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Consider the general semilinear elliptic second-order PDE $$ u_t-\mathcal L u=f\left(t,x,u,\nabla u\right) $$ where $\mathcal L$ is an elliptic linear operator (like minus the Laplace operator), $t \...

Let $U$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$. Set $N = \frac{2n}{n-2}$. I am interested in the following equation: $$ -\Delta \phi + R \phi + \phi^{N-1} = 0 $$ ...

Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...

Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled? The boundary condition is that the ...