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Table of contents

Sauer, E. Scholz, C. Smeenk, J. Stachel, N. Straumann, R. Wald, S. Walter, C. The moment polytope also encodes quantum information about the actions of G. Using the methods of geometric quantization, one can frequently convert this action into a representations, p , of G on a Hilbert space, and in some sense the moment polytope is a diagrammatic picture of the irreducible representations of G which occur as subrepresentations of p. Precise versions of this item of folklore are discussed in Chapters 3 and 4. Also, midway through Chapter 2 a more complicated object is discussed: the Duistermaat-Heckman measure, and the author explains in Chapter 4 how one can read off from this measure the approximate multiplicities with which the irreducible representations of G occur in p.

This gives an excuse to touch on some results which are in themselves of great current interest: the Duistermaat-Heckman theorem, the localization theorems in equivariant cohomology of Atiyah-Bott and Berline-Vergne and the recent extremely exciting generalizations of these results by Witten, Jeffrey-Kirwan, Lalkman, and others. The last two chapters of this book are a self-contained and somewhat unorthodox treatment of the theory of toric varieties in which the usual hierarchal relation of complex to symplectic is reversed.

This book is addressed to researchers. Account Options Sign in. Top charts. New arrivals. The subject matter of this work is an area of Lorentzian geometry which has not been heretofore much investigated: Do there exist Lorentzian manifolds all of whose light-like geodesics are periodic? This book is concerned with the deformation theory of M2,1 which furnishes almost all the known examples of these objects.

It also has a section describing conformal invariants of these objects, the most interesting being the determinant of a two dimensional "Floquet operator," invented by Paneitz and Segal. Reviews Review Policy. Published on. Original pages. Best For. Web, Tablet. Content Protection. Learn More. Flag as inappropriate. It syncs automatically with your account and allows you to read online or offline wherever you are.

Please follow the detailed Help center instructions to transfer the files to supported eReaders. More related to cosmology. See more. Geometry, Fields and Cosmology: Techniques and Applications. Book This series of Schools have been carefully planned to provide a sound background and preparation for students embarking on research in these and related topics. Consequently, the contents of these lectures have been meticulously selected and arranged.

The topics in the present volume offer a firm mathematical foundation for a number of subjects to be de veloped later. The style of the book is pedagogical and should appeal to students and research workers attempt ing to learn the modern techniques involved. A number of specially selected problems with hints and solutions have been included to assist the reader in achieving mastery of the topics.

We decided to bring out this volume containing the lecture notes since we felt that they would be useful to a wider community of research workers, many of whom could not participate in the school. We thank all the lecturers for their meticulous lectures, the enthusiasm they brought to the discussions and for kindly writing up their lecture notes. It is a pleasure to thank G.

Manjunatha for his meticulous assistence over a long period, in preparing this volume for publication. Fuchsian reduction is a method for representing solutions of nonlinear PDEs near singularities. The technique has multiple applications including soliton theory, Einstein's equations and cosmology, stellar models, laser collapse, conformal geometry and combustion.

Developed in the s for semilinear wave equations, Fuchsian reduction research has grown in response to those problems in pure and applied mathematics where numerical computations fail. Michael Tsamparlis. Writing a new book on the classic subject of Special Relativity, on which numerous important physicists have contributed and many books have already been written, can be like adding another epicycle to the Ptolemaic cosmology.

Furthermore, it is our belief that if a book has no new elements, but simply repeats what is written in the existing literature, perhaps with a different style, then this is not enough to justify its publication. However, after having spent a number of years, both in class and research with relativity, I have come to the conclusion that there exists a place for a new book.

Moreover current trends encourage the application of techniques in producing quick results and not tedious conceptual approaches resulting in long-lasting reasoning. As a result of the above, a major aim in the writing of this book has been the distinction between the mathematics of Minkowski space and the physics of r- ativity.

In conjunction, geometers introduced numerous innovations adapted to or designed for solving problems that arose from space-time physics. This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site.

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To understand the properties of these functions, it became important to study their behavior near their singularities in the complex plane. For linear equations, two cases were distinguished: the Fuchsian case, in which all formal solutions converge, and the non-Fuchsian case. Elliptic problems in corner domains and problems with double characteristics also led to further generalizations.

This development was considered as fairly mature in the s; it was realized that some problems required complicated expansions with logarithms and variable powers, beyond the scope of existing results, but viii Preface it was assumed that this behavior was nongeneric. Nonlinear problems were practically ignored.

Numerical studies of such space-times led to spiky behavior: were these spikes artefacts? Other problems seemed unrelated to Fuchsian PDEs. Outside mathematics, we may mention laser collapse and the weak detonation problem. Also, the theory of solitons has provided, from on, a plethora of formal series solutions for completely integrable PDEs, of which one would like to know whether they represent actual solutions. Do these series have any relevance to nearly integrable problems?

The method of Fuchsian reduction, or reduction for short, has provided answers to the above questions. The function v determines the regular part of u. This representation has the same advantages as an exact solution, because one can prove that the remainder T m v is indeed negligible for T small. The right-hand side may involve derivatives of v.

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Since v is typically obtained from u by subtracting its singularities and dividing by a power of T , v will be called the renormalized unknown. Even though L is scale-invariant, s may not have power-like behavior.

Also, in many cases, it is possible to give a geometric interpretation of the terms that make up s. The introduction, Chapter 1, outlines the main steps of the method in algorithmic form. Part I describes a systematic strategy for achieving reduction. A few general principles that govern the search for a reduced form are given. The list of examples of equations amenable to reduction presented in this volume is not meant to be exhaustive.

In fact, every new application of reduction so far has led to a new class of PDEs to which these ideas apply. Part II develops variants of several existence results for hyperbolic and elliptic problems in order to solve the reduced Fuchsian problem, since the transformed problem is generally not amenable to classical results on singular PDEs. Part III presents applications. It should be accessible after an upperundergraduate course in analysis, and to nonmathematicians, provided they take for granted the proofs and the theorems from the other parts.

Indeed, the discussion of ideas and applications has been clearly separated from statements of theorems and proofs, to enable the volume to be read at various levels. Together with the computations worked out in the solutions to the problems, the volume is meant to be self-contained. Most chapters contain a problem section. The solutions worked out at the end of the volume may be taken as further prototypes of application of reduction techniques. A number of forerunners of reduction may be mentioned.


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It has remained a part of complex analysis. In fact, the catalogue of possible singularities in this limited framework is still not complete in many respects. Most of the equations arising in applications are not covered by this analysis. The regularization of collisions in the N -body problem. This line of thought has gradually waned, perhaps because of the smallness of the radius of convergence of the series in some cases, and again because the relevance to nonanalytic problems was not pursued systematically.

This viewpoint, from the present perspective, puts expansions at the main focus of interest; all relevant information is derived from them. For this approach to be relevant beyond complex analysis, it was necessary to understand which aspects of the Weierstrass viewpoint admit a generalization to nonanalytic problems with nonlinearities—and this generalization required a mature theory of nonlinear PDEs which was developed relatively recently. The existence of a mature theory of elliptic and hyperbolic PDEs, which could be generalized to singular problems.

The failure of the search for a weak functional setting that would include blowup singularities for the simplest nonlinear wave equations. The rediscovery of complex analysis stimulated by the emergence of soliton theory. Preface xi On a more personal note, a number of mathematicians have, directly or indirectly, helped the author in the emergence of reduction techniques: D. Aronson, C. Bardos, L. Boutet de Monvel, P. Garrett, P. Lax, W. Littman, L. Nirenberg, P. Olver, W. Strauss, D. Sattinger, A. Tannenbaum, E. In fact, my indebtedness extends to many other mathematicians whom I have met or read, including the anonymous referees.

I am also grateful to him for welcoming this volume in this series, and to A. Kostant and A. The technical aspects of the theory are developed in the subsequent chapters. We then describe the main steps of the reduction process in general terms, and show in concrete situations how this reduction is achieved. We close this introduction with a survey of the impact of reduction on applications.

Taking singularity locus as one of the parameters, one obtains a parameterization without redundancy, unlike the parameterization by t0 and the Cauchy data at time t0. The square brackets indicate that F may depend on u and its derivatives, as well as on independent variables. First reduction and formal solutions. Second reduction and characterization of solutions. Invertibility and stability of solutions. Many other types of leading behavior arise in applications, including logarithms and variable powers. They will be discussed in due time. If it is possible to transform a problem into this form by a change of variables and unknowns, we say that it admits of reduction.

General results from Chap. The set of arbitrary functions, together with the equation of the singular set, form the singularity data. In some problems, the singular set is prescribed at the outset, and is not a free parameter; the singularity data consist then only of the arbitrary functions or parameters in the expansion. Remark 1. One then appeals to one of the general results of Chaps.

We now turn to the fourth step of the reduction process. If, on the other hand, we have another way of parameterizing solutions, we need to compare these two parameterizations. At this stage, we know how singularity data vary: perturbation of Cauchy data merely displaces the singular set or changes the arbitrary parameters in the expansion, or both. Thus, the main technical point is the reduction to Fuchsian form and its exploitation. For this reason, we now give a few very simple illustrations of the process leading to Fuchsian form.

The roots of P are called resonances or Fuchs indices, and they are again associated to solutions with power leading behavior. Unlike Fuchs—Frobenius theory, the right-hand side and the solution may not have a continuation to a full complex neighborhood of the origin. A treatment by reduction of some equations with irregular singular points is given in Problem 4. Choose the expansion variable T. Step B. List all possible leading terms and choose one. Step D. Step E.

Determine the form of the solution. Step F. Compute the second reduced equation. Step G. Show that formal solutions are associated to actual solutions. Step H. Determine whether the solutions of step G are stable, by inverting the mapping from singularity data to solutions. We also work out completely a simple example of analysis of blowup, and outline another, which introduces the need for logarithmic terms. This is a Fuchsian equation. This is not yet a Fuchsian equation, because the right-hand side is not divisible by T.

The stable manifold is parameterized by a. That z admits an expansion in powers of d is a consequence of Schauder theory. Now, such operators arise naturally from the asymptotic analysis of geometric problems leading to boundary blowup. We develop an appropriate regularity theory in Chap.

Since these solutions may be investigated by standard means, we do not pursue their study any further. Upon multiplication by T 2 , equation 1. The general results of Chap. The second reduction is now complete: given any pair a, b , there is a unique solution u t; a, b of equation 1. To sum up, we have proved the following theorem: Theorem 1. If a becomes large, it is conceivable that the solution has another singularity between 0 and a. The appropriate stability statement, which involves setting up a correspondence between singularity data at the two singularities, is left to the reader.

However, this leads to higher and higher powers of ln T if the computation is pushed further. Theorem 1. There is a family of solutions of 1. This family is a local representation of the general solution: the parameters describing the asymptotics are smooth functions of the Cauchy data at a nearby regular point. Here, v is a power series in two variables T and T ln T , entirely determined by its constant term. The singularity data are a, b. We conclude with the stability analysis.

Even though v exhibits branching because of the logarithm, it is obtained from a single-valued function of two variables by performing a multivalued substitution. In other words, this representation is a uniformization of the solution. Since the manipulations involved are typical of those required for all the applications to nonlinear waves, we write out the computations in detail. This equation is the n-dimensional Liouville equation. In one space dimension, this equation is exactly solvable; see Sect. It was this exact solution that suggested 1.

The objective is to show that near singularities, the equation is not governed by the wave operator, but by an operator for which the singular set is characteristic. This suggests that blowup singularities for nonlinear wave equations are not due to the focusing of rays for the wave operator, and that the correct results on the propagation of singularities must be based on this Fuchsian principal part rather than the wave operator.

This statement will be substantiated in Chap. It is convenient to put coordinate indices as exponents, and to use primed indices to denote derivatives with respect to the coordinates X, T. Lemma 1. The result follows. However, it is not possible to continue the expansion with a term v 2 T 2 : substitution into the equation shows that v 2 does not contribute any term of degree 0 to the equation. In fact, v 2 is arbitrary, and we must include a term in R1 X T 2 ln T in the expansion. In the analytic case, the existence theorems from Chap.

In this case, one can say more: Theorem 1. For the proof, see Problem 3. For other nonlinearities, the no-logarithm condition may also involve the second fundamental form of the blowup surface. An example of such computations will be outlined in Chap. Similar results are available in the elliptic case. Taking T to be the distance to the singular surface enables one to give a geometric interpretation for other elements of the expansion of the solution. Stability is proved in Sect. One sees a spot of light. Now move your hand slightly, so that the spot of light moves on the wall.

Clearly, the motion of the spot is not a wave propagation, because the spot does not move by itself, but merely because its source moves. This pattern is a collective result of the evolution of the solution as a whole, for one singularity is not necessarily causally related to nearby singularities in space-time.

The considerations leading to this concept are further elaborated in Sect. Reduction techniques investigate whether it is possible to embed a singular solution into a family of solutions with the maximum number of free functions or parameters; if this is the case, we say that the singular solution is stable. Therefore, it represents large-amplitude waves accurately, a short time before blowup.

In addition to the above advantages explicit formulas, geometric interpretation, substitute for numerics , reduction explains how to interpret rigorously the linearized solutions that are more singular than the solution of the detonation problem. As in the previous application, Reduction shows that blowup leads to the formation of a pattern—as opposed to a wave: the various points on the blowup surface are not causally related to one another, but nearby points on this surface have nearby domains of dependence. The explicit character of reduction accounts for its practical usefulness.

The big-bang model has been derived on the assumption that the largescale structure of the universe is spatially isotropic and homogeneous. After many inconclusive attempts, numerics were tried; they were consistent with AVD behavior, except at certain places corresponding to spikes in the output of computation. A detailed analysis of the expansion of the solutions in this case shows that it involves four arbitrary functions, and that the form of the expansion changes if the derivative of one of the arbitrary functions vanishes.

Some spikes observed in computations are not numerical artefacts, but correspond precisely to the extrema of this arbitrary function. Other spikes, due to a poor choice of coordinates, may also be analyzed. This work has been extended to other types of matter terms.

Kichenassamy, Satyanad 1963-

Thus, angles between curves are well determined, but length scales may vary from point to point. The consideration of minimizing sequences for R is the simplest strategy to prove the Riemann mapping theorem. The mapping radius was extensively studied in the twentieth century, and has several other applications that require understanding the boundary behavior of the mapping radius; see the review article [8]. Reduction leads to a proof of this result, without assuming the domain to be simply connected.

It turns out Problem 9. Equation 1. The latter property holds in higher dimensions, and for large classes of superlinear monotone nonlinearities, in non-simply-connected domains as well. As a consequence, solutions to the Liouville equation satisfy an interior a priori bound involving only the distance to the boundary and not the boundary values at all. Keller and Rademacher also studied this equation in three 18 1 Introduction dimensions, which is relevant to electrohydrodynamics.

The minima of the radius function also occur as points of concentration of minimizing sequences in variational problems of recent interest. This and many other applications require a detailed knowledge of v [8]. For these reasons, it is desirable to know the boundary behavior of v. We seek to obtain C a classical solution of this equation. By contrast, u cannot be interpreted as a distribution solution of 1. For this reason, we now allow n to be a real parameter, unrelated to the space dimension.

Therefore, the Liouville and Loewner—Nirenberg equations admit of reduction, and the regularity of the hyperbolic radius is equivalent to the extension of Schauder theory to the Fuchsian, degenerate elliptic equation 1. In fact, the result cannot be the sole consequence of ellipticity; simple examples show that the result is false if one does not take the form of lower-order terms into account. This issue is familiar in the theory of PDEs with degenerate quadratic form, such as the so-called Keldysh or Fichera problems, but the estimates we need do not follow from these Lp results.

One example of a problem with linear degeneracy has been worked out [71], but the method does not apply to quadratic degeneracy, such as in our case. There are no symmetry assumptions on the metric. They solved this problem for the case of n odd; we have solved the problem in full generality.

This seems to be useful in the so-called holographic representation Witten. Part I Fuchsian Reduction 2 Formal Series The purpose of this chapter is to construct formal solutions for equations or systems of the general form 1. The space F S should be taken large enough to automatically contain all solutions of 1. An example of a set of series with variable exponents is discussed next [16]; further examples are left to the exercises. Theorem 2. Let q0 be constant. The action of R D on the space of polynomials in L of degree at most m is therefore represented by a nonsingular triangular matrix.

But the degree of Q1 cannot be less than m. Both are of course closely related. As for systems, we have the following theorem. Let A be a matrix and q a vector, both independent of T and L; let m be a nonnegative integer. The result follows in this case. If A is singular, we decompose the space on which A acts into the direct sum of a space on which it is invertible and one on which it is nilpotent. Therefore, all the uk for k 3 This follows from the Jordan decomposition theorem.

Recall that A is nilpotent if some power of A vanishes. Still larger spaces of series are considered in the problems. The same notation will be used for vector-valued spaces of series if no confusion ensues. Remark 2. Thus, a polynomial or power series P t is inessential if P t, t ln t,. One repeats this operation for every monomial occurring in u. This concludes the proof. Similarly, N must annihilate terms that do not contain t. This operator has the desired properties. The result amounts to the identity D[w T, T ln T,. We will need a more precise result.

Lemma 2. The decomposition of the lemma is not unique in general. We turn to such results. Equation 2. They can be computed recursively using Theorem 2. This is equivalent to the condition given in the theorem. If we replace 2. We record a simple special case. Corollary 2.

For generalizations of these spaces to other sets of exponents, see Problem 2. We turn to the main solvability result relative to the operator N. We close this section with a few results on still more general spaces of series. The following is easily checked. Proposition 2. We now turn to solvability results for Fuchsian equations in AL,s. The unknown u may have several components. Let p and r be integers equal to at least 1. Let g be a polynomial of total degree in u at most p.

By assumption, T r g is a sum of expressions to which Lemma 2. By Theorem 2. We may now state the main result. More generally, we may allow g to be a formal power series in u, A1 u,. The second shows how to perform a second reduction directly if a formal solution of high order is already at hand. How to increase the eigenvalues of A Since the existence theorems require the eigenvalues of A to have nonnegative real parts, it is useful to be able to transform the equation at hand so as to achieve this.

The Fuchsian form, thanks to Lemma 2. Given a Fuchsian system 2. Any solution of 2. By Lemma 2. Thus, we have replaced 2. How to make use of an approximate solution to high order Given the existence of an approximate solution to a very high order, one can directly obtain a second reduced equation; in addition to clarifying the structure of the argument, this technique is useful when it is not convenient to write out the details of the formal solution.

Situations of this type are relevant for the applications in Chap. We cannot work with power series in t and tx because the operator D is not invertible on this space. Using 2. This is a direct consequence of Proposition 2. We have, using 2. Also, Proposition 2. By Proposition 2. The parameters c and d are real.

We prove a and b together. Let us j seek solutions of 2. From the results of Sect. If neither c nor d is a nonnegative integer, we see by induction that there is a solution with uj independent of t: u is a solution in pure powers of t. If precisely one of them is a nonnegative integer, k j equals 2 except for precisely one value j0 of j. This group therefore 40 2 Formal Series serves as a symmetry for singularity analysis of very general classes of PDEs without nontrivial point symmetries.

This is the only section of the book that requires some knowledge of representation theory, and will not be needed in the sequel. For further information on invariant theory, see [81]. We consider polynomial invariants and covariants. These operators are related to SL 2 in the following way. The other commutation relations follow easily from these. Let u be a sum of monomials of the same degree g and weight p, in the variables t0 ,. Conversely, if u is inessential, so is M u. In particular, any monomial in t0 ,.

The other cases, as well as the converse, are proved similarly. Write v t, t ln t,. This is the desired result. Problems 2. Study separately the case of real and complex solutions. In particular, show that in the real case, the appropriate space of formal series should contain all expressions of the form q L, C1 , C2 ,. Extend the result to more general equations.

Fuchsian Reduction: Applications to Geometry, Cosmology and Mathematical Physics

This is stronger than the triangle inequality. They are adapted to applications to nonlinear waves, and cover in particular all the applications to soliton theory. We limit ourselves to fairly common situations, and do not strive for the best hypotheses on the nonlinearities. Results are taken from [] and []. Let m be the order of the equation, which will also be assumed to be the order of the highest time derivative. Thus, the singularity surface is assumed to be noncharacteristic. Since Q does not involve derivatives with respect to x, Q is a polynomial, and its roots, which maydepend on x, are the resonances.

Solutions corresponding to the same 3. Let us summarize the results: Theorem 3. Remark 3. We further require that the second most singular terms also not involve space derivatives. However, many examples do not require this more complete treatment. Fix u0 among the roots of P. We prove that the substitution 3. Theorem 3. Resonances could vary with x. However, in many cases, even if u0 is not constant, the resonances are. Step 1: First change of unknown. Step 2: Introduction of logarithms and second change of unknown.

Step 3: Substitution into 3. We now substitute this result into 3. Consider each term fa ua separately. We therefore need to consider only two types of terms: 50 3 General Reduction Methods 1. Combining these equations, we reach the desired assertion. Step 4: Choice of k0 and hq.

The arguments of Sect. It holds for translation-invariant PDEs. Compute the action of D on each of the new unknowns, taking 3. We have 3. Using Theorem 2. This completes the second reduction. Again, for simplicity, we limit ourselves to a simple setup with algebraic leading behavior and constant resonances. Four assumptions are now described and motivated. Under assumptions 3. Since by 3. We obtain, using 3. This completes the reduction. We give here an estimate of the optimal i. We begin by viewing the solution as a function of x, t0 ,.

Thus, we may replace 3. We now state the results: Theorem 3. Then there are inessential polynomials I0 ,. Specializing to the case of simple resonances, we obtain the following corollary: Corollary 3. More precisely, there is a formal solution of 3. To say that u is a solution of 3. We therefore consider the most general series solution of this equation and show that its essential part is independent of Jq. We then compute the formal solution to some high order, and introduce the Iq.

We then show that one may introduce inessential polynomials Iq into the equation in such a way that the resulting equation will have a solution in which the inessential part is identically zero. We argue by induction on g. The operator M has been studied in Sect. We merely need to check that the right-hand side has the desired form, since ug will be uniquely determined. Indeed, R N is invertible on the space of polynomials in t0 ,. Since inessential functions are stable by product with other functions i. Step 2: General case. The earlier results about the form of Gq still hold.

Also, the essential part of ug involves k arbitrary functions of x, because case 1 of Theorem 2. In practice, we have 3. Step 3: Introduction of Iq. Note that vg contains arbitrary functions of x corresponding to each resonance. This completes the proof of Theorem 3. Step 4: Proof of Corollary 3. If all resonances are simple and greater than 1, or if 1 is a simple and compatible resonance i. If we assume that g is a simple resonance, and that for j Remark 3. This property may fail if 1 is a resonance. The values of r leading to a nontrivial solution are the resonances. While it is easy to prove this result in the general situations described in the previous chapter, it is as easy to check it afresh in new situations.

Therefore, the practical procedure is, with the notation of Sect. If the problem admits a group action, we may obtain solutions of the linearization from any solution that is not invariant by the group. This result, which is a special case of Theorem 3. A simpler example is provided by Problem 3. It is possible for the simple pole to split into two or more poles by perturbation. Equation 3. The transformed solution has, in general, a circular natural boundary.

Results The results below actually apply to any equation that admits the transformation formula 3. This assumption is clear for ODEs; it allows one to extend 3. The resonance structure will be derived on the sole basis of the representation formula. This is a special case of the transformation 3. Our results are as follows. The restriction that the solution y x be analytic is essential. One can rephrase the assumption in Theorem 3. Theorems 3. These results therefore hold for any third-order autonomous equation, hence for both equations 3.

It follows from Theorem 3. A result similar to Theorem 3. An illustration of Theorem 3. We conclude, without computing the symmetry group of the equation, that this equation does not admit the transformation law 3. Proof of Theorem 3. This proves Theorem 3. Solving the Cauchy problem, we can construct solutions to which each of Theorems 3.

If these three 3. For any constant a, equation 3. Using transformations 3. For equation 3. Let a be a constant. These parameters are in correspondence with the Cauchy data at a nearby regular point. These three theorems are proved in Problem 3. We close this chapter with a general result on stability of solutions.

This result makes rigorous the idea that a series with as many free parameters as there are Cauchy data must represent the general solution locally. It is the simplest case in which Step H of the general program can be carried out. To prove the result, one must show that these parameters are not redundant. We achieve this by a reduction to the implicit function theorem. Applying the inverse function theorem to this map near x0 , 0,.

These data are clearly redundant. In the case of 3. Problems 3. See Problem Prove Theorems 3.

Prove Theorem 1. In Sect. Thanks to them, as soon as the second reduction has been achieved, one can immediately conclude that the singularity data determine a unique singular solution, and that the formal asymptotics of Part I are valid.

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The method of proof is based on an iteration in a Banach space, as in the modern approach to the Cauchy—Kovalevskaya theorem. We review some basic facts, without proofs [50, 23]. The converse is not true see Problem 4. One often calls the origin a regular singular point if there is a fundamental solution of the form S z z P , where S has at most a pole at the origin, and P is constant; non-Fuchsian ODEs may admit the origin as a regular singular point.

A more direct approach is to construct a fundamental matrix. This tallies with the developments of Chap. In Chap. It does not seem to be applicable to nonlinear situations and is therefore not further discussed. Remark 4. If the equation contains parameters, it is often useful to view them as new space variables, whose derivatives do not enter into f. In this way, analytic dependence with respect to parameters is obtained as a byproduct of the proof. The nonlinearity f is assumed to preserve analyticity in space and continuity in time, and to be Lipschitz in u and u x whenever u is bounded.

Theorem 4. The system 4. Thus, the solution will be bounded on a domain that shrinks with time; this is consistent with the behavior of the domain of dependence of solutions of the Cauchy problem. This can always be achieved since we are allowed to take R very large. We show that this sequence converges in the a-norm if a is small. One may, by introducing new dependent variables, assume, as we will, that f is linear in Dz.

We prove the following result. If A has no eigenvalue with negative real part, 4. The solution of 4. Equation 4. One checks directly, using the Cauchy— Riemann equations, that this does provide an analytic solution to the problem. Step 3 We estimate the s-norm of Hu in terms of the a-norm of u. Let us label tr0 , tr1 ,. If we choose m such that 0 4. An abstract convergence result for linear ODEs may be found in [99].

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Kichenassamy, Satyanad [WorldCat Identities]

For classical applications, see for instance [23]. The results for PDEs have a long history. In fact, it is possible to prove the result using the contraction mapping principle, but as the reader can see, the choice of norm is non trivial, motivated as it is by the proofs based on iteration in scales of Banach spaces. The modern interest in Fuchsian PDEs seems to go back to [77, 11], although the stage had been prepared to some extent by work on the Euler—Poisson— Darboux equation; see [, ].

Generalized Fuchsian problems are necessary to deal with in order to take logarithmic terms into account []. As far as reduction is concerned, the iteration used in the proof of the existence theorem generates automatically the expansion of the solution: it is not necessary to know beforehand the form of this expansion. Problems 4. Recover the Cauchy—Kovalevskaya theorem for equations linear in the derivatives from the results of this chapter.

This problem shows how to recover asymptotics of solutions of linear equations with irregular, or non-Fuchsian, singularities by Fuchsian reduction. Results are taken from [91]. Give an example of a non-Fuchsian system with a regular singular point. The results are adapted to application to Fuchsian equations obtained by the second reduction, see Sect.

Recall that Sect. The simplest result, which parallels the Cauchy—Lipschitz theorem, is easy to state and to prove. We begin this chapter with a more precise theorem, Theorem 5. We then turn to the case of PDEs, in which we allow several Fuchsian variables, to cope with the type of equations that may arise in applications. The development is modeled on the theory of symmetric-hyperbolic systems [, ], which it contains as a special case. The same holds for the partial derivatives fV of f with respect to the components of V.

Indeed, 5. Under assumptions H1 — H4 , problem 5. Any solution of 5. If assumption H5 holds, we have the following Theorem. Theorem 5. Assume that the eigenvalues of A have nonnegative real parts, and that the eigenvalues with zero real part have Jordan blocks of maximal size M. Assume further that 5. Then H1 , H2 , and H4 hold. The proof is given in Problem 5. A simple corollary is the following: Theorem 5. Take A as in Theorem 5. The assumptions of Theorem 5. The results follow. Remark 5. The optimality of Theorem 5. None of these solutions is O 1.

Let E be a Banach space. Let A be a bounded operator and f as above. Equation 5. Such problems arise from nonlinear wave equations by second reduction. Our assumptions on Q, A, M , B, and f are as follows. Since we are interested in solutions that vanish initially, this truncation is reasonable. The introduction of V should not be confused with the change of scalar product commonly encountered in the theory of symmetric systems []: it is due here to the fact that Au, u may change sign, even if all the eigenvalues of A are nonnegative.

The second part of A4 is used to prove the estimates on the time derivatives of the solution. We may now state the result. Let s be an integer. One then passes to the limit. They will be applied to regularized equations, where B will be a smooth approximation to B.

The result follows, for small t. We seek an L2 estimate on v. Since the nonlinear term is bounded and sublinear on L2 , we may apply the procedure of 1 to derive an L2 estimate of v. The argument continues as before. Approximate equation The strategy consists in approximating B by bounded operators. We establish here a priori estimates. We check the assumptions of Theorem 5. Lemma 5. The lemma is therefore proved. Using Lemma 5. Now [, pp. By application of Theorem 5. This will play the role of A2 in the sequel.

Only A3 requires a separate argument. Applying the procedure of Sect. Let us now turn to systems 5. We now have a solution w of 5. For such a system, standard results [] show that solutions that start in H s are continuous in time, with values in H s , and therefore w x, t,. We have therefore proved Theorem 5. We outline two issues: i expressions involving the solution, such as tu x, t , may be more regular than u; ii the optimal regularity may not be obtained by taking the arbitrary functions to be smooth.

These issues are illustrated on examples; no general statement seems to be available at this time. The second issue pertains to the regularity of the arbitrary functions. By contrast, for Fuchsian equations, 98 5 Fuchsian Initial-Value Problems in Sobolev Spaces the correct regularity of the arbitrary functions may require going beyond Sobolev spaces. We analyze in detail the situation for the one-dimensional model problem 5. Here, u0 occurs both as an initial condition, and in the right-hand side.

The structure of solutions is given by the following theorem. If u0 is not analytic, the expansion remains valid as an asymptotic expansion. Solutions are uniquely determined by the choice of a1. Before giving the proof, let us state the main point: taking a1 very smooth does not generate the H s solutions in the theorem: Theorem 5. We now turn to the proof of these results. Proof of Theorem 5. This equation has no solution of class C 1 unless u0 is constant. On the other hand, from the study of the analytic case, if u0 is analytic, then tv is an analytic function of x, t, and t ln t.

The convergence follows from Theorem 4. Uniqueness follows from the above argument. The result is proved by direct substitution. This completes the proof of Theorem 5. We now turn to the estimation of the regularity of a1 for solutions in H s. This provides an expression for the most general solution in H s.

Observe that k is not a classical symbol. The restriction on the arbitrary function occurring in the general solution is missed by the formal calculation in powers of t and t ln t. They depend on one parameter. We give a typical example of this reduction. Consider the solution u of the n-dimensional Liouville equation considered in Sect.

Math Mornings at Yale: Higher Dimensional Space and the Things In It

Furthermore, v can be computed from w, and u from w. The system for u is simply the symmetric-hyperbolic system associated with the Liouville equation. The following system implies In This matrix has eigenvalues 0, 3, and 1, with multiplicities 1, 1, n. Its null space is generated by 1, 2, 0,. We conclude by showing that the function w does determine a solution of the Liouville equation for u. Thus, we have proved that w determines precisely one solution of 5.

This is noteworthy, since not all solutions of the equation for u correspond to solutions of Problems 5. Can one replace continuity with respect to t in assumption H3 , Sect. Prove Theorem 5. Improve Theorem 5. Remark 6.