41

Besse (1978, p.202) has the spectra of compact rank 1 symmetric spaces (CROSSes). In addition to $\mathrm S^n$ due apparently to Heine (1863, §19; 1878, §128), this gives $\mathbf{RP}^n$, $\mathbf{CP}^n$, $\mathbf{HP}^n$ and $\mathbf{OP}^2$. Edit: Also, for $M$ a compact semisimple Lie group it is well known (due apparently to Freudenthal (1954))1 that the ...


22

It does not hold for hyperbolic plane. It follows since the volume growth of the hyperbolic plane is exponential, while volume growth of $\mathbb{R}^N$ is polynomial.


20

You can compute the eigenvalues explicitly for flat tori $\mathbb{R}^n/\Gamma$, where $\mathbb{R}^n$ has the standard Euclidean metric and $\Gamma$ is a lattice. The eigenvectors all have the form $e^{2 \pi i \langle v, w \rangle}$ where $v \in \mathbb{R}^n$ and $w \in \Gamma^{\vee}$ lies in the dual lattice. The corresponding eigenvalue is $-4 \pi^2 \| w \|^...


19

There is a general result of Seeley which states that if $A$ is an elliptic, selfadjoint positive scalar pseudo-differential operator of order $k$ on a compact Riemann manifold, then for any $\newcommand{\bR}{\mathbb{R}}$$s\in\bR$ the operator $A^s$ defined by functional calculus is in fact described by an elliptic selfadoint pseudo-differential ...


14

Observe that if $u$ is homogeneous of degree $\alpha$, then $N(r) \equiv \alpha$. (After integrating by parts, the numerator becomes $r \int_{\partial B_r} u u_{\nu} = \alpha \int_{\partial B_r} u^2$.) In addition, harmonic functions that are homogeneous of degree $\alpha$ tend to oscillate over distance $\sim \alpha^{-1}$ on $\partial B_1$ (think $r^{\alpha}...


13

The answer is No, and this is an interesting question. As Terry Tao commented, the test case is when $B$ is replaced by a half-space $H$, so let us consider the latter case for a moment and fields $u$ with compact support in $\bar B$ (i.e. not vanishing at the boundary). Because $H$ is dilation invariant, a Garding inequality implies actually an ...


13

Jeffrey Weeks has computed the spectra of homogeneous elliptic manifolds. For arithmetic hyperbolic manifolds, the spectrum is in principle computable in the sense that one may define a Selberg zeta function arithmetically, whose roots give the spectrum. Certain other homogeneous Heisenberg manifolds have their spectra computed.


12

That doesn't work because $H_0^1$ functions are small near the boundary, so testing against them won't detect bad behavior of $u$ near $\partial\Omega$. For a concrete example, take $\Omega$ as the unit ball and $u(x)=1/(1-|x|)\notin L^1$. Then $$ \int |uv|\, dx \le \left( \int \frac{v^2\, dx}{(1-|x|)^{3/2}} \int \frac{dx}{(1-|x|)^{1/2}} \right)^{1/2} . $$ ...


12

This is actually a system of first-order PDEs, of $2n$ (the dimension of $M$) equations. To see that it is elliptic, let us consider the symplest case of $M={\mathbb R}^{2n}$ with the standard symplectic structure, $H\equiv0$ and $$J=\begin{pmatrix} 0_n & I_n \\ -I_n & 0_n \end{pmatrix}.$$ then the system writes blockwise as $$\partial_sv+\...


11

There are lots of nontrivial solutions for any negative $\lambda$. Here's how to construct them all. First, we can decompose an arbitrary symmetric $2$-tensor $h$ as $h=fg+u$, where $f$ is a scalar function and $u$ is trace-free. It follows that $\operatorname{tr} h = mf$, and $\delta_g\delta_g h = -\Delta f + \delta_g\delta_g u$. After some simplification,...


11

Perhaps to resolve this issue it helps to work out a simple example. Take a region $D$ consisting of the strip $|x|<1$, $0<y<1$, and a $y$-independent conductivity profile $$\sigma(x)=\begin{cases} 1 &\text{for} -1<x<0,\\ 2& \text{for}\;\;\;\; 0<x<1. \end{cases} $$ The solution of the Poisson equation $\nabla \cdot (\sigma\nabla ...


10

Because it is not true. For example, suppose that $f: D^n \to D^n$ (the unit disk in $R^n$ with Euclidean metric) is harmonic -- i.e. the components are bounded harmonic functions in the ordinary sense satisfying $f_1^2 + \ldots + f_n^2 < 1$. Let $\phi: D^n \to D^n$ be the identity map, where the domain has the Euclidean and the range has the hyperbolic ...


10

I made a trip to the library and scanned the relevant pages from Miranda's 1955 book: page 152-153 and page 154-155 the references are: [3] J. Leray, J.Math. pures et appl. 17, 89-104 (1938) [8] R. Cacciopoli, Rend.Acc.Lincei 22, 305-310 and 376-379 (1935). [11] R. Cacciopoli, Giorn.Mat.Battaglini, 80, 186-212 (1950-51)


10

Here is a simple way of producing first order elliptic operators $\newcommand{\bR}{\mathbb{R}}$ $C^\infty(\mathbb{R}^n, W)\to C^\infty(\bR^n, W)$ with constant coefficients. Denote by $L(W)$ the space of linear operators $W\to W$. Consider a map $\newcommand{\si}{\sigma}$ $$\si :\bR^n\to L(W), $$ such that $\si(x)$ is invertible for any $x\in\mathbb{R}^...


10

1.If $U$ satisfies LAP then there exists a $V$ such that $(U,V)$ satisfies CR. In fact, $V$ is unique up to the addition of a term of the form $a + bx + cy + d xy$, where $a$, $b$, $c$, and $d$ are constants. This is an elementary consequence of the Frobenius Theorem. 2.You need to specify what you mean by 'algebra'. The space of functions $F = U + i V$ ...


10

If it exists, the inverse of an elliptic operator $P$ is its Green's operator. In general, an inverse does not exist, but a parametrix does. A parametrix is an operator $Q$ such that $PQ-I$ and $QP-I$ are compact operators. (I am assuming that $P$ is an elliptic operator on a closed manifold and acts between sections of vector bundles.) The existence of a ...


10

This question requires an articulated answer, since the topic dealt is complex and ramified. A fundamental solution for a not necessarily divergence form $2$nd order elliptic system with $C^{2,h}$ coefficients was first constructed by Georges Giraud in 1932 ([5]) by using the theory of multidimensional singular integrals he developed at the same time and ...


9

Perhaps you would be interested in Witten's proof of the Poincare-Hopf theorem. Given a smooth nondegenerate vector field $V$ on a smooth closed manifold $M$, the theorem asserts that the Euler characteristic of $M$ is equal to the sum of the signs of the critical points of $V$. Perhaps this isn't as interesting as the dynamical behavior that you mentioned ...


9

Let $A$ be this matrix. Because of the formula $$\int_D\sigma\nabla u\cdot\nabla v\, dx=\sum_{i,j}a_{ij}U_iI_j,$$ ($U$ for voltages of $u$, $I$ for currents of $v$), we see three necessary conditions: the matrix must be symmetric, it must be positive semi-definite, and $A{\bf1}=0$. Actually, if $U\ne \mu{\bf1}$, then $u$ is not constant and the choice $v=u$...


9

Dini continuity may be what you are looking for: if $f = \Delta u$ is Dini continuous, that is, $$\int_0^1 \frac{\omega_f(t)}{t} \, dt < \infty,$$ then $u$ is $C^2$. This is a rather old result, but I do not know the reference. A quick Google search leads to Poisson's equation by T. Gantumur, see Theorem 42 and Corollary 43 there.


8

Certainly not: think of $\Omega$ the unit disk, and $u$ the harmonic extension to $\Omega$ of any continuous, nowhere differentiable function on $\partial \Omega$.


8

Why is it called this name? Diamagnetism is a quantum effect in which material under an applied external magnetic field generate their own "opposite" magnetic field. The most prominent example happens in superconductors. A consequence of this is that matter under an applied external magnetic field "gain" energy (or rather, removing magnetic fields ...


8

Not an precise answer : As far as I remember this is not really an issue. One has to replace by an other version of divergence theorem ( for less regular domain) using geometric measure theory . The key point I think is that the set where the nodal lines are not regular (often called the singular set $\{u=0\}\cap\{\nabla u=0\}$ is of dimension at most $n-...


8

Start with the linear problem: Let $K \subset \Gamma(E)$ denote the kernel of $P$ and $\hat{K} \subset \Gamma(F)$ the cokernel. Assume that there are inner products on the vector bundles $E$ and $F$ and let $K^\perp$ and $\hat{K}^\perp$ the orthogonal complements with respect to the induced $L^2$ inner product on $\Gamma(E)$ and $\Gamma(F)$. Then the ...


8

This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on elliptic PDEs): Given a bounded open domain $\Omega \subset \mathbb{R}^n$, there exists $C>0$ such that for any function (or even just a distribution) $u$ compactly ...


8

I don't know the literature, but it maybe helpful to understand exactly what this notion of differentiability gains for us. First, unlike the Sobolev notions, this does not handle functions which belong in Holder classes. For example, based on the definition the absolute value function is not $L^p$ differentiable for any $p$, but it is certainly (locally) ...


7

To begn with, your Boundary-Value Problem (BVP) is under-determined, because it lacks one boundary condition: because the PDE is elliptic and fourth-order, you need two boundary conditions, not only one. Because you insist on working with $H^2\cap H^1_0$, it seems that the hidden BC is $$u=0\qquad\hbox{on }\partial\Omega.$$ So let us assume that your BVP ...


7

Careful. C^\infty is not dense in C^{\alpha}. Nonetheless you can first produce an L^2 solution then use Schauder estimates to show that the solution is C^{2,\alpha}. This last step is localizable -- you can multiply by a smooth cutoff function (and use some preliminary estimates to say that the solution is C^2, for example) and transfer to a coordinate ...


7

De Giorgi solved Hilbert's 19th problem (http://en.wikipedia.org/wiki/Hilbert's_nineteenth_problem) Nash independently and almost simultaneously obtained the parabolic version of the same result. Nash's result implies that all quasilinear parabolic equations, under some very reasonable assumptions, have smooth solutions. Both De Giorgi's proof and Nash's ...


7

A precise statement and proof of the relationship between stability and the existence of Calabi-Yau metrics is in: arXiv:1302.0282, Xiuxiong Chen, Simon Donaldson, Song Sun, Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof arXiv:1212.4714, Xiuxiong Chen, Simon Donaldson, Song Sun, Kahler-...


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