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Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

2 votes
1 answer
126 views

Spectral analysis for nonlocal elliptic operator

Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to $$-\Delta u=f,\qua …
4 votes
1 answer
659 views

Poincare Inequality for $H^2$ function satisfying homogeneous Robin boundary conditions

Let $\Omega\subset\mathbb{R}^3$ be a bounded smooth domain. In general, for a Poincare inequality of the type $$\|u\|_{L^2}\le C \|\nabla u\|_{L^2}$$ to hold for all $u\in X\subset H^1(\Omega)$ and $C …
1 vote
1 answer
269 views

Poisson equation in a periodic strip

Consider the periodic strip $\Omega=\mathbb{T}\times[0,1]$ where $\mathbb{T}$ is the 1D torus with period 1. We consider the mixed Dirichlet/Neumann problem $$-\Delta u=f$$ with boundary conditions $$ …