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All approaches I am familiar with are based on some Tauberian theorem: one obtains informations about various weighted counts of eigenvalues and then one removes the weights via some Tauberian theor …

The result is true with some caveats. Under your assumptions we have the following results. 1. If $u\in W^{2k,2}(M)$ and $Lu\in C^{j,\alpha}(M)$, then $u\in C^{2k+j,\alpha}(M)$. 2. There exists $C> …

When $L$ has analytic coefficients, any solution of $Lu=0$ in $\mathbb{R}^n$ is automatically analytic. You should be asking about conditions guaranteeing that the only solution of this equation is $ …

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 s …

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 $\newc …

Try this old book by Guido Stampacchia Équations elliptiques du second ordre à coefficients discontinus, Presses de l'Université de Montréal 1966. If you cannot find this try this clasic by Olga La …

I assume that you require $f\in C^\infty(U)$. You do not need regularity of the boundary of $U\subset \mathbb{R}^N$. The condition $u\in H^m_{loc}(U)$ is equivalent with $\widetilde{\phi u}\in H^m(\ …

Standard regularity theory guarantees that $u$ is smooth. If you want to use Cauchy-Kovaleskaya you would need to know that both $g$ and $\frac{\partial u}{\partial \nu}$ are real analytic: these a …

First of all, there should be a constraint on the exponent $\tau$ to guarantee certain integrability conditions. Define $$n^*:=\frac{2(n+\lambda)}{n-2}, $$ where $n$ denotes the dimension of the ba …

In principle, one can compute the spectrum of any homogeneous compact Riemannian manifold because in this case the problem is essentially representation theoretic. However, performing this computation …

If you are on a compact domain $\Omega$in $\mathbb{R}^n$ you need to impose some elliptic boundary conditions or it is difficult to make predictions. Assume for simlicity that $f$ satisfies the D …

The unique continuation is valid for generalized Laplacians. This follows from Hörmander's result in Hörmander, Lars, Uniqueness theorems for second order elliptic differential equations, Commun. Pa …

$\DeclareMathOperator{\dist}{dist}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\pa}{\partial}$ Suppose that $N=2$ and $\Omega$ is is the unit disk. Choose $$ u= -1+ar^4+br^5\in C^2(\overline{\Omeg …