Questions tagged [elliptic-pde]
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
1,263 questions
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System of divergence free vector fields
Let $\Omega \subset \mathbb R^p$ be a convex, bounded domain with a smooth boundary. Let $a_{ij} : \Omega \to \mathbb R_+$ be a non-negative smooth function for $i, j \in \{1, 2\}.$ I am interested in ...
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Bochner identity for general linear equations
I am looking to prove the following: Let $\lambda \leq A(x) \leq \Lambda$ be a symmetric, positive definite matrix. Let $u$ solve $\text{div} A(x) \nabla u = 0$. Then I want to show $$ \Lambda \Delta ...
- 569
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Non-attainment of infimum energy among smooth competitors in Dirichlet problem
Let $M$ be a compact Riemannian manifold with boundary $\partial M \ne \emptyset$ and dimension at least $2$, and let $N$ be a compact Riemannian manifold with no boundary of dimension at least $2$ ...
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Higher-order vanishing near the boundary for solutions to $- \Delta u = f$
I was considering a certain equivalence of norms on $H^{- 1}(\Omega)$, which led me to the following question: here, let $\Omega \subseteq \mathbb{R}^d$ be a bounded open set, with $C^{\infty}$ ...
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The first eigenvalue of linear elliptic operator which is not uniformly elliptic
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary (the smoothness assumption may be relaxed). Consider a linear differential operator
$$D=-a_{ij}(x)\partial_i\partial_j +b_i(x)\...
- 23.3k
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Existence of stable or finite Morse index solutions for the 1D Liouville equation
I'm interested in solutions of the Liouville equation $$-\Delta u = e^u$$ in $\mathbb{R}^N$.
In this article, Farina proves that for $2\leq N \leq 9$ there is no stable solution. He also remarks that ...
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Reference request: potential theory for general uniformly elliptic operators
I am looking for textbooks/references that discuss the potential theory (in particular Green's functions) of general uniformly elliptic (linear) operators. Ideally, it should include:
existence of ...
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regularity of the kernel $K(x,y) $?
Let $L $ be a differential operator on $ L^2(\mathbb{R}^n) $, self-adjoint with spectrum $ \mathbb{R}^+ $. For $ \lambda \notin \mathbb{R}^+ $, the resolvent $ R_\lambda = (L - \lambda I)^{-1} $ is ...
- 11
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Comparison principle between different elliptic operators
Let $\Omega$ be a sufficiently smooth bounded domain in $\mathbb{R}^n$. Let $u_i$ be non-negative functions defined in $\overline{\Omega}$ and consider elliptic linear operators in divergence form $L :...
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Asymptotic behavior of the Green function of $(-\Delta_{g_{\mathbb{S}^3}}+1)^{1/2}$ on $\mathbb{S}^3$
Let $A=\left(-\Delta_{g_{\mathbb{S}^3}}+1\right)^{1 / 2}$. I expect the Green function on $(\mathbb{S}^3,g_{\mathbb{S}^3})$ has the asymptotic behavior $G(x, y) \sim c d(x, y)^{-2}$ as $x \rightarrow ...
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Morse index of mountain pass solution for fractional Laplacian
Suppose $u\in H^s_{0}(\Omega)$ is a classical solution of
\begin{equation}\begin{cases}
(-\Delta)^s u = f(u) & \text{in }\Omega,\\
u=0 & \text{in }\Omega^c.
\end{cases}\end{equation}
with $N\...
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Simplicity of elliptic eigenvalue problem
I had asked a (related question) a few months ago - but now I have a different question in the same setting:
Consider the generalized eigenvalue problem:
$$ [- \nabla \cdot (D(\mathbf{x}) \nabla) + \...
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Abstract higher-order parabolic problems (reference request)
I'm currently reading "Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems" by H. Amann and I have some doubts on how this work may be generalized to higher ...
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Regularity of the Leray projection on the diffeomorphism group
Let $(M, \langle \cdot, \cdot \rangle)$ be a closed $C^\infty$ Riemannian manifold with $\mathrm{dim}(M) = m$ and let $\mathrm{vol}$ denote the renormalized volume measure on $M$. Let also $s > m/2$...
- 325
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Spectral theory for the boundary operator $P_{g}^{3,b}$ arising from the Paneitz operator on the 4-ball
I am working with the following eigenvalue problem on the 4–ball $(\mathbb{B}^4,g)$, associated with the Paneitz operator $(P_g^4)$ and its boundary operator $P_g^{3,b}$ on $(\mathbb{S}^3,\hat g)$:
\...
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