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We consider a smooth compact manifold with boundary, M,  embedded in a smooth manifold of the same dimension on which an amenable group \(\Gamma\) acts by isometries. We do not assume M to be invariant under \(\Gamma\). This results in a partial action of \(\Gamma\) on: For $g\in \Gamma$ we let \(M^\circ_g = g(M^\circ)\cap M^\circ\) and obtain diffeomorphisms \(g:M^\circ_{g^{-1}} \to

M^\circ_g

\).

We assume that any two images of \(\partial M\) under \(\Gamma\) either coincide or are disjoint and that only finitely many lie in M. The spherical blow-up of these images of \(\partial M\) in M yields a

manifold Y with  boundary consisting of finitely many components. Moreover, Y inherits a partial action of \(\Gamma\).

We can then define the C*-algebra \(\mathcal

A=\overline{\Psi_\Gamma(Y,\partial Y)}\) of operators on \(L^2(Y)\oplus

L^2(\partial Y)\), generated by the algebra \(\Psi(Y,\partial Y)\) of operators of order and type zero in Boutet de Monvel's calculus on Y and partial isometries associated with the partial action. Denote by

\(\Sigma=\overline{\Psi(Y,\partial Y)}/\mathcal K

\) the symbol space. If the partial action of \(\Gamma\) on Prim(\(\Sigma\)) is topologically

free, we find a criterion for the Fredholm property of the operators in \(\overline{\Psi_\Gamma(Y,\partial Y)}\).

Moreover, we obtain the classification of the elliptic elements in

\(\overline{\Psi_\Gamma(Y,\partial Y)}\) modulo stable homotopies: For \(\mathcal A_0= C(Y\sqcup \partial Y)\rtimes\Gamma\)

\(Ell(\mathcal A_0,\mathcal A)\cong K_0(C_0(T^*Y^\circ)\rtimes\Gamma

)\oplus K_0(C(\partial Y)\rtimes \Gamma).\)

If \(\Gamma\) is finitely generated and of polynomial growth, then the elements associated with the second summand do not contribute to the index.

Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics

Let G be a compact Lie group that acts smoothly on a closed manifold M. Using a general Simonenko principle, we derive a novel criterion for the Fredholm property of G-pseudodifferential operators acting on Sobolev spaces of sections of vector bundles over M. In case the group is finite, we obtain a further characterization of the Fredholm property of G-pseudodifferential operators in terms of the invertibility of suitable symbols.

Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics

This is an introduction to the analysis of nonlinear evolution equations on manifolds with conical singularities via maximal regularity techniques. We address the specific difficulties due to the singularities, in particular the choice of extensions of the conic Laplacian that guarantee the existence of a bounded \(H_\infty\)-calculus. We introduce the relevant technical tools and survey, as main examples, applications to the porous medium equation, the fractional porous medium equation, the Yamabe flow, and the Cahn-Hilliard equation.

Related project(s):
30Nonlinear evolution equations on singular manifolds

We show that, on a manifold with conical singularities, the asymptotics of the solutions to the porous medium equation near the conical points are determined by the spectrum of the Laplacian on the cross-section of the cone. The key to this result is a precise description of the maximal domain of the cone Laplacian.

Related project(s):
30Nonlinear evolution equations on singular manifolds

A Calderón projector for an elliptic operator $P$ on a manifold with boundary $X$ is a projection from general boundary data to the set of boundary data of solutions $u$ of $Pu=0$. Seeley proved in 1966 that for compact $X$ and for $P$ uniformly elliptic up to the boundary there is a Calder\'on projector which is a pseudodifferential operator on $\partial X$. We generalize this result to the setting of fibred cusp operators, a class of elliptic operators on certain non-compact manifolds having a special fibred structure at infinity.

This applies, for example, to the Laplacian on certain locally symmetric spaces or on particular singular spaces, such as a domain with cusp singularity or the complement of two touching smooth strictly convex domains in Euclidean space. Our main technical tool is the $\phi$-pseudodifferential calculus introduced by Mazzeo and Melrose.

In our presentation we provide a setting that may  be useful for doing analogous constructions for other types of singularities.

Related project(s):
13Analysis on spaces with fibred cusps49Analysis on spaces with fibred cusps II

We study the Fredholm solvability for a new class of nonlocal boundary value problems associated with group actions on smooth manifolds. Namely, we consider the case in which the group action is defined on an ambient manifold without boundary and does not preserve the manifold with boundary on which the problem is stated. In particular, the group action does not map the boundary to itself. The orbits of the boundary under the group action split the manifold into subdomains, and this decomposition, being combined with the C*-algebra techniques, plays an important role in our approach to the analysis of the problem.

Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics

In this paper, we study curve shortening flow on Riemann surfaces with singular metrics. It turns out that this flow is governed by a degenerate quasilinear parabolic equation. Under natural geometric assumptions, we prove short-time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and prove some collapsing and convergence results.

Related project(s):
23Spectral geometry, index theory and geometric flows on singular spaces30Nonlinear evolution equations on singular manifolds

We show \(R\)-sectoriality for the fractional powers of possibly non-invertible \(R\)-sectorial operators. Applications concern existence, uniqueness and maximal \(L^{q}\)-regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided and its relation to the local geometry is established. Our method extends the freezing-of-coefficients method to the case of non-local operators that are expressed as linear combinations of terms in the form of a product of a function and a fractional power of a local operator.

Related project(s):
30Nonlinear evolution equations on singular manifolds

Realizations of differential operators subject to differential boundary conditions on manifolds with conical singularities are shown to have a bounded \(H_{\infty}\)-calculus in appropriate \(L_{p}\)-Sobolev spaces provided suitable conditions of parameter-ellipticity are satisfied. Applications concern the Dirichlet and Neumann Laplacian and the porous medium equation.

Related project(s):
30Nonlinear evolution equations on singular manifolds

We consider the Cahn-Hilliard equation on manifolds with conical singularities. For appropriate initial data we show that the solution exists in the maximal \(L^q\)-regularity space for all times and becomes instantaneously smooth in space and time, where the maximal \(L^q\)-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we provide precise information concerning the asymptotic behavior of the solution close to the conical tips in terms of the local geometry.

Related project(s):
30Nonlinear evolution equations on singular manifolds

It is observed that in Banach spaces, sectorial operators having bounded imaginary powers satisfy a Heinz-Kato inequality.

Related project(s):
30Nonlinear evolution equations on singular manifolds

We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross-section we show well posedness of the short time solution in the \(L^q\)-setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asymptotic expansion of the evolving metric close to the boundary in terms of the initial local geometry. Due to the blow up of the scalar curvature close to the singularities we use maximal \(L^q\)-regularity theory for conically degenerate operators.

Related project(s):
30Nonlinear evolution equations on singular manifolds

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal \(L^q\)-regularity space for all times and is instantaneously smooth in space and time, where the maximal \(L^q\)-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.

Related project(s):
30Nonlinear evolution equations on singular manifolds