Filtern
Erscheinungsjahr
Dokumenttyp
- Wissenschaftlicher Artikel (40)
- Preprint (35)
- Monographie/Sammelband (27)
- Rezension (1)
Sprache
- Englisch (103)
Gehört zur Bibliographie
- ja (103) (entfernen)
Schlagworte
- Fredholm property (5)
- index (5)
- Cauchy problem (4)
- Navier-Stokes equations (4)
- Toeplitz operators (4)
- Dirac operator (3)
- Quasilinear equations (3)
- Riemann-Hilbert problem (3)
- classical solution (3)
- star product (3)
- Cauchy data spaces (2)
- Clifford algebra (2)
- Dirichlet to Neumann operator (2)
- Euler-Lagrange equations (2)
- Fredholm operators (2)
- Heat equation (2)
- Hodge theory (2)
- Laplace-Beltrami operator (2)
- Neumann problem (2)
- Removable sets (2)
- singular perturbation (2)
- trace (2)
- Analytic continuation (1)
- Analytic extension (1)
- Angular derivatives (1)
- Beltrami equation (1)
- Boundary value problem (1)
- Boundary value problems for first order systems (1)
- Carleman formulas (1)
- Composition operators (1)
- De Rham complex (1)
- Differential invariant (1)
- Dirichlet problem (1)
- Discontinuous Robin condition (1)
- Electromagnetic waves (1)
- Elliptic complex (1)
- Elliptic complexes (1)
- Equivalence (1)
- Euler equations (1)
- Extremal problem (1)
- Fischer-Riesz equations (1)
- Fredholm operator (1)
- Green formula (1)
- Green formulas (1)
- Holomorphic map (1)
- Holomorphic mappings (1)
- Lagrangian system (1)
- Lamé system (1)
- Lefschetz number (1)
- Lipschitz domain (1)
- Lipschitz domains (1)
- Newton method (1)
- Non-coercive problem (1)
- Non-linear semigroups (1)
- Nonlinear Laplace operator (1)
- Perturbed complexes (1)
- Porous medium equation (1)
- Primary: 47B35 (1)
- Root function (1)
- Scattering (1)
- Secondary: 47L80 (1)
- Semigroup (1)
- Stratton-Chu formulas (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 problems (1)
- characteristic boundary point (1)
- characteristic points (1)
- cohomology (1)
- composition operator (1)
- curvature (1)
- cusp (1)
- dbar-Neumann problem (1)
- discontinuous Robin condition (1)
- division of spaces (1)
- dynamical system (1)
- elliptic complex (1)
- elliptic complexes (1)
- ellipticity with parameter (1)
- equivalence (1)
- evolution equation (1)
- first boundary value problem (1)
- heat equation (1)
- inegral formulas (1)
- integral 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)
- mapping degree (1)
- mixed problems (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)
- pseudodifferential operator (1)
- quasiconformal mapping (1)
- quasilinear Fredholm operator (1)
- quasilinear equation (1)
- regular figures (1)
- regularisation (1)
- regularization (1)
- removable set (1)
- removable sets (1)
- root functions (1)
- singular integral equations (1)
- singular point (1)
- spectral kernel function (1)
- spectral theorem (1)
- spirallike function (1)
- strongly pseudoconvex domains (1)
- symplectic manifold (1)
- the Dirichlet problem (1)
- the first boundary value problem (1)
- weak boundary values (1)
- weighted Hölder spaces (1)
We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.
We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems.
We investigate nonlinear problems which appear as Euler-Lagrange equations for a variational problem. They include in particular variational boundary value problems for nonlinear elliptic equations studied by F. Browder in the 1960s. We establish a solvability criterion of such problems and elaborate an efficient orthogonal projection method for constructing approximate solutions.
We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed
problems, and construct an explicit formula for approximate solutions.
We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed problems, and construct an explicit formula for approximate solutions.
Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point
(2015)
The Dirichlet problem for the heat equation in a bounded domain aS, a"e (n+1) is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane t = c, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to the characteristic point were established by Petrovskii (1934) under the assumption that the Dirichlet data are continuous. The appearance of Petrovskii's paper was stimulated by the existing interest to the investigation of general boundary-value problems for parabolic equations in bounded domains. We contribute to the study of this problem by finding a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal characteristic boundary point and analyzing its asymptotic behavior.
Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point
(2012)
The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character.