Institut für Mathematik
Refine
Year of publication
Document Type
- Preprint (88)
- Article (36)
- Monograph/Edited Volume (24)
- Review (1)
Language
- English (149)
Keywords
- index (9)
- manifolds with singularities (6)
- Fredholm property (5)
- Toeplitz operators (5)
- Hodge theory (4)
- Navier-Stokes equations (4)
- pseudodifferential operators (4)
- star product (4)
- 'eta' invariant (3)
- Cauchy problem (3)
- Dirichlet to Neumann operator (3)
- Neumann problem (3)
- Quasilinear equations (3)
- boundary value problems (3)
- classical solution (3)
- differential operators (3)
- elliptic complexes (3)
- Beltrami equation (2)
- Cauchy data spaces (2)
- Dirac operator (2)
- Euler-Lagrange equations (2)
- Heat equation (2)
- Laplace-Beltrami operator (2)
- Lefschetz number (2)
- Nonlinear Laplace operator (2)
- Removable sets (2)
- Riemann-Hilbert problem (2)
- cohomology (2)
- elliptic operators (2)
- integral formulas (2)
- monodromy matrix (2)
- pseudodifferential operator (2)
- root functions (2)
- singular perturbation (2)
- trace (2)
- Analytic continuation (1)
- Analytic extension (1)
- Angular derivatives (1)
- Boundary value problem (1)
- Boundary value problems for first order systems (1)
- Capture into resonance (1)
- Carleman formulas (1)
- Clifford algebra (1)
- Composition operators (1)
- De Rham complex (1)
- Differential invariant (1)
- Dirichlet problem (1)
- Discontinuous Robin condition (1)
- Elliptic complexes (1)
- Equivalence (1)
- Euler equations (1)
- Extremal problem (1)
- Fischer-Riesz equations (1)
- Fredholm operator (1)
- Fredholm operators (1)
- G-index (1)
- G-trace (1)
- Green formula (1)
- Holomorphic map (1)
- Holomorphic mappings (1)
- Lagrangian system (1)
- Lamé system (1)
- Lipschitz domain (1)
- Lipschitz domains (1)
- Newton method (1)
- Non-coercive problem (1)
- Non-linear semigroups (1)
- Perturbed complexes (1)
- Porous medium equation (1)
- Primary: 47B35 (1)
- Pseudodifferential operators (1)
- Quasiconformal mapping (1)
- Root function (1)
- Second order elliptic equations (1)
- Secondary: 47L80 (1)
- Semigroup (1)
- Sturm-Liouville problem (1)
- Sturm-Liouville problems (1)
- Symplectic manifold (1)
- Unit disk (1)
- WKB method (1)
- Zeta-function (1)
- analytic continuation (1)
- asymptotic expansion (1)
- asymptotic methods (1)
- asymptotics (1)
- boundary layer (1)
- boundary value problem (1)
- boundary values problems (1)
- characteristic boundary point (1)
- characteristic points (1)
- composition operator (1)
- curvature (1)
- cusp (1)
- dbar-Neumann problem (1)
- de Rham complex (1)
- discontinuous Robin condition (1)
- division of spaces (1)
- divisors (1)
- domains with singularities (1)
- dynamical system (1)
- elliptic complex (1)
- ellipticity with parameter (1)
- equivalence (1)
- evolution equation (1)
- first boundary value problem (1)
- fundamental solution (1)
- geometric optics approximation (1)
- heat equation (1)
- inegral formulas (1)
- integral representation method (1)
- invariant (1)
- inverse problems (1)
- lattice packing and covering (1)
- lattice point (1)
- logarithmic residue (1)
- manifold with boundary (1)
- manifolds with cusps (1)
- manifolds with edges (1)
- mapping degree (1)
- matching of asymptotic expansions (1)
- meromorphic family (1)
- mixed problems (1)
- non-coercive boundary conditions (1)
- nonlinear equations (1)
- nonlinear semigroup (1)
- nonsmooth curves (1)
- p-Laplace Operator (1)
- p-Laplace equation (1)
- p-Laplace operator (1)
- polyhedra and polytopes (1)
- porous medium equation (1)
- quasiconformal mapping (1)
- quasilinear Fredholm operator (1)
- quasilinear equation (1)
- regular figures (1)
- regularisation (1)
- regularization (1)
- relative cohomology (1)
- removable set (1)
- removable sets (1)
- residue (1)
- singular integral equations (1)
- singular point (1)
- small parameter (1)
- spectral kernel function (1)
- spectral theorem (1)
- spirallike function (1)
- strongly pseudoconvex domains (1)
- symmetry group (1)
- symplectic manifold (1)
- the Dirichlet problem (1)
- the first boundary value problem (1)
- weak boundary values (1)
- weighted Hölder spaces (1)
- weighted spaces (1)
- ∂-operator (1)
Institute
- Institut für Mathematik (149) (remove)
We consider the Cauchy problem for the heat equation in a cylinder C (T) = X x (0, T) over a domain X in R (n) , with data on a strip lying on the lateral surface. The strip is of the form S x (0, T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S, we derive an explicit formula for solutions of this problem.
This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces.
We prove that if u is a locally Lipschitz continuous function on an open set chi subset of Rn + 1 satisfying the nonlinear heat equation partial derivative(t)u = Delta(vertical bar u vertical bar(p-1) u), p > 1, weakly away from the zero set u(-1) (0) in chi, then u is a weak solution to this equation in all of chi.
We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam´e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.
We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer.
We describe a natural construction of deformation quantization on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.
Let A be a nonlinear differential operator on an open set X subset of R-n and S a closed subset of X. Given a class F of functions in X, the set S is said to be removable for F relative to A if any weak solution of A(u) = 0 in XS of class F satisfies this equation weakly in all of X. For the most extensively studied classes F, we show conditions on S which guarantee that S is removable for F relative to A.
In a bounded domain with smooth boundary in R^3 we consider the stationary Maxwell equations
for a function u with values in R^3 subject to a nonhomogeneous condition
(u,v)_x = u_0 on
the boundary, where v is a given vector field and u_0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.