Volume 15, 2003
Proceedings of the International Conference on "Partial Differential Equations on
Spaces with Geometric Singularities",
GUEST EDITORS: T. Gramchev, P. Popivanov
Section 1:
Pseudodifferential Operators Calculus and Applications
Nicola, F. and Rodino, L. SG Pseudodifferential Operators and Weak Hyperbolicity
(pp. 5-20)
Schulze, B.-W. Crack Theory with Singularities on the Boundary
(pp. 21-66)
Section 2: Local Solvability
and Cauchy Problems for Weakly Hyperbolic Equations
De Donno, G. Hypoellipticity of Anisotropic Partial Differential
Equations (pp. 67-84)
Dreher, M. Anomalous Singularities for Hyperbolic Equations with Degeneracy of Infinite
Order (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)
Section 3: Evolution
PDE in Mathematical Physics
Cadeddu, L. and Gramchev, T. Nonlinear Estimates in Anisotropic Gevrey
Spaces (pp. 149-160)
Fabricant, A., Kutev, N. and Rangelov, T. On Principle Eigenvalue for Linear Second Order
Elliptic Equations in Divergence Form (pp. 161-170)
Kappeler, T. and Topalov, P. Ricatti Representation for Elements in
H-1(T) and its Applications (pp. 171-188)
Mishev, D. and Petrova, Z. On the Zeros of the Solutions to Nonlinear Hyperbolic
Equations with Delays (pp. 189-200)
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
Rn. 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: 35S15, 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 Gl 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
Gscr in the
(¶a u)-variables
for a critical index
scr. For this class of equations we prove the local solvability in
Gevrey classes.
About strictly
hyperbolic operators with non-regular coefficients
Michael Reissig
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.