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We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus.
We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus.
The inhomogeneous ∂-equations is an inexhaustible source of locally unsolvable equations, subelliptic estimates and other phenomena in partial differential equations. Loosely speaking, for the anaysis on complex manifolds with boundary nonelliptic problems are typical rather than elliptic ones. Using explicit integral representations we assign a Fredholm complex to the Dolbeault complex over an arbitrary bounded domain in C up(n).
We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted Lp-spaces over B,1 < p < ∞. Under suitable ellipticity assumptions we can define a family of complex powers A up(z), z ∈ C. We also obtain sufficient information on the resolvent of A to show the boundedness of the pure imaginary powers. Examples concern unique solvability and maximal regularity of the solution of the Cauchy problem u' - Δu = f, u(0) = 0, for the Laplacian on conical manifolds.
Boundary value problems for pseudodifferential operators (with or without the transmission property) are characterised as a substructure of the edge pseudodifferential calculus with constant discrete asymptotics. The boundary in this case is the edge and the inner normal the model cone of local wedges. Elliptic boundary value problems for non-integer powers of the Laplace symbol belong to the examples as well as problems for the identity in the interior with a prescribed number of trace and potential conditions. Transmission operators are characterised as smoothing Mellin and Green operators with meromorphic symbols.
Contents: Introduction 1 Edge calculus with parameters 1.1 Cone asymptotics and Green symbols 1.2 Mellin edge symbols 1.3 The edge symbol algebra 1.4 Operators on a manifold with edges 2 Corner symbols and iterated asymptotics 2.1 Holomorphic corner symbols 2.2 Meromorphic corner symbols and ellipicity 2.3 Weighted corner Sobolev spaces 2.4 Iterated asymptotics 3 The edge corner algebra with trace and potential conditions 3.1 Green corner operators 3.2 Smoothing Mellin corner operators 3.3 The edge corner algebra 3.4 Ellipicity and regularity with asymptotics 3.5 Examples and remarks
A function has vanishing mean oscillation (VMO) on R up(n) if its mean oscillation - the local average of its pointwise deviation from its mean value - both is uniformly bounded over all cubes within R up(n) and converges to zero with the volume of the cube. The more restrictive class of functions with vanishing lower oscillation (VLO) arises when the mean value is replaced by the minimum value in this definition. It is shown here that each VMO function is the difference of two functions in VLO.
It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptoic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schuze's notion of asymptotic type for conormal asymptotics close to a conical point is refined. This in turn allows to perform explicit calculations on asymptotic types - modulo the resolution of the spectral problem for determining the singular exponents in the asmptotic expansions.