arXiv subjects

Agnieszka Kałamajska

Publications and source records attributed to Agnieszka Kałamajska.

11 recordsLinked to original sources

arXiv2026

Singular semilinear elliptic equations in nondivergence form

We study the singular semilinear equation $-Pu = \frac{f}{u^γ}$ on a bounded domain $Ω$ with Dirichlet condition $u \equiv 0$ on $\partial Ω$ , where $P$ is a second-order elliptic differential operator in nondivergence form. We obtain the existence of a solution under the assumptions that $Ω\in C^{1,1}$ and $P$ has $C^1$ coefficients, as well as the uniqueness of solutions in $L^1(Ω)$, under the assumptions that $Ω\in C^2$ and $P$ has $C^2$ coefficients. Our proofs are based on a novel combination of tools, such as recently obtained nonlinear variants of Gagliardo--Nirenberg inequalities, estimates of Green functions, and new variants of Kato-type inequalities.

math.AP↗

arXiv2025

Non-linear Gagliardo--Nirenberg inequality involving a second-order elliptic operator in non-divergent form

We obtain the inequalities of the form $$\int_Ω|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_Ω \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +Θ,$$ where $Ω\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{\rm loc}(Ω)$ is non-negative, $P$ is a uniformly elliptic operator in non-divergent form, ${\cal T}_{H}(\cdot )$ is certain transformation of the monotone $C^1$ function $H(\cdot)$, which is the primitive of the weight $h(\cdot)$, and $Θ$ is the boundary term which depends on boundary values of $u$ and $\nabla u$, which hold under some additional assumptions. Our results are linked to some results from probability and potential theories, e.g.~to some variants of the Douglas formulae.

math.AP↗

arXiv2022

Asymptotics, trace, and density results for weighted Dirichlet spaces defined on the halfline

We give analytic description for the completion of $C_0^\infty ( \mathbf{R}_+)$ in Dirichlet space $D^{1,p}(\mathbf{R}_+, ω):= \{ u:\mathbf{R}_+\rightarrow \mathbf{R}: u\ \hbox{ is locally absolutely continuous on} \ \mathbf{R}_+ \ {\rm and}\ \| u^{'}\|_{L^p(\mathbf{R}_+, ω)}<\infty \}$, for given continuous weight $ω$, in terms of the local $B_p$ conditions due to Kufner and Opic, where $1<p<\infty$. Moreover, we propose applications of our results to: analysis of Hardy inequalities, boundary value problems, complex interpolation theory, and to derivation of new Morrey type inequalities.

math.FA↗

arXiv2022

Density results and trace operator in weighted Sobolev Spaces defined on the half line equipped with power weights

We study properties of $W_0^{1,p}(\mathbb{R}_+,t^β)$ - the completion of $C_0^\infty(\mathbb{R}_+)$ in the power-weighted Sobolev spaces $W^{1,p}(\mathbb{R}_+,t^β)$, where $β\in\mathbb{R}$. Among other results, we obtain the analytic characterization of $W_0^{1,p}(\mathbb{R}_+,t^β)$ for all $β\in \mathbb{R}$. Our analysis is based on the precise study the two trace operators: $Tr^{0}(u):= \lim_{t\to 0} u(t)$ and $Tr^{\infty}(u):= \lim_{t\to \infty} u(t)$, which leads to the analysis of the asymptotic behavior of functions from $W_0^{1,p}(\mathbb{R}_+,t^β)$ near zero or infinity. The obtained statements can contribute to the proper formulation of Boundary Value Problems in ODE's, or PDE's with the radial symmetries. We can also apply our results to some questions in the complex interpolation theory, raised by M. Cwikel and A. Einav in 2019, which we discuss within the particular case of Sobolev spaces $W^{1,p}(\mathbb{R}_+,t^β)$.

math.CA↗

arXiv2021

Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted $L^q$ spaces on metric spaces

We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into $L^q$ spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences of covering families and local Poincaré type inequalities. We show how to construct such suitable coverings and Poincaré inequalities. For locally doubling measures, we prove a self-improvement property for two-weighted Poincaré inequalities, which applies also to lower-dimensional measures. We simultaneously treat various Sobolev spaces, such as the Newtonian, fractional Hajłasz and Poincaré type spaces, for rather general measures and sets, including fractals and domains with fractal boundaries. By considering lower-dimensional measures on the boundaries of such domains, we obtain trace embeddings for the above spaces. In the case of Newtonian spaces we exactly characterize when embeddings into $L^q$ spaces with respect to another measure are compact. Our tools are illustrated by concrete examples. For measures satisfying suitable dimension conditions, we recover several classical embedding theorems on domains and fractal sets in ${\bf R}^n$.

math.FA↗

arXiv2019

On maximum principles for radial solutions to nonlinear elliptic PDE's via Opial-type inequalities

We consider degenerated nonlinear PDE of elliptic type: $$ - \mathrm{div}(a(|x|)|\nabla w(x)|^{p-2} \nabla w(x)) + h(|x|,w(x),\langle\nabla w(x),\frac{x}{|x|}\rangle)=ϕ(w(x)), $$ where $x$ belongs to the ball in $\bf{R}^n$. Using the argument based on Opial-type inequalities, we investigate qualitative properties of their radial solutions, like e.g. maximum principles, monotonicity, as well as nonexistence of the nontrivial solutions.

math.AP↗

arXiv2018

On one variant of strongly nonlinear Gagliardo-Nirenberg inequality involving Laplace operator with application to nonlinear elliptic problems

We obtain the inequality $$\int_Ω|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_Ω \left( \sqrt{ |Δu(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $Ω\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{loc}(Ω)$ is postive and obeys some additional assumptions, $Δu$ is the Laplace operator, ${\cal T}_{h,C}(\cdot )$ is certain transformation of the continuous function $h(\cdot)$. We also explain how to apply such inequality to deduce regularity for solutions of nonlinear eigenvalue problems of elliptic type for degenerated PDEs, with the illustration within the model of electrostatic micromechanical systems (MEMS).

math.AP↗

arXiv2017

Weak lower semicontinuity by means of anisotropic parametrized measures

It is well known that besides oscillations, sequences bounded only in $L^1$ can also develop concentrations, and if the latter occurs, we can at most hope for weak$^*$ convergence in the sense of measures. Here we derive a new tool to handle mutual interferences of an oscillating and concentrating sequence with another weakly converging sequence. We introduce a couple of explicit examples showing a variety of possible kinds of behavior and outline some applications in Sobolev spaces.

math.AP↗

arXiv2016

On certain variant of strongly nonlinear interpolation inequality in dimension n

We obtain the inequality $$\int_Ω|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_Ω \left( \sqrt{ |\nabla^{(2)} u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $Ω\subseteq {\bf R}^n$ and $n\ge 2$, $u:Ω\rightarrow {\bf R}$ is in certain subset in second order Sobolev space $W^{2,1}_{loc}(Ω)$, $\nabla^{(2)} u$ is the Hessian matrix of $u$, ${\cal T}_{h,C}(u)$ is certain transformation of the continuous function $h(\cdot)$. Such inequality is the generalization of similar inequality holding in one dimension, obtained earlier by second author and Peszek.

math.AP↗

arXiv2016

Caccioppoli--type estimates and Hardy--type inequalities derived from degenerated $p$--harmonic problems

We obtain Caccioppoli--type estimates for nontrivial and nonnegative solutions to the anticoercive partial differential inequalities of elliptic type involving degenerated $p$--Laplacian: $-Δ_{p,a} u:= -\mathrm{div}(a(x)|\na u|^{p-2}\na u)\ge b(x)Φ(u)$, where $u$ is defined in a domain $Ω$. Using Caccioppoli--type estimates, we obtain several variants of Hardy--type inequalities in weighted Sobolev spaces.

math.AP↗

arXiv2011

On some nonlinear extensions of the Gagliardo-Nirenberg inequality with applications to nonlinear eigenvalue problems

We derive inequality [\int_{\r} |f^{'}(x)|^ph(f(x))dx \le (\sqrt{p-1})^p\int_{\r}(\sqrt{|f^{"}(x){\cal T}_h(f(x))|})^ph(f(x))dx,] where $f$ belongs locally to Sobolev space $W^{2,1}$ and $f^{'}$ has bounded support. Here $h(...)$ is a given function and ${\cal T}_h(...)$ is its given transform, it is independent of $p$. In case when $h\equiv 1$ we retrieve the well known inequality: (\int_{\r} |f^{'}(x)|^pdx \le (\sqrt{p-1})^p \int_{\r}(\sqrt{|f^{"}(x)f(x)|})^pdx.) Our inequalities have form similar to the classical second order Oppial inequalites. They also extend certain class of inequalities due to Mazya, used to obtain second order isoperimetric inequalities and capacitary estimates. We apply them to obtain new apriori estimates for nonlinear eigenvalue problems.

math.AP↗