/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

http://ieeexplore.ieee.org/iel5/5610941/5624686/05624745.pdf

Power electronics

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.

Lévy processes and infinitely divisible distributions

19세기말 애국적 담론과 新小說의 정체성

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The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

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}.

The Dirichlet problem for the fractional Laplacian: regularity up to the boundary

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

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The Pohozaev identity for the fractional Laplacian

In this paper we survey some results on the Dirichlet problem ( Lu = f in u = g in R n n for nonlocal operators of the form Lu(x) = PV Z Rn u(x) u(x + y) K(y)dy: We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. In order to include some natural operators L in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the eld. 2010 Mathematics Subject Classication: 47G20, 60G52, 35B65.

http://mat.uab.cat/pubmat/fitxers/download/FileType:pdf/FolderName:v60(1)/FileName:60116_01.pdf

Nonlocal elliptic equations in bounded domains: a survey

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $(-\Delta)^s u = f(u)$ in $\Omega$, $u \equiv 0$ in $\mathbb R^n\setminus\Omega$. Here, $s\in(0,1)$, $(-\Delta)^s$ is the fractional Laplacian in $\mathbb R^n$, and $\Omega$ is a bounded $C^{1,1}$ domain. 
To establish the identity we use, among other things, that if $u$ is a bounded solution then $u/\delta^s|_{\Omega}$ is $C^\alpha$ up to the boundary $\partial\Omega$, where $\delta(x)={\rm dist}(x,\partial\Omega)$. In the fractional Pohozaev identity, the function $u/\delta^s|_{\partial\Omega}$ plays the role that $\partial u/\partial\nu$ plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over $\partial\Omega$) which is completely local. 
As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.

Xavier Ros-Oton

Papers

The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

The Dirichlet problem for the fractional Laplacian: regularity up to the boundary

The Pohozaev identity for the fractional Laplacian

Nonlocal elliptic equations in bounded domains: a survey

The Pohozaev identity for the fractional Laplacian