Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1705.05525

Boundary regularity, Pohozaev identities, and nonexistence results

Xavier Ros-Oton
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this expository paper we survey some recent results on Dirichlet problems of the form $Lu=f(x,u)$ in $Ω$, $u\equiv0$ in $\mathbb R^n\backslashΩ$. We first discuss in detail the boundary regularity of solutions, stating the main known results of Grubb and of the author and Serra. We also give a simplified proof of one of such results, focusing on the main ideas and on the blow-up techniques that we developed in \cite{RS-K,RS-stable}. After this, we present the Pohozaev identities established in \cite{RS-Poh,RSV,Grubb-Poh} and give a sketch of their proofs, which use strongly the fine boundary regularity results discussed previously. Finally, we show how these Pohozaev identities can be used to deduce nonexistence of solutions or unique continuation properties. The operators $L$ under consideration are integro-differential operator of order $2s$, $s\in(0,1)$, the model case being the fractional Laplacian $L=(-Δ)^s$.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xavier Ros-Oton. 2017-05-16. Boundary regularity, Pohozaev identities, and nonexistence results. https://arxiv.org/abs/1705.05525

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation

We study blow-up dynamics for the $L^2$-critical cubic Zakharov--Kuznetsov equation in two dimensions, \[ \partial_tu+\partial_{x_1}(Δu+u^3)=0 \qquad\text{on }\mathbb R^2. \] For a class of localized $H^1$ perturbations of the ground state $Q$, we establish a trichotomy near the soliton manifold: exit from a small $L^2$-tube, global asymptotic stability, or finite-time blow-up. In the stable blow-up regime, the solution concentrates a single bubble and \[ λ(t)\sim \ell_0(T-t)^{1/(3-c)}, \] where $\ell_0>0$ depends on the initial datum and $c\in(1,2)$ is an explicit constant determined by the transverse tail of the first-order approximate profile. Consequently, \[ \|\nabla u(t)\|_{L^2} \sim \frac{\|\nabla Q\|_{L^2}} {\ell_0(T-t)^{1/(3-c)}}. \] After subtraction of the concentrating soliton, the radiation converges strongly in $L^p(\mathbb R^2)$ for every $2\leq p<\infty$ to a common nonzero profile $u^*$, while \[ u^*\notin H^s(\mathbb R^2) \qquad\text{for every }s\geq\frac c2. \] The stable blow-up branch is open in the relative $H^1$ topology of the localized class. Finally, every non-soliton datum in this class with non-positive energy blows up in finite time. Interval-arithmetic computer-assisted proofs certify the numerical inputs to the virial coercivity argument. They also yield a rigorous enclosure of $c$, justifying the polynomial moment of order $21$ imposed on the initial data.

math.AP↗
arXiv2026

Propagation of wave packets close to conical intersections

In this paper, we study the propagation of wave packets close to conical intersections with respect to a system of two Schr{ö}dinger equations presenting a codimension 2 crossing. We focus on the dynamics that occur when the wave packets pass through an area close to the crossing, and our main results provide an explicit formula for the outgoing wave packet in terms of the incoming one, with a complete description of its phase and of the classical trajectories it follows, including a drift.

math.AP↗
arXiv2026

A Volterra Calculus for Lie Groupoids

A pseudodifferential Volterra calculus for inverting parabolic differential equations on Lie groupoids is introduced. This enables the study of fundamental solutions of various cases of heat flows on singular manifolds with corners with non-resonant boundary indicial symbols, such as the $b$-manifolds, as well as other geometric bisection covariant heat flows. We also establish the short time asymptotic expansion for the heat kernel of a positive, elliptic differential operator on a Lie groupoid that acts on suitable Sobolev Hilbert modules and is positive definite with respect to the appropriate $L^2$ inner product.

math.AP↗