arXiv subjects

Michał Borowski

Publications and source records attributed to Michał Borowski.

9 recordsLinked to original sources

arXiv2025

Carleson-type removability for $p$-parabolic equations

We characterize removable sets for Hölder continuous solutions to degenerate parabolic equations of $p$-growth. A sufficient and necessary condition for a set to be removable is given in terms of an intrinsic parabolic Hausdorff measure, which depends on the considered Hölder exponent. We present a new method to prove the sufficient condition, which relies only on fundamental properties of the obstacle problem and supersolutions, and applies to a general class of operators. For the necessity of the condition, we establish the Hölder continuity of solutions with measure data, provided the measure satisfies a suitable decay property. The techniques developed in this article provide a new point of view even in the case $p=2$.

math.AP↗

arXiv2025

Smoothness of weight sharply discards Lavrentiev's gap for double phase functionals

We show that the smoother the weight, the broader the range of exponents for which the Lavrentiev's gap is absent for the double phase functionals, i.e., $u \mapsto \int_Ω \left(|\nabla u|^p + a(x)|\nabla u|^q\right)\,dx\,, \quad 1 \leq p \leq q < \infty,\, a(\cdot) \geq 0\,.$ In particular, if $a \in C^\infty$, then no additional restrictions are required on $p$ and $q$. For $a \in C^{k, α}$, we establish the optimal range of exponents, which reads $q \leq p + (k + α)\max(1, p/N)$. Thereby, we extend previously known results which consider Hölder continuous $a$ (i.e., $q \leq p + α\max(1, p/N)$), showing that the range of exponents extends naturally upon imposing more regularity on $a$.

math.FA↗

arXiv2025

Discarding Lavrentiev's Gap in Non-autonomous and Non-Convex Variational Problems

We establish that the Lavrentiev gap between Sobolev and Lipschitz maps does not occur for a scalar variational problem of the form: \[ \textrm{to minimize} \qquad u \mapsto \int_Ωf(x,u,\nabla u)\,dx \,, \] under a Dirichlet boundary condition. Here, $Ω$ is a bounded Lipschitz open set in $\rn$, $N\geq 1$ and the function $f$ is required to be measurable with respect to the spatial variable, continuous with respect to the second one, and convex with respect to the last variable. Under these assumptions alone, Lavrentiev gaps may occur, as illustrated by classical examples in the literature. We identify an additional natural condition on $f$ to discard such phenomena, that can be interpreted as a balance between the variations with respect to the first variable and the growth with respect to the last one. This unifies most of the structural assumptions that have been introduced so far to prevent the occurence of Lavrentiev gaps. Remarkably, typical assumptions that are usually imposed on $f$ in this setting are dropped here: we do not require $f$ to be bounded or convex with respect to the second variable, nor impose any condition of $Δ_2$-kind with respect to the last variable.

math.AP↗

arXiv2025

Composition operators in Orlicz-Sobolev spaces

The continuity of the Nemytskii operator between Orlicz-Sobolev spaces is investigated. Natural Orlicz-Sobolev versions of classical results for standard Sobolev are established. The results presented not only extend the latter, but also improve them in borderline situations. Anisotropic Orlicz-Sobolev spaces are included in our analysis. The results offered for this class of spaces are new even for customary anisotropic Sobolev spaces.

math.FA↗

arXiv2025

Convergence of projected stochastic approximation algorithm

We study the Robbins-Monro stochastic approximation algorithm with projections on a hyperrectangle and prove its convergence. This work fills a gap in the convergence proof of the classic book by Kushner and Yin. Using the ODE method, we show that the algorithm converges to stationary points of a related projected ODE. Our results provide a better theoretical foundation for stochastic optimization techniques, including stochastic gradient descent and its proximal version. These results extend the algorithm's applicability and relax some assumptions of previous research.

math.OC↗

arXiv2023

Boundedness of Wolff-type potentials and applications to PDEs

We provide a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff--Havin--Maz'ya type. As a consequence, we prove a reduction principle for that integral operators, that is, a characterization of those rearrangement invariant spaces between which the potentials are bounded via a one-dimensional inequality of Hardy-type. Since the special case of the mentioned potential is known to control precisely very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, we infer the local regularity properties of the solutions in rearrangement invariant spaces for prescribed classes of data.

math.AP↗

arXiv2023

Absence and presence of Lavrentiev's phenomenon for double phase functionals upon every choice of exponents

We study classes of weights ensuring the absence and presence of the Lavrentiev's phenomenon for double phase functionals upon every choice of exponents. We introduce a new sharp scale for weights for which there is no Lavrentiev's phenomenon up to a counterexample we provide. This scale embraces the sharp range for $α$-Hölder continuous weights. Moreover, it allows excluding the gap for every choice of exponents $q,p>1$.

math.AP↗

arXiv2022

Absence of Lavrentiev's gap for anisotropic functionals

We establish the absence of the Lavrentiev gap between Sobolev and smooth maps for a non-autonomous variational problem of a general structure, where the integrand is assumed to be controlled by a function which is convex and anisotropic with respect to the last variable. This fact results from new results on good approximation properties of the natural underlying unconventional function space. Scalar and vector-valued problems are studied.

math.AP↗

arXiv2021

Controlling monotonicity of nonlinear operators

Controlling the monotonicity and growth of Leray--Lions' operators including the $p$-Laplacian plays a fundamental role in the theory of existence and regularity of solutions to second order nonlinear PDE. We collect, correct, and supply known estimates including the discussion on the constants. Moreover, we provide a comprehensive treatment of related results for operators with Orlicz growth. We pay special attention to exposition of the proofs and the use of elementary arguments.

math.AP↗