Spectral Analysis of a Family of Second-Order Elliptic Operators with Nonlocal Boundary Condition Indexed by a Probabilty Measure

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dc.creatorAri, Iddo Ben
dc.creatorPinsky, Ross
dc.date2007-07-04
dc.date.accessioned2026-07-07T08:13:57Z
dc.date.available2026-07-07T08:13:57Z
dc.descriptionLet $D\subset R^d$ be a bounded domain and let \[ L=\frac12\nabla\cdot a\nabla +b\cdot\nabla \] %\[ %L=\frac12\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}+\sum_{i=1}^db_i\frac{\partial}{\partial x_i}, %\] be a second order elliptic operator on $D$. Let $ν$ be a probability measure on $D$. Denote by ${\mathcal L}$ the differential operator whose domain is specified by the following non-local boundary condition: $$ {\mathcal D_{\mathcal L}}=\{f\in C^2(\ol{D}): \int_D f dν= f|_{\partial D}\}, $$ and which coincides with $L$ on its domain. It is known that $\mathcal L$ possesses an infinite sequence of eigenvalues, and that with the exception of the zero eigenvalue, all eigenvalues have negative real part. Define the spectral gap of $\mathcal {L}$, indexed by $ν$, by γ_1(ν)\equiv\sup\{\re λ:0\neq λis an eigenvalue for {\mathcal L}\}. In this paper we investigate the eigenvalues of $\mathcal L$ in general and the spectral gap $γ_1(ν)$ in particular. The operator $\mathcal L$ is the generator of a diffusion process with random jumps from the boundary, and $γ_1(ν)$ measures the exponential rate of convergence of this process to its invariant measure.
dc.descriptionTo appear in the Journal of Functional Analysis
dc.identifierhttps://arxiv.org/abs/0707.0612
dc.identifierhttp://arxiv.org/abs/0707.0612
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/132955
dc.subjectSpectral Theory
dc.subjectAnalysis of PDEs
dc.subject35P15, 35P25
dc.titleSpectral Analysis of a Family of Second-Order Elliptic Operators with Nonlocal Boundary Condition Indexed by a Probabilty Measure
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