Author
Tomasz Grzywny
Bio: Tomasz Grzywny is an academic researcher from Wrocław University of Technology. The author has contributed to research in topics: Heat kernel & Exponent. The author has an hindex of 19, co-authored 63 publications receiving 1280 citations.
Papers published on a yearly basis
Papers
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TL;DR: For the isotropic unimodal probability convolutional semigroups, this article gave sharp bounds for their Levy-Khintchine exponent with Matuszewska indices strictly between 0 and 2.
172 citations
TL;DR: In this paper, the authors give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.
Abstract: We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.
129 citations
TL;DR: In this paper, the scale invariant Harnack inequality and regularity properties for harmonic functions with respect to an isotropic unimodal Levy process with the characteristic exponent ψ satisfying some scaling condition were proved.
Abstract: We prove the scale invariant Harnack inequality and regularity properties for harmonic functions with respect to an isotropic unimodal Levy process with the characteristic exponent ψ satisfying some scaling condition. We derive sharp estimates of the potential measure and capacity of balls, and further, under the assumption that ψ satisfies the lower scaling condition, sharp estimates of the potential kernel of the underlying process. This allows us to establish the Krylov–Safonov type estimate, which is the key ingredient in the approach of Bass and Levin, that we follow.
121 citations
TL;DR: In this article, the authors give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.
Abstract: We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.
99 citations
TL;DR: In this paper, the authors give superharmonic functions and derive sharp bounds for the expected exit time and probability of survival for isotropic unimodal Levy processes in smooth domains.
Abstract: We give superharmonic functions and derive sharp bounds for the expected exit time and probability of survival for isotropic unimodal Levy processes in smooth domains.
63 citations
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Book•
01 Jan 2013
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.
1,957 citations
TL;DR: In this article, the Pohozaev identity up to the boundary of the Dirichlet problem for the fractional Laplacian was shown to hold for the case of ( − Δ ) s u = g in Ω, u ≡ 0 in R n \ Ω, for some s ∈ ( 0, 1 ) and g ∈ L ∞ ( Ω ), then u is C s ( R n ) and u / δ s | Ω is C α up to boundary ∂Ω for some α ∈( 0
804 citations
Posted Content•
TL;DR: In this article, the authors studied the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian and developed a fractional analog of the Krylov boundary Harnack method.
Abstract: We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if $u$ is a solution of $(-\Delta)^s u = g$ in $\Omega$, $u \equiv 0$ in $\R^n\setminus\Omega$, for some $s\in(0,1)$ and $g \in L^\infty(\Omega)$, then $u$ is $C^s(\R^n)$ and $u/\delta^s|_{\Omega}$ is $C^\alpha$ up to the boundary $\partial\Omega$ for some $\alpha\in(0,1)$, where $\delta(x)={\rm dist}(x,\partial\Omega)$. For this, we develop a fractional analog of the Krylov boundary Harnack method.
Moreover, under further regularity assumptions on $g$ we obtain higher order H\"older estimates for $u$ and $u/\delta^s$. Namely, the $C^\beta$ norms of $u$ and $u/\delta^s$ in the sets $\{x\in\Omega : \delta(x)\geq\rho\}$ are controlled by $C\rho^{s-\beta}$ and $C\rho^{\alpha-\beta}$, respectively.
These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian \cite{RS-CRAS,RS}.
427 citations
TL;DR: In this paper, the authors develop the theory in L p Sobolev spaces (1 p ∞ ) in a modern setting. But they do not cover complex powers of the Laplacian ( − Δ ) μ with μ ∉ Z, which are not covered in this paper.
311 citations
TL;DR: In this article, the Pohozaev identity for the semilinear Dirichlet problem has been proved for a non-local version of the problem with a boundary term (an integral over ∂Ω) which is completely local.
Abstract: In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem \({(-\Delta)^s u =f(u)}\) in \({\Omega, u\equiv0}\) in \({{\mathbb R}^n\backslash\Omega}\) Here, \({s\in(0,1)}\) , (−Δ)s is the fractional Laplacian in \({\mathbb{R}^n}\) , and Ω is a bounded C1,1 domain To establish the identity we use, among other things, that if u is a bounded solution then \({u/\delta^s|_{\Omega}}\) is Cα up to the boundary ∂Ω, where δ(x) = dist(x,∂Ω) In the fractional Pohozaev identity, the function \({u/\delta^s|_{\partial\Omega}}\) plays the role that ∂u/∂ν plays in the classical one Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over ∂Ω) which is completely local As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities
298 citations