

We present results and ideas on flexibility and its teaching for university mathematicians based on research and our experience in professional development programs. We believe this could provide food for thought for instructors of calculus courses and beyond.

math.HO↗
arXiv2023

A revised condition for harmonic analysis in generalized Orlicz spaces on unbounded domains

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.

math.FA↗
arXiv2023

Regularity theory for non-autonomous problems with a priori assumptions

We study weak solutions and minimizers $u$ of the non-autonomous problems $\operatorname{div} A(x, Du)=0$ and $\min_v \int_ΩF(x,Dv)\,dx$ with quasi-isotropic $(p, q)$-growth. We consider the case that $u$ is bounded, Hölder continuous or lies in a Lebesgue space and establish a sharp connection between assumptions on $A$ or $F$ and the corresponding norm of $u$. We prove a Sobolev--Poincaré inequality, higher integrability and the Hölder continuity of $u$ and $Du$. Our proofs are optimized and streamlined versions of earlier research that can more readily be further extended to other settings. Connections between assumptions on $A$ or $F$ and assumptions on $u$ are known for the double phase energy $F(x, ξ)=|ξ|^p + a(x)|ξ|^q$. We obtain slightly better results even in this special case. Furthermore, we also cover perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, triple phase, double variable exponent as well as variable exponent double phase energies and the results are new in most of these special cases.

math.AP↗
arXiv2023

Convergence of generalized Orlicz norms with lower growth rate tending to infinity

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.

math.AP↗
arXiv2022

Bounded variation spaces with generalized Orlicz growth related to image denoising

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.

math.FA↗
arXiv2022

Bloch estimates in non-doubling generalized Orlicz spaces

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.

math.AP↗
arXiv2022

Hölder continuity of $ω$-minimizers of functionals with generalized Orlicz growth

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.

math.AP↗
arXiv2022

Regularity theory for non-autonomous partial differential equations without Uhlenbeck structure

We establish maximal local regularity results of weak solutions or local minimizers of \[ \operatorname{div} A(x, Du)=0 \quad\text{and}\quad \min_u \int_ΩF(x,Du)\,dx, \] providing new ellipticity and continuity assumptions on $A$ or $F$ with general $(p,q)$-growth. Optimal regularity theory for the above non-autonomous problems is a long-standing issue; the classical approach by Giaquinta and Giusti involves assuming that the nonlinearity $F$ satisfies a structure condition. This means that the growth and ellipticity conditions depend on a given special function, such as $t^p$, $φ(t)$, $t^{p(x)}$, $t^p+a(x)t^q$, and not only $F$ but also the given function is assumed to satisfy suitable continuity conditions. Hence these regularity conditions depend on given special functions. In this paper we study the problem without recourse to special function structure and without assuming Uhlenbeck structure. We introduce a new ellipticity condition using $A$ or $F$ only, which entails that the function is quasi-isotropic, i.e.\ it may depend on the direction, but only up to a multiplicative constant. Moreover, we formulate the continuity condition on $A$ or $F$ without specific structure and without direct restriction on the ratio $\frac qp$ of the parameters from the $(p,q)$-growth condition. We establish local $C^{1,α}$-regularity for some $α\in(0,1)$ and $C^α$-regularity for any $α\in(0,1)$ of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases.

math.AP↗
arXiv2022

Riesz spaces with generalized Orlicz growth

We consider a Riesz $ϕ$-variation for functions $f$ defined on the real line when $φ:Ω\times[0,\infty)\to[0,\infty)$ is a generalized $Φ$-function. We show that it generates a quasi-Banach space and derive an explicit formula for the modular when the function $f$ has bounded variation. The resulting $BV$-type energy has previously appeared in image restoration models. We generalize and improve previous results in the variable exponent and Orlicz cases and answer a question regarding the Riesz--Medvedev variation by Appell, Banaś and Merentes [\emph{Bounded Variation and Around}, Studies in Nonlinear Analysis and Applications, Vol. 17, De Gruyter, Berlin/Boston, 2014].

math.FA↗
arXiv2021

Minimizers of abstract generalized Orlicz--bounded variation energy

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.

math.AP↗
arXiv2021

A proof of Hall's conjecture on length of ray images under starlike mappings of order $α$

Assume that $f$ lies in the class of starlike functions of order $α\in [0,1)$, that is, which are regular and univalent for $|z|<1$ and such that $${\rm Re} \left (\frac{zf'(z)}{f(z)} \right ) > α~\mbox{ for } |z|<1. $$ In this paper we show that for each $α\in [0,1)$, the following sharp inequality holds: $$ |f(re^{iθ})|^{-1} \int_{0}^{r}|f'(ue^{iθ})| du \leq \frac{Γ(\frac12)Γ(2-α)}{Γ(\frac32-α)} ~\mbox {for every $r<1$ and $θ$}. $$ This settles the conjecture of Hall (1980).

math.CV↗
arXiv2020

Maximal regularity for local minimizers of non-autonomous functionals

We establish local $C^{1,α}$-regularity for some $α\in(0,1)$ and $C^α$-regularity for any $α\in(0,1)$ of local minimizers of the functional \[ v\ \mapsto\ \int_Ωϕ(x,|Dv|)\,dx, \] where $ϕ$ satisfies a $(p,q)$-growth condition. Establishing such a regularity theory with sharp, general conditions has been an open problem since the 1980s. In contrast to previous results, we formulate the continuity requirement on $ϕ$ in terms of a single condition for the map $(x,t)\mapsto ϕ(x,t)$, rather than separately in the $x$- and $t$-directions. Thus we can obtain regularity results for functionals without assuming that the gap $\frac qp$ between the upper and lower growth bounds is close to $1$. Moreover, for $ϕ(x,t)$ with particular structure, including $p$-, Orlicz-, $p(x)$- and double phase-growth, our single condition implies known, essentially optimal, regularity conditions. Hence, we handle regularity theory for the above functional in a universal way.

math.AP↗