Tomasz Klimsiak

activity

20092026

most citedDirichlet forms and semilinear elliptic equations with measure data

53 citations · 217 across the 49 of their papers we have counts for

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33 papers · 1 filter

math.AP2026

Stability of Solutions to Semilinear Exterior Dirichlet Problems under Convergence of Lévy Symbols

Tomasz Klimsiak, Andrzej Rozkosz, Leszek Słomiński

We study stability of semilinear exterior Dirichlet problems driven by possibly nonsymmetric Levy operators. Let $L^{n}$ and $L$ have Lévy-Khintchine symbols $ψ_{n}$ and $ψ$ , respectiv…

math.AP2026

Isolated Singularities and Measure Data Problems for Semilinear Equations Driven by Stable Lévy Operators

Kamil Dunst, Tomasz Klimsiak

We investigate positive solutions of semilinear equations driven by uniformly elliptic strictly $2 s$ -stable Lévy operators, where $s \in (0, 1)$ . We first prove that every positive d…

math.AP2026

Positive Solutions for Sublinear Equations with Compact Positivity-Improving Resolvent

Tomasz Klimsiak

We establish a Brézis-Oswald-type spectral principle for semilinear equations \[ -Lu=f(x,u), \] where $L$ is the generator of a positive $C_{0}$ -semigroup on $L^{p}$ with compact posit…

math.AP2025

Nonlinear Hardy-Stein type identities for harmonic functions relative to symmetric integro-differential operators

Tomasz Klimsiak, Andrzej Rozkosz

We show identities of Hardy-Stein type for harmonic functions relative to integro-differential operators corresponding to general symmetric regular Dirichlet forms satisfying the a…

math.AP2025

Bôcher type theorem for elliptic equations with drift perturbed Lévy operator

Tomasz Klimsiak

A classical Bôcher's theorem asserts that any positive harmonic function (with respect to the Laplacian) in the punctured unit ball can be expressed, up to the multiplication const…

math.AP2024

Poisson equation with measure data, reconstruction formula and Doob classes of processes

Andrzej Rozkosz, Tomasz Klimsiak

We consider the Dirichlet problem for equation involving a general operator associated with a symmetric transient regular Dirichlet form and bounded Borel measure on the right-hand…

math.AP2023

Generalized solutions to semilinear elliptic equations with measure data

Tomasz Klimsiak

We address an open problem posed by H. Brezis, M. Marcus and A.C. Ponce in: Nonlinear elliptic equations with measures revisited. In: Mathematical Aspects of Nonlinear Dispersive E…

math.AP2023★ 6 cited

Dirichlet problem for semilinear partial integro-differential equations: the method of orthogonal projection

Tomasz Klimsiak, Andrzej Rozkosz

We study the Dirichlet problem for semilinear equations on general open sets with measure data on the right-hand side and irregular boundary data. For this purpose we develop the c…

math.AP2021

Nonlinear elliptic equations with integro-differential divergence form operators and measure data under sign condition on the nonlinearity

Tomasz Klimsiak

We study existence problem for semilinear equations with Borel measure data and operator generated by a symmetric Markov semigroup. We assume merely that the nonlinear part satisfi…

math.AP2020

On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

Tomasz Klimsiak, Tomasz Komorowski

In the present paper we prove estimates on {subsolutions of the equation $- A v + c (x) v = 0$ }, $x \in D$ , where $D \subset R^{d}$ is a domain (i.e. an open and connected set) and $A$ is a…

math.AP2020

On uniqueness and structure of renormalized solutions to integro-differential equations with general measure data

Tomasz Klimsiak

We propose a new definition of renormalized solution to linear equation with self-adjoint operator generating a Markov semigroup and bounded Borel measure on the right-hand side. W…

math.AP2019

Location of zeros of non-trivial positive supersolutions to Schrödinger equations

Tomasz Klimsiak

We study Schrödinger operators on $L^{2} (E; m)$ of the form $- A + V$ with singular potentials $V$ . We address the question posed by H. Brezis about the structure of the set ${u = 0}$ fo…

math.AP2019

Schrödinger equations with smooth measure potential and general measure data

Tomasz Klimsiak

We study equations driven by Schrödinger operators consisting of a self-adjoint Dirichlet operator and a singular potential, which belongs to a class of positive Borel measures abs…

math.AP2019

Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator

Tomasz Klimsiak

We study the asymptotics of solutions of logistic type equations with fractional Laplacian as time goes to infinity and as the exponent in nonlinear part goes to infinity. We prove…

math.AP2019

Uniqueness for an obstacle problem arising from logistic-type equations with fractional Laplacian

Tomasz Klimsiak

We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under considerati…

math.AP2018

Reflected BSDEs and obstacle problem for semilinear PDEs in divergence form

Tomasz Klimsiak

We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and…

math.AP2018★ 2 cited

Smooth measures and capacities associated with nonlocal parabolic operators

Tomasz Klimsiak, Andrzej Rozkosz

We consider a family ${L_{t}, t \in [0, T]}$ of closed operators generated by a family of regular (non-symmetric) Dirichlet forms ${(B^{(t)}, V), t \in [0, T]}$ on $L^{2} (E; m)$ . We show…

math.AP2018

On the structure of diffuse measures for parabolic capacities

Tomasz Klimsiak, Andrzej Rozkosz

Let $Q = (0, T) \times Ω$ , where $Ω$ is a bounded open subset of $R^{d}$ . We consider the parabolic $p$ -capacity on $Q$ naturally associated with the usual $p$ -Laplacian. Droniou…

math.AP2018★ 3 cited

On semilinear elliptic equations with diffuse measures

Tomasz Klimsiak, Andrzej Rozkosz

We consider semilinear equation of the form $- Lu = f (x, u) + μ$ , where $L$ is the operator corresponding to a transient symmetric regular Dirichlet form $E$ , $μ$ is a diffuse…

math.AP2017

Trace operator and the Dirichlet problem for elliptic equations on arbitrary bounded open sets

Tomasz Klimsiak

We consider the Dirichlet problem on general, possibly nonsmooth bounded domain, for elliptic linear equation with uniformly elliptic divergence form operator. We investigate caref…

math.AP2016

Semilinear elliptic equations with Dirichlet operator and singular nonlinearities

Tomasz Klimsiak

In the paper we consider elliptic equations of the form $- A u = u^{- γ} \cdot μ$ , where $A$ is the operator associated with a regular symmetric Dirichlet form, $μ$ is a positive nontrivi…

math.AP2016★ 15 cited

Reduced measures for semilinear elliptic equations involving Dirichlet operators

Tomasz Klimsiak

We consider elliptic equations of the form (E) $- A u = f (x, u) + μ$ , where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincrea…

math.AP2016

Obstacle problem for evolution equations involving measure data and operator corresponding to semi-Dirichlet form

Tomasz Klimsiak

In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-D…

math.AP2016★ 4 cited

Renormalized solutions of semilinear elliptic equations with general measure data

Tomasz Klimsiak, Andrzej Rozkosz

In the paper, we first propose a definition of renormalized solution of semilinear elliptic equation involving operator corresponding to a general (possibly nonlocal) symmetric reg…