sessione speciale su Partial Differential Equations of Mixed 
Elliptic-Hyperbolic Type
and Applications".

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\centerline{\large \bf Hypergeometric Distributions and Tricomi Operators}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{J.\ Barros-Neto}
\centerline{\it Rutgers University, USA}
\centerline{e-mail {\tt jbn@math.rutgers.edu}}

%%% in case of several authors:
\smallskip
\centerline{Fernando Cardoso}
\centerline{\it Universidade Federal de Pernambuco, Brazil}
\centerline{e-mail {\tt fernando@dmath.ufpe.br}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par
%   Text of the abstract
Consider the generalized Tricomi operator
$$
{\cal T}_g=y\Delta_x +\frac{\partial^2}{\partial y^2},
$$
with $\Delta_x=\sum_{j=1}^n{\partial^2}/{\partial x_j^2},$ $n>1.$ To
each of these operators corresponds a certain hypergeometric
distribution, in the sense of S.\ Delache \& J.\ Leray and I.\ M.\
Gelfand, that plays a fundamental role in finding fundamental
solutions for ${\cal T}_g.$ With these, we can strengthen and extend
the results of our paper ``Bessel Integrals and Fundamental Solutions
for a Generalized Tricomi Operator.''

\bigskip

%%% in case of several authors:
{\bf  Presented by J.\ Barros-Neto}
%%%

\end{document}

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\centerline{\large \bf Regular Reflection of Weak Shocks:}
\centerline{\large \bf A Global  Existence Result}
\bigskip

\centerline{Sun\v{c}ica \v{C}ani\'{c} }
\centerline{\it University of Houston, USA}
\centerline{e-mail (canic@math.uh.edu)}

%%in case of several authors:
\smallskip
\centerline{Barbara L. Keyfitz}
\centerline{\it University of Houston, USA}
\centerline{e-mail (blk@math.uh.edu)}
\smallskip
\centerline{Eun Ehui Kim } \centerline{\it California State
University at Long Beach, USA} \centerline{e-mail
ehkim@math.uh.edu)}
%%%

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par

% Text of the abstract (1000-2000 words).

When a supersonic shock wave hits a corner of a wedge,
a reflected wave forms. For certain values of the Mach number and
wedge angles the reflected wave meets the incident shock at a point
on the wedge forming the  "regular reflection" wave pattern.
Depending on the parameters in the problem the flow behind the reflected
wave in regular reflection can be either (a) subsonic everywhere, or
(b) there is a region of supersonic flow located  behind the reflected wave near the wall.

We study existence and stability of these two patterns in the situations  when
the incident shock is weak and the wedge angle is small.
The corresponding asymptotic equations that govern the flow are
the two-dimensional Burgers equations or the unsteady transonic small disturbance equations,
valid in a bounded region near the foot of the incident shock.
We prove the existence of a global solution in  each case (a) and (b).
We note that although the reduced equations have an unbounded subsonic region
(which is an artefact of the asymptotic reduction), our  proof holds
only  in a finite (not small), bounded region around the foot of the incident shock
(where the reduced equations are valid).
The existence proof is based on the study of a free-boundary problem
for a (degenerate) quasilinear elliptic equation.
Main ideas of the proof (based on the elliptic regularization and
 compactness arguments)
will be presented.

\bigskip

%%% in case of several authors:
{\bf Presented by Sun\v{c}ica \v{C}ani\'{c}.}
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%\include{FBPold}
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\centerline{\large \bf Multidimensional Transonic Shocks and Free
Boundary Problems} \centerline{\large \bf for the Euler Equations
for Potential Fluids in Unbounded Domains}



\bigskip


\centerline{Gui-Qiang Chen} \centerline{\it Northwestern
University, USA} \centerline{e-mail {\tt
gqchen@math.northwestern.edu}}


%%% in case of several authors:
\smallskip
\centerline{Mikhail Feldman} \centerline{\it University of
Wisconsin at Madison, USA} \centerline{e-mail {\tt
feldman@math.wisc.edu}}
%%%


\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\par
%   Text of the abstract


We will first discuss our recent results on the existence and
stability of multidimensional transonic shocks
(hyperbolic-elliptic shocks) for the Euler equations for steady
potential fluids in infinite cylinders. The Euler equations,
consisting of the conservation law of mass and the Bernoulli law
for velocity, can be written as a second-order, nonlinear
elliptic-hyperbolic equation of mixed type for the velocity
potential. The transonic shock problem in an infinite cylinder can
be formulated into the following free boundary problem: The free
boundary is the location of the multidimensional transonic shock
which divides two regions of smooth flow in the infinite cylinder,
and the equation is hyperbolic in the upstream region where the
smooth perturbed flow is supersonic. We will introduce a nonlinear
approach to deal with such a free boundary problem in order to
solve the transonic shock problem in unbounded domains. Our
results indicate that there exists a unique solution of the free
boundary problem such that the equation is always elliptic in the
unbounded downstream region, the velocity state at infinity in the
downstream direction is uniquely determined by the given
hyperbolic phase, and the free boundary is smooth, provided that
the hyperbolic phase is close to a uniform flow; and the free
boundary is also stable under the steady perturbation of the
hyperbolic phase. We will also present some further results in
this direction, including the results on the existence and
stability of multidimensional transonic shocks near spherical or
circular transonic shocks in unbounded domains.



\bigskip
%%% in case of several authors:
{\bf  Presented by Gui-Qiang Chen}
%%%


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\baselineskip=15pt

\centerline{\large \bf Doubling properties of the harmonic measure}

\centerline{\large \bf associated with degenerate elliptic operators}

\bigskip

\centerline{Fausto Ferrari}
\centerline{\it University of Bologna, Italy}
\centerline{e-mail {\tt ferrari@dm.unibo.it}}

%%% in case of several authors:
\smallskip
\centerline{Bruno Franchi}
\centerline{\it University of Bologna, Italy}
\centerline{e-mail {\tt franchib@dm.unibo.it}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par
Let $X=\{X_1,\dots,X_p\}$ be a family of smooth vector fields in
$\mathbb R^n$, and  let $\Omega\subset \mathbb R^n$ be a connected
open subset with sufficiently regular boundary $\partial\Omega$
(for instance, take $\partial \Omega$ of class $C^{1,\alpha}$,
$0<\alpha \le 1$). If $X$ satisfies the so-called H\"ormander rank
condition, i.e. if the rank of the Lie algebra generated by $X$
equals $n$ at each point of a neighborhood of $\bar\Omega$,
  very few results have
been proved so far concerning the behavior up to the boundary of
positive solutions of
boundary value problems associated with the
degenerate elliptic operator $\mathcal L=
\sum_{j=1}^pX_j^*X_j$. This is basically due to
        the presence of {\sl characteristic points} of $\partial\Omega$,
  that behave  -- in a sense -- as cusps do for the (usual
elliptic) Laplace operator.

So far, the sharpest results in the literature concerning
boundary behavior of  solutions of
$\mathcal L u=0$, and that are the exact counterpart
of the elliptic theory, have been proved for NTA-domains related to
the  Carnot-Carath\'edory metric associated
with
$X$. On the other hand, the characterization of NTA-domains is a very
delicate point
already for the simplest examples, since it is related to the geometry
of  metric balls.

Thus, we
are interested in seeking
  what we can say when the structure
of Carnot-Carath\'edory balls is too complicated to be handled explicitly.
In fact we stress
that, already in simple step 3 Carnot groups like the so-called
Engel group, the geometry of the CC-balls and its interaction
with that of the boundary becomes essentially more complicated.
  In other words, our aim is
to describe as precisely as we can what happens when  no
structure assumptions
are made on the vector fields besides H\"ormander's condition.

More precisely, we prove a (in general not scale-invariant)
doubling property for the ``harmonic measure''
associated with $\mathcal L$, and a (in general not scale-invariant)
boundary Harnack principle for a large class of domains.

\bigskip

%%% in case of several authors:
{\bf  Presented by Bruno Franchi}
%%%

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\baselineskip=15pt

\centerline{\large \bf  Steady States for Quantum Hydrodynamic models: Existence of positive solutions
}
\centerline{\large \bf and
 limiting behavior of  dispersion/diffusion singular perturbations}

\bigskip

\centerline{Irene M. Gamba}
\centerline{\it The University of Texas at Austin, USA}
\centerline{e-mail {\tt gamba@math.utexas.edu}}

%%% in case of several authors:
\smallskip
\centerline{Ansgar Juengel}
\centerline{\it Universit\"at Konstanz, Germany}
\centerline{e-mail {\tt juengel@fmi.uni-konstanz.de}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par
%   Text of the abstract

We analyze a quantum trajectory model given by
a steady-state hydrodynamic system for quantum fluids with positive constant
temperature  in bounded domains for arbitrary large data. This system
is a dispersive perturbation of transonic flow models.
The momentum equation can be written as a dispersive third-order equation
for the particle density where viscous effects may be  incorporated.

The phenomena that admit positivity of the
solutions are studied. The cases, one space dimensional  dispersive  or
 non-dispersive, viscous or
non-viscous, are thoroughly analyzed with respect to
positivity and existence or non-existence of solutions, all depending
on the constitutive
relation for the pressure law. We distinguish between isothermal (linear)
 and isentropic (power law) pressure functions of the
density. It is proven that in the dispersive, non-viscous
model, a classical positive solution only exists for ``small'' (positive)
particle current densities, both for   the isentropic and isothermal case.
Uniqueness is also shown in the isentropic subsonic case, when the
pressure law is strictly convex.
However, we prove that no weak isentropic solution can exist for ``large''
current densities. The dispersive, viscous
problem admits a classical positive solution for all current densities,
both for the isentropic and
isothermal case, with an ``hyper-diffusion" condition.

The proofs are based on a reformulation of the equations as a
singular elliptic second-order problem and on a variant of the Stampacchia
truncation technique. Some of the  results are extended to general third-order
equations
in any space dimension.

Finally, the semi-classical and the inviscid limit
are rigorously performed in the one-dimensional case.  It is shown that the semi-classical and inviscid
limit commute for sufficiently small data (i.e.\ current density)
corresponding to subsonic states, were the inviscid non-dispersive solution is regular.
In addition, we show
these limits do {\em not} commute in general. The proofs are based on a
reformulation of the problem as a singular second-order elliptic system
and on elliptic and $W^{1,1}$ estimates.

\bigskip

%%% in case of several authors:
{\bf  Presented by Irene M. Gamba}
%%%

\end{document}

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%% For labels, try to write : \label{yourname01},        %%
%% \label{yourname02}...                                 %%
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\documentclass{article}
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\centerline{\large \bf Weak shock reflection}


\bigskip

\centerline{John K. Hunter}
\centerline{\it University of California, Davis, U.S.A.}
\centerline{e-mail {\tt jkhunter@ucdavis.edu}}

%%% in case of several authors:
\smallskip
\centerline{Allen M. Tesdall}
\centerline{\it University of California, Davis, U.S.A.}
\centerline{e-mail {\tt amtesdal@ucdavis.edu}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par
%   Text of the abstract


We present numerical solutions of a two-dimensional
Riemann problem for the unsteady transonic
small disturbance equations
that provides an asymptotic description of the Mach reflection
of weak shock waves.  In self-similar coordinates,
the solution satisfies a nonlinear
mixed-type system of conservation laws with source terms.
We develop a new numerical scheme to solve the self-similar
equations, and use local grid refinement to resolve
the solution in the reflection region.
The solutions contain a remarkably complex
structure: there is a sequence of triple points
and tiny supersonic patches immediately behind the leading
triple point that is formed by the reflection of weak shocks
and expansion waves between the sonic line and the Mach shock.
An expansion fan originates at each triple point,
thus resolving the von Neumann paradox of weak shock reflection.
The numerical solutions raise the question of whether there is an
infinite sequence of triple points in an inviscid
weak shock Mach reflection. It seems likely that the kind of
behavior observed here also occurs in many other mixed-type systems of
conservation laws.

\bigskip

%%% in case of several authors:
{\bf  Presented by John K. Hunter}
%%%
\end{document}

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%%
%%
%% For labels, try to write : \label{yourname01},        %%
%% \label{yourname02}...                                 %%
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\documentclass{article}
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\baselineskip=15pt

\centerline{\large \bf Mixed Hyperbolic-Elliptic Systems in Self-Similar Flows}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Sun\v{c}ica \v{C}ani\'{c}}
\centerline{\it University of Houston, USA}
\centerline{e-mail {\tt canic@math.uh.edu}}

%%% in case of several authors:
\smallskip
\centerline{Barbara Lee Keyfitz }
\centerline{\it University of Houston, USA}
\centerline{e-mail {\tt blk@math.uh.edu}}
%%%

\smallskip
\centerline{Eun Heui Kim}
\centerline{\it  California State
University at Long Beach, USA}
\centerline{e-mail {\tt ekim4@csubl.edu}}
%%%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par
%   Text of the abstract
>From the observation that self-similar solutions of conservation laws
in two space dimensions change type, it follows that for systems of
more than two equations, such as the equations of gas dynamics, the
reduced systems will be of mixed hyperbolic-%
elliptic type, in some regions of space.
We derive  mixed systems for the isentropic and
adiabatic equations of compressible gas dynamics and
show that the mixed systems which arise exhibit complicated
nonlinear dependence.
In a prototype system, the nonlinear wave system, this behavior
is much simplified, and we outline the solution to some typical
Riemann problems.
\bigskip

%%% in case of several authors:
{\bf  Presented by Barbara Lee Keyfitz}
%%%

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\centerline{\large \bf A Modification of Frankl's Problem }


\centerline{\large \bf Associated with the Airfoil Design }


\bigskip


\centerline{A.G.Kuz'min} \centerline{\it St. Petersburg State
University, Russia} \centerline{e-mail \, {\tt
alexander.kuzmin@pobox.spbu.ru}}


\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\par


Frankl (1947) considered transonic flow with local supersonic
regions over a symmetric airfoil and suggested physical arguments
in favour of a problem in which the Neumann condition was
prescribed on the axis of symmetry and on the airfoil except for
an arc $\Gamma_a$. The uniqueness of the smooth solution to
Frankl's problem was established by Morawetz (1956), who employed
the hodograph plane and used an auxiliary function that was
monotonous along the characteristics in the supersonic region.
Cook (1978) proved a similar theorem directly in the physical
plane. The important consequence of the uniqueness is that if a
smooth solution to the Frankl problem exists, then it determines
uniquely the shockless shape of the arc $\Gamma_a$. Thus, the
Frankl problem is actually associated with airfoil adaptation to
changing flow conditions.


In this work, we study a modification of the Frankl problem in
which the arc $\Gamma_a$ is free of boundary conditions, and the
Neumann condition is replaced by the Dirichlet one along a portion
of the airfoil. We demonstrate that the above mentioned result on
the uniqueness remains valid in this case. From the practical
point of view, the problem at hand is concerned with the shockless
airfoil design under a given target velocity on a portion of the
airfoil. The obtained result shows that the numerical methods, in
which the slip condition / flow velocity distribution are
prescribed over the full airfoil, cannot guarantee the absence of
shock waves from the flow.
\medskip


\noindent {\Large References}
\smallskip


\noindent [1] Frankl F I (1947) \,
   Prikladnaya Mat i Mekh. {\bf 11}, no.1: 199--202.


\noindent [2] Morawetz C S (1956) \,
   Comm. Pure Appl. Math. {\bf 9}: 45--68.


\noindent [3] Cook L P (1978) \,
   Indiana University Math J. {\bf 27}, no. 1: 51--71.


\end{document}


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%% to a large audience and to avoid difficulties with    %%
%% the handling of your file, please limit the use of    %%
%% formulas as much as possible.
%%
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%% your abstract. In case that you must have references, %%
%% then do not use \cite commands.                       %%
%%
%%
%% For labels, try to write : \label{yourname01},        %%
%% \label{yourname02}...                                 %%
%% Your abstract should be no more than one page long.
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass{article}
\usepackage{amsthm,amsmath,amssymb,latexsym}
\pagestyle{empty} \textwidth=12.8cm \textheight=21.7cm
\hoffset=-0.3in \voffset=-0.6in
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\begin{document}
\baselineskip=15pt

\centerline{\large \bf Critical exponents for equations of mixed
elliptic-hyperbolic type}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Daniela E.\ Lupo} \centerline{\it Dipartimento di
Matematica, Politecnico di Milano, Italy} \centerline{e-mail {\tt
danlup@mate.polimi.it}}

%%% in case of several authors:
\smallskip
\centerline{Kevin R.\ Payne} \centerline{\it Dipartimento di
Matematica, Universit\`a di Milano, Italy} \centerline{e-mail {\tt
payne@mat.unimi.it}}
%%%

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\par
%   Text of the abstract
For semilinear Gellerstedt equations with Tricomi, Goursat or
Dirichlet boundary conditions we prove Pohozaev type identities
and derive non existence results that exploit an invariance of the
linear part with respect to certain non homogeneous dilations. A
critical exponent phenomenon of power type in the nonlinearity is
exhibited in these mixed elliptic hyperbolic or degenerate
settings where the power is one less than the critical exponent in
a relevant Sobolev imbedding.
\bigskip

%%% in case of several authors:
{\bf  Presented by Daniela E.\ Lupo}
%%%

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%%
%% For labels, try to write : \label{yourname01},        %%
%% \label{yourname02}...                                 %%
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\documentclass{article}
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\baselineskip=15pt

\centerline{\large \bf Two Singular Problems for the Transonic Equation}

\centerline{\large \bf }

\bigskip

\centerline{Cathleen S. Morawetz} \centerline{\it Courant
Institute of Mathematical Sciences, USA} \centerline{e-mail {\tt
morawetz@cims.nyu.edu}}


\bigskip
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\par

The author proposes singular Dirichlet problems for Tricomi-like
equations. Their solution would clarify how shocks are initiated in
steady transonic flows.

\bigskip


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\centerline{\large \bf Conservation laws for equations of mixed
elliptic-hyperbolic type}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Daniela E.\ Lupo} \centerline{\it Dipartimento di
Matematica, Politecnico di Milano, Italy} \centerline{e-mail {\tt
danlup@mate.polimi.it}}

%%% in case of several authors:
\smallskip
\centerline{Kevin R.\ Payne} \centerline{\it Dipartimento di
Matematica, Universit\`a di Milano, Italy} \centerline{e-mail {\tt
payne@mat.unimi.it}}
%%%

\bigskip
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%%
\par
%   Text of the abstract
For equations mixed elliptic-hyperbolic type such as the Tricomi
equation invariances in the space of solutions with respect to
various local groups of transformations is examined. To each
invariance property a corresponding conservation law is derivable
either via Noether's theorem since the equations are in divergence
form or by the method of multipliers. The resulting conservation
laws are applied to prove uniqueness theorems and to generate {\em
a priori} \ estimates for the original equation.

\bigskip

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\centerline{\large \bf  3D Protter problems for mixed type equations }

\bigskip

\centerline{Nedyu Popivanov} \centerline{\it Department of Mathematics and
Informatics, University of Sofia, Bulgaria} \centerline{e-mail
{nedyu@fmi.uni-sofia.bg}}

\bigskip

%   Text of the abstract
At a conference of AMS in New York in 1952 M.H.Protter formulated and
studied some boundary value problems for the wave equation in a $3D$ domain $%
\Omega _{0}$, bounded by two characteristic cones and a plane domain, which
are three-dimensional analogues of Darboux-problems (or Cauchy-Goursat
problems) on the plane. In 1954 he initiated the study of such 3-dimensional
problems for the mixed type equation\vspace{0in}

\begin{equation}
{Lu:=K(t)\left( \ \ u_{x_{1}x_{1}}+u_{x_{2}x_{2}}\right) \ \
-u_{tt}=f,\qquad tK(t)>0,\text{ \ }t\neq 0}  \tag{1}
\end{equation}
in domain $\Omega $ bounded by two characteristic cones in the hyperbolic
halfspace $\{t>0\}$ and an elliptic part in $\{t<0\}.$ These problems have
been formulated by M. Protter as three-dimensional analogues of a plane
problem given by C. Morawetz and examined by C. Morawetz, P. Lax and R.
Phillips.

What is the situation around both these problems now, 50 years later?

1. Many authors studied these problems using different methods, like:
Wiener-Hopf method, special Legendre functions, a priori estimates, nonlocal
regularization and others. In the case of the wave equation it is shown that
for any $n\in N$ there exists a $C^{n}(\bar{\Omega}_{0})$ - function, for
which the corresponding unique \textit{generalized solution }belongs to $%
C^{n}(\bar{\Omega}_{0}\backslash O),$ but it has a strong power-type
singularity $\left( x_{1}^{2}+x_{2}^{2}+t^{2}\right) ^{-n/2}$ at the point $O
$. This singularity is isolated only at the vertex $O$ of the characteristic
cone and does not propagate along the cone. We will present here some final
results for the exact behavior of the singular solutions at the point $O.$
Also, we will give some necessary and sufficient conditions for the function
$f,$ under which only classical solution exists. Finally, some weight a
priori estimates are stated.

2. According to the mixed type equation (1) in $\Omega ,$ the situation is
still away from the final results. Some uniqueness results have been
obtained by many authors. But up to now there are no general existence
results for thes problem in $\Omega .$ This situation can be interpreted in
terms of improperly posed (or ill-posed) problems. Using Friedrichs' theory
of symmetric positive operators, we find and investigate a non-local
problems which are regularisers, in some sence, of these ill-posed problems.

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\centerline{\large \bf Approaching a partial differential equation
of mixed elliptic-hyperbolic type}



\bigskip


\centerline{Giorgio Talenti} \centerline{\it Dipartimento di
Matematica U.~Dini, Universit\`a di Firenze, viale Morgagni 67/A,
50134 Firenze, Italy.} \centerline{ e-mail {\tt
talenti@math.unifi.it}}



%%% in case of several authors:
%\smallskip
%\centerline{Mikhail Feldman} \centerline{\it University of
%Wisconsin at Madison, USA} \centerline{e-mail {\tt
%feldman@math.wisc.edu}}
%%%


\bigskip
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\par
%   Text of the abstract


Suppose an isotropic, non-conducting, non-dissipative medium and a
mono\-chro\-matic electromagnetic field interact in absence of
electric charges. Let $n$ and $\nu$ denote the refractive index
and the wave number, respectively. Here $n$ is a scalar
real-valued field, whose reciprocal is proportional to the
relevant velocity of propagation through the medium, and $\nu$ is
a large positive parameter, whose reciprocal is proportional to
the length of waves involved. The following Helmholtz equation
\begin{equation}
\label{helm} \Delta U +\nu^2 n^2 U=0
\end{equation}
is an archetype of those partial differential equations that ensue
from Max\-well's system and model the subject matter
mathematically. A distinctive feature of (\ref{helm}) is {\it
stiffness} --- the order of magnitude of $\nu$ is significantly
greater than that of the other coefficients involved.

An expansion, which represents solutions to (\ref{helm})
asymptotically as $\nu\to +\infty,$ originates from WKBJ method
and reads thus
%N.B. in originale \` scritto sums
\begin{equation}
\label{asympt} U\simeq \exp(i \nu S) \sums_{k=0}^\infty A_k\cdot
(i \nu)^{-k}.
\end{equation}
Here $S$ and $A_k$ are scalar fields, independent on $\nu.$ The
former, named {\it eikonal,} is real-valued and governed by $$
|\nabla S|^2 =n^2; $$ the latter is complex-valued and governed by
the so-called {\it transport equations.}

Inference built upon expansion (\ref{asympt}) amounts to
geometrical op\-tics.\break Though successful in describing both
the propagation of light and the concurrence of caustics via the
mechanism of rays, geometrical optics is inherently unable to
account for those phenomena, such as the development of evanescent
waves, that take place beyond a caustic.

A more powerful asymptotic expansion, which is apt to represent
solutions to (\ref{helm}) on {\it both sides of a caustic,}
simultaneously in the region covered by geometric optical rays and
in the opposite region where geometrical optics breaks down, is
provided by a theory of Kravtsov and Ludwig. In case the caustic
involved is smooth and convex, such an expansion reads
\begin{equation}
\label{krav} U\simeq e^{i \nu v} \Bigl\{\ai(\nu^{2/3} u)
\sums_{k=0}^\infty A_k\!\cdot\! (i\nu)^{-k} + i \nu^{-1/3}
\ai'(\nu^{2/3} u) \sum_{k=0}^\infty B_k\!\cdot\!
(i\nu)^{-k}\Bigr\}.
\end{equation}
Here $u, v, A_k, B_k$ are scalar fields, independent on $\nu;$ $u$
and $v$ are real-valued, $A_k$ and $B_k$ are complex-valued; $\ai$
denotes the {\it Airy function}.

Properties of $\ai$ inform us that the right-hand side of
(\ref{krav}) oscillates rapidly where $u$ is negative, approaches
smoothly a limit if $u$ approaches $0,$ quenches fast where $u$ is
positive. Therefore (\ref{krav}) matches geometrical optics in the
region where $u$ is negative and predicts the occurrence of damped
waves in the region where $u$ is positive; a caustic take place on
the level surface where $u=0.$

Assembling (\ref{helm}) and (\ref{krav}) results in
\begin{equation}
\label{syst}
\begin{array}{lll}
&&u\ |\na u|^2 -|\na v|^2 +n^2=0\\ &&\na u\cdot\na v=0
\end{array}
\end{equation}
--- a fully nonlinear, first-order partial differential system
governing $u$ and $v.$

In the present paper we sketch some lineaments of (\ref{syst}) in
the case where the space dimension equals $2,$ i.e. we let $x$ and
$y$ denote rectangular coordinates in the Euclidean plane and
investigate the following system
\begin{equation}
\label{syst2}
\begin{array}{lllll}
&&u\ (u_x^2+u_y^2)-v_x^2-v_y^2 +n^2(x,y)&=&0\\ &&u_x v_x+u_y
v_y&=&0.
\end{array}
\end{equation}



\bigskip
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%{\bf  Presented by Author}
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