Pliska Studia Mathematica Bulgarica
Volume 15, 2003
Proceedings of the International Conference on "Partial Differential Equations on Spaces with Geometric Singularities", Sofia, September 2-8, 2002
GUEST EDITORS: T. Gramchev, P. Popivanov
Sofia, 2003
C O N T E N T S
- Nicola, F., Rodino, L. SG Pseudodifferential Operators and Weak Hyperbolicity. (pp. 5-20)
- Schulze, B.-W. Crack Theory with Singularities on the Boundary. (pp. 21-66)
- De Donno, G. Hypoellipticity of Anisotropic Partial Differential Equations. (pp. 67-84)
- Dreher, M. Anomalous Singularities for Hyperbolic Equations with Degeneracy of Infinite Order. (pp. 85-92)
- Oliaro, A. Gevrey Local Solvability for Semilinear Partial Differential Equations. (pp. 93-104)
- Reissig, M. About Strictly Hyperbolic Operators with Nonregular Coefficients. (pp. 105-130)
- Vaillant, J. Diagonalizable Complex Systems, Reduced Dimension and Hermitian Systems, II. (pp. 131-148)
- Cadeddu, L., Gramchev, T. Nonlinear Estimates in Anisotropic Gevrey Spaces. (pp. 149-160)
- Fabricant, A., Kutev, N., Rangelov, T. On Principle Eigenvalue for Linear Second Order Elliptic Equations in Divergence Form. (pp. 161-170)
- Kappeler, T., Topalov, P. Ricatti Representation for Elements in H-1(T) and its Applications. (pp. 171-188)
- Mishev, D., Petrova, Z. On the Zeros of the Solutions to Nonlinear Hyperbolic Equations with Delays. (pp. 189-200)
A B S T R A C T S
SG Pseudodifferential Operators and Weak Hyperbolicity
F. Nicola nicola@dm.unito.it
L. Rodino rodino@dm.unito.it
2000 Math. Subj. Classification:35S05, 47G30, 58J42.
Key words: Pseudo-differential operators, trace functionals, weak hyperbolicity.
We consider a class
of pseudo-differential operators globally defined in R^{n}. For them we
discuss trace functionals, distribution of eigenvalues, essential spectrum and
weak hyperbolicity.
Crack Theory with Singularities at the Boundary
B.-W. Schulze schulze@math.uni-potsdam.de
2000 Math. Subj. Classification:5S15, 35J70, 35J40, 38J40.
Key words: Pseudo-differential boundary problems, mixed
elliptic problems with singular interfaces, corner operators, weighted
corner.
We investigate crack problems, where the crack boundary has conical singularities. Elliptic
operators with two-sided elliptic boundary conditions on the plus and minus
sides of the crack will be interpreted as elements of a corner algebra of
boundary value problems. The
corresponding operators will be completed by extra edge conditions on the crack
boundary to Fredholm operators in corner Sobolev spaces with double weights, and
there are parametrices within the calculus.
Hypoellipticity of Anisotropic Partial Differential Equations
Giuseppe De Donno
2000 Math. Subj. Classification:35S05.
Key words: Partial differential equations, hypoellipticity, Gevrey spaces.
We propose an approach based on methods from microlocal analysis, for characterizing the
hypoellipticity in C^{¥} and Gevrey G^{l} classes of semilinear anisotropic partial
differential operators with multiple characteristics, in dimension n ³ 3. Conditions are imposed on the lower order terms of the
linear part of the operator; we also consider
C^{¥} nonlinear perturbations, see Theorem 1.1
and Theorem 1.4 below.
Anomalous Singularities for Hyperbolic Equations with Degeneracy of Infinite
Order
Michael Dreher
2000 Math. Subj. Classification: 35L80.
Key words: Weakly hyperbolic equations, propagation of
singularities, lacunas, loss or regularity.
We consider weakly
hyperbolic operators with degeneracy of infinite order and study the Sobolev
regularity of solutions to semi-linear Cauchy problems in the
lacunas.
Gevrey Local
Solvability for Semilinear Partial Differential Equations
Alessandro Oliaro
2000 Math. Subj. Classification: 35S05
Key words: Operators with multiple characteristics,
Gevrey classes, local solvability.
In this paper we
deal with a class of semilinear anisotropic partial differential equations. The
nonlinearity is allowed to be Gevrey of a certain order both in x and ¶^{a} u, with an additional
condition when it is G^{scr} in the (¶^{a} u)-variables for a
critical index s_{cr}. For this class of equations we prove the local
solvability in Gevrey classes.
About Strictly
Hyperbolic Operators with Non-regular Coefficients
Michael Reissig nicola@dm.unito.it
2000 Math. Subj.
Classification:35L15,
35L80, 35S05, 35S30.
Key words: Strictly hyperbolic Cauchy problems,
non-Lipschitz coefficients in time, classes of well-posedness, construction of
parametrix, refined perfect diagonalization procedure, regularization
techniques, sharp G\aa rding's
inequality.
Diagonalizable
Complex Systems, Reduced Dimension and Hermitian Systems II
Jean Vaillant
2000 Math. Subj.
Classification: 35L40.
Key words: Strong hyperbolicity, symmetric, hermitian
systems.
We consider a first order differential
system. If its principal part a(x,\xi) is hyperbolic – that means that the
characteristic roots are real for every (x,\xi) - and if it is symmetric or
hermitian, it is usual to construct an energy inequality; if the system is
linear and C^{\infty}, the Cauchy problem is C^{\infty} is well-posed, for any
zero order terms; in some non-linear cases, we have existence theorem. Moreover
in the case of constant coefficients, the theorem by Kasahara and Yamaguti
states the equivalence between strong hyperbolicity and uniformly (real)
diagonalizability. So it is natural to study systems whose the principal part is
diagonalizable or uniformly diagonalizable for each value of the variable x and
to seek for conditions of symmetry or hermiticity. P. D. Lax in [12] gave an
example of 3 \times 3 system with constant coefficients, strongly hyperbolic and
not equivalent to a symmetric system. G. Strang [7] stated that for 2 \times 2
systems with constant coefficients, strong hyperbolicity and symmetry of the
system in a convenient basis are equivalent. In [13] J. Vaillant defined the
reduced dimension of a real a(\xi); this definition is such that the reduced
dimension of the system is equal to the reduced dimension of the determinant, if
the system is diagonalizable; the reduced dimension of a polynomial was defined
by Atiyah Bott and G\aa{}rding; in [13] it was stated that, if the reduced
dimension of the principal part of the system is more than
m\left(\frac{m+1}{2}\right) and if the system is diagonalizable (some additional
condition, in fact implied by the two first ones, as it will be proved by T.
Nishitani [3], was satisfied), then the principal part is, in fact, symmetric in
an convenient basis; we denote that the system is presymmetric: there exists T
such that T^{-1}a(\xi)T is symmetric, for every \xi; the analogous result, in
the case of complex coefficients, was obtained in the third cycle thesis of D.
Schiltz.
Y. Oshime [6], in a series of papers studied
completely the 3 \times 3 diagonalizable real and complex system and
characterized symmetric and hermitian system. In [3] T. Nishitani improved the
result [13] and stated that, if the dimension m \geq 3, if the reduced dimension
d \geq m\left(\frac{m+1}{2}\right)-1 and if the system is diagonalizable, it is
presymmetric; for m = 3 this result is optimum, by [6]. In [8] J. Vaillant
stated for m=4 and in [9] for
general m \geq 4, that, if the system is strongly hyperbolic and if d
\geq m\left(\frac{m+1}{2}\right) -2, it is presymmetric.
T. Nishitani and J.
Vaillant [4] stated in the case of
variable coefficients, that, if for every x the previous conditions are
satisfied, then the principal part is regularly presymmetric (that means there
exists a regular-the same regularity as the coefficients-matrix T(x) such that
T^{-1}(x)a(x,\xi)T(x) is symmetric for every (x,\xi); in fact they stated that,
if d \geq m\left(\frac{m+1}{2}\right) - \left[\frac{m}{2} \right]
and
if for every x,
a(x,\xi) is presymmetric then it is regularly presymmetric; that implies, thanks to the result with
constant coefficients, the precedent result.
Then, J. Vaillant
states in the case of complex coefficients that if the reduced dimension (in the
real) d_R(a) \geq m^2-2 and if the system is diagonalizable, then it is
prehermitian. The schedule of the proof will be published in the Proceedings of
the Cortona colloquium
(2001) and in the present paper.
We conjecture also
that, if d_R \geq m^2-3, m \geq 4, and if the system is strongly hyperbolic,
then the principal part is prehermitian; the result is, at the moment, is
obtained for m =4 (to appear).
Nonlinear Estimates
in Anisotropic Gevrey Spaces
Lucio Cadeddu cadeddu@unica.it
Todor Gramchev todor@unica.it
2000 Math. Subj. Classification: 35G20,
47H30.
Key words: Anisotropic Gevrey classes, nonlinear
composition estimates, evolution PDE.
We introduce scales
of Banach spaces of anisotropic Gevrey functions depending on multidimensional
parameters. We prove estimates in such spaces for composition maps and
nonlinearities in conservative forms. Applications for
solvability and regularity of solutions of nonlinear PDEs are
outlined.
On Principle
Eigenvalue for Linear Second Order Elliptic Equations in Divergence Form
A. Fabricant
N. Kutev
T. Rangelov
2000 Math. Subj. Classification: 35J15,
35J25, 35B05, 35B50.
Key words: Elliptic equations, maximum and comparison
principle, eigenvalue problem.
The principle
eigenvalue and the maximum principle for second-order elliptic equations is
studied. New necessary and sufficient conditions for symmetric and nonsymmetric
operators are obtained. Applications for the estimation of the first eigenvalue
are given.
Riccati Representation for Elements in H^{-1}(\T) and its Applications
Thomas Kappeler
Peter Topalov
2000 Math. Subj. Classification: 35L05,
34L15, 35D05, 35Q53.
Key words: Hill operator, Riccati transform, Birkhoff
coordinates, KdV.
This paper is
concerned with the spectral properties of the Schr{\"o}dinger operator
L_q\eqdef-\frac{d^2}{dx^2}+q with periodic potential q from the Sobolev space
H^{-1}(\T). We obtain asymptotic formulas and a priori estimates for the
periodic and Dirichlet
eigenvalues which generalize known results for the case of potentials q\in
L^2_0(\T). The key idea is to reduce the problem to a known one -- the spectrum
of the impedance operator - via a nonlinear analytic isomorphism of the Sobolev
spaces H^{-1}_0(\T) and
L^2_0(\T).
On the Zeros of the
Solutions to Nonlinear Hyperbolic Equations with Delays
Z. A. Petrova
D. P. Mishev
2000 Math. Subj. Classification: 35B05,
35L15.
Key words: Hyperbolic equation, characteristic initial
value problem, oscillation, eventually positive solution, eventually negative
solution
We consider the
nonlinear hyperbolic equation with delays u_{xy} + \lambda
u_{xy}(x-\sigma,y-\tau) + c(x,y,u,u_{x},u_{y}) = f(x,y).
We obtain
sufficient conditions for oscillation of the solutions of
problems
of Goursat in the case, where \lambda \ge 0.