sessione speciale su Partial Differential Equations of Mixed Elliptic-Hyperbolic Type and Applications". --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="barros-abs.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The abstract must be written in LaTeX2e %% %% Make sure that your abstract can be compiled before %% %% you send it to the organizer. %% %% Do not use personal macros in your TeX file, and, %% %% for the sake of making your abstract understandable %% %% to a large audience and to avoid difficulties with %% %% the handling of your file, please limit the use of %% %% formulas as much as possible. %% %% We suggest that you do not include references in %% %% 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 \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \begin{document} \baselineskip=15pt \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} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="canic-abs.tex" \documentstyle{article} \pagestyle{empty} \textwidth=12.8cm \textheight=21.7cm \hoffset=-0.3in \voffset=-0.6in \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \begin{document} \baselineskip=15pt %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}.} %%% %\include{FBPold} \end{document} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="chen-abs.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass{article} \usepackage{amsthm,amsmath,amssymb,latexsym} \pagestyle{empty} \textwidth=12.8cm \textheight=21.7cm \hoffset=-0.3in \voffset=-0.6in \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \begin{document} \baselineskip=15pt \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} %%% \end{document} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="franchi-abs.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The abstract must be written in LaTeX2e %% %% Make sure that your abstract can be compiled before %% %% you send it to the organizer. %% %% Do not use personal macros in your TeX file, and, %% %% for the sake of making your abstract understandable %% %% to a large audience and to avoid difficulties with %% %% the handling of your file, please limit the use of %% %% formulas as much as possible. %% %% We suggest that you do not include references in %% %% 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 \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \begin{document} \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} %%% \end{document} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="gamba-abs.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The abstract must be written in LaTeX2e %% %% Make sure that your abstract can be compiled before %% %% you send it to the organizer. %% %% Do not use personal macros in your TeX file, and, %% %% for the sake of making your abstract understandable %% %% to a large audience and to avoid difficulties with %% %% the handling of your file, please limit the use of %% %% formulas as much as possible. %% %% We suggest that you do not include references in %% %% 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 \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \begin{document} \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} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="hunter-abs.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The abstract must be written in LaTeX2e %% %% Make sure that your abstract can be compiled before %% %% you send it to the organizer. %% %% Do not use personal macros in your TeX file, and, %% %% for the sake of making your abstract understandable %% %% to a large audience and to avoid difficulties with %% %% the handling of your file, please limit the use of %% %% formulas as much as possible. %% %% We suggest that you do not include references in %% %% 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 \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \begin{document} \baselineskip=15pt \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} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="keyfitz-abs.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The abstract must be written in LaTeX2e %% %% Make sure that your abstract can be compiled before %% %% you send it to the organizer. %% %% Do not use personal macros in your TeX file, and, %% %% for the sake of making your abstract understandable %% %% to a large audience and to avoid difficulties with %% %% the handling of your file, please limit the use of %% %% formulas as much as possible. %% %% We suggest that you do not include references in %% %% 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 \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \begin{document} \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} %%% \end{document} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="kuzmin-abs.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass{article} \usepackage{amsthm,amsmath,amssymb,latexsym} \pagestyle{empty} \textwidth=12.8cm \textheight=21.7cm \hoffset=-0.3in \voffset=-0.6in \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \begin{document} \baselineskip=15pt \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="lupo-abstract.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The abstract must be written in LaTeX2e %% %% Make sure that your abstract can be compiled before %% %% you send it to the organizer. %% %% Do not use personal macros in your TeX file, and, %% %% for the sake of making your abstract understandable %% %% to a large audience and to avoid difficulties with %% %% the handling of your file, please limit the use of %% %% formulas as much as possible. %% %% We suggest that you do not include references in %% %% 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 \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \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} %%% \end{document} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="morawetz-abs.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The abstract must be written in LaTeX2e %% %% Make sure that your abstract can be compiled before %% %% you send it to the organizer. %% %% Do not use personal macros in your TeX file, and, %% %% for the sake of making your abstract understandable %% %% to a large audience and to avoid difficulties with %% %% the handling of your file, please limit the use of %% %% formulas as much as possible. %% %% We suggest that you do not include references in %% %% 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 \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \begin{document} \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \par The author proposes singular Dirichlet problems for Tricomi-like equations. Their solution would clarify how shocks are initiated in steady transonic flows. \bigskip \end{document} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="payne-abstract.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The abstract must be written in LaTeX2e %% %% Make sure that your abstract can be compiled before %% %% you send it to the organizer. %% %% Do not use personal macros in your TeX file, and, %% %% for the sake of making your abstract understandable %% %% to a large audience and to avoid difficulties with %% %% the handling of your file, please limit the use of %% %% formulas as much as possible. %% %% We suggest that you do not include references in %% %% 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 \parskip=6pt \lineskip=18pt %------------------------------------------------------------ \begin{document} \baselineskip=15pt \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% \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 %%% in case of several authors: {\bf Presented by Kevin R.\ Payne} %%% \end{document} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="popivanov-abs.tex" %------------------------------------------------------------ \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsthm,amsmath,amssymb,latexsym} %TCIDATA{OutputFilter=Latex.dll} %TCIDATA{LastRevised=Fri Mar 15 15:17:51 2002} %TCIDATA{} %TCIDATA{CSTFile=article.cst} \pagestyle{empty} \textwidth=12.8cm \textheight=21.7cm \hoffset=-0.3in \voffset=-0.6in \parskip=6pt \lineskip=18pt %\input{tcilatex} \begin{document} \baselineskip=15pt \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. \end{document} --=====================_30420872==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="talenti-abs.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass{article} \usepackage{amsthm,amsmath,amssymb,latexsym} \pagestyle{empty} \textwidth=12.8cm \textheight=21.7cm \hoffset=-0.3in \voffset=-0.6in \parskip=6pt \lineskip=18pt %newcommands di talenti \newcommand{\na} {\nabla} \newcommand{\sums} {\sum\limits} \newcommand{\ai} {\mbox{\rm Ai}} %------------------------------------------------------------ \begin{document} \baselineskip=15pt \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 %%% in case of several authors: %{\bf Presented by Author} %%% \end{document}