# Questions tagged [elliptic-operators]

For questions about elliptic differential operators.

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### Approaches to Atiayh Singer index theorem

Soon i will have to choose a scientific adviser. Mathematicians in my university almost explicitly work on theory of ( partial ) differential equations, which i do not really like. But there is one ...
• 11
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### Approximation with bounded function

Let $\mathbb{D}$ be the unit disc, and let $B(o,r) \subset B(o,r')$ be two balls contained in $\mathbb{D}$. Assume that we have a $C^{\infty}$ function $f: \mathbb{D} \to [a,b]$ which has all its ...
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1 vote
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### Equivariant Multiplicative Formula for Sphere Bundles in the Index Theorem

I am reading through the proof of the Atiyah-Singer index theorem, out of Lawson-Michelson, and I'm a bit confused about the proof of multiplicative property for indices of sphere bundles, where I ...
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### DeRham complex for $ad(P)$-valued forms

Let $M$ be a smooth manifold. It is well known that the exterior differential $d:\Omega^r(M)\to \Omega^r(M)$ determines an elliptic complex. Let $(P,M,\pi)$ be a principal $G$-bundle over an oriented ...
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1 vote
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### On elliptic operators with non-continuous coefficients

Let $E, F \rightarrow M$ be two smooth vector bundles over a closed smooth manifold $M$, and $P: \Gamma(E) \rightarrow \Gamma(F)$ be a linear elliptic differential operator of order 1 with ...
• 321
1 vote
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### Regularity resultado for elliptic operator

Let $\Omega$ a smooth bounded domain in $\mathbb{R}^N$. Consider the following operator in the divergent form $$L(u) = \sum_{i,j=1}^{n}( a_{i,j} u_{x_i})_{x_j} + \sum_{i=1}^{n} b_i u_{x_i} + cu.$$ ...
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### A question on Hopf Lemma (strong maximum principle)

I'm following Evans here. The statement is as follows: take a subsolution $u\in C^2(U)\cap C^1(\overline{U})$ of a 2nd order elliptic operator $L$ in non-divergence form. Suppose also $U$ satisfies ...
• 1,795
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### Hessian of a functional

The question may be very basic, but I am a bit confused about the concepts, so it would be nice if you can clarify them for me and/or suggest some good references to fully understand them. I am ...
• 1,253
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### Formula for the inverse of Laplacian plus constant in a ball\semiball

Let $\lambda_1$ be the first positive eigenvalue of the following problem in the unit semiball $\mathbb{B}_+^n = \{x \in \mathbb{R}^n : \vert x \vert \leq 1 \text{ and } x_n \geq 0 \}$: \begin{cases} \...
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1 vote
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### Motivations for the eigenvalue problem of an elliptic operator

I was looking at the chapter 6.5 of Evans’ book about the eigenvalue problem for (anti-)symmetric elliptic operators, and I was wondering what were the motivations for such a problem. I guess there ...
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