Joaquim Serra

Bio: Joaquim Serra is an academic researcher from ETH Zurich. The author has contributed to research in topics: Boundary (topology) & Order (ring theory). The author has an hindex of 25, co-authored 65 publications receiving 3084 citations. Previous affiliations of Joaquim Serra include Polytechnic University of Catalonia.

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

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}.

The Pohozaev identity for the fractional Laplacian

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

The Pohozaev identity for the fractional Laplacian

Abstract: 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 $\partial u/\partial u$ 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.

Lévy processes and infinitely divisible distributions

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.