Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coercivity, essential norms, and the Galerkin method for second-kind
  integral equations on polyhedral and Lipschitz domains   It is well known that, with a particular choice of norm, the classical
double-layer potential operator $D$ has essential norm $<1/2$ as an operator on
the natural trace space $H^{1/2}(\Gamma)$ whenever $\Gamma$ is the boundary of
a bounded Lipschitz domain. This implies, for the standard second-kind boundary
integral equations for the interior and exterior Dirichlet and Neumann problems
in potential theory, convergence of the Galerkin method in $H^{1/2}(\Gamma)$
for any sequence of finite-dimensional subspaces $(\mathcal{H}_N)_{N=1}^\infty$
that is asymptotically dense in $H^{1/2}(\Gamma)$. Long-standing open questions
are whether the essential norm is also $<1/2$ for $D$ as an operator on
$L^2(\Gamma)$ for all Lipschitz $\Gamma$ in 2-d; or whether, for all Lipschitz
$\Gamma$ in 2-d and 3-d, or at least for the smaller class of Lipschitz
polyhedra in 3-d, the weaker condition holds that the operators $\pm
\frac{1}{2}I+D$ are compact perturbations of coercive operators -- this a
necessary and sufficient condition for the convergence of the Galerkin method
for every sequence of subspaces $(\mathcal{H}_N)_{N=1}^\infty$ that is
asymptotically dense in $L^2(\Gamma)$. We settle these open questions
negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz
constant equal to one for which the essential norm of $D$ is $\geq 1/2$, and
examples with Lipschitz constant two for which the operators $\pm \frac{1}{2}I
+D$ are not coercive plus compact. We also give, for every $C>0$, examples of
Lipschitz polyhedra for which the essential norm is $\geq C$ and for which
$\lambda I+D$ is not a compact perturbation of a coercive operator for any real
or complex $\lambda$ with $|\lambda|\leq C$. Finally, we resolve negatively a
related open question in the convergence theory for collocation methods.

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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int